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Theorem ovolicc2lem2 25267
Description: Lemma for ovolicc2 25271. (Contributed by Mario Carneiro, 14-Jun-2014.)
Hypotheses
Ref Expression
ovolicc.1 (πœ‘ β†’ 𝐴 ∈ ℝ)
ovolicc.2 (πœ‘ β†’ 𝐡 ∈ ℝ)
ovolicc.3 (πœ‘ β†’ 𝐴 ≀ 𝐡)
ovolicc2.4 𝑆 = seq1( + , ((abs ∘ βˆ’ ) ∘ 𝐹))
ovolicc2.5 (πœ‘ β†’ 𝐹:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ)))
ovolicc2.6 (πœ‘ β†’ π‘ˆ ∈ (𝒫 ran ((,) ∘ 𝐹) ∩ Fin))
ovolicc2.7 (πœ‘ β†’ (𝐴[,]𝐡) βŠ† βˆͺ π‘ˆ)
ovolicc2.8 (πœ‘ β†’ 𝐺:π‘ˆβŸΆβ„•)
ovolicc2.9 ((πœ‘ ∧ 𝑑 ∈ π‘ˆ) β†’ (((,) ∘ 𝐹)β€˜(πΊβ€˜π‘‘)) = 𝑑)
ovolicc2.10 𝑇 = {𝑒 ∈ π‘ˆ ∣ (𝑒 ∩ (𝐴[,]𝐡)) β‰  βˆ…}
ovolicc2.11 (πœ‘ β†’ 𝐻:π‘‡βŸΆπ‘‡)
ovolicc2.12 ((πœ‘ ∧ 𝑑 ∈ 𝑇) β†’ if((2nd β€˜(πΉβ€˜(πΊβ€˜π‘‘))) ≀ 𝐡, (2nd β€˜(πΉβ€˜(πΊβ€˜π‘‘))), 𝐡) ∈ (π»β€˜π‘‘))
ovolicc2.13 (πœ‘ β†’ 𝐴 ∈ 𝐢)
ovolicc2.14 (πœ‘ β†’ 𝐢 ∈ 𝑇)
ovolicc2.15 𝐾 = seq1((𝐻 ∘ 1st ), (β„• Γ— {𝐢}))
ovolicc2.16 π‘Š = {𝑛 ∈ β„• ∣ 𝐡 ∈ (πΎβ€˜π‘›)}
Assertion
Ref Expression
ovolicc2lem2 ((πœ‘ ∧ (𝑁 ∈ β„• ∧ Β¬ 𝑁 ∈ π‘Š)) β†’ (2nd β€˜(πΉβ€˜(πΊβ€˜(πΎβ€˜π‘)))) ≀ 𝐡)
Distinct variable groups:   𝑑,𝑛,𝑒,𝐴   𝐡,𝑛,𝑑,𝑒   𝑑,𝐻   𝐢,𝑛,𝑑   𝑛,𝐹,𝑑   𝑛,𝐾,𝑑,𝑒   𝑛,𝐺,𝑑   𝑛,π‘Š   πœ‘,𝑛,𝑑   𝑇,𝑛,𝑑   𝑛,𝑁,𝑑,𝑒   π‘ˆ,𝑛,𝑑,𝑒
Allowed substitution hints:   πœ‘(𝑒)   𝐢(𝑒)   𝑆(𝑒,𝑑,𝑛)   𝑇(𝑒)   𝐹(𝑒)   𝐺(𝑒)   𝐻(𝑒,𝑛)   π‘Š(𝑒,𝑑)
Proof of Theorem ovolicc2lem2
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 ovolicc.2 . . . . . 6 (πœ‘ β†’ 𝐡 ∈ ℝ)
21adantr 479 . . . . 5 ((πœ‘ ∧ 𝑁 ∈ β„•) β†’ 𝐡 ∈ ℝ)
3 ovolicc2.5 . . . . . . . . 9 (πœ‘ β†’ 𝐹:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ)))
4 inss2 4228 . . . . . . . . 9 ( ≀ ∩ (ℝ Γ— ℝ)) βŠ† (ℝ Γ— ℝ)
5 fss 6733 . . . . . . . . 9 ((𝐹:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ)) ∧ ( ≀ ∩ (ℝ Γ— ℝ)) βŠ† (ℝ Γ— ℝ)) β†’ 𝐹:β„•βŸΆ(ℝ Γ— ℝ))
63, 4, 5sylancl 584 . . . . . . . 8 (πœ‘ β†’ 𝐹:β„•βŸΆ(ℝ Γ— ℝ))
76adantr 479 . . . . . . 7 ((πœ‘ ∧ 𝑁 ∈ β„•) β†’ 𝐹:β„•βŸΆ(ℝ Γ— ℝ))
8 ovolicc2.8 . . . . . . . . 9 (πœ‘ β†’ 𝐺:π‘ˆβŸΆβ„•)
98adantr 479 . . . . . . . 8 ((πœ‘ ∧ 𝑁 ∈ β„•) β†’ 𝐺:π‘ˆβŸΆβ„•)
10 nnuz 12869 . . . . . . . . . . . 12 β„• = (β„€β‰₯β€˜1)
11 ovolicc2.15 . . . . . . . . . . . 12 𝐾 = seq1((𝐻 ∘ 1st ), (β„• Γ— {𝐢}))
12 1zzd 12597 . . . . . . . . . . . 12 (πœ‘ β†’ 1 ∈ β„€)
13 ovolicc2.14 . . . . . . . . . . . 12 (πœ‘ β†’ 𝐢 ∈ 𝑇)
14 ovolicc2.11 . . . . . . . . . . . 12 (πœ‘ β†’ 𝐻:π‘‡βŸΆπ‘‡)
1510, 11, 12, 13, 14algrf 16514 . . . . . . . . . . 11 (πœ‘ β†’ 𝐾:β„•βŸΆπ‘‡)
1615ffvelcdmda 7085 . . . . . . . . . 10 ((πœ‘ ∧ 𝑁 ∈ β„•) β†’ (πΎβ€˜π‘) ∈ 𝑇)
17 ineq1 4204 . . . . . . . . . . . 12 (𝑒 = (πΎβ€˜π‘) β†’ (𝑒 ∩ (𝐴[,]𝐡)) = ((πΎβ€˜π‘) ∩ (𝐴[,]𝐡)))
1817neeq1d 2998 . . . . . . . . . . 11 (𝑒 = (πΎβ€˜π‘) β†’ ((𝑒 ∩ (𝐴[,]𝐡)) β‰  βˆ… ↔ ((πΎβ€˜π‘) ∩ (𝐴[,]𝐡)) β‰  βˆ…))
19 ovolicc2.10 . . . . . . . . . . 11 𝑇 = {𝑒 ∈ π‘ˆ ∣ (𝑒 ∩ (𝐴[,]𝐡)) β‰  βˆ…}
2018, 19elrab2 3685 . . . . . . . . . 10 ((πΎβ€˜π‘) ∈ 𝑇 ↔ ((πΎβ€˜π‘) ∈ π‘ˆ ∧ ((πΎβ€˜π‘) ∩ (𝐴[,]𝐡)) β‰  βˆ…))
2116, 20sylib 217 . . . . . . . . 9 ((πœ‘ ∧ 𝑁 ∈ β„•) β†’ ((πΎβ€˜π‘) ∈ π‘ˆ ∧ ((πΎβ€˜π‘) ∩ (𝐴[,]𝐡)) β‰  βˆ…))
2221simpld 493 . . . . . . . 8 ((πœ‘ ∧ 𝑁 ∈ β„•) β†’ (πΎβ€˜π‘) ∈ π‘ˆ)
239, 22ffvelcdmd 7086 . . . . . . 7 ((πœ‘ ∧ 𝑁 ∈ β„•) β†’ (πΊβ€˜(πΎβ€˜π‘)) ∈ β„•)
247, 23ffvelcdmd 7086 . . . . . 6 ((πœ‘ ∧ 𝑁 ∈ β„•) β†’ (πΉβ€˜(πΊβ€˜(πΎβ€˜π‘))) ∈ (ℝ Γ— ℝ))
25 xp2nd 8010 . . . . . 6 ((πΉβ€˜(πΊβ€˜(πΎβ€˜π‘))) ∈ (ℝ Γ— ℝ) β†’ (2nd β€˜(πΉβ€˜(πΊβ€˜(πΎβ€˜π‘)))) ∈ ℝ)
2624, 25syl 17 . . . . 5 ((πœ‘ ∧ 𝑁 ∈ β„•) β†’ (2nd β€˜(πΉβ€˜(πΊβ€˜(πΎβ€˜π‘)))) ∈ ℝ)
272, 26ltnled 11365 . . . 4 ((πœ‘ ∧ 𝑁 ∈ β„•) β†’ (𝐡 < (2nd β€˜(πΉβ€˜(πΊβ€˜(πΎβ€˜π‘)))) ↔ Β¬ (2nd β€˜(πΉβ€˜(πΊβ€˜(πΎβ€˜π‘)))) ≀ 𝐡))
28 simprl 767 . . . . . 6 ((πœ‘ ∧ (𝑁 ∈ β„• ∧ 𝐡 < (2nd β€˜(πΉβ€˜(πΊβ€˜(πΎβ€˜π‘)))))) β†’ 𝑁 ∈ β„•)
291adantr 479 . . . . . . 7 ((πœ‘ ∧ (𝑁 ∈ β„• ∧ 𝐡 < (2nd β€˜(πΉβ€˜(πΊβ€˜(πΎβ€˜π‘)))))) β†’ 𝐡 ∈ ℝ)
3021adantrr 713 . . . . . . . . . 10 ((πœ‘ ∧ (𝑁 ∈ β„• ∧ 𝐡 < (2nd β€˜(πΉβ€˜(πΊβ€˜(πΎβ€˜π‘)))))) β†’ ((πΎβ€˜π‘) ∈ π‘ˆ ∧ ((πΎβ€˜π‘) ∩ (𝐴[,]𝐡)) β‰  βˆ…))
3130simprd 494 . . . . . . . . 9 ((πœ‘ ∧ (𝑁 ∈ β„• ∧ 𝐡 < (2nd β€˜(πΉβ€˜(πΊβ€˜(πΎβ€˜π‘)))))) β†’ ((πΎβ€˜π‘) ∩ (𝐴[,]𝐡)) β‰  βˆ…)
32 n0 4345 . . . . . . . . 9 (((πΎβ€˜π‘) ∩ (𝐴[,]𝐡)) β‰  βˆ… ↔ βˆƒπ‘₯ π‘₯ ∈ ((πΎβ€˜π‘) ∩ (𝐴[,]𝐡)))
3331, 32sylib 217 . . . . . . . 8 ((πœ‘ ∧ (𝑁 ∈ β„• ∧ 𝐡 < (2nd β€˜(πΉβ€˜(πΊβ€˜(πΎβ€˜π‘)))))) β†’ βˆƒπ‘₯ π‘₯ ∈ ((πΎβ€˜π‘) ∩ (𝐴[,]𝐡)))
34 xp1st 8009 . . . . . . . . . . . 12 ((πΉβ€˜(πΊβ€˜(πΎβ€˜π‘))) ∈ (ℝ Γ— ℝ) β†’ (1st β€˜(πΉβ€˜(πΊβ€˜(πΎβ€˜π‘)))) ∈ ℝ)
3524, 34syl 17 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑁 ∈ β„•) β†’ (1st β€˜(πΉβ€˜(πΊβ€˜(πΎβ€˜π‘)))) ∈ ℝ)
3635adantrr 713 . . . . . . . . . 10 ((πœ‘ ∧ (𝑁 ∈ β„• ∧ 𝐡 < (2nd β€˜(πΉβ€˜(πΊβ€˜(πΎβ€˜π‘)))))) β†’ (1st β€˜(πΉβ€˜(πΊβ€˜(πΎβ€˜π‘)))) ∈ ℝ)
3736adantr 479 . . . . . . . . 9 (((πœ‘ ∧ (𝑁 ∈ β„• ∧ 𝐡 < (2nd β€˜(πΉβ€˜(πΊβ€˜(πΎβ€˜π‘)))))) ∧ π‘₯ ∈ ((πΎβ€˜π‘) ∩ (𝐴[,]𝐡))) β†’ (1st β€˜(πΉβ€˜(πΊβ€˜(πΎβ€˜π‘)))) ∈ ℝ)
38 simpr 483 . . . . . . . . . . . . 13 (((πœ‘ ∧ (𝑁 ∈ β„• ∧ 𝐡 < (2nd β€˜(πΉβ€˜(πΊβ€˜(πΎβ€˜π‘)))))) ∧ π‘₯ ∈ ((πΎβ€˜π‘) ∩ (𝐴[,]𝐡))) β†’ π‘₯ ∈ ((πΎβ€˜π‘) ∩ (𝐴[,]𝐡)))
39 elin 3963 . . . . . . . . . . . . 13 (π‘₯ ∈ ((πΎβ€˜π‘) ∩ (𝐴[,]𝐡)) ↔ (π‘₯ ∈ (πΎβ€˜π‘) ∧ π‘₯ ∈ (𝐴[,]𝐡)))
4038, 39sylib 217 . . . . . . . . . . . 12 (((πœ‘ ∧ (𝑁 ∈ β„• ∧ 𝐡 < (2nd β€˜(πΉβ€˜(πΊβ€˜(πΎβ€˜π‘)))))) ∧ π‘₯ ∈ ((πΎβ€˜π‘) ∩ (𝐴[,]𝐡))) β†’ (π‘₯ ∈ (πΎβ€˜π‘) ∧ π‘₯ ∈ (𝐴[,]𝐡)))
4140simprd 494 . . . . . . . . . . 11 (((πœ‘ ∧ (𝑁 ∈ β„• ∧ 𝐡 < (2nd β€˜(πΉβ€˜(πΊβ€˜(πΎβ€˜π‘)))))) ∧ π‘₯ ∈ ((πΎβ€˜π‘) ∩ (𝐴[,]𝐡))) β†’ π‘₯ ∈ (𝐴[,]𝐡))
42 ovolicc.1 . . . . . . . . . . . . 13 (πœ‘ β†’ 𝐴 ∈ ℝ)
43 elicc2 13393 . . . . . . . . . . . . 13 ((𝐴 ∈ ℝ ∧ 𝐡 ∈ ℝ) β†’ (π‘₯ ∈ (𝐴[,]𝐡) ↔ (π‘₯ ∈ ℝ ∧ 𝐴 ≀ π‘₯ ∧ π‘₯ ≀ 𝐡)))
4442, 1, 43syl2anc 582 . . . . . . . . . . . 12 (πœ‘ β†’ (π‘₯ ∈ (𝐴[,]𝐡) ↔ (π‘₯ ∈ ℝ ∧ 𝐴 ≀ π‘₯ ∧ π‘₯ ≀ 𝐡)))
4544ad2antrr 722 . . . . . . . . . . 11 (((πœ‘ ∧ (𝑁 ∈ β„• ∧ 𝐡 < (2nd β€˜(πΉβ€˜(πΊβ€˜(πΎβ€˜π‘)))))) ∧ π‘₯ ∈ ((πΎβ€˜π‘) ∩ (𝐴[,]𝐡))) β†’ (π‘₯ ∈ (𝐴[,]𝐡) ↔ (π‘₯ ∈ ℝ ∧ 𝐴 ≀ π‘₯ ∧ π‘₯ ≀ 𝐡)))
4641, 45mpbid 231 . . . . . . . . . 10 (((πœ‘ ∧ (𝑁 ∈ β„• ∧ 𝐡 < (2nd β€˜(πΉβ€˜(πΊβ€˜(πΎβ€˜π‘)))))) ∧ π‘₯ ∈ ((πΎβ€˜π‘) ∩ (𝐴[,]𝐡))) β†’ (π‘₯ ∈ ℝ ∧ 𝐴 ≀ π‘₯ ∧ π‘₯ ≀ 𝐡))
4746simp1d 1140 . . . . . . . . 9 (((πœ‘ ∧ (𝑁 ∈ β„• ∧ 𝐡 < (2nd β€˜(πΉβ€˜(πΊβ€˜(πΎβ€˜π‘)))))) ∧ π‘₯ ∈ ((πΎβ€˜π‘) ∩ (𝐴[,]𝐡))) β†’ π‘₯ ∈ ℝ)
481ad2antrr 722 . . . . . . . . 9 (((πœ‘ ∧ (𝑁 ∈ β„• ∧ 𝐡 < (2nd β€˜(πΉβ€˜(πΊβ€˜(πΎβ€˜π‘)))))) ∧ π‘₯ ∈ ((πΎβ€˜π‘) ∩ (𝐴[,]𝐡))) β†’ 𝐡 ∈ ℝ)
4940simpld 493 . . . . . . . . . . 11 (((πœ‘ ∧ (𝑁 ∈ β„• ∧ 𝐡 < (2nd β€˜(πΉβ€˜(πΊβ€˜(πΎβ€˜π‘)))))) ∧ π‘₯ ∈ ((πΎβ€˜π‘) ∩ (𝐴[,]𝐡))) β†’ π‘₯ ∈ (πΎβ€˜π‘))
5030simpld 493 . . . . . . . . . . . . 13 ((πœ‘ ∧ (𝑁 ∈ β„• ∧ 𝐡 < (2nd β€˜(πΉβ€˜(πΊβ€˜(πΎβ€˜π‘)))))) β†’ (πΎβ€˜π‘) ∈ π‘ˆ)
51 ovolicc.3 . . . . . . . . . . . . . 14 (πœ‘ β†’ 𝐴 ≀ 𝐡)
52 ovolicc2.4 . . . . . . . . . . . . . 14 𝑆 = seq1( + , ((abs ∘ βˆ’ ) ∘ 𝐹))
53 ovolicc2.6 . . . . . . . . . . . . . 14 (πœ‘ β†’ π‘ˆ ∈ (𝒫 ran ((,) ∘ 𝐹) ∩ Fin))
54 ovolicc2.7 . . . . . . . . . . . . . 14 (πœ‘ β†’ (𝐴[,]𝐡) βŠ† βˆͺ π‘ˆ)
55 ovolicc2.9 . . . . . . . . . . . . . 14 ((πœ‘ ∧ 𝑑 ∈ π‘ˆ) β†’ (((,) ∘ 𝐹)β€˜(πΊβ€˜π‘‘)) = 𝑑)
5642, 1, 51, 52, 3, 53, 54, 8, 55ovolicc2lem1 25266 . . . . . . . . . . . . 13 ((πœ‘ ∧ (πΎβ€˜π‘) ∈ π‘ˆ) β†’ (π‘₯ ∈ (πΎβ€˜π‘) ↔ (π‘₯ ∈ ℝ ∧ (1st β€˜(πΉβ€˜(πΊβ€˜(πΎβ€˜π‘)))) < π‘₯ ∧ π‘₯ < (2nd β€˜(πΉβ€˜(πΊβ€˜(πΎβ€˜π‘)))))))
5750, 56syldan 589 . . . . . . . . . . . 12 ((πœ‘ ∧ (𝑁 ∈ β„• ∧ 𝐡 < (2nd β€˜(πΉβ€˜(πΊβ€˜(πΎβ€˜π‘)))))) β†’ (π‘₯ ∈ (πΎβ€˜π‘) ↔ (π‘₯ ∈ ℝ ∧ (1st β€˜(πΉβ€˜(πΊβ€˜(πΎβ€˜π‘)))) < π‘₯ ∧ π‘₯ < (2nd β€˜(πΉβ€˜(πΊβ€˜(πΎβ€˜π‘)))))))
5857adantr 479 . . . . . . . . . . 11 (((πœ‘ ∧ (𝑁 ∈ β„• ∧ 𝐡 < (2nd β€˜(πΉβ€˜(πΊβ€˜(πΎβ€˜π‘)))))) ∧ π‘₯ ∈ ((πΎβ€˜π‘) ∩ (𝐴[,]𝐡))) β†’ (π‘₯ ∈ (πΎβ€˜π‘) ↔ (π‘₯ ∈ ℝ ∧ (1st β€˜(πΉβ€˜(πΊβ€˜(πΎβ€˜π‘)))) < π‘₯ ∧ π‘₯ < (2nd β€˜(πΉβ€˜(πΊβ€˜(πΎβ€˜π‘)))))))
5949, 58mpbid 231 . . . . . . . . . 10 (((πœ‘ ∧ (𝑁 ∈ β„• ∧ 𝐡 < (2nd β€˜(πΉβ€˜(πΊβ€˜(πΎβ€˜π‘)))))) ∧ π‘₯ ∈ ((πΎβ€˜π‘) ∩ (𝐴[,]𝐡))) β†’ (π‘₯ ∈ ℝ ∧ (1st β€˜(πΉβ€˜(πΊβ€˜(πΎβ€˜π‘)))) < π‘₯ ∧ π‘₯ < (2nd β€˜(πΉβ€˜(πΊβ€˜(πΎβ€˜π‘))))))
6059simp2d 1141 . . . . . . . . 9 (((πœ‘ ∧ (𝑁 ∈ β„• ∧ 𝐡 < (2nd β€˜(πΉβ€˜(πΊβ€˜(πΎβ€˜π‘)))))) ∧ π‘₯ ∈ ((πΎβ€˜π‘) ∩ (𝐴[,]𝐡))) β†’ (1st β€˜(πΉβ€˜(πΊβ€˜(πΎβ€˜π‘)))) < π‘₯)
6146simp3d 1142 . . . . . . . . 9 (((πœ‘ ∧ (𝑁 ∈ β„• ∧ 𝐡 < (2nd β€˜(πΉβ€˜(πΊβ€˜(πΎβ€˜π‘)))))) ∧ π‘₯ ∈ ((πΎβ€˜π‘) ∩ (𝐴[,]𝐡))) β†’ π‘₯ ≀ 𝐡)
6237, 47, 48, 60, 61ltletrd 11378 . . . . . . . 8 (((πœ‘ ∧ (𝑁 ∈ β„• ∧ 𝐡 < (2nd β€˜(πΉβ€˜(πΊβ€˜(πΎβ€˜π‘)))))) ∧ π‘₯ ∈ ((πΎβ€˜π‘) ∩ (𝐴[,]𝐡))) β†’ (1st β€˜(πΉβ€˜(πΊβ€˜(πΎβ€˜π‘)))) < 𝐡)
6333, 62exlimddv 1936 . . . . . . 7 ((πœ‘ ∧ (𝑁 ∈ β„• ∧ 𝐡 < (2nd β€˜(πΉβ€˜(πΊβ€˜(πΎβ€˜π‘)))))) β†’ (1st β€˜(πΉβ€˜(πΊβ€˜(πΎβ€˜π‘)))) < 𝐡)
64 simprr 769 . . . . . . 7 ((πœ‘ ∧ (𝑁 ∈ β„• ∧ 𝐡 < (2nd β€˜(πΉβ€˜(πΊβ€˜(πΎβ€˜π‘)))))) β†’ 𝐡 < (2nd β€˜(πΉβ€˜(πΊβ€˜(πΎβ€˜π‘)))))
6542, 1, 51, 52, 3, 53, 54, 8, 55ovolicc2lem1 25266 . . . . . . . 8 ((πœ‘ ∧ (πΎβ€˜π‘) ∈ π‘ˆ) β†’ (𝐡 ∈ (πΎβ€˜π‘) ↔ (𝐡 ∈ ℝ ∧ (1st β€˜(πΉβ€˜(πΊβ€˜(πΎβ€˜π‘)))) < 𝐡 ∧ 𝐡 < (2nd β€˜(πΉβ€˜(πΊβ€˜(πΎβ€˜π‘)))))))
6650, 65syldan 589 . . . . . . 7 ((πœ‘ ∧ (𝑁 ∈ β„• ∧ 𝐡 < (2nd β€˜(πΉβ€˜(πΊβ€˜(πΎβ€˜π‘)))))) β†’ (𝐡 ∈ (πΎβ€˜π‘) ↔ (𝐡 ∈ ℝ ∧ (1st β€˜(πΉβ€˜(πΊβ€˜(πΎβ€˜π‘)))) < 𝐡 ∧ 𝐡 < (2nd β€˜(πΉβ€˜(πΊβ€˜(πΎβ€˜π‘)))))))
6729, 63, 64, 66mpbir3and 1340 . . . . . 6 ((πœ‘ ∧ (𝑁 ∈ β„• ∧ 𝐡 < (2nd β€˜(πΉβ€˜(πΊβ€˜(πΎβ€˜π‘)))))) β†’ 𝐡 ∈ (πΎβ€˜π‘))
68 fveq2 6890 . . . . . . . 8 (𝑛 = 𝑁 β†’ (πΎβ€˜π‘›) = (πΎβ€˜π‘))
6968eleq2d 2817 . . . . . . 7 (𝑛 = 𝑁 β†’ (𝐡 ∈ (πΎβ€˜π‘›) ↔ 𝐡 ∈ (πΎβ€˜π‘)))
70 ovolicc2.16 . . . . . . 7 π‘Š = {𝑛 ∈ β„• ∣ 𝐡 ∈ (πΎβ€˜π‘›)}
7169, 70elrab2 3685 . . . . . 6 (𝑁 ∈ π‘Š ↔ (𝑁 ∈ β„• ∧ 𝐡 ∈ (πΎβ€˜π‘)))
7228, 67, 71sylanbrc 581 . . . . 5 ((πœ‘ ∧ (𝑁 ∈ β„• ∧ 𝐡 < (2nd β€˜(πΉβ€˜(πΊβ€˜(πΎβ€˜π‘)))))) β†’ 𝑁 ∈ π‘Š)
7372expr 455 . . . 4 ((πœ‘ ∧ 𝑁 ∈ β„•) β†’ (𝐡 < (2nd β€˜(πΉβ€˜(πΊβ€˜(πΎβ€˜π‘)))) β†’ 𝑁 ∈ π‘Š))
7427, 73sylbird 259 . . 3 ((πœ‘ ∧ 𝑁 ∈ β„•) β†’ (Β¬ (2nd β€˜(πΉβ€˜(πΊβ€˜(πΎβ€˜π‘)))) ≀ 𝐡 β†’ 𝑁 ∈ π‘Š))
7574con1d 145 . 2 ((πœ‘ ∧ 𝑁 ∈ β„•) β†’ (Β¬ 𝑁 ∈ π‘Š β†’ (2nd β€˜(πΉβ€˜(πΊβ€˜(πΎβ€˜π‘)))) ≀ 𝐡))
7675impr 453 1 ((πœ‘ ∧ (𝑁 ∈ β„• ∧ Β¬ 𝑁 ∈ π‘Š)) β†’ (2nd β€˜(πΉβ€˜(πΊβ€˜(πΎβ€˜π‘)))) ≀ 𝐡)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1085   = wceq 1539  βˆƒwex 1779   ∈ wcel 2104   β‰  wne 2938  {crab 3430   ∩ cin 3946   βŠ† wss 3947  βˆ…c0 4321  ifcif 4527  π’« cpw 4601  {csn 4627  βˆͺ cuni 4907   class class class wbr 5147   Γ— cxp 5673  ran crn 5676   ∘ ccom 5679  βŸΆwf 6538  β€˜cfv 6542  (class class class)co 7411  1st c1st 7975  2nd c2nd 7976  Fincfn 8941  β„cr 11111  1c1 11113   + caddc 11115   < clt 11252   ≀ cle 11253   βˆ’ cmin 11448  β„•cn 12216  (,)cioo 13328  [,]cicc 13331  seqcseq 13970  abscabs 15185
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-n0 12477  df-z 12563  df-uz 12827  df-ioo 13332  df-icc 13335  df-fz 13489  df-seq 13971
This theorem is referenced by:  ovolicc2lem3  25268  ovolicc2lem4  25269
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