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Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  poimirlem32 Structured version   Visualization version   GIF version
Theorem poimirlem32 37010
Description: Lemma for poimir 37011, combining poimirlem28 37006, poimirlem30 37008, and poimirlem31 37009 to get Equation (1) of [Kulpa] p. 547. (Contributed by Brendan Leahy, 21-Aug-2020.)
Hypotheses
Ref Expression
poimir.0 (πœ‘ β†’ 𝑁 ∈ β„•)
poimir.i 𝐼 = ((0[,]1) ↑m (1...𝑁))
poimir.r 𝑅 = (∏tβ€˜((1...𝑁) Γ— {(topGenβ€˜ran (,))}))
poimir.1 (πœ‘ β†’ 𝐹 ∈ ((𝑅 β†Ύt 𝐼) Cn 𝑅))
poimir.2 ((πœ‘ ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑧 ∈ 𝐼 ∧ (π‘§β€˜π‘›) = 0)) β†’ ((πΉβ€˜π‘§)β€˜π‘›) ≀ 0)
poimir.3 ((πœ‘ ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑧 ∈ 𝐼 ∧ (π‘§β€˜π‘›) = 1)) β†’ 0 ≀ ((πΉβ€˜π‘§)β€˜π‘›))
Assertion
Ref Expression
poimirlem32 (πœ‘ β†’ βˆƒπ‘ ∈ 𝐼 βˆ€π‘› ∈ (1...𝑁)βˆ€π‘£ ∈ (𝑅 β†Ύt 𝐼)(𝑐 ∈ 𝑣 β†’ βˆ€π‘Ÿ ∈ { ≀ , β—‘ ≀ }βˆƒπ‘§ ∈ 𝑣 0π‘Ÿ((πΉβ€˜π‘§)β€˜π‘›)))
Distinct variable groups:   𝑧,𝑛,πœ‘   𝑛,𝐹   𝑛,𝑁   πœ‘,𝑧   𝑧,𝐹   𝑧,𝑁   𝑛,𝑐,π‘Ÿ,𝑣,𝑧,πœ‘   𝐹,𝑐,π‘Ÿ,𝑣   𝐼,𝑐,𝑛,π‘Ÿ,𝑣,𝑧   𝑁,𝑐,π‘Ÿ,𝑣   𝑅,𝑐,𝑛,π‘Ÿ,𝑣,𝑧
Proof of Theorem poimirlem32
Dummy variables 𝑓 𝑖 𝑗 π‘˜ π‘š 𝑝 π‘ž 𝑠 𝑔 π‘Ž 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 poimir.0 . . . . . . 7 (πœ‘ β†’ 𝑁 ∈ β„•)
21adantr 480 . . . . . 6 ((πœ‘ ∧ π‘˜ ∈ β„•) β†’ 𝑁 ∈ β„•)
3 fvoveq1 7424 . . . . . . . . . . . . 13 (𝑝 = ((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) β†’ (πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜}))) = (πΉβ€˜(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {π‘˜}))))
43fveq1d 6883 . . . . . . . . . . . 12 (𝑝 = ((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) β†’ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) = ((πΉβ€˜(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘))
54breq2d 5150 . . . . . . . . . . 11 (𝑝 = ((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) β†’ (0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ↔ 0 ≀ ((πΉβ€˜(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘)))
6 fveq1 6880 . . . . . . . . . . . 12 (𝑝 = ((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) β†’ (π‘β€˜π‘) = (((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})))β€˜π‘))
76neeq1d 2992 . . . . . . . . . . 11 (𝑝 = ((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) β†’ ((π‘β€˜π‘) β‰  0 ↔ (((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})))β€˜π‘) β‰  0))
85, 7anbi12d 630 . . . . . . . . . 10 (𝑝 = ((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) β†’ ((0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0) ↔ (0 ≀ ((πΉβ€˜(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})))β€˜π‘) β‰  0)))
98ralbidv 3169 . . . . . . . . 9 (𝑝 = ((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) β†’ (βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0) ↔ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})))β€˜π‘) β‰  0)))
109rabbidv 3432 . . . . . . . 8 (𝑝 = ((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) β†’ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)} = {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})))β€˜π‘) β‰  0)})
1110uneq2d 4155 . . . . . . 7 (𝑝 = ((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) β†’ ({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)}) = ({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})))β€˜π‘) β‰  0)}))
1211supeq1d 9437 . . . . . 6 (𝑝 = ((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) β†’ sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)}), ℝ, < ) = sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})))β€˜π‘) β‰  0)}), ℝ, < ))
131nnnn0d 12529 . . . . . . . . . . 11 (πœ‘ β†’ 𝑁 ∈ β„•0)
14 0elfz 13595 . . . . . . . . . . 11 (𝑁 ∈ β„•0 β†’ 0 ∈ (0...𝑁))
1513, 14syl 17 . . . . . . . . . 10 (πœ‘ β†’ 0 ∈ (0...𝑁))
1615snssd 4804 . . . . . . . . 9 (πœ‘ β†’ {0} βŠ† (0...𝑁))
17 ssrab2 4069 . . . . . . . . . . 11 {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)} βŠ† (1...𝑁)
18 fz1ssfz0 13594 . . . . . . . . . . 11 (1...𝑁) βŠ† (0...𝑁)
1917, 18sstri 3983 . . . . . . . . . 10 {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)} βŠ† (0...𝑁)
2019a1i 11 . . . . . . . . 9 (πœ‘ β†’ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)} βŠ† (0...𝑁))
2116, 20unssd 4178 . . . . . . . 8 (πœ‘ β†’ ({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)}) βŠ† (0...𝑁))
22 ltso 11291 . . . . . . . . 9 < Or ℝ
23 snfi 9040 . . . . . . . . . . 11 {0} ∈ Fin
24 fzfi 13934 . . . . . . . . . . . 12 (1...𝑁) ∈ Fin
25 rabfi 9265 . . . . . . . . . . . 12 ((1...𝑁) ∈ Fin β†’ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)} ∈ Fin)
2624, 25ax-mp 5 . . . . . . . . . . 11 {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)} ∈ Fin
27 unfi 9168 . . . . . . . . . . 11 (({0} ∈ Fin ∧ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)} ∈ Fin) β†’ ({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)}) ∈ Fin)
2823, 26, 27mp2an 689 . . . . . . . . . 10 ({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)}) ∈ Fin
29 c0ex 11205 . . . . . . . . . . . 12 0 ∈ V
3029snid 4656 . . . . . . . . . . 11 0 ∈ {0}
31 elun1 4168 . . . . . . . . . . 11 (0 ∈ {0} β†’ 0 ∈ ({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)}))
32 ne0i 4326 . . . . . . . . . . 11 (0 ∈ ({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)}) β†’ ({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)}) β‰  βˆ…)
3330, 31, 32mp2b 10 . . . . . . . . . 10 ({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)}) β‰  βˆ…
34 0red 11214 . . . . . . . . . . . . 13 ((πœ‘ β†’ 𝑁 ∈ β„•) β†’ 0 ∈ ℝ)
3534snssd 4804 . . . . . . . . . . . 12 ((πœ‘ β†’ 𝑁 ∈ β„•) β†’ {0} βŠ† ℝ)
361, 35ax-mp 5 . . . . . . . . . . 11 {0} βŠ† ℝ
37 elfzelz 13498 . . . . . . . . . . . . . 14 (𝑛 ∈ (1...𝑁) β†’ 𝑛 ∈ β„€)
3837ssriv 3978 . . . . . . . . . . . . 13 (1...𝑁) βŠ† β„€
39 zssre 12562 . . . . . . . . . . . . 13 β„€ βŠ† ℝ
4038, 39sstri 3983 . . . . . . . . . . . 12 (1...𝑁) βŠ† ℝ
4117, 40sstri 3983 . . . . . . . . . . 11 {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)} βŠ† ℝ
4236, 41unssi 4177 . . . . . . . . . 10 ({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)}) βŠ† ℝ
4328, 33, 423pm3.2i 1336 . . . . . . . . 9 (({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)}) ∈ Fin ∧ ({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)}) β‰  βˆ… ∧ ({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)}) βŠ† ℝ)
44 fisupcl 9460 . . . . . . . . 9 (( < Or ℝ ∧ (({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)}) ∈ Fin ∧ ({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)}) β‰  βˆ… ∧ ({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)}) βŠ† ℝ)) β†’ sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)}), ℝ, < ) ∈ ({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)}))
4522, 43, 44mp2an 689 . . . . . . . 8 sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)}), ℝ, < ) ∈ ({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)})
46 ssel 3967 . . . . . . . 8 (({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)}) βŠ† (0...𝑁) β†’ (sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)}), ℝ, < ) ∈ ({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)}) β†’ sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)}), ℝ, < ) ∈ (0...𝑁)))
4721, 45, 46mpisyl 21 . . . . . . 7 (πœ‘ β†’ sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)}), ℝ, < ) ∈ (0...𝑁))
4847ad2antrr 723 . . . . . 6 (((πœ‘ ∧ π‘˜ ∈ β„•) ∧ 𝑝:(1...𝑁)⟢(0...π‘˜)) β†’ sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)}), ℝ, < ) ∈ (0...𝑁))
49 elfznn 13527 . . . . . . . . 9 (𝑛 ∈ (1...𝑁) β†’ 𝑛 ∈ β„•)
50 nngt0 12240 . . . . . . . . . . . 12 (𝑛 ∈ β„• β†’ 0 < 𝑛)
5150adantr 480 . . . . . . . . . . 11 ((𝑛 ∈ β„• ∧ (π‘β€˜π‘›) = 0) β†’ 0 < 𝑛)
52 simpr 484 . . . . . . . . . . . . . 14 ((0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0) β†’ (π‘β€˜π‘) β‰  0)
5352ralimi 3075 . . . . . . . . . . . . 13 (βˆ€π‘ ∈ (1...𝑠)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0) β†’ βˆ€π‘ ∈ (1...𝑠)(π‘β€˜π‘) β‰  0)
54 elfznn 13527 . . . . . . . . . . . . . 14 (𝑠 ∈ (1...𝑁) β†’ 𝑠 ∈ β„•)
55 nnre 12216 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ β„• β†’ 𝑛 ∈ ℝ)
56 nnre 12216 . . . . . . . . . . . . . . . . . . . 20 (𝑠 ∈ β„• β†’ 𝑠 ∈ ℝ)
57 lenlt 11289 . . . . . . . . . . . . . . . . . . . 20 ((𝑛 ∈ ℝ ∧ 𝑠 ∈ ℝ) β†’ (𝑛 ≀ 𝑠 ↔ Β¬ 𝑠 < 𝑛))
5855, 56, 57syl2an 595 . . . . . . . . . . . . . . . . . . 19 ((𝑛 ∈ β„• ∧ 𝑠 ∈ β„•) β†’ (𝑛 ≀ 𝑠 ↔ Β¬ 𝑠 < 𝑛))
59 elfz1b 13567 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 ∈ (1...𝑠) ↔ (𝑛 ∈ β„• ∧ 𝑠 ∈ β„• ∧ 𝑛 ≀ 𝑠))
6059biimpri 227 . . . . . . . . . . . . . . . . . . . 20 ((𝑛 ∈ β„• ∧ 𝑠 ∈ β„• ∧ 𝑛 ≀ 𝑠) β†’ 𝑛 ∈ (1...𝑠))
61603expia 1118 . . . . . . . . . . . . . . . . . . 19 ((𝑛 ∈ β„• ∧ 𝑠 ∈ β„•) β†’ (𝑛 ≀ 𝑠 β†’ 𝑛 ∈ (1...𝑠)))
6258, 61sylbird 260 . . . . . . . . . . . . . . . . . 18 ((𝑛 ∈ β„• ∧ 𝑠 ∈ β„•) β†’ (Β¬ 𝑠 < 𝑛 β†’ 𝑛 ∈ (1...𝑠)))
63 fveq2 6881 . . . . . . . . . . . . . . . . . . . . 21 (𝑏 = 𝑛 β†’ (π‘β€˜π‘) = (π‘β€˜π‘›))
6463eqeq1d 2726 . . . . . . . . . . . . . . . . . . . 20 (𝑏 = 𝑛 β†’ ((π‘β€˜π‘) = 0 ↔ (π‘β€˜π‘›) = 0))
6564rspcev 3604 . . . . . . . . . . . . . . . . . . 19 ((𝑛 ∈ (1...𝑠) ∧ (π‘β€˜π‘›) = 0) β†’ βˆƒπ‘ ∈ (1...𝑠)(π‘β€˜π‘) = 0)
6665expcom 413 . . . . . . . . . . . . . . . . . 18 ((π‘β€˜π‘›) = 0 β†’ (𝑛 ∈ (1...𝑠) β†’ βˆƒπ‘ ∈ (1...𝑠)(π‘β€˜π‘) = 0))
6762, 66sylan9 507 . . . . . . . . . . . . . . . . 17 (((𝑛 ∈ β„• ∧ 𝑠 ∈ β„•) ∧ (π‘β€˜π‘›) = 0) β†’ (Β¬ 𝑠 < 𝑛 β†’ βˆƒπ‘ ∈ (1...𝑠)(π‘β€˜π‘) = 0))
6867an32s 649 . . . . . . . . . . . . . . . 16 (((𝑛 ∈ β„• ∧ (π‘β€˜π‘›) = 0) ∧ 𝑠 ∈ β„•) β†’ (Β¬ 𝑠 < 𝑛 β†’ βˆƒπ‘ ∈ (1...𝑠)(π‘β€˜π‘) = 0))
69 nne 2936 . . . . . . . . . . . . . . . . . 18 (Β¬ (π‘β€˜π‘) β‰  0 ↔ (π‘β€˜π‘) = 0)
7069rexbii 3086 . . . . . . . . . . . . . . . . 17 (βˆƒπ‘ ∈ (1...𝑠) Β¬ (π‘β€˜π‘) β‰  0 ↔ βˆƒπ‘ ∈ (1...𝑠)(π‘β€˜π‘) = 0)
71 rexnal 3092 . . . . . . . . . . . . . . . . 17 (βˆƒπ‘ ∈ (1...𝑠) Β¬ (π‘β€˜π‘) β‰  0 ↔ Β¬ βˆ€π‘ ∈ (1...𝑠)(π‘β€˜π‘) β‰  0)
7270, 71bitr3i 277 . . . . . . . . . . . . . . . 16 (βˆƒπ‘ ∈ (1...𝑠)(π‘β€˜π‘) = 0 ↔ Β¬ βˆ€π‘ ∈ (1...𝑠)(π‘β€˜π‘) β‰  0)
7368, 72imbitrdi 250 . . . . . . . . . . . . . . 15 (((𝑛 ∈ β„• ∧ (π‘β€˜π‘›) = 0) ∧ 𝑠 ∈ β„•) β†’ (Β¬ 𝑠 < 𝑛 β†’ Β¬ βˆ€π‘ ∈ (1...𝑠)(π‘β€˜π‘) β‰  0))
7473con4d 115 . . . . . . . . . . . . . 14 (((𝑛 ∈ β„• ∧ (π‘β€˜π‘›) = 0) ∧ 𝑠 ∈ β„•) β†’ (βˆ€π‘ ∈ (1...𝑠)(π‘β€˜π‘) β‰  0 β†’ 𝑠 < 𝑛))
7554, 74sylan2 592 . . . . . . . . . . . . 13 (((𝑛 ∈ β„• ∧ (π‘β€˜π‘›) = 0) ∧ 𝑠 ∈ (1...𝑁)) β†’ (βˆ€π‘ ∈ (1...𝑠)(π‘β€˜π‘) β‰  0 β†’ 𝑠 < 𝑛))
7653, 75syl5 34 . . . . . . . . . . . 12 (((𝑛 ∈ β„• ∧ (π‘β€˜π‘›) = 0) ∧ 𝑠 ∈ (1...𝑁)) β†’ (βˆ€π‘ ∈ (1...𝑠)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0) β†’ 𝑠 < 𝑛))
7776ralrimiva 3138 . . . . . . . . . . 11 ((𝑛 ∈ β„• ∧ (π‘β€˜π‘›) = 0) β†’ βˆ€π‘  ∈ (1...𝑁)(βˆ€π‘ ∈ (1...𝑠)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0) β†’ 𝑠 < 𝑛))
78 ralunb 4183 . . . . . . . . . . . 12 (βˆ€π‘  ∈ ({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)})𝑠 < 𝑛 ↔ (βˆ€π‘  ∈ {0}𝑠 < 𝑛 ∧ βˆ€π‘  ∈ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)}𝑠 < 𝑛))
79 breq1 5141 . . . . . . . . . . . . . 14 (𝑠 = 0 β†’ (𝑠 < 𝑛 ↔ 0 < 𝑛))
8029, 79ralsn 4677 . . . . . . . . . . . . 13 (βˆ€π‘  ∈ {0}𝑠 < 𝑛 ↔ 0 < 𝑛)
81 oveq2 7409 . . . . . . . . . . . . . . 15 (π‘Ž = 𝑠 β†’ (1...π‘Ž) = (1...𝑠))
8281raleqdv 3317 . . . . . . . . . . . . . 14 (π‘Ž = 𝑠 β†’ (βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0) ↔ βˆ€π‘ ∈ (1...𝑠)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)))
8382ralrab 3681 . . . . . . . . . . . . 13 (βˆ€π‘  ∈ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)}𝑠 < 𝑛 ↔ βˆ€π‘  ∈ (1...𝑁)(βˆ€π‘ ∈ (1...𝑠)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0) β†’ 𝑠 < 𝑛))
8480, 83anbi12i 626 . . . . . . . . . . . 12 ((βˆ€π‘  ∈ {0}𝑠 < 𝑛 ∧ βˆ€π‘  ∈ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)}𝑠 < 𝑛) ↔ (0 < 𝑛 ∧ βˆ€π‘  ∈ (1...𝑁)(βˆ€π‘ ∈ (1...𝑠)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0) β†’ 𝑠 < 𝑛)))
8578, 84bitri 275 . . . . . . . . . . 11 (βˆ€π‘  ∈ ({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)})𝑠 < 𝑛 ↔ (0 < 𝑛 ∧ βˆ€π‘  ∈ (1...𝑁)(βˆ€π‘ ∈ (1...𝑠)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0) β†’ 𝑠 < 𝑛)))
8651, 77, 85sylanbrc 582 . . . . . . . . . 10 ((𝑛 ∈ β„• ∧ (π‘β€˜π‘›) = 0) β†’ βˆ€π‘  ∈ ({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)})𝑠 < 𝑛)
87 breq1 5141 . . . . . . . . . . 11 (𝑠 = sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)}), ℝ, < ) β†’ (𝑠 < 𝑛 ↔ sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)}), ℝ, < ) < 𝑛))
8887rspcva 3602 . . . . . . . . . 10 ((sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)}), ℝ, < ) ∈ ({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)}) ∧ βˆ€π‘  ∈ ({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)})𝑠 < 𝑛) β†’ sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)}), ℝ, < ) < 𝑛)
8945, 86, 88sylancr 586 . . . . . . . . 9 ((𝑛 ∈ β„• ∧ (π‘β€˜π‘›) = 0) β†’ sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)}), ℝ, < ) < 𝑛)
9049, 89sylan 579 . . . . . . . 8 ((𝑛 ∈ (1...𝑁) ∧ (π‘β€˜π‘›) = 0) β†’ sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)}), ℝ, < ) < 𝑛)
91903adant2 1128 . . . . . . 7 ((𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟢(0...π‘˜) ∧ (π‘β€˜π‘›) = 0) β†’ sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)}), ℝ, < ) < 𝑛)
9291adantl 481 . . . . . 6 (((πœ‘ ∧ π‘˜ ∈ β„•) ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟢(0...π‘˜) ∧ (π‘β€˜π‘›) = 0)) β†’ sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)}), ℝ, < ) < 𝑛)
9337zred 12663 . . . . . . . . . . 11 (𝑛 ∈ (1...𝑁) β†’ 𝑛 ∈ ℝ)
94933ad2ant1 1130 . . . . . . . . . 10 ((𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟢(0...π‘˜) ∧ (π‘β€˜π‘›) = π‘˜) β†’ 𝑛 ∈ ℝ)
9594adantl 481 . . . . . . . . 9 (((πœ‘ ∧ π‘˜ ∈ β„•) ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟢(0...π‘˜) ∧ (π‘β€˜π‘›) = π‘˜)) β†’ 𝑛 ∈ ℝ)
96 simpr1 1191 . . . . . . . . . . 11 (((πœ‘ ∧ π‘˜ ∈ β„•) ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟢(0...π‘˜) ∧ (π‘β€˜π‘›) = π‘˜)) β†’ 𝑛 ∈ (1...𝑁))
97 simpll 764 . . . . . . . . . . . . 13 (((πœ‘ ∧ π‘˜ ∈ β„•) ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟢(0...π‘˜) ∧ (π‘β€˜π‘›) = π‘˜)) β†’ πœ‘)
98 simplr 766 . . . . . . . . . . . . . . . . 17 (((πœ‘ ∧ π‘˜ ∈ β„•) ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟢(0...π‘˜))) β†’ π‘˜ ∈ β„•)
99 elfzelz 13498 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑖 ∈ (0...π‘˜) β†’ 𝑖 ∈ β„€)
10099zred 12663 . . . . . . . . . . . . . . . . . . . . . 22 (𝑖 ∈ (0...π‘˜) β†’ 𝑖 ∈ ℝ)
101 nndivre 12250 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ ℝ ∧ π‘˜ ∈ β„•) β†’ (𝑖 / π‘˜) ∈ ℝ)
102100, 101sylan 579 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 ∈ (0...π‘˜) ∧ π‘˜ ∈ β„•) β†’ (𝑖 / π‘˜) ∈ ℝ)
103 elfzle1 13501 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑖 ∈ (0...π‘˜) β†’ 0 ≀ 𝑖)
104100, 103jca 511 . . . . . . . . . . . . . . . . . . . . . 22 (𝑖 ∈ (0...π‘˜) β†’ (𝑖 ∈ ℝ ∧ 0 ≀ 𝑖))
105 nnrp 12982 . . . . . . . . . . . . . . . . . . . . . . 23 (π‘˜ ∈ β„• β†’ π‘˜ ∈ ℝ+)
106105rpregt0d 13019 . . . . . . . . . . . . . . . . . . . . . 22 (π‘˜ ∈ β„• β†’ (π‘˜ ∈ ℝ ∧ 0 < π‘˜))
107 divge0 12080 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑖 ∈ ℝ ∧ 0 ≀ 𝑖) ∧ (π‘˜ ∈ ℝ ∧ 0 < π‘˜)) β†’ 0 ≀ (𝑖 / π‘˜))
108104, 106, 107syl2an 595 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 ∈ (0...π‘˜) ∧ π‘˜ ∈ β„•) β†’ 0 ≀ (𝑖 / π‘˜))
109 elfzle2 13502 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑖 ∈ (0...π‘˜) β†’ 𝑖 ≀ π‘˜)
110109adantr 480 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (0...π‘˜) ∧ π‘˜ ∈ β„•) β†’ 𝑖 ≀ π‘˜)
111100adantr 480 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑖 ∈ (0...π‘˜) ∧ π‘˜ ∈ β„•) β†’ 𝑖 ∈ ℝ)
112 1red 11212 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑖 ∈ (0...π‘˜) ∧ π‘˜ ∈ β„•) β†’ 1 ∈ ℝ)
113105adantl 481 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑖 ∈ (0...π‘˜) ∧ π‘˜ ∈ β„•) β†’ π‘˜ ∈ ℝ+)
114111, 112, 113ledivmuld 13066 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (0...π‘˜) ∧ π‘˜ ∈ β„•) β†’ ((𝑖 / π‘˜) ≀ 1 ↔ 𝑖 ≀ (π‘˜ Β· 1)))
115 nncn 12217 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (π‘˜ ∈ β„• β†’ π‘˜ ∈ β„‚)
116115mulridd 11228 . . . . . . . . . . . . . . . . . . . . . . . . 25 (π‘˜ ∈ β„• β†’ (π‘˜ Β· 1) = π‘˜)
117116breq2d 5150 . . . . . . . . . . . . . . . . . . . . . . . 24 (π‘˜ ∈ β„• β†’ (𝑖 ≀ (π‘˜ Β· 1) ↔ 𝑖 ≀ π‘˜))
118117adantl 481 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 ∈ (0...π‘˜) ∧ π‘˜ ∈ β„•) β†’ (𝑖 ≀ (π‘˜ Β· 1) ↔ 𝑖 ≀ π‘˜))
119114, 118bitrd 279 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ (0...π‘˜) ∧ π‘˜ ∈ β„•) β†’ ((𝑖 / π‘˜) ≀ 1 ↔ 𝑖 ≀ π‘˜))
120110, 119mpbird 257 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 ∈ (0...π‘˜) ∧ π‘˜ ∈ β„•) β†’ (𝑖 / π‘˜) ≀ 1)
121 elicc01 13440 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 / π‘˜) ∈ (0[,]1) ↔ ((𝑖 / π‘˜) ∈ ℝ ∧ 0 ≀ (𝑖 / π‘˜) ∧ (𝑖 / π‘˜) ≀ 1))
122102, 108, 120, 121syl3anbrc 1340 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (0...π‘˜) ∧ π‘˜ ∈ β„•) β†’ (𝑖 / π‘˜) ∈ (0[,]1))
123122ancoms 458 . . . . . . . . . . . . . . . . . . 19 ((π‘˜ ∈ β„• ∧ 𝑖 ∈ (0...π‘˜)) β†’ (𝑖 / π‘˜) ∈ (0[,]1))
124 elsni 4637 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 ∈ {π‘˜} β†’ 𝑗 = π‘˜)
125124oveq2d 7417 . . . . . . . . . . . . . . . . . . . 20 (𝑗 ∈ {π‘˜} β†’ (𝑖 / 𝑗) = (𝑖 / π‘˜))
126125eleq1d 2810 . . . . . . . . . . . . . . . . . . 19 (𝑗 ∈ {π‘˜} β†’ ((𝑖 / 𝑗) ∈ (0[,]1) ↔ (𝑖 / π‘˜) ∈ (0[,]1)))
127123, 126syl5ibrcom 246 . . . . . . . . . . . . . . . . . 18 ((π‘˜ ∈ β„• ∧ 𝑖 ∈ (0...π‘˜)) β†’ (𝑗 ∈ {π‘˜} β†’ (𝑖 / 𝑗) ∈ (0[,]1)))
128127impr 454 . . . . . . . . . . . . . . . . 17 ((π‘˜ ∈ β„• ∧ (𝑖 ∈ (0...π‘˜) ∧ 𝑗 ∈ {π‘˜})) β†’ (𝑖 / 𝑗) ∈ (0[,]1))
12998, 128sylan 579 . . . . . . . . . . . . . . . 16 ((((πœ‘ ∧ π‘˜ ∈ β„•) ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟢(0...π‘˜))) ∧ (𝑖 ∈ (0...π‘˜) ∧ 𝑗 ∈ {π‘˜})) β†’ (𝑖 / 𝑗) ∈ (0[,]1))
130 simprr 770 . . . . . . . . . . . . . . . 16 (((πœ‘ ∧ π‘˜ ∈ β„•) ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟢(0...π‘˜))) β†’ 𝑝:(1...𝑁)⟢(0...π‘˜))
131 vex 3470 . . . . . . . . . . . . . . . . . 18 π‘˜ ∈ V
132131fconst 6767 . . . . . . . . . . . . . . . . 17 ((1...𝑁) Γ— {π‘˜}):(1...𝑁)⟢{π‘˜}
133132a1i 11 . . . . . . . . . . . . . . . 16 (((πœ‘ ∧ π‘˜ ∈ β„•) ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟢(0...π‘˜))) β†’ ((1...𝑁) Γ— {π‘˜}):(1...𝑁)⟢{π‘˜})
134 fzfid 13935 . . . . . . . . . . . . . . . 16 (((πœ‘ ∧ π‘˜ ∈ β„•) ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟢(0...π‘˜))) β†’ (1...𝑁) ∈ Fin)
135 inidm 4210 . . . . . . . . . . . . . . . 16 ((1...𝑁) ∩ (1...𝑁)) = (1...𝑁)
136129, 130, 133, 134, 134, 135off 7681 . . . . . . . . . . . . . . 15 (((πœ‘ ∧ π‘˜ ∈ β„•) ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟢(0...π‘˜))) β†’ (𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})):(1...𝑁)⟢(0[,]1))
137 poimir.i . . . . . . . . . . . . . . . . 17 𝐼 = ((0[,]1) ↑m (1...𝑁))
138137eleq2i 2817 . . . . . . . . . . . . . . . 16 ((𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})) ∈ 𝐼 ↔ (𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})) ∈ ((0[,]1) ↑m (1...𝑁)))
139 ovex 7434 . . . . . . . . . . . . . . . . 17 (0[,]1) ∈ V
140 ovex 7434 . . . . . . . . . . . . . . . . 17 (1...𝑁) ∈ V
141139, 140elmap 8861 . . . . . . . . . . . . . . . 16 ((𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})) ∈ ((0[,]1) ↑m (1...𝑁)) ↔ (𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})):(1...𝑁)⟢(0[,]1))
142138, 141bitri 275 . . . . . . . . . . . . . . 15 ((𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})) ∈ 𝐼 ↔ (𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})):(1...𝑁)⟢(0[,]1))
143136, 142sylibr 233 . . . . . . . . . . . . . 14 (((πœ‘ ∧ π‘˜ ∈ β„•) ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟢(0...π‘˜))) β†’ (𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})) ∈ 𝐼)
1441433adantr3 1168 . . . . . . . . . . . . 13 (((πœ‘ ∧ π‘˜ ∈ β„•) ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟢(0...π‘˜) ∧ (π‘β€˜π‘›) = π‘˜)) β†’ (𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})) ∈ 𝐼)
145 3anass 1092 . . . . . . . . . . . . . . . 16 ((𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟢(0...π‘˜) ∧ (π‘β€˜π‘›) = π‘˜) ↔ (𝑛 ∈ (1...𝑁) ∧ (𝑝:(1...𝑁)⟢(0...π‘˜) ∧ (π‘β€˜π‘›) = π‘˜)))
146 ancom 460 . . . . . . . . . . . . . . . 16 ((𝑛 ∈ (1...𝑁) ∧ (𝑝:(1...𝑁)⟢(0...π‘˜) ∧ (π‘β€˜π‘›) = π‘˜)) ↔ ((𝑝:(1...𝑁)⟢(0...π‘˜) ∧ (π‘β€˜π‘›) = π‘˜) ∧ 𝑛 ∈ (1...𝑁)))
147145, 146bitri 275 . . . . . . . . . . . . . . 15 ((𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟢(0...π‘˜) ∧ (π‘β€˜π‘›) = π‘˜) ↔ ((𝑝:(1...𝑁)⟢(0...π‘˜) ∧ (π‘β€˜π‘›) = π‘˜) ∧ 𝑛 ∈ (1...𝑁)))
148 ffn 6707 . . . . . . . . . . . . . . . . . 18 (𝑝:(1...𝑁)⟢(0...π‘˜) β†’ 𝑝 Fn (1...𝑁))
149148ad2antrl 725 . . . . . . . . . . . . . . . . 17 (((πœ‘ ∧ π‘˜ ∈ β„•) ∧ (𝑝:(1...𝑁)⟢(0...π‘˜) ∧ (π‘β€˜π‘›) = π‘˜)) β†’ 𝑝 Fn (1...𝑁))
150 fnconstg 6769 . . . . . . . . . . . . . . . . . 18 (π‘˜ ∈ V β†’ ((1...𝑁) Γ— {π‘˜}) Fn (1...𝑁))
151131, 150mp1i 13 . . . . . . . . . . . . . . . . 17 (((πœ‘ ∧ π‘˜ ∈ β„•) ∧ (𝑝:(1...𝑁)⟢(0...π‘˜) ∧ (π‘β€˜π‘›) = π‘˜)) β†’ ((1...𝑁) Γ— {π‘˜}) Fn (1...𝑁))
152 fzfid 13935 . . . . . . . . . . . . . . . . 17 (((πœ‘ ∧ π‘˜ ∈ β„•) ∧ (𝑝:(1...𝑁)⟢(0...π‘˜) ∧ (π‘β€˜π‘›) = π‘˜)) β†’ (1...𝑁) ∈ Fin)
153 simplrr 775 . . . . . . . . . . . . . . . . 17 ((((πœ‘ ∧ π‘˜ ∈ β„•) ∧ (𝑝:(1...𝑁)⟢(0...π‘˜) ∧ (π‘β€˜π‘›) = π‘˜)) ∧ 𝑛 ∈ (1...𝑁)) β†’ (π‘β€˜π‘›) = π‘˜)
154131fvconst2 7197 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ (1...𝑁) β†’ (((1...𝑁) Γ— {π‘˜})β€˜π‘›) = π‘˜)
155154adantl 481 . . . . . . . . . . . . . . . . 17 ((((πœ‘ ∧ π‘˜ ∈ β„•) ∧ (𝑝:(1...𝑁)⟢(0...π‘˜) ∧ (π‘β€˜π‘›) = π‘˜)) ∧ 𝑛 ∈ (1...𝑁)) β†’ (((1...𝑁) Γ— {π‘˜})β€˜π‘›) = π‘˜)
156149, 151, 152, 152, 135, 153, 155ofval 7674 . . . . . . . . . . . . . . . 16 ((((πœ‘ ∧ π‘˜ ∈ β„•) ∧ (𝑝:(1...𝑁)⟢(0...π‘˜) ∧ (π‘β€˜π‘›) = π‘˜)) ∧ 𝑛 ∈ (1...𝑁)) β†’ ((𝑝 ∘f / ((1...𝑁) Γ— {π‘˜}))β€˜π‘›) = (π‘˜ / π‘˜))
157156anasss 466 . . . . . . . . . . . . . . 15 (((πœ‘ ∧ π‘˜ ∈ β„•) ∧ ((𝑝:(1...𝑁)⟢(0...π‘˜) ∧ (π‘β€˜π‘›) = π‘˜) ∧ 𝑛 ∈ (1...𝑁))) β†’ ((𝑝 ∘f / ((1...𝑁) Γ— {π‘˜}))β€˜π‘›) = (π‘˜ / π‘˜))
158147, 157sylan2b 593 . . . . . . . . . . . . . 14 (((πœ‘ ∧ π‘˜ ∈ β„•) ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟢(0...π‘˜) ∧ (π‘β€˜π‘›) = π‘˜)) β†’ ((𝑝 ∘f / ((1...𝑁) Γ— {π‘˜}))β€˜π‘›) = (π‘˜ / π‘˜))
159 nnne0 12243 . . . . . . . . . . . . . . . 16 (π‘˜ ∈ β„• β†’ π‘˜ β‰  0)
160115, 159dividd 11985 . . . . . . . . . . . . . . 15 (π‘˜ ∈ β„• β†’ (π‘˜ / π‘˜) = 1)
161160ad2antlr 724 . . . . . . . . . . . . . 14 (((πœ‘ ∧ π‘˜ ∈ β„•) ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟢(0...π‘˜) ∧ (π‘β€˜π‘›) = π‘˜)) β†’ (π‘˜ / π‘˜) = 1)
162158, 161eqtrd 2764 . . . . . . . . . . . . 13 (((πœ‘ ∧ π‘˜ ∈ β„•) ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟢(0...π‘˜) ∧ (π‘β€˜π‘›) = π‘˜)) β†’ ((𝑝 ∘f / ((1...𝑁) Γ— {π‘˜}))β€˜π‘›) = 1)
163 ovex 7434 . . . . . . . . . . . . . 14 (𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})) ∈ V
164 eleq1 2813 . . . . . . . . . . . . . . . . 17 (𝑧 = (𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})) β†’ (𝑧 ∈ 𝐼 ↔ (𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})) ∈ 𝐼))
165 fveq1 6880 . . . . . . . . . . . . . . . . . 18 (𝑧 = (𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})) β†’ (π‘§β€˜π‘›) = ((𝑝 ∘f / ((1...𝑁) Γ— {π‘˜}))β€˜π‘›))
166165eqeq1d 2726 . . . . . . . . . . . . . . . . 17 (𝑧 = (𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})) β†’ ((π‘§β€˜π‘›) = 1 ↔ ((𝑝 ∘f / ((1...𝑁) Γ— {π‘˜}))β€˜π‘›) = 1))
167164, 1663anbi23d 1435 . . . . . . . . . . . . . . . 16 (𝑧 = (𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})) β†’ ((𝑛 ∈ (1...𝑁) ∧ 𝑧 ∈ 𝐼 ∧ (π‘§β€˜π‘›) = 1) ↔ (𝑛 ∈ (1...𝑁) ∧ (𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})) ∈ 𝐼 ∧ ((𝑝 ∘f / ((1...𝑁) Γ— {π‘˜}))β€˜π‘›) = 1)))
168167anbi2d 628 . . . . . . . . . . . . . . 15 (𝑧 = (𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})) β†’ ((πœ‘ ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑧 ∈ 𝐼 ∧ (π‘§β€˜π‘›) = 1)) ↔ (πœ‘ ∧ (𝑛 ∈ (1...𝑁) ∧ (𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})) ∈ 𝐼 ∧ ((𝑝 ∘f / ((1...𝑁) Γ— {π‘˜}))β€˜π‘›) = 1))))
169 fveq2 6881 . . . . . . . . . . . . . . . . 17 (𝑧 = (𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})) β†’ (πΉβ€˜π‘§) = (πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜}))))
170169fveq1d 6883 . . . . . . . . . . . . . . . 16 (𝑧 = (𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})) β†’ ((πΉβ€˜π‘§)β€˜π‘›) = ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘›))
171170breq2d 5150 . . . . . . . . . . . . . . 15 (𝑧 = (𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})) β†’ (0 ≀ ((πΉβ€˜π‘§)β€˜π‘›) ↔ 0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘›)))
172168, 171imbi12d 344 . . . . . . . . . . . . . 14 (𝑧 = (𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})) β†’ (((πœ‘ ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑧 ∈ 𝐼 ∧ (π‘§β€˜π‘›) = 1)) β†’ 0 ≀ ((πΉβ€˜π‘§)β€˜π‘›)) ↔ ((πœ‘ ∧ (𝑛 ∈ (1...𝑁) ∧ (𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})) ∈ 𝐼 ∧ ((𝑝 ∘f / ((1...𝑁) Γ— {π‘˜}))β€˜π‘›) = 1)) β†’ 0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘›))))
173 poimir.3 . . . . . . . . . . . . . 14 ((πœ‘ ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑧 ∈ 𝐼 ∧ (π‘§β€˜π‘›) = 1)) β†’ 0 ≀ ((πΉβ€˜π‘§)β€˜π‘›))
174163, 172, 173vtocl 3538 . . . . . . . . . . . . 13 ((πœ‘ ∧ (𝑛 ∈ (1...𝑁) ∧ (𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})) ∈ 𝐼 ∧ ((𝑝 ∘f / ((1...𝑁) Γ— {π‘˜}))β€˜π‘›) = 1)) β†’ 0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘›))
17597, 96, 144, 162, 174syl13anc 1369 . . . . . . . . . . . 12 (((πœ‘ ∧ π‘˜ ∈ β„•) ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟢(0...π‘˜) ∧ (π‘β€˜π‘›) = π‘˜)) β†’ 0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘›))
176 simpr 484 . . . . . . . . . . . . 13 ((πœ‘ ∧ π‘˜ ∈ β„•) β†’ π‘˜ ∈ β„•)
177 simp3 1135 . . . . . . . . . . . . 13 ((𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟢(0...π‘˜) ∧ (π‘β€˜π‘›) = π‘˜) β†’ (π‘β€˜π‘›) = π‘˜)
178 neeq1 2995 . . . . . . . . . . . . . . 15 ((π‘β€˜π‘›) = π‘˜ β†’ ((π‘β€˜π‘›) β‰  0 ↔ π‘˜ β‰  0))
179159, 178syl5ibrcom 246 . . . . . . . . . . . . . 14 (π‘˜ ∈ β„• β†’ ((π‘β€˜π‘›) = π‘˜ β†’ (π‘β€˜π‘›) β‰  0))
180179imp 406 . . . . . . . . . . . . 13 ((π‘˜ ∈ β„• ∧ (π‘β€˜π‘›) = π‘˜) β†’ (π‘β€˜π‘›) β‰  0)
181176, 177, 180syl2an 595 . . . . . . . . . . . 12 (((πœ‘ ∧ π‘˜ ∈ β„•) ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟢(0...π‘˜) ∧ (π‘β€˜π‘›) = π‘˜)) β†’ (π‘β€˜π‘›) β‰  0)
182 vex 3470 . . . . . . . . . . . . 13 𝑛 ∈ V
183 fveq2 6881 . . . . . . . . . . . . . . 15 (𝑏 = 𝑛 β†’ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) = ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘›))
184183breq2d 5150 . . . . . . . . . . . . . 14 (𝑏 = 𝑛 β†’ (0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ↔ 0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘›)))
18563neeq1d 2992 . . . . . . . . . . . . . 14 (𝑏 = 𝑛 β†’ ((π‘β€˜π‘) β‰  0 ↔ (π‘β€˜π‘›) β‰  0))
186184, 185anbi12d 630 . . . . . . . . . . . . 13 (𝑏 = 𝑛 β†’ ((0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0) ↔ (0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘›) ∧ (π‘β€˜π‘›) β‰  0)))
187182, 186ralsn 4677 . . . . . . . . . . . 12 (βˆ€π‘ ∈ {𝑛} (0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0) ↔ (0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘›) ∧ (π‘β€˜π‘›) β‰  0))
188175, 181, 187sylanbrc 582 . . . . . . . . . . 11 (((πœ‘ ∧ π‘˜ ∈ β„•) ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟢(0...π‘˜) ∧ (π‘β€˜π‘›) = π‘˜)) β†’ βˆ€π‘ ∈ {𝑛} (0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0))
18937zcnd 12664 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ (1...𝑁) β†’ 𝑛 ∈ β„‚)
190 1cnd 11206 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ (1...𝑁) β†’ 1 ∈ β„‚)
191189, 190subeq0ad 11578 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ (1...𝑁) β†’ ((𝑛 βˆ’ 1) = 0 ↔ 𝑛 = 1))
192191biimpcd 248 . . . . . . . . . . . . . . . . 17 ((𝑛 βˆ’ 1) = 0 β†’ (𝑛 ∈ (1...𝑁) β†’ 𝑛 = 1))
193 1z 12589 . . . . . . . . . . . . . . . . . . . . 21 1 ∈ β„€
194 fzsn 13540 . . . . . . . . . . . . . . . . . . . . 21 (1 ∈ β„€ β†’ (1...1) = {1})
195193, 194ax-mp 5 . . . . . . . . . . . . . . . . . . . 20 (1...1) = {1}
196 oveq2 7409 . . . . . . . . . . . . . . . . . . . 20 (𝑛 = 1 β†’ (1...𝑛) = (1...1))
197 sneq 4630 . . . . . . . . . . . . . . . . . . . 20 (𝑛 = 1 β†’ {𝑛} = {1})
198195, 196, 1973eqtr4a 2790 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 1 β†’ (1...𝑛) = {𝑛})
199198raleqdv 3317 . . . . . . . . . . . . . . . . . 18 (𝑛 = 1 β†’ (βˆ€π‘ ∈ (1...𝑛)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0) ↔ βˆ€π‘ ∈ {𝑛} (0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)))
200199biimprd 247 . . . . . . . . . . . . . . . . 17 (𝑛 = 1 β†’ (βˆ€π‘ ∈ {𝑛} (0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0) β†’ βˆ€π‘ ∈ (1...𝑛)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)))
201192, 200syl6 35 . . . . . . . . . . . . . . . 16 ((𝑛 βˆ’ 1) = 0 β†’ (𝑛 ∈ (1...𝑁) β†’ (βˆ€π‘ ∈ {𝑛} (0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0) β†’ βˆ€π‘ ∈ (1...𝑛)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0))))
202 ralun 4184 . . . . . . . . . . . . . . . . . . . 20 ((βˆ€π‘ ∈ (1...(𝑛 βˆ’ 1))(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0) ∧ βˆ€π‘ ∈ {𝑛} (0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)) β†’ βˆ€π‘ ∈ ((1...(𝑛 βˆ’ 1)) βˆͺ {𝑛})(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0))
203 npcan1 11636 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑛 ∈ β„‚ β†’ ((𝑛 βˆ’ 1) + 1) = 𝑛)
204189, 203syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑛 ∈ (1...𝑁) β†’ ((𝑛 βˆ’ 1) + 1) = 𝑛)
205 elfzuz 13494 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑛 ∈ (1...𝑁) β†’ 𝑛 ∈ (β„€β‰₯β€˜1))
206204, 205eqeltrd 2825 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑛 ∈ (1...𝑁) β†’ ((𝑛 βˆ’ 1) + 1) ∈ (β„€β‰₯β€˜1))
207 peano2zm 12602 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑛 ∈ β„€ β†’ (𝑛 βˆ’ 1) ∈ β„€)
208 uzid 12834 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑛 βˆ’ 1) ∈ β„€ β†’ (𝑛 βˆ’ 1) ∈ (β„€β‰₯β€˜(𝑛 βˆ’ 1)))
209 peano2uz 12882 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑛 βˆ’ 1) ∈ (β„€β‰₯β€˜(𝑛 βˆ’ 1)) β†’ ((𝑛 βˆ’ 1) + 1) ∈ (β„€β‰₯β€˜(𝑛 βˆ’ 1)))
21037, 207, 208, 2094syl 19 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑛 ∈ (1...𝑁) β†’ ((𝑛 βˆ’ 1) + 1) ∈ (β„€β‰₯β€˜(𝑛 βˆ’ 1)))
211204, 210eqeltrrd 2826 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑛 ∈ (1...𝑁) β†’ 𝑛 ∈ (β„€β‰₯β€˜(𝑛 βˆ’ 1)))
212 fzsplit2 13523 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝑛 βˆ’ 1) + 1) ∈ (β„€β‰₯β€˜1) ∧ 𝑛 ∈ (β„€β‰₯β€˜(𝑛 βˆ’ 1))) β†’ (1...𝑛) = ((1...(𝑛 βˆ’ 1)) βˆͺ (((𝑛 βˆ’ 1) + 1)...𝑛)))
213206, 211, 212syl2anc 583 . . . . . . . . . . . . . . . . . . . . . 22 (𝑛 ∈ (1...𝑁) β†’ (1...𝑛) = ((1...(𝑛 βˆ’ 1)) βˆͺ (((𝑛 βˆ’ 1) + 1)...𝑛)))
214204oveq1d 7416 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑛 ∈ (1...𝑁) β†’ (((𝑛 βˆ’ 1) + 1)...𝑛) = (𝑛...𝑛))
215 fzsn 13540 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑛 ∈ β„€ β†’ (𝑛...𝑛) = {𝑛})
21637, 215syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑛 ∈ (1...𝑁) β†’ (𝑛...𝑛) = {𝑛})
217214, 216eqtrd 2764 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑛 ∈ (1...𝑁) β†’ (((𝑛 βˆ’ 1) + 1)...𝑛) = {𝑛})
218217uneq2d 4155 . . . . . . . . . . . . . . . . . . . . . 22 (𝑛 ∈ (1...𝑁) β†’ ((1...(𝑛 βˆ’ 1)) βˆͺ (((𝑛 βˆ’ 1) + 1)...𝑛)) = ((1...(𝑛 βˆ’ 1)) βˆͺ {𝑛}))
219213, 218eqtrd 2764 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 ∈ (1...𝑁) β†’ (1...𝑛) = ((1...(𝑛 βˆ’ 1)) βˆͺ {𝑛}))
220219raleqdv 3317 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ (1...𝑁) β†’ (βˆ€π‘ ∈ (1...𝑛)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0) ↔ βˆ€π‘ ∈ ((1...(𝑛 βˆ’ 1)) βˆͺ {𝑛})(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)))
221202, 220imbitrrid 245 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ (1...𝑁) β†’ ((βˆ€π‘ ∈ (1...(𝑛 βˆ’ 1))(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0) ∧ βˆ€π‘ ∈ {𝑛} (0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)) β†’ βˆ€π‘ ∈ (1...𝑛)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)))
222221expd 415 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ (1...𝑁) β†’ (βˆ€π‘ ∈ (1...(𝑛 βˆ’ 1))(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0) β†’ (βˆ€π‘ ∈ {𝑛} (0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0) β†’ βˆ€π‘ ∈ (1...𝑛)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0))))
223222com12 32 . . . . . . . . . . . . . . . . 17 (βˆ€π‘ ∈ (1...(𝑛 βˆ’ 1))(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0) β†’ (𝑛 ∈ (1...𝑁) β†’ (βˆ€π‘ ∈ {𝑛} (0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0) β†’ βˆ€π‘ ∈ (1...𝑛)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0))))
224223adantl 481 . . . . . . . . . . . . . . . 16 (((𝑛 βˆ’ 1) ∈ (1...𝑁) ∧ βˆ€π‘ ∈ (1...(𝑛 βˆ’ 1))(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)) β†’ (𝑛 ∈ (1...𝑁) β†’ (βˆ€π‘ ∈ {𝑛} (0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0) β†’ βˆ€π‘ ∈ (1...𝑛)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0))))
225201, 224jaoi 854 . . . . . . . . . . . . . . 15 (((𝑛 βˆ’ 1) = 0 ∨ ((𝑛 βˆ’ 1) ∈ (1...𝑁) ∧ βˆ€π‘ ∈ (1...(𝑛 βˆ’ 1))(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0))) β†’ (𝑛 ∈ (1...𝑁) β†’ (βˆ€π‘ ∈ {𝑛} (0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0) β†’ βˆ€π‘ ∈ (1...𝑛)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0))))
226225imdistand 570 . . . . . . . . . . . . . 14 (((𝑛 βˆ’ 1) = 0 ∨ ((𝑛 βˆ’ 1) ∈ (1...𝑁) ∧ βˆ€π‘ ∈ (1...(𝑛 βˆ’ 1))(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0))) β†’ ((𝑛 ∈ (1...𝑁) ∧ βˆ€π‘ ∈ {𝑛} (0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)) β†’ (𝑛 ∈ (1...𝑁) ∧ βˆ€π‘ ∈ (1...𝑛)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0))))
227226com12 32 . . . . . . . . . . . . 13 ((𝑛 ∈ (1...𝑁) ∧ βˆ€π‘ ∈ {𝑛} (0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)) β†’ (((𝑛 βˆ’ 1) = 0 ∨ ((𝑛 βˆ’ 1) ∈ (1...𝑁) ∧ βˆ€π‘ ∈ (1...(𝑛 βˆ’ 1))(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0))) β†’ (𝑛 ∈ (1...𝑁) ∧ βˆ€π‘ ∈ (1...𝑛)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0))))
228 elun 4140 . . . . . . . . . . . . . 14 ((𝑛 βˆ’ 1) ∈ ({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)}) ↔ ((𝑛 βˆ’ 1) ∈ {0} ∨ (𝑛 βˆ’ 1) ∈ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)}))
229 ovex 7434 . . . . . . . . . . . . . . . 16 (𝑛 βˆ’ 1) ∈ V
230229elsn 4635 . . . . . . . . . . . . . . 15 ((𝑛 βˆ’ 1) ∈ {0} ↔ (𝑛 βˆ’ 1) = 0)
231 oveq2 7409 . . . . . . . . . . . . . . . . 17 (π‘Ž = (𝑛 βˆ’ 1) β†’ (1...π‘Ž) = (1...(𝑛 βˆ’ 1)))
232231raleqdv 3317 . . . . . . . . . . . . . . . 16 (π‘Ž = (𝑛 βˆ’ 1) β†’ (βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0) ↔ βˆ€π‘ ∈ (1...(𝑛 βˆ’ 1))(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)))
233232elrab 3675 . . . . . . . . . . . . . . 15 ((𝑛 βˆ’ 1) ∈ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)} ↔ ((𝑛 βˆ’ 1) ∈ (1...𝑁) ∧ βˆ€π‘ ∈ (1...(𝑛 βˆ’ 1))(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)))
234230, 233orbi12i 911 . . . . . . . . . . . . . 14 (((𝑛 βˆ’ 1) ∈ {0} ∨ (𝑛 βˆ’ 1) ∈ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)}) ↔ ((𝑛 βˆ’ 1) = 0 ∨ ((𝑛 βˆ’ 1) ∈ (1...𝑁) ∧ βˆ€π‘ ∈ (1...(𝑛 βˆ’ 1))(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0))))
235228, 234bitri 275 . . . . . . . . . . . . 13 ((𝑛 βˆ’ 1) ∈ ({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)}) ↔ ((𝑛 βˆ’ 1) = 0 ∨ ((𝑛 βˆ’ 1) ∈ (1...𝑁) ∧ βˆ€π‘ ∈ (1...(𝑛 βˆ’ 1))(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0))))
236 oveq2 7409 . . . . . . . . . . . . . . 15 (π‘Ž = 𝑛 β†’ (1...π‘Ž) = (1...𝑛))
237236raleqdv 3317 . . . . . . . . . . . . . 14 (π‘Ž = 𝑛 β†’ (βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0) ↔ βˆ€π‘ ∈ (1...𝑛)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)))
238237elrab 3675 . . . . . . . . . . . . 13 (𝑛 ∈ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)} ↔ (𝑛 ∈ (1...𝑁) ∧ βˆ€π‘ ∈ (1...𝑛)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)))
239227, 235, 2383imtr4g 296 . . . . . . . . . . . 12 ((𝑛 ∈ (1...𝑁) ∧ βˆ€π‘ ∈ {𝑛} (0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)) β†’ ((𝑛 βˆ’ 1) ∈ ({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)}) β†’ 𝑛 ∈ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)}))
240 elun2 4169 . . . . . . . . . . . 12 (𝑛 ∈ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)} β†’ 𝑛 ∈ ({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)}))
241239, 240syl6 35 . . . . . . . . . . 11 ((𝑛 ∈ (1...𝑁) ∧ βˆ€π‘ ∈ {𝑛} (0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)) β†’ ((𝑛 βˆ’ 1) ∈ ({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)}) β†’ 𝑛 ∈ ({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)})))
24296, 188, 241syl2anc 583 . . . . . . . . . 10 (((πœ‘ ∧ π‘˜ ∈ β„•) ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟢(0...π‘˜) ∧ (π‘β€˜π‘›) = π‘˜)) β†’ ((𝑛 βˆ’ 1) ∈ ({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)}) β†’ 𝑛 ∈ ({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)})))
243 fimaxre2 12156 . . . . . . . . . . . . 13 ((({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)}) βŠ† ℝ ∧ ({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)}) ∈ Fin) β†’ βˆƒπ‘– ∈ ℝ βˆ€π‘— ∈ ({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)})𝑗 ≀ 𝑖)
24442, 28, 243mp2an 689 . . . . . . . . . . . 12 βˆƒπ‘– ∈ ℝ βˆ€π‘— ∈ ({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)})𝑗 ≀ 𝑖
24542, 33, 2443pm3.2i 1336 . . . . . . . . . . 11 (({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)}) βŠ† ℝ ∧ ({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)}) β‰  βˆ… ∧ βˆƒπ‘– ∈ ℝ βˆ€π‘— ∈ ({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)})𝑗 ≀ 𝑖)
246245suprubii 12186 . . . . . . . . . 10 (𝑛 ∈ ({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)}) β†’ 𝑛 ≀ sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)}), ℝ, < ))
247242, 246syl6 35 . . . . . . . . 9 (((πœ‘ ∧ π‘˜ ∈ β„•) ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟢(0...π‘˜) ∧ (π‘β€˜π‘›) = π‘˜)) β†’ ((𝑛 βˆ’ 1) ∈ ({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)}) β†’ 𝑛 ≀ sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)}), ℝ, < )))
248 ltm1 12053 . . . . . . . . . 10 (𝑛 ∈ ℝ β†’ (𝑛 βˆ’ 1) < 𝑛)
249 peano2rem 11524 . . . . . . . . . . 11 (𝑛 ∈ ℝ β†’ (𝑛 βˆ’ 1) ∈ ℝ)
25042, 45sselii 3971 . . . . . . . . . . . 12 sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)}), ℝ, < ) ∈ ℝ
251 ltletr 11303 . . . . . . . . . . . 12 (((𝑛 βˆ’ 1) ∈ ℝ ∧ 𝑛 ∈ ℝ ∧ sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)}), ℝ, < ) ∈ ℝ) β†’ (((𝑛 βˆ’ 1) < 𝑛 ∧ 𝑛 ≀ sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)}), ℝ, < )) β†’ (𝑛 βˆ’ 1) < sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)}), ℝ, < )))
252250, 251mp3an3 1446 . . . . . . . . . . 11 (((𝑛 βˆ’ 1) ∈ ℝ ∧ 𝑛 ∈ ℝ) β†’ (((𝑛 βˆ’ 1) < 𝑛 ∧ 𝑛 ≀ sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)}), ℝ, < )) β†’ (𝑛 βˆ’ 1) < sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)}), ℝ, < )))
253249, 252mpancom 685 . . . . . . . . . 10 (𝑛 ∈ ℝ β†’ (((𝑛 βˆ’ 1) < 𝑛 ∧ 𝑛 ≀ sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)}), ℝ, < )) β†’ (𝑛 βˆ’ 1) < sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)}), ℝ, < )))
254248, 253mpand 692 . . . . . . . . 9 (𝑛 ∈ ℝ β†’ (𝑛 ≀ sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)}), ℝ, < ) β†’ (𝑛 βˆ’ 1) < sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)}), ℝ, < )))
25595, 247, 254sylsyld 61 . . . . . . . 8 (((πœ‘ ∧ π‘˜ ∈ β„•) ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟢(0...π‘˜) ∧ (π‘β€˜π‘›) = π‘˜)) β†’ ((𝑛 βˆ’ 1) ∈ ({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)}) β†’ (𝑛 βˆ’ 1) < sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)}), ℝ, < )))
256250ltnri 11320 . . . . . . . . . 10 Β¬ sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)}), ℝ, < ) < sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)}), ℝ, < )
257 breq1 5141 . . . . . . . . . 10 (sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)}), ℝ, < ) = (𝑛 βˆ’ 1) β†’ (sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)}), ℝ, < ) < sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)}), ℝ, < ) ↔ (𝑛 βˆ’ 1) < sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)}), ℝ, < )))
258256, 257mtbii 326 . . . . . . . . 9 (sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)}), ℝ, < ) = (𝑛 βˆ’ 1) β†’ Β¬ (𝑛 βˆ’ 1) < sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)}), ℝ, < ))
259258necon2ai 2962 . . . . . . . 8 ((𝑛 βˆ’ 1) < sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)}), ℝ, < ) β†’ sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)}), ℝ, < ) β‰  (𝑛 βˆ’ 1))
260255, 259syl6 35 . . . . . . 7 (((πœ‘ ∧ π‘˜ ∈ β„•) ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟢(0...π‘˜) ∧ (π‘β€˜π‘›) = π‘˜)) β†’ ((𝑛 βˆ’ 1) ∈ ({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)}) β†’ sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)}), ℝ, < ) β‰  (𝑛 βˆ’ 1)))
261 eleq1 2813 . . . . . . . . 9 (sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)}), ℝ, < ) = (𝑛 βˆ’ 1) β†’ (sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)}), ℝ, < ) ∈ ({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)}) ↔ (𝑛 βˆ’ 1) ∈ ({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)})))
26245, 261mpbii 232 . . . . . . . 8 (sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)}), ℝ, < ) = (𝑛 βˆ’ 1) β†’ (𝑛 βˆ’ 1) ∈ ({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)}))
263262necon3bi 2959 . . . . . . 7 (Β¬ (𝑛 βˆ’ 1) ∈ ({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)}) β†’ sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)}), ℝ, < ) β‰  (𝑛 βˆ’ 1))
264260, 263pm2.61d1 180 . . . . . 6 (((πœ‘ ∧ π‘˜ ∈ β„•) ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟢(0...π‘˜) ∧ (π‘β€˜π‘›) = π‘˜)) β†’ sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(𝑝 ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (π‘β€˜π‘) β‰  0)}), ℝ, < ) β‰  (𝑛 βˆ’ 1))
2652, 12, 48, 92, 264, 176poimirlem28 37006 . . . . 5 ((πœ‘ ∧ π‘˜ ∈ β„•) β†’ βˆƒπ‘  ∈ (((0..^π‘˜) ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)})βˆ€π‘– ∈ (0...𝑁)βˆƒπ‘— ∈ (0...𝑁)𝑖 = sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})))β€˜π‘) β‰  0)}), ℝ, < ))
266 nn0ex 12475 . . . . . . . . . . . 12 β„•0 ∈ V
267 fzo0ssnn0 13710 . . . . . . . . . . . 12 (0..^π‘˜) βŠ† β„•0
268 mapss 8879 . . . . . . . . . . . 12 ((β„•0 ∈ V ∧ (0..^π‘˜) βŠ† β„•0) β†’ ((0..^π‘˜) ↑m (1...𝑁)) βŠ† (β„•0 ↑m (1...𝑁)))
269266, 267, 268mp2an 689 . . . . . . . . . . 11 ((0..^π‘˜) ↑m (1...𝑁)) βŠ† (β„•0 ↑m (1...𝑁))
270 xpss1 5685 . . . . . . . . . . 11 (((0..^π‘˜) ↑m (1...𝑁)) βŠ† (β„•0 ↑m (1...𝑁)) β†’ (((0..^π‘˜) ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)}) βŠ† ((β„•0 ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)}))
271269, 270ax-mp 5 . . . . . . . . . 10 (((0..^π‘˜) ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)}) βŠ† ((β„•0 ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)})
272271sseli 3970 . . . . . . . . 9 (𝑠 ∈ (((0..^π‘˜) ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)}) β†’ 𝑠 ∈ ((β„•0 ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)}))
273 xp1st 8000 . . . . . . . . . 10 (𝑠 ∈ (((0..^π‘˜) ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)}) β†’ (1st β€˜π‘ ) ∈ ((0..^π‘˜) ↑m (1...𝑁)))
274 elmapi 8839 . . . . . . . . . 10 ((1st β€˜π‘ ) ∈ ((0..^π‘˜) ↑m (1...𝑁)) β†’ (1st β€˜π‘ ):(1...𝑁)⟢(0..^π‘˜))
275 frn 6714 . . . . . . . . . 10 ((1st β€˜π‘ ):(1...𝑁)⟢(0..^π‘˜) β†’ ran (1st β€˜π‘ ) βŠ† (0..^π‘˜))
276273, 274, 2753syl 18 . . . . . . . . 9 (𝑠 ∈ (((0..^π‘˜) ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)}) β†’ ran (1st β€˜π‘ ) βŠ† (0..^π‘˜))
277272, 276jca 511 . . . . . . . 8 (𝑠 ∈ (((0..^π‘˜) ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)}) β†’ (𝑠 ∈ ((β„•0 ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)}) ∧ ran (1st β€˜π‘ ) βŠ† (0..^π‘˜)))
278277anim1i 614 . . . . . . 7 ((𝑠 ∈ (((0..^π‘˜) ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)}) ∧ βˆ€π‘– ∈ (0...𝑁)βˆƒπ‘— ∈ (0...𝑁)𝑖 = sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})))β€˜π‘) β‰  0)}), ℝ, < )) β†’ ((𝑠 ∈ ((β„•0 ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)}) ∧ ran (1st β€˜π‘ ) βŠ† (0..^π‘˜)) ∧ βˆ€π‘– ∈ (0...𝑁)βˆƒπ‘— ∈ (0...𝑁)𝑖 = sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})))β€˜π‘) β‰  0)}), ℝ, < )))
279 anass 468 . . . . . . 7 (((𝑠 ∈ ((β„•0 ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)}) ∧ ran (1st β€˜π‘ ) βŠ† (0..^π‘˜)) ∧ βˆ€π‘– ∈ (0...𝑁)βˆƒπ‘— ∈ (0...𝑁)𝑖 = sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})))β€˜π‘) β‰  0)}), ℝ, < )) ↔ (𝑠 ∈ ((β„•0 ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)}) ∧ (ran (1st β€˜π‘ ) βŠ† (0..^π‘˜) ∧ βˆ€π‘– ∈ (0...𝑁)βˆƒπ‘— ∈ (0...𝑁)𝑖 = sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})))β€˜π‘) β‰  0)}), ℝ, < ))))
280278, 279sylib 217 . . . . . 6 ((𝑠 ∈ (((0..^π‘˜) ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)}) ∧ βˆ€π‘– ∈ (0...𝑁)βˆƒπ‘— ∈ (0...𝑁)𝑖 = sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})))β€˜π‘) β‰  0)}), ℝ, < )) β†’ (𝑠 ∈ ((β„•0 ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)}) ∧ (ran (1st β€˜π‘ ) βŠ† (0..^π‘˜) ∧ βˆ€π‘– ∈ (0...𝑁)βˆƒπ‘— ∈ (0...𝑁)𝑖 = sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})))β€˜π‘) β‰  0)}), ℝ, < ))))
281280reximi2 3071 . . . . 5 (βˆƒπ‘  ∈ (((0..^π‘˜) ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)})βˆ€π‘– ∈ (0...𝑁)βˆƒπ‘— ∈ (0...𝑁)𝑖 = sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})))β€˜π‘) β‰  0)}), ℝ, < ) β†’ βˆƒπ‘  ∈ ((β„•0 ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)})(ran (1st β€˜π‘ ) βŠ† (0..^π‘˜) ∧ βˆ€π‘– ∈ (0...𝑁)βˆƒπ‘— ∈ (0...𝑁)𝑖 = sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})))β€˜π‘) β‰  0)}), ℝ, < )))
282265, 281syl 17 . . . 4 ((πœ‘ ∧ π‘˜ ∈ β„•) β†’ βˆƒπ‘  ∈ ((β„•0 ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)})(ran (1st β€˜π‘ ) βŠ† (0..^π‘˜) ∧ βˆ€π‘– ∈ (0...𝑁)βˆƒπ‘— ∈ (0...𝑁)𝑖 = sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})))β€˜π‘) β‰  0)}), ℝ, < )))
283282ralrimiva 3138 . . 3 (πœ‘ β†’ βˆ€π‘˜ ∈ β„• βˆƒπ‘  ∈ ((β„•0 ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)})(ran (1st β€˜π‘ ) βŠ† (0..^π‘˜) ∧ βˆ€π‘– ∈ (0...𝑁)βˆƒπ‘— ∈ (0...𝑁)𝑖 = sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})))β€˜π‘) β‰  0)}), ℝ, < )))
284 nnex 12215 . . . 4 β„• ∈ V
285140, 266ixpconst 8897 . . . . . . 7 X𝑛 ∈ (1...𝑁)β„•0 = (β„•0 ↑m (1...𝑁))
286 omelon 9637 . . . . . . . . . 10 Ο‰ ∈ On
287 nn0ennn 13941 . . . . . . . . . . 11 β„•0 β‰ˆ β„•
288 nnenom 13942 . . . . . . . . . . 11 β„• β‰ˆ Ο‰
289287, 288entr2i 9001 . . . . . . . . . 10 Ο‰ β‰ˆ β„•0
290 isnumi 9937 . . . . . . . . . 10 ((Ο‰ ∈ On ∧ Ο‰ β‰ˆ β„•0) β†’ β„•0 ∈ dom card)
291286, 289, 290mp2an 689 . . . . . . . . 9 β„•0 ∈ dom card
292291rgenw 3057 . . . . . . . 8 βˆ€π‘› ∈ (1...𝑁)β„•0 ∈ dom card
293 finixpnum 36963 . . . . . . . 8 (((1...𝑁) ∈ Fin ∧ βˆ€π‘› ∈ (1...𝑁)β„•0 ∈ dom card) β†’ X𝑛 ∈ (1...𝑁)β„•0 ∈ dom card)
29424, 292, 293mp2an 689 . . . . . . 7 X𝑛 ∈ (1...𝑁)β„•0 ∈ dom card
295285, 294eqeltrri 2822 . . . . . 6 (β„•0 ↑m (1...𝑁)) ∈ dom card
296140, 140mapval 8828 . . . . . . . . 9 ((1...𝑁) ↑m (1...𝑁)) = {𝑓 ∣ 𝑓:(1...𝑁)⟢(1...𝑁)}
297 mapfi 9344 . . . . . . . . . 10 (((1...𝑁) ∈ Fin ∧ (1...𝑁) ∈ Fin) β†’ ((1...𝑁) ↑m (1...𝑁)) ∈ Fin)
29824, 24, 297mp2an 689 . . . . . . . . 9 ((1...𝑁) ↑m (1...𝑁)) ∈ Fin
299296, 298eqeltrri 2822 . . . . . . . 8 {𝑓 ∣ 𝑓:(1...𝑁)⟢(1...𝑁)} ∈ Fin
300 f1of 6823 . . . . . . . . 9 (𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁) β†’ 𝑓:(1...𝑁)⟢(1...𝑁))
301300ss2abi 4055 . . . . . . . 8 {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)} βŠ† {𝑓 ∣ 𝑓:(1...𝑁)⟢(1...𝑁)}
302 ssfi 9169 . . . . . . . 8 (({𝑓 ∣ 𝑓:(1...𝑁)⟢(1...𝑁)} ∈ Fin ∧ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)} βŠ† {𝑓 ∣ 𝑓:(1...𝑁)⟢(1...𝑁)}) β†’ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)} ∈ Fin)
303299, 301, 302mp2an 689 . . . . . . 7 {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)} ∈ Fin
304 finnum 9939 . . . . . . 7 ({𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)} ∈ Fin β†’ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)} ∈ dom card)
305303, 304ax-mp 5 . . . . . 6 {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)} ∈ dom card
306 xpnum 9942 . . . . . 6 (((β„•0 ↑m (1...𝑁)) ∈ dom card ∧ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)} ∈ dom card) β†’ ((β„•0 ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)}) ∈ dom card)
307295, 305, 306mp2an 689 . . . . 5 ((β„•0 ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)}) ∈ dom card
308 ssrab2 4069 . . . . . . . 8 {𝑠 ∈ ((β„•0 ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)}) ∣ (ran (1st β€˜π‘ ) βŠ† (0..^π‘˜) ∧ βˆ€π‘– ∈ (0...𝑁)βˆƒπ‘— ∈ (0...𝑁)𝑖 = sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})))β€˜π‘) β‰  0)}), ℝ, < ))} βŠ† ((β„•0 ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)})
309308rgenw 3057 . . . . . . 7 βˆ€π‘˜ ∈ β„• {𝑠 ∈ ((β„•0 ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)}) ∣ (ran (1st β€˜π‘ ) βŠ† (0..^π‘˜) ∧ βˆ€π‘– ∈ (0...𝑁)βˆƒπ‘— ∈ (0...𝑁)𝑖 = sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})))β€˜π‘) β‰  0)}), ℝ, < ))} βŠ† ((β„•0 ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)})
310 ss2iun 5005 . . . . . . 7 (βˆ€π‘˜ ∈ β„• {𝑠 ∈ ((β„•0 ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)}) ∣ (ran (1st β€˜π‘ ) βŠ† (0..^π‘˜) ∧ βˆ€π‘– ∈ (0...𝑁)βˆƒπ‘— ∈ (0...𝑁)𝑖 = sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})))β€˜π‘) β‰  0)}), ℝ, < ))} βŠ† ((β„•0 ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)}) β†’ βˆͺ π‘˜ ∈ β„• {𝑠 ∈ ((β„•0 ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)}) ∣ (ran (1st β€˜π‘ ) βŠ† (0..^π‘˜) ∧ βˆ€π‘– ∈ (0...𝑁)βˆƒπ‘— ∈ (0...𝑁)𝑖 = sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})))β€˜π‘) β‰  0)}), ℝ, < ))} βŠ† βˆͺ π‘˜ ∈ β„• ((β„•0 ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)}))
311309, 310ax-mp 5 . . . . . 6 βˆͺ π‘˜ ∈ β„• {𝑠 ∈ ((β„•0 ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)}) ∣ (ran (1st β€˜π‘ ) βŠ† (0..^π‘˜) ∧ βˆ€π‘– ∈ (0...𝑁)βˆƒπ‘— ∈ (0...𝑁)𝑖 = sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})))β€˜π‘) β‰  0)}), ℝ, < ))} βŠ† βˆͺ π‘˜ ∈ β„• ((β„•0 ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)})
312 1nn 12220 . . . . . . 7 1 ∈ β„•
313 ne0i 4326 . . . . . . 7 (1 ∈ β„• β†’ β„• β‰  βˆ…)
314 iunconst 4996 . . . . . . 7 (β„• β‰  βˆ… β†’ βˆͺ π‘˜ ∈ β„• ((β„•0 ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)}) = ((β„•0 ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)}))
315312, 313, 314mp2b 10 . . . . . 6 βˆͺ π‘˜ ∈ β„• ((β„•0 ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)}) = ((β„•0 ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)})
316311, 315sseqtri 4010 . . . . 5 βˆͺ π‘˜ ∈ β„• {𝑠 ∈ ((β„•0 ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)}) ∣ (ran (1st β€˜π‘ ) βŠ† (0..^π‘˜) ∧ βˆ€π‘– ∈ (0...𝑁)βˆƒπ‘— ∈ (0...𝑁)𝑖 = sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})))β€˜π‘) β‰  0)}), ℝ, < ))} βŠ† ((β„•0 ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)})
317 ssnum 10030 . . . . 5 ((((β„•0 ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)}) ∈ dom card ∧ βˆͺ π‘˜ ∈ β„• {𝑠 ∈ ((β„•0 ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)}) ∣ (ran (1st β€˜π‘ ) βŠ† (0..^π‘˜) ∧ βˆ€π‘– ∈ (0...𝑁)βˆƒπ‘— ∈ (0...𝑁)𝑖 = sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})))β€˜π‘) β‰  0)}), ℝ, < ))} βŠ† ((β„•0 ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)})) β†’ βˆͺ π‘˜ ∈ β„• {𝑠 ∈ ((β„•0 ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)}) ∣ (ran (1st β€˜π‘ ) βŠ† (0..^π‘˜) ∧ βˆ€π‘– ∈ (0...𝑁)βˆƒπ‘— ∈ (0...𝑁)𝑖 = sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})))β€˜π‘) β‰  0)}), ℝ, < ))} ∈ dom card)
318307, 316, 317mp2an 689 . . . 4 βˆͺ π‘˜ ∈ β„• {𝑠 ∈ ((β„•0 ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)}) ∣ (ran (1st β€˜π‘ ) βŠ† (0..^π‘˜) ∧ βˆ€π‘– ∈ (0...𝑁)βˆƒπ‘— ∈ (0...𝑁)𝑖 = sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})))β€˜π‘) β‰  0)}), ℝ, < ))} ∈ dom card
319 fveq2 6881 . . . . . . . 8 (𝑠 = (π‘”β€˜π‘˜) β†’ (1st β€˜π‘ ) = (1st β€˜(π‘”β€˜π‘˜)))
320319rneqd 5927 . . . . . . 7 (𝑠 = (π‘”β€˜π‘˜) β†’ ran (1st β€˜π‘ ) = ran (1st β€˜(π‘”β€˜π‘˜)))
321320sseq1d 4005 . . . . . 6 (𝑠 = (π‘”β€˜π‘˜) β†’ (ran (1st β€˜π‘ ) βŠ† (0..^π‘˜) ↔ ran (1st β€˜(π‘”β€˜π‘˜)) βŠ† (0..^π‘˜)))
322 fveq2 6881 . . . . . . . . . . . . . . . . . . . . 21 (𝑠 = (π‘”β€˜π‘˜) β†’ (2nd β€˜π‘ ) = (2nd β€˜(π‘”β€˜π‘˜)))
323322imaeq1d 6048 . . . . . . . . . . . . . . . . . . . 20 (𝑠 = (π‘”β€˜π‘˜) β†’ ((2nd β€˜π‘ ) β€œ (1...𝑗)) = ((2nd β€˜(π‘”β€˜π‘˜)) β€œ (1...𝑗)))
324323xpeq1d 5695 . . . . . . . . . . . . . . . . . . 19 (𝑠 = (π‘”β€˜π‘˜) β†’ (((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) = (((2nd β€˜(π‘”β€˜π‘˜)) β€œ (1...𝑗)) Γ— {1}))
325322imaeq1d 6048 . . . . . . . . . . . . . . . . . . . 20 (𝑠 = (π‘”β€˜π‘˜) β†’ ((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) = ((2nd β€˜(π‘”β€˜π‘˜)) β€œ ((𝑗 + 1)...𝑁)))
326325xpeq1d 5695 . . . . . . . . . . . . . . . . . . 19 (𝑠 = (π‘”β€˜π‘˜) β†’ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}) = (((2nd β€˜(π‘”β€˜π‘˜)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))
327324, 326uneq12d 4156 . . . . . . . . . . . . . . . . . 18 (𝑠 = (π‘”β€˜π‘˜) β†’ ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})) = ((((2nd β€˜(π‘”β€˜π‘˜)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘˜)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})))
328319, 327oveq12d 7419 . . . . . . . . . . . . . . . . 17 (𝑠 = (π‘”β€˜π‘˜) β†’ ((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) = ((1st β€˜(π‘”β€˜π‘˜)) ∘f + ((((2nd β€˜(π‘”β€˜π‘˜)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘˜)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))))
329328fvoveq1d 7423 . . . . . . . . . . . . . . . 16 (𝑠 = (π‘”β€˜π‘˜) β†’ (πΉβ€˜(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {π‘˜}))) = (πΉβ€˜(((1st β€˜(π‘”β€˜π‘˜)) ∘f + ((((2nd β€˜(π‘”β€˜π‘˜)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘˜)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {π‘˜}))))
330329fveq1d 6883 . . . . . . . . . . . . . . 15 (𝑠 = (π‘”β€˜π‘˜) β†’ ((πΉβ€˜(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) = ((πΉβ€˜(((1st β€˜(π‘”β€˜π‘˜)) ∘f + ((((2nd β€˜(π‘”β€˜π‘˜)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘˜)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘))
331330breq2d 5150 . . . . . . . . . . . . . 14 (𝑠 = (π‘”β€˜π‘˜) β†’ (0 ≀ ((πΉβ€˜(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ↔ 0 ≀ ((πΉβ€˜(((1st β€˜(π‘”β€˜π‘˜)) ∘f + ((((2nd β€˜(π‘”β€˜π‘˜)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘˜)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘)))
332328fveq1d 6883 . . . . . . . . . . . . . . 15 (𝑠 = (π‘”β€˜π‘˜) β†’ (((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})))β€˜π‘) = (((1st β€˜(π‘”β€˜π‘˜)) ∘f + ((((2nd β€˜(π‘”β€˜π‘˜)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘˜)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})))β€˜π‘))
333332neeq1d 2992 . . . . . . . . . . . . . 14 (𝑠 = (π‘”β€˜π‘˜) β†’ ((((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})))β€˜π‘) β‰  0 ↔ (((1st β€˜(π‘”β€˜π‘˜)) ∘f + ((((2nd β€˜(π‘”β€˜π‘˜)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘˜)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})))β€˜π‘) β‰  0))
334331, 333anbi12d 630 . . . . . . . . . . . . 13 (𝑠 = (π‘”β€˜π‘˜) β†’ ((0 ≀ ((πΉβ€˜(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})))β€˜π‘) β‰  0) ↔ (0 ≀ ((πΉβ€˜(((1st β€˜(π‘”β€˜π‘˜)) ∘f + ((((2nd β€˜(π‘”β€˜π‘˜)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘˜)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (((1st β€˜(π‘”β€˜π‘˜)) ∘f + ((((2nd β€˜(π‘”β€˜π‘˜)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘˜)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})))β€˜π‘) β‰  0)))
335334ralbidv 3169 . . . . . . . . . . . 12 (𝑠 = (π‘”β€˜π‘˜) β†’ (βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})))β€˜π‘) β‰  0) ↔ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(((1st β€˜(π‘”β€˜π‘˜)) ∘f + ((((2nd β€˜(π‘”β€˜π‘˜)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘˜)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (((1st β€˜(π‘”β€˜π‘˜)) ∘f + ((((2nd β€˜(π‘”β€˜π‘˜)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘˜)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})))β€˜π‘) β‰  0)))
336335rabbidv 3432 . . . . . . . . . . 11 (𝑠 = (π‘”β€˜π‘˜) β†’ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})))β€˜π‘) β‰  0)} = {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(((1st β€˜(π‘”β€˜π‘˜)) ∘f + ((((2nd β€˜(π‘”β€˜π‘˜)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘˜)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (((1st β€˜(π‘”β€˜π‘˜)) ∘f + ((((2nd β€˜(π‘”β€˜π‘˜)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘˜)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})))β€˜π‘) β‰  0)})
337336uneq2d 4155 . . . . . . . . . 10 (𝑠 = (π‘”β€˜π‘˜) β†’ ({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})))β€˜π‘) β‰  0)}) = ({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(((1st β€˜(π‘”β€˜π‘˜)) ∘f + ((((2nd β€˜(π‘”β€˜π‘˜)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘˜)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (((1st β€˜(π‘”β€˜π‘˜)) ∘f + ((((2nd β€˜(π‘”β€˜π‘˜)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘˜)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})))β€˜π‘) β‰  0)}))
338337supeq1d 9437 . . . . . . . . 9 (𝑠 = (π‘”β€˜π‘˜) β†’ sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})))β€˜π‘) β‰  0)}), ℝ, < ) = sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(((1st β€˜(π‘”β€˜π‘˜)) ∘f + ((((2nd β€˜(π‘”β€˜π‘˜)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘˜)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (((1st β€˜(π‘”β€˜π‘˜)) ∘f + ((((2nd β€˜(π‘”β€˜π‘˜)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘˜)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})))β€˜π‘) β‰  0)}), ℝ, < ))
339338eqeq2d 2735 . . . . . . . 8 (𝑠 = (π‘”β€˜π‘˜) β†’ (𝑖 = sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})))β€˜π‘) β‰  0)}), ℝ, < ) ↔ 𝑖 = sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(((1st β€˜(π‘”β€˜π‘˜)) ∘f + ((((2nd β€˜(π‘”β€˜π‘˜)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘˜)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (((1st β€˜(π‘”β€˜π‘˜)) ∘f + ((((2nd β€˜(π‘”β€˜π‘˜)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘˜)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})))β€˜π‘) β‰  0)}), ℝ, < )))
340339rexbidv 3170 . . . . . . 7 (𝑠 = (π‘”β€˜π‘˜) β†’ (βˆƒπ‘— ∈ (0...𝑁)𝑖 = sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})))β€˜π‘) β‰  0)}), ℝ, < ) ↔ βˆƒπ‘— ∈ (0...𝑁)𝑖 = sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(((1st β€˜(π‘”β€˜π‘˜)) ∘f + ((((2nd β€˜(π‘”β€˜π‘˜)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘˜)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (((1st β€˜(π‘”β€˜π‘˜)) ∘f + ((((2nd β€˜(π‘”β€˜π‘˜)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘˜)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})))β€˜π‘) β‰  0)}), ℝ, < )))
341340ralbidv 3169 . . . . . 6 (𝑠 = (π‘”β€˜π‘˜) β†’ (βˆ€π‘– ∈ (0...𝑁)βˆƒπ‘— ∈ (0...𝑁)𝑖 = sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})))β€˜π‘) β‰  0)}), ℝ, < ) ↔ βˆ€π‘– ∈ (0...𝑁)βˆƒπ‘— ∈ (0...𝑁)𝑖 = sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(((1st β€˜(π‘”β€˜π‘˜)) ∘f + ((((2nd β€˜(π‘”β€˜π‘˜)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘˜)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (((1st β€˜(π‘”β€˜π‘˜)) ∘f + ((((2nd β€˜(π‘”β€˜π‘˜)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘˜)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})))β€˜π‘) β‰  0)}), ℝ, < )))
342321, 341anbi12d 630 . . . . 5 (𝑠 = (π‘”β€˜π‘˜) β†’ ((ran (1st β€˜π‘ ) βŠ† (0..^π‘˜) ∧ βˆ€π‘– ∈ (0...𝑁)βˆƒπ‘— ∈ (0...𝑁)𝑖 = sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})))β€˜π‘) β‰  0)}), ℝ, < )) ↔ (ran (1st β€˜(π‘”β€˜π‘˜)) βŠ† (0..^π‘˜) ∧ βˆ€π‘– ∈ (0...𝑁)βˆƒπ‘— ∈ (0...𝑁)𝑖 = sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(((1st β€˜(π‘”β€˜π‘˜)) ∘f + ((((2nd β€˜(π‘”β€˜π‘˜)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘˜)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (((1st β€˜(π‘”β€˜π‘˜)) ∘f + ((((2nd β€˜(π‘”β€˜π‘˜)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘˜)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})))β€˜π‘) β‰  0)}), ℝ, < ))))
343342ac6num 10470 . . . 4 ((β„• ∈ V ∧ βˆͺ π‘˜ ∈ β„• {𝑠 ∈ ((β„•0 ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)}) ∣ (ran (1st β€˜π‘ ) βŠ† (0..^π‘˜) ∧ βˆ€π‘– ∈ (0...𝑁)βˆƒπ‘— ∈ (0...𝑁)𝑖 = sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})))β€˜π‘) β‰  0)}), ℝ, < ))} ∈ dom card ∧ βˆ€π‘˜ ∈ β„• βˆƒπ‘  ∈ ((β„•0 ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)})(ran (1st β€˜π‘ ) βŠ† (0..^π‘˜) ∧ βˆ€π‘– ∈ (0...𝑁)βˆƒπ‘— ∈ (0...𝑁)𝑖 = sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})))β€˜π‘) β‰  0)}), ℝ, < ))) β†’ βˆƒπ‘”(𝑔:β„•βŸΆ((β„•0 ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)}) ∧ βˆ€π‘˜ ∈ β„• (ran (1st β€˜(π‘”β€˜π‘˜)) βŠ† (0..^π‘˜) ∧ βˆ€π‘– ∈ (0...𝑁)βˆƒπ‘— ∈ (0...𝑁)𝑖 = sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(((1st β€˜(π‘”β€˜π‘˜)) ∘f + ((((2nd β€˜(π‘”β€˜π‘˜)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘˜)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (((1st β€˜(π‘”β€˜π‘˜)) ∘f + ((((2nd β€˜(π‘”β€˜π‘˜)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘˜)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})))β€˜π‘) β‰  0)}), ℝ, < ))))
344284, 318, 343mp3an12 1447 . . 3 (βˆ€π‘˜ ∈ β„• βˆƒπ‘  ∈ ((β„•0 ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)})(ran (1st β€˜π‘ ) βŠ† (0..^π‘˜) ∧ βˆ€π‘– ∈ (0...𝑁)βˆƒπ‘— ∈ (0...𝑁)𝑖 = sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (((1st β€˜π‘ ) ∘f + ((((2nd β€˜π‘ ) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜π‘ ) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})))β€˜π‘) β‰  0)}), ℝ, < )) β†’ βˆƒπ‘”(𝑔:β„•βŸΆ((β„•0 ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)}) ∧ βˆ€π‘˜ ∈ β„• (ran (1st β€˜(π‘”β€˜π‘˜)) βŠ† (0..^π‘˜) ∧ βˆ€π‘– ∈ (0...𝑁)βˆƒπ‘— ∈ (0...𝑁)𝑖 = sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(((1st β€˜(π‘”β€˜π‘˜)) ∘f + ((((2nd β€˜(π‘”β€˜π‘˜)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘˜)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (((1st β€˜(π‘”β€˜π‘˜)) ∘f + ((((2nd β€˜(π‘”β€˜π‘˜)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘˜)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})))β€˜π‘) β‰  0)}), ℝ, < ))))
345283, 344syl 17 . 2 (πœ‘ β†’ βˆƒπ‘”(𝑔:β„•βŸΆ((β„•0 ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)}) ∧ βˆ€π‘˜ ∈ β„• (ran (1st β€˜(π‘”β€˜π‘˜)) βŠ† (0..^π‘˜) ∧ βˆ€π‘– ∈ (0...𝑁)βˆƒπ‘— ∈ (0...𝑁)𝑖 = sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(((1st β€˜(π‘”β€˜π‘˜)) ∘f + ((((2nd β€˜(π‘”β€˜π‘˜)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘˜)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (((1st β€˜(π‘”β€˜π‘˜)) ∘f + ((((2nd β€˜(π‘”β€˜π‘˜)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘˜)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})))β€˜π‘) β‰  0)}), ℝ, < ))))
3461ad2antrr 723 . . . 4 (((πœ‘ ∧ 𝑔:β„•βŸΆ((β„•0 ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)})) ∧ βˆ€π‘˜ ∈ β„• (ran (1st β€˜(π‘”β€˜π‘˜)) βŠ† (0..^π‘˜) ∧ βˆ€π‘– ∈ (0...𝑁)βˆƒπ‘— ∈ (0...𝑁)𝑖 = sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(((1st β€˜(π‘”β€˜π‘˜)) ∘f + ((((2nd β€˜(π‘”β€˜π‘˜)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘˜)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (((1st β€˜(π‘”β€˜π‘˜)) ∘f + ((((2nd β€˜(π‘”β€˜π‘˜)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘˜)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})))β€˜π‘) β‰  0)}), ℝ, < ))) β†’ 𝑁 ∈ β„•)
347 poimir.r . . . 4 𝑅 = (∏tβ€˜((1...𝑁) Γ— {(topGenβ€˜ran (,))}))
348 poimir.1 . . . . 5 (πœ‘ β†’ 𝐹 ∈ ((𝑅 β†Ύt 𝐼) Cn 𝑅))
349348ad2antrr 723 . . . 4 (((πœ‘ ∧ 𝑔:β„•βŸΆ((β„•0 ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)})) ∧ βˆ€π‘˜ ∈ β„• (ran (1st β€˜(π‘”β€˜π‘˜)) βŠ† (0..^π‘˜) ∧ βˆ€π‘– ∈ (0...𝑁)βˆƒπ‘— ∈ (0...𝑁)𝑖 = sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(((1st β€˜(π‘”β€˜π‘˜)) ∘f + ((((2nd β€˜(π‘”β€˜π‘˜)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘˜)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (((1st β€˜(π‘”β€˜π‘˜)) ∘f + ((((2nd β€˜(π‘”β€˜π‘˜)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘˜)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})))β€˜π‘) β‰  0)}), ℝ, < ))) β†’ 𝐹 ∈ ((𝑅 β†Ύt 𝐼) Cn 𝑅))
350 eqid 2724 . . . 4 ((πΉβ€˜(((1st β€˜(π‘”β€˜π‘)) ∘f + ((((2nd β€˜(π‘”β€˜π‘)) β€œ (1...π‘š)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘)) β€œ ((π‘š + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {𝑝})))β€˜π‘›) = ((πΉβ€˜(((1st β€˜(π‘”β€˜π‘)) ∘f + ((((2nd β€˜(π‘”β€˜π‘)) β€œ (1...π‘š)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘)) β€œ ((π‘š + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {𝑝})))β€˜π‘›)
351 simplr 766 . . . 4 (((πœ‘ ∧ 𝑔:β„•βŸΆ((β„•0 ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)})) ∧ βˆ€π‘˜ ∈ β„• (ran (1st β€˜(π‘”β€˜π‘˜)) βŠ† (0..^π‘˜) ∧ βˆ€π‘– ∈ (0...𝑁)βˆƒπ‘— ∈ (0...𝑁)𝑖 = sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(((1st β€˜(π‘”β€˜π‘˜)) ∘f + ((((2nd β€˜(π‘”β€˜π‘˜)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘˜)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (((1st β€˜(π‘”β€˜π‘˜)) ∘f + ((((2nd β€˜(π‘”β€˜π‘˜)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘˜)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})))β€˜π‘) β‰  0)}), ℝ, < ))) β†’ 𝑔:β„•βŸΆ((β„•0 ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)}))
352 simpl 482 . . . . . . 7 ((ran (1st β€˜(π‘”β€˜π‘˜)) βŠ† (0..^π‘˜) ∧ βˆ€π‘– ∈ (0...𝑁)βˆƒπ‘— ∈ (0...𝑁)𝑖 = sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(((1st β€˜(π‘”β€˜π‘˜)) ∘f + ((((2nd β€˜(π‘”β€˜π‘˜)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘˜)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (((1st β€˜(π‘”β€˜π‘˜)) ∘f + ((((2nd β€˜(π‘”β€˜π‘˜)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘˜)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})))β€˜π‘) β‰  0)}), ℝ, < )) β†’ ran (1st β€˜(π‘”β€˜π‘˜)) βŠ† (0..^π‘˜))
353352ralimi 3075 . . . . . 6 (βˆ€π‘˜ ∈ β„• (ran (1st β€˜(π‘”β€˜π‘˜)) βŠ† (0..^π‘˜) ∧ βˆ€π‘– ∈ (0...𝑁)βˆƒπ‘— ∈ (0...𝑁)𝑖 = sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(((1st β€˜(π‘”β€˜π‘˜)) ∘f + ((((2nd β€˜(π‘”β€˜π‘˜)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘˜)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (((1st β€˜(π‘”β€˜π‘˜)) ∘f + ((((2nd β€˜(π‘”β€˜π‘˜)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘˜)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})))β€˜π‘) β‰  0)}), ℝ, < )) β†’ βˆ€π‘˜ ∈ β„• ran (1st β€˜(π‘”β€˜π‘˜)) βŠ† (0..^π‘˜))
354353adantl 481 . . . . 5 (((πœ‘ ∧ 𝑔:β„•βŸΆ((β„•0 ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)})) ∧ βˆ€π‘˜ ∈ β„• (ran (1st β€˜(π‘”β€˜π‘˜)) βŠ† (0..^π‘˜) ∧ βˆ€π‘– ∈ (0...𝑁)βˆƒπ‘— ∈ (0...𝑁)𝑖 = sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(((1st β€˜(π‘”β€˜π‘˜)) ∘f + ((((2nd β€˜(π‘”β€˜π‘˜)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘˜)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (((1st β€˜(π‘”β€˜π‘˜)) ∘f + ((((2nd β€˜(π‘”β€˜π‘˜)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘˜)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})))β€˜π‘) β‰  0)}), ℝ, < ))) β†’ βˆ€π‘˜ ∈ β„• ran (1st β€˜(π‘”β€˜π‘˜)) βŠ† (0..^π‘˜))
355 2fveq3 6886 . . . . . . . 8 (π‘˜ = 𝑝 β†’ (1st β€˜(π‘”β€˜π‘˜)) = (1st β€˜(π‘”β€˜π‘)))
356355rneqd 5927 . . . . . . 7 (π‘˜ = 𝑝 β†’ ran (1st β€˜(π‘”β€˜π‘˜)) = ran (1st β€˜(π‘”β€˜π‘)))
357 oveq2 7409 . . . . . . 7 (π‘˜ = 𝑝 β†’ (0..^π‘˜) = (0..^𝑝))
358356, 357sseq12d 4007 . . . . . 6 (π‘˜ = 𝑝 β†’ (ran (1st β€˜(π‘”β€˜π‘˜)) βŠ† (0..^π‘˜) ↔ ran (1st β€˜(π‘”β€˜π‘)) βŠ† (0..^𝑝)))
359358rspccva 3603 . . . . 5 ((βˆ€π‘˜ ∈ β„• ran (1st β€˜(π‘”β€˜π‘˜)) βŠ† (0..^π‘˜) ∧ 𝑝 ∈ β„•) β†’ ran (1st β€˜(π‘”β€˜π‘)) βŠ† (0..^𝑝))
360354, 359sylan 579 . . . 4 ((((πœ‘ ∧ 𝑔:β„•βŸΆ((β„•0 ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)})) ∧ βˆ€π‘˜ ∈ β„• (ran (1st β€˜(π‘”β€˜π‘˜)) βŠ† (0..^π‘˜) ∧ βˆ€π‘– ∈ (0...𝑁)βˆƒπ‘— ∈ (0...𝑁)𝑖 = sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(((1st β€˜(π‘”β€˜π‘˜)) ∘f + ((((2nd β€˜(π‘”β€˜π‘˜)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘˜)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (((1st β€˜(π‘”β€˜π‘˜)) ∘f + ((((2nd β€˜(π‘”β€˜π‘˜)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘˜)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})))β€˜π‘) β‰  0)}), ℝ, < ))) ∧ 𝑝 ∈ β„•) β†’ ran (1st β€˜(π‘”β€˜π‘)) βŠ† (0..^𝑝))
361 simpll 764 . . . . . 6 (((πœ‘ ∧ 𝑔:β„•βŸΆ((β„•0 ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)})) ∧ βˆ€π‘˜ ∈ β„• (ran (1st β€˜(π‘”β€˜π‘˜)) βŠ† (0..^π‘˜) ∧ βˆ€π‘– ∈ (0...𝑁)βˆƒπ‘— ∈ (0...𝑁)𝑖 = sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(((1st β€˜(π‘”β€˜π‘˜)) ∘f + ((((2nd β€˜(π‘”β€˜π‘˜)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘˜)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (((1st β€˜(π‘”β€˜π‘˜)) ∘f + ((((2nd β€˜(π‘”β€˜π‘˜)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘˜)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})))β€˜π‘) β‰  0)}), ℝ, < ))) β†’ πœ‘)
362 poimir.2 . . . . . 6 ((πœ‘ ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑧 ∈ 𝐼 ∧ (π‘§β€˜π‘›) = 0)) β†’ ((πΉβ€˜π‘§)β€˜π‘›) ≀ 0)
363361, 362sylan 579 . . . . 5 ((((πœ‘ ∧ 𝑔:β„•βŸΆ((β„•0 ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)})) ∧ βˆ€π‘˜ ∈ β„• (ran (1st β€˜(π‘”β€˜π‘˜)) βŠ† (0..^π‘˜) ∧ βˆ€π‘– ∈ (0...𝑁)βˆƒπ‘— ∈ (0...𝑁)𝑖 = sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(((1st β€˜(π‘”β€˜π‘˜)) ∘f + ((((2nd β€˜(π‘”β€˜π‘˜)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘˜)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (((1st β€˜(π‘”β€˜π‘˜)) ∘f + ((((2nd β€˜(π‘”β€˜π‘˜)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘˜)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})))β€˜π‘) β‰  0)}), ℝ, < ))) ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑧 ∈ 𝐼 ∧ (π‘§β€˜π‘›) = 0)) β†’ ((πΉβ€˜π‘§)β€˜π‘›) ≀ 0)
364 eqid 2724 . . . . 5 ((1st β€˜(π‘”β€˜π‘)) ∘f + ((((2nd β€˜(π‘”β€˜π‘)) β€œ (1...π‘š)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘)) β€œ ((π‘š + 1)...𝑁)) Γ— {0}))) = ((1st β€˜(π‘”β€˜π‘)) ∘f + ((((2nd β€˜(π‘”β€˜π‘)) β€œ (1...π‘š)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘)) β€œ ((π‘š + 1)...𝑁)) Γ— {0})))
365 simpr 484 . . . . . . . 8 ((ran (1st β€˜(π‘”β€˜π‘˜)) βŠ† (0..^π‘˜) ∧ βˆ€π‘– ∈ (0...𝑁)βˆƒπ‘— ∈ (0...𝑁)𝑖 = sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(((1st β€˜(π‘”β€˜π‘˜)) ∘f + ((((2nd β€˜(π‘”β€˜π‘˜)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘˜)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (((1st β€˜(π‘”β€˜π‘˜)) ∘f + ((((2nd β€˜(π‘”β€˜π‘˜)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘˜)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})))β€˜π‘) β‰  0)}), ℝ, < )) β†’ βˆ€π‘– ∈ (0...𝑁)βˆƒπ‘— ∈ (0...𝑁)𝑖 = sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(((1st β€˜(π‘”β€˜π‘˜)) ∘f + ((((2nd β€˜(π‘”β€˜π‘˜)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘˜)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (((1st β€˜(π‘”β€˜π‘˜)) ∘f + ((((2nd β€˜(π‘”β€˜π‘˜)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘˜)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})))β€˜π‘) β‰  0)}), ℝ, < ))
366365ralimi 3075 . . . . . . 7 (βˆ€π‘˜ ∈ β„• (ran (1st β€˜(π‘”β€˜π‘˜)) βŠ† (0..^π‘˜) ∧ βˆ€π‘– ∈ (0...𝑁)βˆƒπ‘— ∈ (0...𝑁)𝑖 = sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(((1st β€˜(π‘”β€˜π‘˜)) ∘f + ((((2nd β€˜(π‘”β€˜π‘˜)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘˜)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (((1st β€˜(π‘”β€˜π‘˜)) ∘f + ((((2nd β€˜(π‘”β€˜π‘˜)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘˜)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})))β€˜π‘) β‰  0)}), ℝ, < )) β†’ βˆ€π‘˜ ∈ β„• βˆ€π‘– ∈ (0...𝑁)βˆƒπ‘— ∈ (0...𝑁)𝑖 = sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(((1st β€˜(π‘”β€˜π‘˜)) ∘f + ((((2nd β€˜(π‘”β€˜π‘˜)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘˜)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (((1st β€˜(π‘”β€˜π‘˜)) ∘f + ((((2nd β€˜(π‘”β€˜π‘˜)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘˜)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})))β€˜π‘) β‰  0)}), ℝ, < ))
367366adantl 481 . . . . . 6 (((πœ‘ ∧ 𝑔:β„•βŸΆ((β„•0 ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)})) ∧ βˆ€π‘˜ ∈ β„• (ran (1st β€˜(π‘”β€˜π‘˜)) βŠ† (0..^π‘˜) ∧ βˆ€π‘– ∈ (0...𝑁)βˆƒπ‘— ∈ (0...𝑁)𝑖 = sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(((1st β€˜(π‘”β€˜π‘˜)) ∘f + ((((2nd β€˜(π‘”β€˜π‘˜)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘˜)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (((1st β€˜(π‘”β€˜π‘˜)) ∘f + ((((2nd β€˜(π‘”β€˜π‘˜)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘˜)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})))β€˜π‘) β‰  0)}), ℝ, < ))) β†’ βˆ€π‘˜ ∈ β„• βˆ€π‘– ∈ (0...𝑁)βˆƒπ‘— ∈ (0...𝑁)𝑖 = sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(((1st β€˜(π‘”β€˜π‘˜)) ∘f + ((((2nd β€˜(π‘”β€˜π‘˜)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘˜)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (((1st β€˜(π‘”β€˜π‘˜)) ∘f + ((((2nd β€˜(π‘”β€˜π‘˜)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘˜)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})))β€˜π‘) β‰  0)}), ℝ, < ))
368 2fveq3 6886 . . . . . . . . . . . . . . . . . . . . . 22 (π‘˜ = 𝑝 β†’ (2nd β€˜(π‘”β€˜π‘˜)) = (2nd β€˜(π‘”β€˜π‘)))
369368imaeq1d 6048 . . . . . . . . . . . . . . . . . . . . 21 (π‘˜ = 𝑝 β†’ ((2nd β€˜(π‘”β€˜π‘˜)) β€œ (1...𝑗)) = ((2nd β€˜(π‘”β€˜π‘)) β€œ (1...𝑗)))
370369xpeq1d 5695 . . . . . . . . . . . . . . . . . . . 20 (π‘˜ = 𝑝 β†’ (((2nd β€˜(π‘”β€˜π‘˜)) β€œ (1...𝑗)) Γ— {1}) = (((2nd β€˜(π‘”β€˜π‘)) β€œ (1...𝑗)) Γ— {1}))
371368imaeq1d 6048 . . . . . . . . . . . . . . . . . . . . 21 (π‘˜ = 𝑝 β†’ ((2nd β€˜(π‘”β€˜π‘˜)) β€œ ((𝑗 + 1)...𝑁)) = ((2nd β€˜(π‘”β€˜π‘)) β€œ ((𝑗 + 1)...𝑁)))
372371xpeq1d 5695 . . . . . . . . . . . . . . . . . . . 20 (π‘˜ = 𝑝 β†’ (((2nd β€˜(π‘”β€˜π‘˜)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}) = (((2nd β€˜(π‘”β€˜π‘)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))
373370, 372uneq12d 4156 . . . . . . . . . . . . . . . . . . 19 (π‘˜ = 𝑝 β†’ ((((2nd β€˜(π‘”β€˜π‘˜)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘˜)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})) = ((((2nd β€˜(π‘”β€˜π‘)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})))
374355, 373oveq12d 7419 . . . . . . . . . . . . . . . . . 18 (π‘˜ = 𝑝 β†’ ((1st β€˜(π‘”β€˜π‘˜)) ∘f + ((((2nd β€˜(π‘”β€˜π‘˜)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘˜)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) = ((1st β€˜(π‘”β€˜π‘)) ∘f + ((((2nd β€˜(π‘”β€˜π‘)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))))
375 sneq 4630 . . . . . . . . . . . . . . . . . . 19 (π‘˜ = 𝑝 β†’ {π‘˜} = {𝑝})
376375xpeq2d 5696 . . . . . . . . . . . . . . . . . 18 (π‘˜ = 𝑝 β†’ ((1...𝑁) Γ— {π‘˜}) = ((1...𝑁) Γ— {𝑝}))
377374, 376oveq12d 7419 . . . . . . . . . . . . . . . . 17 (π‘˜ = 𝑝 β†’ (((1st β€˜(π‘”β€˜π‘˜)) ∘f + ((((2nd β€˜(π‘”β€˜π‘˜)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘˜)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {π‘˜})) = (((1st β€˜(π‘”β€˜π‘)) ∘f + ((((2nd β€˜(π‘”β€˜π‘)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {𝑝})))
378377fveq2d 6885 . . . . . . . . . . . . . . . 16 (π‘˜ = 𝑝 β†’ (πΉβ€˜(((1st β€˜(π‘”β€˜π‘˜)) ∘f + ((((2nd β€˜(π‘”β€˜π‘˜)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘˜)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {π‘˜}))) = (πΉβ€˜(((1st β€˜(π‘”β€˜π‘)) ∘f + ((((2nd β€˜(π‘”β€˜π‘)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {𝑝}))))
379378fveq1d 6883 . . . . . . . . . . . . . . 15 (π‘˜ = 𝑝 β†’ ((πΉβ€˜(((1st β€˜(π‘”β€˜π‘˜)) ∘f + ((((2nd β€˜(π‘”β€˜π‘˜)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘˜)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) = ((πΉβ€˜(((1st β€˜(π‘”β€˜π‘)) ∘f + ((((2nd β€˜(π‘”β€˜π‘)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {𝑝})))β€˜π‘))
380379breq2d 5150 . . . . . . . . . . . . . 14 (π‘˜ = 𝑝 β†’ (0 ≀ ((πΉβ€˜(((1st β€˜(π‘”β€˜π‘˜)) ∘f + ((((2nd β€˜(π‘”β€˜π‘˜)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘˜)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ↔ 0 ≀ ((πΉβ€˜(((1st β€˜(π‘”β€˜π‘)) ∘f + ((((2nd β€˜(π‘”β€˜π‘)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {𝑝})))β€˜π‘)))
381374fveq1d 6883 . . . . . . . . . . . . . . 15 (π‘˜ = 𝑝 β†’ (((1st β€˜(π‘”β€˜π‘˜)) ∘f + ((((2nd β€˜(π‘”β€˜π‘˜)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘˜)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})))β€˜π‘) = (((1st β€˜(π‘”β€˜π‘)) ∘f + ((((2nd β€˜(π‘”β€˜π‘)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})))β€˜π‘))
382381neeq1d 2992 . . . . . . . . . . . . . 14 (π‘˜ = 𝑝 β†’ ((((1st β€˜(π‘”β€˜π‘˜)) ∘f + ((((2nd β€˜(π‘”β€˜π‘˜)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘˜)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})))β€˜π‘) β‰  0 ↔ (((1st β€˜(π‘”β€˜π‘)) ∘f + ((((2nd β€˜(π‘”β€˜π‘)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})))β€˜π‘) β‰  0))
383380, 382anbi12d 630 . . . . . . . . . . . . 13 (π‘˜ = 𝑝 β†’ ((0 ≀ ((πΉβ€˜(((1st β€˜(π‘”β€˜π‘˜)) ∘f + ((((2nd β€˜(π‘”β€˜π‘˜)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘˜)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (((1st β€˜(π‘”β€˜π‘˜)) ∘f + ((((2nd β€˜(π‘”β€˜π‘˜)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘˜)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})))β€˜π‘) β‰  0) ↔ (0 ≀ ((πΉβ€˜(((1st β€˜(π‘”β€˜π‘)) ∘f + ((((2nd β€˜(π‘”β€˜π‘)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {𝑝})))β€˜π‘) ∧ (((1st β€˜(π‘”β€˜π‘)) ∘f + ((((2nd β€˜(π‘”β€˜π‘)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})))β€˜π‘) β‰  0)))
384383ralbidv 3169 . . . . . . . . . . . 12 (π‘˜ = 𝑝 β†’ (βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(((1st β€˜(π‘”β€˜π‘˜)) ∘f + ((((2nd β€˜(π‘”β€˜π‘˜)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘˜)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (((1st β€˜(π‘”β€˜π‘˜)) ∘f + ((((2nd β€˜(π‘”β€˜π‘˜)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘˜)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})))β€˜π‘) β‰  0) ↔ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(((1st β€˜(π‘”β€˜π‘)) ∘f + ((((2nd β€˜(π‘”β€˜π‘)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {𝑝})))β€˜π‘) ∧ (((1st β€˜(π‘”β€˜π‘)) ∘f + ((((2nd β€˜(π‘”β€˜π‘)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})))β€˜π‘) β‰  0)))
385384rabbidv 3432 . . . . . . . . . . 11 (π‘˜ = 𝑝 β†’ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(((1st β€˜(π‘”β€˜π‘˜)) ∘f + ((((2nd β€˜(π‘”β€˜π‘˜)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘˜)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (((1st β€˜(π‘”β€˜π‘˜)) ∘f + ((((2nd β€˜(π‘”β€˜π‘˜)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘˜)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})))β€˜π‘) β‰  0)} = {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(((1st β€˜(π‘”β€˜π‘)) ∘f + ((((2nd β€˜(π‘”β€˜π‘)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {𝑝})))β€˜π‘) ∧ (((1st β€˜(π‘”β€˜π‘)) ∘f + ((((2nd β€˜(π‘”β€˜π‘)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})))β€˜π‘) β‰  0)})
386385uneq2d 4155 . . . . . . . . . 10 (π‘˜ = 𝑝 β†’ ({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(((1st β€˜(π‘”β€˜π‘˜)) ∘f + ((((2nd β€˜(π‘”β€˜π‘˜)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘˜)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (((1st β€˜(π‘”β€˜π‘˜)) ∘f + ((((2nd β€˜(π‘”β€˜π‘˜)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘˜)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})))β€˜π‘) β‰  0)}) = ({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(((1st β€˜(π‘”β€˜π‘)) ∘f + ((((2nd β€˜(π‘”β€˜π‘)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {𝑝})))β€˜π‘) ∧ (((1st β€˜(π‘”β€˜π‘)) ∘f + ((((2nd β€˜(π‘”β€˜π‘)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})))β€˜π‘) β‰  0)}))
387386supeq1d 9437 . . . . . . . . 9 (π‘˜ = 𝑝 β†’ sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(((1st β€˜(π‘”β€˜π‘˜)) ∘f + ((((2nd β€˜(π‘”β€˜π‘˜)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘˜)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (((1st β€˜(π‘”β€˜π‘˜)) ∘f + ((((2nd β€˜(π‘”β€˜π‘˜)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘˜)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})))β€˜π‘) β‰  0)}), ℝ, < ) = sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(((1st β€˜(π‘”β€˜π‘)) ∘f + ((((2nd β€˜(π‘”β€˜π‘)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {𝑝})))β€˜π‘) ∧ (((1st β€˜(π‘”β€˜π‘)) ∘f + ((((2nd β€˜(π‘”β€˜π‘)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})))β€˜π‘) β‰  0)}), ℝ, < ))
388387eqeq2d 2735 . . . . . . . 8 (π‘˜ = 𝑝 β†’ (𝑖 = sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(((1st β€˜(π‘”β€˜π‘˜)) ∘f + ((((2nd β€˜(π‘”β€˜π‘˜)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘˜)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (((1st β€˜(π‘”β€˜π‘˜)) ∘f + ((((2nd β€˜(π‘”β€˜π‘˜)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘˜)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})))β€˜π‘) β‰  0)}), ℝ, < ) ↔ 𝑖 = sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(((1st β€˜(π‘”β€˜π‘)) ∘f + ((((2nd β€˜(π‘”β€˜π‘)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {𝑝})))β€˜π‘) ∧ (((1st β€˜(π‘”β€˜π‘)) ∘f + ((((2nd β€˜(π‘”β€˜π‘)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})))β€˜π‘) β‰  0)}), ℝ, < )))
389388rexbidv 3170 . . . . . . 7 (π‘˜ = 𝑝 β†’ (βˆƒπ‘— ∈ (0...𝑁)𝑖 = sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(((1st β€˜(π‘”β€˜π‘˜)) ∘f + ((((2nd β€˜(π‘”β€˜π‘˜)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘˜)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (((1st β€˜(π‘”β€˜π‘˜)) ∘f + ((((2nd β€˜(π‘”β€˜π‘˜)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘˜)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})))β€˜π‘) β‰  0)}), ℝ, < ) ↔ βˆƒπ‘— ∈ (0...𝑁)𝑖 = sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(((1st β€˜(π‘”β€˜π‘)) ∘f + ((((2nd β€˜(π‘”β€˜π‘)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {𝑝})))β€˜π‘) ∧ (((1st β€˜(π‘”β€˜π‘)) ∘f + ((((2nd β€˜(π‘”β€˜π‘)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})))β€˜π‘) β‰  0)}), ℝ, < )))
390 eqeq1 2728 . . . . . . . . 9 (𝑖 = π‘ž β†’ (𝑖 = sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(((1st β€˜(π‘”β€˜π‘)) ∘f + ((((2nd β€˜(π‘”β€˜π‘)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {𝑝})))β€˜π‘) ∧ (((1st β€˜(π‘”β€˜π‘)) ∘f + ((((2nd β€˜(π‘”β€˜π‘)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})))β€˜π‘) β‰  0)}), ℝ, < ) ↔ π‘ž = sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(((1st β€˜(π‘”β€˜π‘)) ∘f + ((((2nd β€˜(π‘”β€˜π‘)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {𝑝})))β€˜π‘) ∧ (((1st β€˜(π‘”β€˜π‘)) ∘f + ((((2nd β€˜(π‘”β€˜π‘)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})))β€˜π‘) β‰  0)}), ℝ, < )))
391390rexbidv 3170 . . . . . . . 8 (𝑖 = π‘ž β†’ (βˆƒπ‘— ∈ (0...𝑁)𝑖 = sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(((1st β€˜(π‘”β€˜π‘)) ∘f + ((((2nd β€˜(π‘”β€˜π‘)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {𝑝})))β€˜π‘) ∧ (((1st β€˜(π‘”β€˜π‘)) ∘f + ((((2nd β€˜(π‘”β€˜π‘)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})))β€˜π‘) β‰  0)}), ℝ, < ) ↔ βˆƒπ‘— ∈ (0...𝑁)π‘ž = sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(((1st β€˜(π‘”β€˜π‘)) ∘f + ((((2nd β€˜(π‘”β€˜π‘)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {𝑝})))β€˜π‘) ∧ (((1st β€˜(π‘”β€˜π‘)) ∘f + ((((2nd β€˜(π‘”β€˜π‘)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})))β€˜π‘) β‰  0)}), ℝ, < )))
392 oveq2 7409 . . . . . . . . . . . . . . . . . . . . . 22 (𝑗 = π‘š β†’ (1...𝑗) = (1...π‘š))
393392imaeq2d 6049 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 = π‘š β†’ ((2nd β€˜(π‘”β€˜π‘)) β€œ (1...𝑗)) = ((2nd β€˜(π‘”β€˜π‘)) β€œ (1...π‘š)))
394393xpeq1d 5695 . . . . . . . . . . . . . . . . . . . 20 (𝑗 = π‘š β†’ (((2nd β€˜(π‘”β€˜π‘)) β€œ (1...𝑗)) Γ— {1}) = (((2nd β€˜(π‘”β€˜π‘)) β€œ (1...π‘š)) Γ— {1}))
395 oveq1 7408 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑗 = π‘š β†’ (𝑗 + 1) = (π‘š + 1))
396395oveq1d 7416 . . . . . . . . . . . . . . . . . . . . . 22 (𝑗 = π‘š β†’ ((𝑗 + 1)...𝑁) = ((π‘š + 1)...𝑁))
397396imaeq2d 6049 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 = π‘š β†’ ((2nd β€˜(π‘”β€˜π‘)) β€œ ((𝑗 + 1)...𝑁)) = ((2nd β€˜(π‘”β€˜π‘)) β€œ ((π‘š + 1)...𝑁)))
398397xpeq1d 5695 . . . . . . . . . . . . . . . . . . . 20 (𝑗 = π‘š β†’ (((2nd β€˜(π‘”β€˜π‘)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}) = (((2nd β€˜(π‘”β€˜π‘)) β€œ ((π‘š + 1)...𝑁)) Γ— {0}))
399394, 398uneq12d 4156 . . . . . . . . . . . . . . . . . . 19 (𝑗 = π‘š β†’ ((((2nd β€˜(π‘”β€˜π‘)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})) = ((((2nd β€˜(π‘”β€˜π‘)) β€œ (1...π‘š)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘)) β€œ ((π‘š + 1)...𝑁)) Γ— {0})))
400399oveq2d 7417 . . . . . . . . . . . . . . . . . 18 (𝑗 = π‘š β†’ ((1st β€˜(π‘”β€˜π‘)) ∘f + ((((2nd β€˜(π‘”β€˜π‘)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) = ((1st β€˜(π‘”β€˜π‘)) ∘f + ((((2nd β€˜(π‘”β€˜π‘)) β€œ (1...π‘š)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘)) β€œ ((π‘š + 1)...𝑁)) Γ— {0}))))
401400fvoveq1d 7423 . . . . . . . . . . . . . . . . 17 (𝑗 = π‘š β†’ (πΉβ€˜(((1st β€˜(π‘”β€˜π‘)) ∘f + ((((2nd β€˜(π‘”β€˜π‘)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {𝑝}))) = (πΉβ€˜(((1st β€˜(π‘”β€˜π‘)) ∘f + ((((2nd β€˜(π‘”β€˜π‘)) β€œ (1...π‘š)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘)) β€œ ((π‘š + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {𝑝}))))
402401fveq1d 6883 . . . . . . . . . . . . . . . 16 (𝑗 = π‘š β†’ ((πΉβ€˜(((1st β€˜(π‘”β€˜π‘)) ∘f + ((((2nd β€˜(π‘”β€˜π‘)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {𝑝})))β€˜π‘) = ((πΉβ€˜(((1st β€˜(π‘”β€˜π‘)) ∘f + ((((2nd β€˜(π‘”β€˜π‘)) β€œ (1...π‘š)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘)) β€œ ((π‘š + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {𝑝})))β€˜π‘))
403402breq2d 5150 . . . . . . . . . . . . . . 15 (𝑗 = π‘š β†’ (0 ≀ ((πΉβ€˜(((1st β€˜(π‘”β€˜π‘)) ∘f + ((((2nd β€˜(π‘”β€˜π‘)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {𝑝})))β€˜π‘) ↔ 0 ≀ ((πΉβ€˜(((1st β€˜(π‘”β€˜π‘)) ∘f + ((((2nd β€˜(π‘”β€˜π‘)) β€œ (1...π‘š)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘)) β€œ ((π‘š + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {𝑝})))β€˜π‘)))
404400fveq1d 6883 . . . . . . . . . . . . . . . 16 (𝑗 = π‘š β†’ (((1st β€˜(π‘”β€˜π‘)) ∘f + ((((2nd β€˜(π‘”β€˜π‘)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})))β€˜π‘) = (((1st β€˜(π‘”β€˜π‘)) ∘f + ((((2nd β€˜(π‘”β€˜π‘)) β€œ (1...π‘š)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘)) β€œ ((π‘š + 1)...𝑁)) Γ— {0})))β€˜π‘))
405404neeq1d 2992 . . . . . . . . . . . . . . 15 (𝑗 = π‘š β†’ ((((1st β€˜(π‘”β€˜π‘)) ∘f + ((((2nd β€˜(π‘”β€˜π‘)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})))β€˜π‘) β‰  0 ↔ (((1st β€˜(π‘”β€˜π‘)) ∘f + ((((2nd β€˜(π‘”β€˜π‘)) β€œ (1...π‘š)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘)) β€œ ((π‘š + 1)...𝑁)) Γ— {0})))β€˜π‘) β‰  0))
406403, 405anbi12d 630 . . . . . . . . . . . . . 14 (𝑗 = π‘š β†’ ((0 ≀ ((πΉβ€˜(((1st β€˜(π‘”β€˜π‘)) ∘f + ((((2nd β€˜(π‘”β€˜π‘)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {𝑝})))β€˜π‘) ∧ (((1st β€˜(π‘”β€˜π‘)) ∘f + ((((2nd β€˜(π‘”β€˜π‘)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})))β€˜π‘) β‰  0) ↔ (0 ≀ ((πΉβ€˜(((1st β€˜(π‘”β€˜π‘)) ∘f + ((((2nd β€˜(π‘”β€˜π‘)) β€œ (1...π‘š)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘)) β€œ ((π‘š + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {𝑝})))β€˜π‘) ∧ (((1st β€˜(π‘”β€˜π‘)) ∘f + ((((2nd β€˜(π‘”β€˜π‘)) β€œ (1...π‘š)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘)) β€œ ((π‘š + 1)...𝑁)) Γ— {0})))β€˜π‘) β‰  0)))
407406ralbidv 3169 . . . . . . . . . . . . 13 (𝑗 = π‘š β†’ (βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(((1st β€˜(π‘”β€˜π‘)) ∘f + ((((2nd β€˜(π‘”β€˜π‘)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {𝑝})))β€˜π‘) ∧ (((1st β€˜(π‘”β€˜π‘)) ∘f + ((((2nd β€˜(π‘”β€˜π‘)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})))β€˜π‘) β‰  0) ↔ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(((1st β€˜(π‘”β€˜π‘)) ∘f + ((((2nd β€˜(π‘”β€˜π‘)) β€œ (1...π‘š)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘)) β€œ ((π‘š + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {𝑝})))β€˜π‘) ∧ (((1st β€˜(π‘”β€˜π‘)) ∘f + ((((2nd β€˜(π‘”β€˜π‘)) β€œ (1...π‘š)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘)) β€œ ((π‘š + 1)...𝑁)) Γ— {0})))β€˜π‘) β‰  0)))
408407rabbidv 3432 . . . . . . . . . . . 12 (𝑗 = π‘š β†’ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(((1st β€˜(π‘”β€˜π‘)) ∘f + ((((2nd β€˜(π‘”β€˜π‘)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {𝑝})))β€˜π‘) ∧ (((1st β€˜(π‘”β€˜π‘)) ∘f + ((((2nd β€˜(π‘”β€˜π‘)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})))β€˜π‘) β‰  0)} = {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(((1st β€˜(π‘”β€˜π‘)) ∘f + ((((2nd β€˜(π‘”β€˜π‘)) β€œ (1...π‘š)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘)) β€œ ((π‘š + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {𝑝})))β€˜π‘) ∧ (((1st β€˜(π‘”β€˜π‘)) ∘f + ((((2nd β€˜(π‘”β€˜π‘)) β€œ (1...π‘š)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘)) β€œ ((π‘š + 1)...𝑁)) Γ— {0})))β€˜π‘) β‰  0)})
409408uneq2d 4155 . . . . . . . . . . 11 (𝑗 = π‘š β†’ ({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(((1st β€˜(π‘”β€˜π‘)) ∘f + ((((2nd β€˜(π‘”β€˜π‘)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {𝑝})))β€˜π‘) ∧ (((1st β€˜(π‘”β€˜π‘)) ∘f + ((((2nd β€˜(π‘”β€˜π‘)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})))β€˜π‘) β‰  0)}) = ({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(((1st β€˜(π‘”β€˜π‘)) ∘f + ((((2nd β€˜(π‘”β€˜π‘)) β€œ (1...π‘š)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘)) β€œ ((π‘š + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {𝑝})))β€˜π‘) ∧ (((1st β€˜(π‘”β€˜π‘)) ∘f + ((((2nd β€˜(π‘”β€˜π‘)) β€œ (1...π‘š)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘)) β€œ ((π‘š + 1)...𝑁)) Γ— {0})))β€˜π‘) β‰  0)}))
410409supeq1d 9437 . . . . . . . . . 10 (𝑗 = π‘š β†’ sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(((1st β€˜(π‘”β€˜π‘)) ∘f + ((((2nd β€˜(π‘”β€˜π‘)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {𝑝})))β€˜π‘) ∧ (((1st β€˜(π‘”β€˜π‘)) ∘f + ((((2nd β€˜(π‘”β€˜π‘)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})))β€˜π‘) β‰  0)}), ℝ, < ) = sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(((1st β€˜(π‘”β€˜π‘)) ∘f + ((((2nd β€˜(π‘”β€˜π‘)) β€œ (1...π‘š)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘)) β€œ ((π‘š + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {𝑝})))β€˜π‘) ∧ (((1st β€˜(π‘”β€˜π‘)) ∘f + ((((2nd β€˜(π‘”β€˜π‘)) β€œ (1...π‘š)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘)) β€œ ((π‘š + 1)...𝑁)) Γ— {0})))β€˜π‘) β‰  0)}), ℝ, < ))
411410eqeq2d 2735 . . . . . . . . 9 (𝑗 = π‘š β†’ (π‘ž = sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(((1st β€˜(π‘”β€˜π‘)) ∘f + ((((2nd β€˜(π‘”β€˜π‘)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {𝑝})))β€˜π‘) ∧ (((1st β€˜(π‘”β€˜π‘)) ∘f + ((((2nd β€˜(π‘”β€˜π‘)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})))β€˜π‘) β‰  0)}), ℝ, < ) ↔ π‘ž = sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(((1st β€˜(π‘”β€˜π‘)) ∘f + ((((2nd β€˜(π‘”β€˜π‘)) β€œ (1...π‘š)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘)) β€œ ((π‘š + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {𝑝})))β€˜π‘) ∧ (((1st β€˜(π‘”β€˜π‘)) ∘f + ((((2nd β€˜(π‘”β€˜π‘)) β€œ (1...π‘š)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘)) β€œ ((π‘š + 1)...𝑁)) Γ— {0})))β€˜π‘) β‰  0)}), ℝ, < )))
412411cbvrexvw 3227 . . . . . . . 8 (βˆƒπ‘— ∈ (0...𝑁)π‘ž = sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(((1st β€˜(π‘”β€˜π‘)) ∘f + ((((2nd β€˜(π‘”β€˜π‘)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {𝑝})))β€˜π‘) ∧ (((1st β€˜(π‘”β€˜π‘)) ∘f + ((((2nd β€˜(π‘”β€˜π‘)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})))β€˜π‘) β‰  0)}), ℝ, < ) ↔ βˆƒπ‘š ∈ (0...𝑁)π‘ž = sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(((1st β€˜(π‘”β€˜π‘)) ∘f + ((((2nd β€˜(π‘”β€˜π‘)) β€œ (1...π‘š)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘)) β€œ ((π‘š + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {𝑝})))β€˜π‘) ∧ (((1st β€˜(π‘”β€˜π‘)) ∘f + ((((2nd β€˜(π‘”β€˜π‘)) β€œ (1...π‘š)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘)) β€œ ((π‘š + 1)...𝑁)) Γ— {0})))β€˜π‘) β‰  0)}), ℝ, < ))
413391, 412bitrdi 287 . . . . . . 7 (𝑖 = π‘ž β†’ (βˆƒπ‘— ∈ (0...𝑁)𝑖 = sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(((1st β€˜(π‘”β€˜π‘)) ∘f + ((((2nd β€˜(π‘”β€˜π‘)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {𝑝})))β€˜π‘) ∧ (((1st β€˜(π‘”β€˜π‘)) ∘f + ((((2nd β€˜(π‘”β€˜π‘)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})))β€˜π‘) β‰  0)}), ℝ, < ) ↔ βˆƒπ‘š ∈ (0...𝑁)π‘ž = sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(((1st β€˜(π‘”β€˜π‘)) ∘f + ((((2nd β€˜(π‘”β€˜π‘)) β€œ (1...π‘š)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘)) β€œ ((π‘š + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {𝑝})))β€˜π‘) ∧ (((1st β€˜(π‘”β€˜π‘)) ∘f + ((((2nd β€˜(π‘”β€˜π‘)) β€œ (1...π‘š)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘)) β€œ ((π‘š + 1)...𝑁)) Γ— {0})))β€˜π‘) β‰  0)}), ℝ, < )))
414389, 413rspc2v 3614 . . . . . 6 ((𝑝 ∈ β„• ∧ π‘ž ∈ (0...𝑁)) β†’ (βˆ€π‘˜ ∈ β„• βˆ€π‘– ∈ (0...𝑁)βˆƒπ‘— ∈ (0...𝑁)𝑖 = sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(((1st β€˜(π‘”β€˜π‘˜)) ∘f + ((((2nd β€˜(π‘”β€˜π‘˜)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘˜)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (((1st β€˜(π‘”β€˜π‘˜)) ∘f + ((((2nd β€˜(π‘”β€˜π‘˜)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘˜)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})))β€˜π‘) β‰  0)}), ℝ, < ) β†’ βˆƒπ‘š ∈ (0...𝑁)π‘ž = sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(((1st β€˜(π‘”β€˜π‘)) ∘f + ((((2nd β€˜(π‘”β€˜π‘)) β€œ (1...π‘š)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘)) β€œ ((π‘š + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {𝑝})))β€˜π‘) ∧ (((1st β€˜(π‘”β€˜π‘)) ∘f + ((((2nd β€˜(π‘”β€˜π‘)) β€œ (1...π‘š)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘)) β€œ ((π‘š + 1)...𝑁)) Γ— {0})))β€˜π‘) β‰  0)}), ℝ, < )))
415367, 414mpan9 506 . . . . 5 ((((πœ‘ ∧ 𝑔:β„•βŸΆ((β„•0 ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)})) ∧ βˆ€π‘˜ ∈ β„• (ran (1st β€˜(π‘”β€˜π‘˜)) βŠ† (0..^π‘˜) ∧ βˆ€π‘– ∈ (0...𝑁)βˆƒπ‘— ∈ (0...𝑁)𝑖 = sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(((1st β€˜(π‘”β€˜π‘˜)) ∘f + ((((2nd β€˜(π‘”β€˜π‘˜)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘˜)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (((1st β€˜(π‘”β€˜π‘˜)) ∘f + ((((2nd β€˜(π‘”β€˜π‘˜)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘˜)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})))β€˜π‘) β‰  0)}), ℝ, < ))) ∧ (𝑝 ∈ β„• ∧ π‘ž ∈ (0...𝑁))) β†’ βˆƒπ‘š ∈ (0...𝑁)π‘ž = sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(((1st β€˜(π‘”β€˜π‘)) ∘f + ((((2nd β€˜(π‘”β€˜π‘)) β€œ (1...π‘š)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘)) β€œ ((π‘š + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {𝑝})))β€˜π‘) ∧ (((1st β€˜(π‘”β€˜π‘)) ∘f + ((((2nd β€˜(π‘”β€˜π‘)) β€œ (1...π‘š)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘)) β€œ ((π‘š + 1)...𝑁)) Γ— {0})))β€˜π‘) β‰  0)}), ℝ, < ))
416346, 137, 347, 349, 363, 364, 351, 360, 415poimirlem31 37009 . . . 4 ((((πœ‘ ∧ 𝑔:β„•βŸΆ((β„•0 ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)})) ∧ βˆ€π‘˜ ∈ β„• (ran (1st β€˜(π‘”β€˜π‘˜)) βŠ† (0..^π‘˜) ∧ βˆ€π‘– ∈ (0...𝑁)βˆƒπ‘— ∈ (0...𝑁)𝑖 = sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(((1st β€˜(π‘”β€˜π‘˜)) ∘f + ((((2nd β€˜(π‘”β€˜π‘˜)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘˜)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (((1st β€˜(π‘”β€˜π‘˜)) ∘f + ((((2nd β€˜(π‘”β€˜π‘˜)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘˜)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})))β€˜π‘) β‰  0)}), ℝ, < ))) ∧ (𝑝 ∈ β„• ∧ 𝑛 ∈ (1...𝑁) ∧ π‘Ÿ ∈ { ≀ , β—‘ ≀ })) β†’ βˆƒπ‘š ∈ (0...𝑁)0π‘Ÿ((πΉβ€˜(((1st β€˜(π‘”β€˜π‘)) ∘f + ((((2nd β€˜(π‘”β€˜π‘)) β€œ (1...π‘š)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘)) β€œ ((π‘š + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {𝑝})))β€˜π‘›))
417346, 137, 347, 349, 350, 351, 360, 416poimirlem30 37008 . . 3 (((πœ‘ ∧ 𝑔:β„•βŸΆ((β„•0 ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)})) ∧ βˆ€π‘˜ ∈ β„• (ran (1st β€˜(π‘”β€˜π‘˜)) βŠ† (0..^π‘˜) ∧ βˆ€π‘– ∈ (0...𝑁)βˆƒπ‘— ∈ (0...𝑁)𝑖 = sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(((1st β€˜(π‘”β€˜π‘˜)) ∘f + ((((2nd β€˜(π‘”β€˜π‘˜)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘˜)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (((1st β€˜(π‘”β€˜π‘˜)) ∘f + ((((2nd β€˜(π‘”β€˜π‘˜)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘˜)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})))β€˜π‘) β‰  0)}), ℝ, < ))) β†’ βˆƒπ‘ ∈ 𝐼 βˆ€π‘› ∈ (1...𝑁)βˆ€π‘£ ∈ (𝑅 β†Ύt 𝐼)(𝑐 ∈ 𝑣 β†’ βˆ€π‘Ÿ ∈ { ≀ , β—‘ ≀ }βˆƒπ‘§ ∈ 𝑣 0π‘Ÿ((πΉβ€˜π‘§)β€˜π‘›)))
418417anasss 466 . 2 ((πœ‘ ∧ (𝑔:β„•βŸΆ((β„•0 ↑m (1...𝑁)) Γ— {𝑓 ∣ 𝑓:(1...𝑁)–1-1-ontoβ†’(1...𝑁)}) ∧ βˆ€π‘˜ ∈ β„• (ran (1st β€˜(π‘”β€˜π‘˜)) βŠ† (0..^π‘˜) ∧ βˆ€π‘– ∈ (0...𝑁)βˆƒπ‘— ∈ (0...𝑁)𝑖 = sup(({0} βˆͺ {π‘Ž ∈ (1...𝑁) ∣ βˆ€π‘ ∈ (1...π‘Ž)(0 ≀ ((πΉβ€˜(((1st β€˜(π‘”β€˜π‘˜)) ∘f + ((((2nd β€˜(π‘”β€˜π‘˜)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘˜)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0}))) ∘f / ((1...𝑁) Γ— {π‘˜})))β€˜π‘) ∧ (((1st β€˜(π‘”β€˜π‘˜)) ∘f + ((((2nd β€˜(π‘”β€˜π‘˜)) β€œ (1...𝑗)) Γ— {1}) βˆͺ (((2nd β€˜(π‘”β€˜π‘˜)) β€œ ((𝑗 + 1)...𝑁)) Γ— {0})))β€˜π‘) β‰  0)}), ℝ, < )))) β†’ βˆƒπ‘ ∈ 𝐼 βˆ€π‘› ∈ (1...𝑁)βˆ€π‘£ ∈ (𝑅 β†Ύt 𝐼)(𝑐 ∈ 𝑣 β†’ βˆ€π‘Ÿ ∈ { ≀ , β—‘ ≀ }βˆƒπ‘§ ∈ 𝑣 0π‘Ÿ((πΉβ€˜π‘§)β€˜π‘›)))
419345, 418exlimddv 1930 1 (πœ‘ β†’ βˆƒπ‘ ∈ 𝐼 βˆ€π‘› ∈ (1...𝑁)βˆ€π‘£ ∈ (𝑅 β†Ύt 𝐼)(𝑐 ∈ 𝑣 β†’ βˆ€π‘Ÿ ∈ { ≀ , β—‘ ≀ }βˆƒπ‘§ ∈ 𝑣 0π‘Ÿ((πΉβ€˜π‘§)β€˜π‘›)))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∨ wo 844   ∧ w3a 1084   = wceq 1533  βˆƒwex 1773   ∈ wcel 2098  {cab 2701   β‰  wne 2932  βˆ€wral 3053  βˆƒwrex 3062  {crab 3424  Vcvv 3466   βˆͺ cun 3938   βŠ† wss 3940  βˆ…c0 4314  {csn 4620  {cpr 4622  βˆͺ ciun 4987   class class class wbr 5138   Or wor 5577   Γ— cxp 5664  β—‘ccnv 5665  dom cdm 5666  ran crn 5667   β€œ cima 5669  Oncon0 6354   Fn wfn 6528  βŸΆwf 6529  β€“1-1-ontoβ†’wf1o 6532  β€˜cfv 6533  (class class class)co 7401   ∘f cof 7661  Ο‰com 7848  1st c1st 7966  2nd c2nd 7967   ↑m cmap 8816  Xcixp 8887   β‰ˆ cen 8932  Fincfn 8935  supcsup 9431  cardccrd 9926  β„‚cc 11104  β„cr 11105  0cc0 11106  1c1 11107   + caddc 11109   Β· cmul 11111   < clt 11245   ≀ cle 11246   βˆ’ cmin 11441   / cdiv 11868  β„•cn 12209  β„•0cn0 12469  β„€cz 12555  β„€β‰₯cuz 12819  β„+crp 12971  (,)cioo 13321  [,]cicc 13324  ...cfz 13481  ..^cfzo 13624   β†Ύt crest 17365  topGenctg 17382  βˆtcpt 17383   Cn ccn 23050
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718  ax-inf2 9632  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183  ax-pre-sup 11184
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-nel 3039  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3959  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-tp 4625  df-op 4627  df-uni 4900  df-int 4941  df-iun 4989  df-iin 4990  df-disj 5104  df-br 5139  df-opab 5201  df-mpt 5222  df-tr 5256  df-id 5564  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-se 5622  df-we 5623  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6290  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-isom 6542  df-riota 7357  df-ov 7404  df-oprab 7405  df-mpo 7406  df-of 7663  df-om 7849  df-1st 7968  df-2nd 7969  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-1o 8461  df-oadd 8465  df-omul 8466  df-er 8699  df-map 8818  df-pm 8819  df-ixp 8888  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-fi 9402  df-sup 9433  df-inf 9434  df-oi 9501  df-dju 9892  df-card 9930  df-acn 9933  df-pnf 11247  df-mnf 11248  df-xr 11249  df-ltxr 11250  df-le 11251  df-sub 11443  df-neg 11444  df-div 11869  df-nn 12210  df-2 12272  df-3 12273  df-n0 12470  df-xnn0 12542  df-z 12556  df-uz 12820  df-q 12930  df-rp 12972  df-xneg 13089  df-xadd 13090  df-xmul 13091  df-ioo 13325  df-icc 13328  df-fz 13482  df-fzo 13625  df-fl 13754  df-seq 13964  df-exp 14025  df-fac 14231  df-bc 14260  df-hash 14288  df-cj 15043  df-re 15044  df-im 15045  df-sqrt 15179  df-abs 15180  df-clim 15429  df-sum 15630  df-dvds 16195  df-rest 17367  df-topgen 17388  df-pt 17389  df-psmet 21220  df-xmet 21221  df-met 21222  df-bl 21223  df-mopn 21224  df-top 22718  df-topon 22735  df-bases 22771  df-cld 22845  df-ntr 22846  df-cls 22847  df-lp 22962  df-cn 23053  df-cnp 23054  df-t1 23140  df-haus 23141  df-cmp 23213  df-tx 23388  df-hmeo 23581  df-hmph 23582  df-ii 24719
This theorem is referenced by:  poimir  37011
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