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Theorem poimirlem19 35570
Description: Lemma for poimir 35584 establishing the vertices of the simplex in poimirlem20 35571. (Contributed by Brendan Leahy, 21-Aug-2020.)
Hypotheses
Ref Expression
poimir.0 (𝜑𝑁 ∈ ℕ)
poimirlem22.s 𝑆 = {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}
poimirlem22.1 (𝜑𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑m (1...𝑁)))
poimirlem22.2 (𝜑𝑇𝑆)
poimirlem22.3 ((𝜑𝑛 ∈ (1...𝑁)) → ∃𝑝 ∈ ran 𝐹(𝑝𝑛) ≠ 0)
poimirlem21.4 (𝜑 → (2nd𝑇) = 𝑁)
Assertion
Ref Expression
poimirlem19 (𝜑𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ((𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))) ∘f + (((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0})))))
Distinct variable groups:   𝑓,𝑗,𝑛,𝑝,𝑡,𝑦   𝜑,𝑗,𝑛,𝑦   𝑗,𝐹,𝑛,𝑦   𝑗,𝑁,𝑛,𝑦   𝑇,𝑗,𝑛,𝑦   𝜑,𝑝,𝑡   𝑓,𝐾,𝑗,𝑛,𝑝,𝑡   𝑓,𝑁,𝑝,𝑡   𝑇,𝑓,𝑝   𝑓,𝐹,𝑝,𝑡   𝑡,𝑇   𝑆,𝑗,𝑛,𝑝,𝑡,𝑦
Allowed substitution hints:   𝜑(𝑓)   𝑆(𝑓)   𝐾(𝑦)

Proof of Theorem poimirlem19
StepHypRef Expression
1 poimirlem22.2 . . 3 (𝜑𝑇𝑆)
2 fveq2 6739 . . . . . . . . . 10 (𝑡 = 𝑇 → (2nd𝑡) = (2nd𝑇))
32breq2d 5082 . . . . . . . . 9 (𝑡 = 𝑇 → (𝑦 < (2nd𝑡) ↔ 𝑦 < (2nd𝑇)))
43ifbid 4479 . . . . . . . 8 (𝑡 = 𝑇 → if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)))
5 2fveq3 6744 . . . . . . . . 9 (𝑡 = 𝑇 → (1st ‘(1st𝑡)) = (1st ‘(1st𝑇)))
6 2fveq3 6744 . . . . . . . . . . . 12 (𝑡 = 𝑇 → (2nd ‘(1st𝑡)) = (2nd ‘(1st𝑇)))
76imaeq1d 5946 . . . . . . . . . . 11 (𝑡 = 𝑇 → ((2nd ‘(1st𝑡)) “ (1...𝑗)) = ((2nd ‘(1st𝑇)) “ (1...𝑗)))
87xpeq1d 5598 . . . . . . . . . 10 (𝑡 = 𝑇 → (((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}))
96imaeq1d 5946 . . . . . . . . . . 11 (𝑡 = 𝑇 → ((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)))
109xpeq1d 5598 . . . . . . . . . 10 (𝑡 = 𝑇 → (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))
118, 10uneq12d 4095 . . . . . . . . 9 (𝑡 = 𝑇 → ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))
125, 11oveq12d 7253 . . . . . . . 8 (𝑡 = 𝑇 → ((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))
134, 12csbeq12dv 3837 . . . . . . 7 (𝑡 = 𝑇if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))
1413mpteq2dv 5168 . . . . . 6 (𝑡 = 𝑇 → (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))))
1514eqeq2d 2750 . . . . 5 (𝑡 = 𝑇 → (𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ↔ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))))
16 poimirlem22.s . . . . 5 𝑆 = {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}
1715, 16elrab2 3620 . . . 4 (𝑇𝑆 ↔ (𝑇 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))))
1817simprbi 500 . . 3 (𝑇𝑆𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))))
191, 18syl 17 . 2 (𝜑𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))))
20 elrabi 3611 . . . . . . . . . . . 12 (𝑇 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} → 𝑇 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
2120, 16eleq2s 2858 . . . . . . . . . . 11 (𝑇𝑆𝑇 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
221, 21syl 17 . . . . . . . . . 10 (𝜑𝑇 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
23 xp1st 7815 . . . . . . . . . 10 (𝑇 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (1st𝑇) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
2422, 23syl 17 . . . . . . . . 9 (𝜑 → (1st𝑇) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
25 xp1st 7815 . . . . . . . . 9 ((1st𝑇) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (1st ‘(1st𝑇)) ∈ ((0..^𝐾) ↑m (1...𝑁)))
2624, 25syl 17 . . . . . . . 8 (𝜑 → (1st ‘(1st𝑇)) ∈ ((0..^𝐾) ↑m (1...𝑁)))
27 elmapfn 8570 . . . . . . . 8 ((1st ‘(1st𝑇)) ∈ ((0..^𝐾) ↑m (1...𝑁)) → (1st ‘(1st𝑇)) Fn (1...𝑁))
2826, 27syl 17 . . . . . . 7 (𝜑 → (1st ‘(1st𝑇)) Fn (1...𝑁))
2928adantr 484 . . . . . 6 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (1st ‘(1st𝑇)) Fn (1...𝑁))
30 1ex 10859 . . . . . . . . . 10 1 ∈ V
31 fnconstg 6629 . . . . . . . . . 10 (1 ∈ V → (((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) Fn ((2nd ‘(1st𝑇)) “ (1...𝑦)))
3230, 31ax-mp 5 . . . . . . . . 9 (((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) Fn ((2nd ‘(1st𝑇)) “ (1...𝑦))
33 c0ex 10857 . . . . . . . . . 10 0 ∈ V
34 fnconstg 6629 . . . . . . . . . 10 (0 ∈ V → (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}) Fn ((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)))
3533, 34ax-mp 5 . . . . . . . . 9 (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}) Fn ((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁))
3632, 35pm3.2i 474 . . . . . . . 8 ((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) Fn ((2nd ‘(1st𝑇)) “ (1...𝑦)) ∧ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}) Fn ((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)))
37 xp2nd 7816 . . . . . . . . . . . . 13 ((1st𝑇) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (2nd ‘(1st𝑇)) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
3824, 37syl 17 . . . . . . . . . . . 12 (𝜑 → (2nd ‘(1st𝑇)) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
39 fvex 6752 . . . . . . . . . . . . 13 (2nd ‘(1st𝑇)) ∈ V
40 f1oeq1 6671 . . . . . . . . . . . . 13 (𝑓 = (2nd ‘(1st𝑇)) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁)))
4139, 40elab 3602 . . . . . . . . . . . 12 ((2nd ‘(1st𝑇)) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ (2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))
4238, 41sylib 221 . . . . . . . . . . 11 (𝜑 → (2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))
43 dff1o3 6689 . . . . . . . . . . . 12 ((2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) ↔ ((2nd ‘(1st𝑇)):(1...𝑁)–onto→(1...𝑁) ∧ Fun (2nd ‘(1st𝑇))))
4443simprbi 500 . . . . . . . . . . 11 ((2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → Fun (2nd ‘(1st𝑇)))
4542, 44syl 17 . . . . . . . . . 10 (𝜑 → Fun (2nd ‘(1st𝑇)))
46 imain 6486 . . . . . . . . . 10 (Fun (2nd ‘(1st𝑇)) → ((2nd ‘(1st𝑇)) “ ((1...𝑦) ∩ ((𝑦 + 1)...𝑁))) = (((2nd ‘(1st𝑇)) “ (1...𝑦)) ∩ ((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁))))
4745, 46syl 17 . . . . . . . . 9 (𝜑 → ((2nd ‘(1st𝑇)) “ ((1...𝑦) ∩ ((𝑦 + 1)...𝑁))) = (((2nd ‘(1st𝑇)) “ (1...𝑦)) ∩ ((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁))))
48 elfznn0 13235 . . . . . . . . . . . . . 14 (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 ∈ ℕ0)
4948nn0red 12181 . . . . . . . . . . . . 13 (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 ∈ ℝ)
5049ltp1d 11792 . . . . . . . . . . . 12 (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 < (𝑦 + 1))
51 fzdisj 13169 . . . . . . . . . . . 12 (𝑦 < (𝑦 + 1) → ((1...𝑦) ∩ ((𝑦 + 1)...𝑁)) = ∅)
5250, 51syl 17 . . . . . . . . . . 11 (𝑦 ∈ (0...(𝑁 − 1)) → ((1...𝑦) ∩ ((𝑦 + 1)...𝑁)) = ∅)
5352imaeq2d 5947 . . . . . . . . . 10 (𝑦 ∈ (0...(𝑁 − 1)) → ((2nd ‘(1st𝑇)) “ ((1...𝑦) ∩ ((𝑦 + 1)...𝑁))) = ((2nd ‘(1st𝑇)) “ ∅))
54 ima0 5963 . . . . . . . . . 10 ((2nd ‘(1st𝑇)) “ ∅) = ∅
5553, 54eqtrdi 2796 . . . . . . . . 9 (𝑦 ∈ (0...(𝑁 − 1)) → ((2nd ‘(1st𝑇)) “ ((1...𝑦) ∩ ((𝑦 + 1)...𝑁))) = ∅)
5647, 55sylan9req 2801 . . . . . . . 8 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((2nd ‘(1st𝑇)) “ (1...𝑦)) ∩ ((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁))) = ∅)
57 fnun 6512 . . . . . . . 8 ((((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) Fn ((2nd ‘(1st𝑇)) “ (1...𝑦)) ∧ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}) Fn ((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁))) ∧ (((2nd ‘(1st𝑇)) “ (1...𝑦)) ∩ ((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁))) = ∅) → ((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})) Fn (((2nd ‘(1st𝑇)) “ (1...𝑦)) ∪ ((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁))))
5836, 56, 57sylancr 590 . . . . . . 7 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})) Fn (((2nd ‘(1st𝑇)) “ (1...𝑦)) ∪ ((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁))))
59 imaundi 6031 . . . . . . . . 9 ((2nd ‘(1st𝑇)) “ ((1...𝑦) ∪ ((𝑦 + 1)...𝑁))) = (((2nd ‘(1st𝑇)) “ (1...𝑦)) ∪ ((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)))
60 nn0p1nn 12159 . . . . . . . . . . . . . . 15 (𝑦 ∈ ℕ0 → (𝑦 + 1) ∈ ℕ)
6148, 60syl 17 . . . . . . . . . . . . . 14 (𝑦 ∈ (0...(𝑁 − 1)) → (𝑦 + 1) ∈ ℕ)
62 nnuz 12507 . . . . . . . . . . . . . 14 ℕ = (ℤ‘1)
6361, 62eleqtrdi 2850 . . . . . . . . . . . . 13 (𝑦 ∈ (0...(𝑁 − 1)) → (𝑦 + 1) ∈ (ℤ‘1))
6463adantl 485 . . . . . . . . . . . 12 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (𝑦 + 1) ∈ (ℤ‘1))
65 poimir.0 . . . . . . . . . . . . . . . 16 (𝜑𝑁 ∈ ℕ)
6665nncnd 11876 . . . . . . . . . . . . . . 15 (𝜑𝑁 ∈ ℂ)
67 npcan1 11287 . . . . . . . . . . . . . . 15 (𝑁 ∈ ℂ → ((𝑁 − 1) + 1) = 𝑁)
6866, 67syl 17 . . . . . . . . . . . . . 14 (𝜑 → ((𝑁 − 1) + 1) = 𝑁)
6968adantr 484 . . . . . . . . . . . . 13 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((𝑁 − 1) + 1) = 𝑁)
70 elfzuz3 13139 . . . . . . . . . . . . . . 15 (𝑦 ∈ (0...(𝑁 − 1)) → (𝑁 − 1) ∈ (ℤ𝑦))
71 peano2uz 12527 . . . . . . . . . . . . . . 15 ((𝑁 − 1) ∈ (ℤ𝑦) → ((𝑁 − 1) + 1) ∈ (ℤ𝑦))
7270, 71syl 17 . . . . . . . . . . . . . 14 (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑁 − 1) + 1) ∈ (ℤ𝑦))
7372adantl 485 . . . . . . . . . . . . 13 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((𝑁 − 1) + 1) ∈ (ℤ𝑦))
7469, 73eqeltrrd 2841 . . . . . . . . . . . 12 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ (ℤ𝑦))
75 fzsplit2 13167 . . . . . . . . . . . 12 (((𝑦 + 1) ∈ (ℤ‘1) ∧ 𝑁 ∈ (ℤ𝑦)) → (1...𝑁) = ((1...𝑦) ∪ ((𝑦 + 1)...𝑁)))
7664, 74, 75syl2anc 587 . . . . . . . . . . 11 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (1...𝑁) = ((1...𝑦) ∪ ((𝑦 + 1)...𝑁)))
7776imaeq2d 5947 . . . . . . . . . 10 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st𝑇)) “ (1...𝑁)) = ((2nd ‘(1st𝑇)) “ ((1...𝑦) ∪ ((𝑦 + 1)...𝑁))))
78 f1ofo 6690 . . . . . . . . . . . 12 ((2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → (2nd ‘(1st𝑇)):(1...𝑁)–onto→(1...𝑁))
79 foima 6660 . . . . . . . . . . . 12 ((2nd ‘(1st𝑇)):(1...𝑁)–onto→(1...𝑁) → ((2nd ‘(1st𝑇)) “ (1...𝑁)) = (1...𝑁))
8042, 78, 793syl 18 . . . . . . . . . . 11 (𝜑 → ((2nd ‘(1st𝑇)) “ (1...𝑁)) = (1...𝑁))
8180adantr 484 . . . . . . . . . 10 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st𝑇)) “ (1...𝑁)) = (1...𝑁))
8277, 81eqtr3d 2781 . . . . . . . . 9 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st𝑇)) “ ((1...𝑦) ∪ ((𝑦 + 1)...𝑁))) = (1...𝑁))
8359, 82eqtr3id 2794 . . . . . . . 8 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((2nd ‘(1st𝑇)) “ (1...𝑦)) ∪ ((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁))) = (1...𝑁))
8483fneq2d 6494 . . . . . . 7 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})) Fn (((2nd ‘(1st𝑇)) “ (1...𝑦)) ∪ ((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁))) ↔ ((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})) Fn (1...𝑁)))
8558, 84mpbid 235 . . . . . 6 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})) Fn (1...𝑁))
86 ovexd 7270 . . . . . 6 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (1...𝑁) ∈ V)
87 inidm 4150 . . . . . 6 ((1...𝑁) ∩ (1...𝑁)) = (1...𝑁)
88 eqidd 2740 . . . . . 6 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) → ((1st ‘(1st𝑇))‘𝑛) = ((1st ‘(1st𝑇))‘𝑛))
89 eqidd 2740 . . . . . 6 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) → (((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛) = (((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛))
9029, 85, 86, 86, 87, 88, 89offval 7499 . . . . 5 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))) = (𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) + (((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛))))
91 elmapi 8554 . . . . . . . . . . . . 13 ((1st ‘(1st𝑇)) ∈ ((0..^𝐾) ↑m (1...𝑁)) → (1st ‘(1st𝑇)):(1...𝑁)⟶(0..^𝐾))
9226, 91syl 17 . . . . . . . . . . . 12 (𝜑 → (1st ‘(1st𝑇)):(1...𝑁)⟶(0..^𝐾))
9392ffvelrnda 6926 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (1...𝑁)) → ((1st ‘(1st𝑇))‘𝑛) ∈ (0..^𝐾))
94 elfzonn0 13317 . . . . . . . . . . 11 (((1st ‘(1st𝑇))‘𝑛) ∈ (0..^𝐾) → ((1st ‘(1st𝑇))‘𝑛) ∈ ℕ0)
9593, 94syl 17 . . . . . . . . . 10 ((𝜑𝑛 ∈ (1...𝑁)) → ((1st ‘(1st𝑇))‘𝑛) ∈ ℕ0)
9695nn0cnd 12182 . . . . . . . . 9 ((𝜑𝑛 ∈ (1...𝑁)) → ((1st ‘(1st𝑇))‘𝑛) ∈ ℂ)
9796adantlr 715 . . . . . . . 8 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) → ((1st ‘(1st𝑇))‘𝑛) ∈ ℂ)
98 ax-1cn 10817 . . . . . . . . . 10 1 ∈ ℂ
99 0cn 10855 . . . . . . . . . 10 0 ∈ ℂ
10098, 99ifcli 4503 . . . . . . . . 9 if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0) ∈ ℂ
101100a1i 11 . . . . . . . 8 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) → if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0) ∈ ℂ)
102 snssi 4738 . . . . . . . . . . 11 (1 ∈ ℂ → {1} ⊆ ℂ)
10398, 102ax-mp 5 . . . . . . . . . 10 {1} ⊆ ℂ
104 snssi 4738 . . . . . . . . . . 11 (0 ∈ ℂ → {0} ⊆ ℂ)
10599, 104ax-mp 5 . . . . . . . . . 10 {0} ⊆ ℂ
106103, 105unssi 4116 . . . . . . . . 9 ({1} ∪ {0}) ⊆ ℂ
10730fconst 6627 . . . . . . . . . . . . 13 ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}):(((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1)))⟶{1}
10833fconst 6627 . . . . . . . . . . . . 13 ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}):(((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁))⟶{0}
109107, 108pm3.2i 474 . . . . . . . . . . . 12 (((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}):(((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1)))⟶{1} ∧ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}):(((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁))⟶{0})
110 simpr 488 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑛 ∈ ((1 + 1)...𝑁)) → 𝑛 ∈ ((1 + 1)...𝑁))
11165nnzd 12311 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑𝑁 ∈ ℤ)
112 1z 12237 . . . . . . . . . . . . . . . . . . . . . . . 24 1 ∈ ℤ
113 peano2z 12248 . . . . . . . . . . . . . . . . . . . . . . . 24 (1 ∈ ℤ → (1 + 1) ∈ ℤ)
114112, 113ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . 23 (1 + 1) ∈ ℤ
115111, 114jctil 523 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → ((1 + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ))
116 elfzelz 13142 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑛 ∈ ((1 + 1)...𝑁) → 𝑛 ∈ ℤ)
117116, 112jctir 524 . . . . . . . . . . . . . . . . . . . . . 22 (𝑛 ∈ ((1 + 1)...𝑁) → (𝑛 ∈ ℤ ∧ 1 ∈ ℤ))
118 fzsubel 13178 . . . . . . . . . . . . . . . . . . . . . 22 ((((1 + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑛 ∈ ℤ ∧ 1 ∈ ℤ)) → (𝑛 ∈ ((1 + 1)...𝑁) ↔ (𝑛 − 1) ∈ (((1 + 1) − 1)...(𝑁 − 1))))
119115, 117, 118syl2an 599 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑛 ∈ ((1 + 1)...𝑁)) → (𝑛 ∈ ((1 + 1)...𝑁) ↔ (𝑛 − 1) ∈ (((1 + 1) − 1)...(𝑁 − 1))))
120110, 119mpbid 235 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑛 ∈ ((1 + 1)...𝑁)) → (𝑛 − 1) ∈ (((1 + 1) − 1)...(𝑁 − 1)))
12198, 98pncan3oi 11124 . . . . . . . . . . . . . . . . . . . . 21 ((1 + 1) − 1) = 1
122121oveq1i 7245 . . . . . . . . . . . . . . . . . . . 20 (((1 + 1) − 1)...(𝑁 − 1)) = (1...(𝑁 − 1))
123120, 122eleqtrdi 2850 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑛 ∈ ((1 + 1)...𝑁)) → (𝑛 − 1) ∈ (1...(𝑁 − 1)))
124123ralrimiva 3108 . . . . . . . . . . . . . . . . . 18 (𝜑 → ∀𝑛 ∈ ((1 + 1)...𝑁)(𝑛 − 1) ∈ (1...(𝑁 − 1)))
125 simpr 488 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑦 ∈ (1...(𝑁 − 1))) → 𝑦 ∈ (1...(𝑁 − 1)))
126 peano2zm 12250 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑁 ∈ ℤ → (𝑁 − 1) ∈ ℤ)
127111, 126syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → (𝑁 − 1) ∈ ℤ)
128127, 112jctil 523 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → (1 ∈ ℤ ∧ (𝑁 − 1) ∈ ℤ))
129 elfzelz 13142 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 ∈ (1...(𝑁 − 1)) → 𝑦 ∈ ℤ)
130129, 112jctir 524 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦 ∈ (1...(𝑁 − 1)) → (𝑦 ∈ ℤ ∧ 1 ∈ ℤ))
131 fzaddel 13176 . . . . . . . . . . . . . . . . . . . . . . 23 (((1 ∈ ℤ ∧ (𝑁 − 1) ∈ ℤ) ∧ (𝑦 ∈ ℤ ∧ 1 ∈ ℤ)) → (𝑦 ∈ (1...(𝑁 − 1)) ↔ (𝑦 + 1) ∈ ((1 + 1)...((𝑁 − 1) + 1))))
132128, 130, 131syl2an 599 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑦 ∈ (1...(𝑁 − 1))) → (𝑦 ∈ (1...(𝑁 − 1)) ↔ (𝑦 + 1) ∈ ((1 + 1)...((𝑁 − 1) + 1))))
133125, 132mpbid 235 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑦 ∈ (1...(𝑁 − 1))) → (𝑦 + 1) ∈ ((1 + 1)...((𝑁 − 1) + 1)))
13468oveq2d 7251 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → ((1 + 1)...((𝑁 − 1) + 1)) = ((1 + 1)...𝑁))
135134adantr 484 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑦 ∈ (1...(𝑁 − 1))) → ((1 + 1)...((𝑁 − 1) + 1)) = ((1 + 1)...𝑁))
136133, 135eleqtrd 2842 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑦 ∈ (1...(𝑁 − 1))) → (𝑦 + 1) ∈ ((1 + 1)...𝑁))
137116zcnd 12313 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑛 ∈ ((1 + 1)...𝑁) → 𝑛 ∈ ℂ)
138129zcnd 12313 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦 ∈ (1...(𝑁 − 1)) → 𝑦 ∈ ℂ)
139 subadd2 11112 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑛 ∈ ℂ ∧ 1 ∈ ℂ ∧ 𝑦 ∈ ℂ) → ((𝑛 − 1) = 𝑦 ↔ (𝑦 + 1) = 𝑛))
14098, 139mp3an2 1451 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑛 ∈ ℂ ∧ 𝑦 ∈ ℂ) → ((𝑛 − 1) = 𝑦 ↔ (𝑦 + 1) = 𝑛))
141 eqcom 2746 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 = (𝑛 − 1) ↔ (𝑛 − 1) = 𝑦)
142 eqcom 2746 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑛 = (𝑦 + 1) ↔ (𝑦 + 1) = 𝑛)
143140, 141, 1423bitr4g 317 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑛 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑦 = (𝑛 − 1) ↔ 𝑛 = (𝑦 + 1)))
144137, 138, 143syl2anr 600 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑦 ∈ (1...(𝑁 − 1)) ∧ 𝑛 ∈ ((1 + 1)...𝑁)) → (𝑦 = (𝑛 − 1) ↔ 𝑛 = (𝑦 + 1)))
145144ralrimiva 3108 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 ∈ (1...(𝑁 − 1)) → ∀𝑛 ∈ ((1 + 1)...𝑁)(𝑦 = (𝑛 − 1) ↔ 𝑛 = (𝑦 + 1)))
146145adantl 485 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑦 ∈ (1...(𝑁 − 1))) → ∀𝑛 ∈ ((1 + 1)...𝑁)(𝑦 = (𝑛 − 1) ↔ 𝑛 = (𝑦 + 1)))
147 reu6i 3658 . . . . . . . . . . . . . . . . . . . 20 (((𝑦 + 1) ∈ ((1 + 1)...𝑁) ∧ ∀𝑛 ∈ ((1 + 1)...𝑁)(𝑦 = (𝑛 − 1) ↔ 𝑛 = (𝑦 + 1))) → ∃!𝑛 ∈ ((1 + 1)...𝑁)𝑦 = (𝑛 − 1))
148136, 146, 147syl2anc 587 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑦 ∈ (1...(𝑁 − 1))) → ∃!𝑛 ∈ ((1 + 1)...𝑁)𝑦 = (𝑛 − 1))
149148ralrimiva 3108 . . . . . . . . . . . . . . . . . 18 (𝜑 → ∀𝑦 ∈ (1...(𝑁 − 1))∃!𝑛 ∈ ((1 + 1)...𝑁)𝑦 = (𝑛 − 1))
150 eqid 2739 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ ((1 + 1)...𝑁) ↦ (𝑛 − 1)) = (𝑛 ∈ ((1 + 1)...𝑁) ↦ (𝑛 − 1))
151150f1ompt 6950 . . . . . . . . . . . . . . . . . 18 ((𝑛 ∈ ((1 + 1)...𝑁) ↦ (𝑛 − 1)):((1 + 1)...𝑁)–1-1-onto→(1...(𝑁 − 1)) ↔ (∀𝑛 ∈ ((1 + 1)...𝑁)(𝑛 − 1) ∈ (1...(𝑁 − 1)) ∧ ∀𝑦 ∈ (1...(𝑁 − 1))∃!𝑛 ∈ ((1 + 1)...𝑁)𝑦 = (𝑛 − 1)))
152124, 149, 151sylanbrc 586 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝑛 ∈ ((1 + 1)...𝑁) ↦ (𝑛 − 1)):((1 + 1)...𝑁)–1-1-onto→(1...(𝑁 − 1)))
153 f1osng 6723 . . . . . . . . . . . . . . . . . 18 ((1 ∈ V ∧ 𝑁 ∈ ℕ) → {⟨1, 𝑁⟩}:{1}–1-1-onto→{𝑁})
15430, 65, 153sylancr 590 . . . . . . . . . . . . . . . . 17 (𝜑 → {⟨1, 𝑁⟩}:{1}–1-1-onto→{𝑁})
15565nnred 11875 . . . . . . . . . . . . . . . . . . . . 21 (𝜑𝑁 ∈ ℝ)
156155ltm1d 11794 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝑁 − 1) < 𝑁)
157127zred 12312 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (𝑁 − 1) ∈ ℝ)
158157, 155ltnled 11009 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ((𝑁 − 1) < 𝑁 ↔ ¬ 𝑁 ≤ (𝑁 − 1)))
159156, 158mpbid 235 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ¬ 𝑁 ≤ (𝑁 − 1))
160 elfzle2 13146 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ (1...(𝑁 − 1)) → 𝑁 ≤ (𝑁 − 1))
161159, 160nsyl 142 . . . . . . . . . . . . . . . . . 18 (𝜑 → ¬ 𝑁 ∈ (1...(𝑁 − 1)))
162 disjsn 4644 . . . . . . . . . . . . . . . . . 18 (((1...(𝑁 − 1)) ∩ {𝑁}) = ∅ ↔ ¬ 𝑁 ∈ (1...(𝑁 − 1)))
163161, 162sylibr 237 . . . . . . . . . . . . . . . . 17 (𝜑 → ((1...(𝑁 − 1)) ∩ {𝑁}) = ∅)
164 1re 10863 . . . . . . . . . . . . . . . . . . . . . 22 1 ∈ ℝ
165164ltp1i 11766 . . . . . . . . . . . . . . . . . . . . 21 1 < (1 + 1)
166114zrei 12212 . . . . . . . . . . . . . . . . . . . . . 22 (1 + 1) ∈ ℝ
167164, 166ltnlei 10983 . . . . . . . . . . . . . . . . . . . . 21 (1 < (1 + 1) ↔ ¬ (1 + 1) ≤ 1)
168165, 167mpbi 233 . . . . . . . . . . . . . . . . . . . 20 ¬ (1 + 1) ≤ 1
169 elfzle1 13145 . . . . . . . . . . . . . . . . . . . 20 (1 ∈ ((1 + 1)...𝑁) → (1 + 1) ≤ 1)
170168, 169mto 200 . . . . . . . . . . . . . . . . . . 19 ¬ 1 ∈ ((1 + 1)...𝑁)
171 disjsn 4644 . . . . . . . . . . . . . . . . . . 19 ((((1 + 1)...𝑁) ∩ {1}) = ∅ ↔ ¬ 1 ∈ ((1 + 1)...𝑁))
172170, 171mpbir 234 . . . . . . . . . . . . . . . . . 18 (((1 + 1)...𝑁) ∩ {1}) = ∅
173 f1oun 6702 . . . . . . . . . . . . . . . . . 18 ((((𝑛 ∈ ((1 + 1)...𝑁) ↦ (𝑛 − 1)):((1 + 1)...𝑁)–1-1-onto→(1...(𝑁 − 1)) ∧ {⟨1, 𝑁⟩}:{1}–1-1-onto→{𝑁}) ∧ ((((1 + 1)...𝑁) ∩ {1}) = ∅ ∧ ((1...(𝑁 − 1)) ∩ {𝑁}) = ∅)) → ((𝑛 ∈ ((1 + 1)...𝑁) ↦ (𝑛 − 1)) ∪ {⟨1, 𝑁⟩}):(((1 + 1)...𝑁) ∪ {1})–1-1-onto→((1...(𝑁 − 1)) ∪ {𝑁}))
174172, 173mpanr1 703 . . . . . . . . . . . . . . . . 17 ((((𝑛 ∈ ((1 + 1)...𝑁) ↦ (𝑛 − 1)):((1 + 1)...𝑁)–1-1-onto→(1...(𝑁 − 1)) ∧ {⟨1, 𝑁⟩}:{1}–1-1-onto→{𝑁}) ∧ ((1...(𝑁 − 1)) ∩ {𝑁}) = ∅) → ((𝑛 ∈ ((1 + 1)...𝑁) ↦ (𝑛 − 1)) ∪ {⟨1, 𝑁⟩}):(((1 + 1)...𝑁) ∪ {1})–1-1-onto→((1...(𝑁 − 1)) ∪ {𝑁}))
175152, 154, 163, 174syl21anc 838 . . . . . . . . . . . . . . . 16 (𝜑 → ((𝑛 ∈ ((1 + 1)...𝑁) ↦ (𝑛 − 1)) ∪ {⟨1, 𝑁⟩}):(((1 + 1)...𝑁) ∪ {1})–1-1-onto→((1...(𝑁 − 1)) ∪ {𝑁}))
176 eleq1 2827 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑛 = 1 → (𝑛 ∈ ((1 + 1)...𝑁) ↔ 1 ∈ ((1 + 1)...𝑁)))
177170, 176mtbiri 330 . . . . . . . . . . . . . . . . . . . . . 22 (𝑛 = 1 → ¬ 𝑛 ∈ ((1 + 1)...𝑁))
178177necon2ai 2973 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 ∈ ((1 + 1)...𝑁) → 𝑛 ≠ 1)
179 ifnefalse 4468 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 ≠ 1 → if(𝑛 = 1, 𝑁, (𝑛 − 1)) = (𝑛 − 1))
180178, 179syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ ((1 + 1)...𝑁) → if(𝑛 = 1, 𝑁, (𝑛 − 1)) = (𝑛 − 1))
181180mpteq2ia 5163 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ ((1 + 1)...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) = (𝑛 ∈ ((1 + 1)...𝑁) ↦ (𝑛 − 1))
182181uneq1i 4090 . . . . . . . . . . . . . . . . . 18 ((𝑛 ∈ ((1 + 1)...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) ∪ {⟨1, 𝑁⟩}) = ((𝑛 ∈ ((1 + 1)...𝑁) ↦ (𝑛 − 1)) ∪ {⟨1, 𝑁⟩})
18330a1i 11 . . . . . . . . . . . . . . . . . . 19 (𝜑 → 1 ∈ V)
18465, 62eleqtrdi 2850 . . . . . . . . . . . . . . . . . . . . 21 (𝜑𝑁 ∈ (ℤ‘1))
185 fzpred 13190 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ (ℤ‘1) → (1...𝑁) = ({1} ∪ ((1 + 1)...𝑁)))
186184, 185syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (1...𝑁) = ({1} ∪ ((1 + 1)...𝑁)))
187 uncom 4084 . . . . . . . . . . . . . . . . . . . 20 ({1} ∪ ((1 + 1)...𝑁)) = (((1 + 1)...𝑁) ∪ {1})
188186, 187eqtr2di 2797 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (((1 + 1)...𝑁) ∪ {1}) = (1...𝑁))
189 iftrue 4462 . . . . . . . . . . . . . . . . . . . 20 (𝑛 = 1 → if(𝑛 = 1, 𝑁, (𝑛 − 1)) = 𝑁)
190189adantl 485 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑛 = 1) → if(𝑛 = 1, 𝑁, (𝑛 − 1)) = 𝑁)
191183, 65, 188, 190fmptapd 7008 . . . . . . . . . . . . . . . . . 18 (𝜑 → ((𝑛 ∈ ((1 + 1)...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) ∪ {⟨1, 𝑁⟩}) = (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))
192182, 191eqtr3id 2794 . . . . . . . . . . . . . . . . 17 (𝜑 → ((𝑛 ∈ ((1 + 1)...𝑁) ↦ (𝑛 − 1)) ∪ {⟨1, 𝑁⟩}) = (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))
19368, 184eqeltrd 2840 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ((𝑁 − 1) + 1) ∈ (ℤ‘1))
194 uzid 12483 . . . . . . . . . . . . . . . . . . . . 21 ((𝑁 − 1) ∈ ℤ → (𝑁 − 1) ∈ (ℤ‘(𝑁 − 1)))
195 peano2uz 12527 . . . . . . . . . . . . . . . . . . . . 21 ((𝑁 − 1) ∈ (ℤ‘(𝑁 − 1)) → ((𝑁 − 1) + 1) ∈ (ℤ‘(𝑁 − 1)))
196127, 194, 1953syl 18 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ((𝑁 − 1) + 1) ∈ (ℤ‘(𝑁 − 1)))
19768, 196eqeltrrd 2841 . . . . . . . . . . . . . . . . . . 19 (𝜑𝑁 ∈ (ℤ‘(𝑁 − 1)))
198 fzsplit2 13167 . . . . . . . . . . . . . . . . . . 19 ((((𝑁 − 1) + 1) ∈ (ℤ‘1) ∧ 𝑁 ∈ (ℤ‘(𝑁 − 1))) → (1...𝑁) = ((1...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)))
199193, 197, 198syl2anc 587 . . . . . . . . . . . . . . . . . 18 (𝜑 → (1...𝑁) = ((1...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)))
20068oveq1d 7250 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (((𝑁 − 1) + 1)...𝑁) = (𝑁...𝑁))
201 fzsn 13184 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℤ → (𝑁...𝑁) = {𝑁})
202111, 201syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝑁...𝑁) = {𝑁})
203200, 202eqtrd 2779 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (((𝑁 − 1) + 1)...𝑁) = {𝑁})
204203uneq2d 4094 . . . . . . . . . . . . . . . . . 18 (𝜑 → ((1...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)) = ((1...(𝑁 − 1)) ∪ {𝑁}))
205199, 204eqtr2d 2780 . . . . . . . . . . . . . . . . 17 (𝜑 → ((1...(𝑁 − 1)) ∪ {𝑁}) = (1...𝑁))
206192, 188, 205f1oeq123d 6677 . . . . . . . . . . . . . . . 16 (𝜑 → (((𝑛 ∈ ((1 + 1)...𝑁) ↦ (𝑛 − 1)) ∪ {⟨1, 𝑁⟩}):(((1 + 1)...𝑁) ∪ {1})–1-1-onto→((1...(𝑁 − 1)) ∪ {𝑁}) ↔ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))):(1...𝑁)–1-1-onto→(1...𝑁)))
207175, 206mpbid 235 . . . . . . . . . . . . . . 15 (𝜑 → (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))):(1...𝑁)–1-1-onto→(1...𝑁))
208 f1oco 6705 . . . . . . . . . . . . . . 15 (((2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) ∧ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))):(1...𝑁)–1-1-onto→(1...𝑁)) → ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))):(1...𝑁)–1-1-onto→(1...𝑁))
20942, 207, 208syl2anc 587 . . . . . . . . . . . . . 14 (𝜑 → ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))):(1...𝑁)–1-1-onto→(1...𝑁))
210 dff1o3 6689 . . . . . . . . . . . . . . 15 (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))):(1...𝑁)–1-1-onto→(1...𝑁) ↔ (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))):(1...𝑁)–onto→(1...𝑁) ∧ Fun ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))))
211210simprbi 500 . . . . . . . . . . . . . 14 (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))):(1...𝑁)–1-1-onto→(1...𝑁) → Fun ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))))
212 imain 6486 . . . . . . . . . . . . . 14 (Fun ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) → (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ ((1...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁))) = ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) ∩ (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁))))
213209, 211, 2123syl 18 . . . . . . . . . . . . 13 (𝜑 → (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ ((1...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁))) = ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) ∩ (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁))))
21461nnred 11875 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ (0...(𝑁 − 1)) → (𝑦 + 1) ∈ ℝ)
215214ltp1d 11792 . . . . . . . . . . . . . . . 16 (𝑦 ∈ (0...(𝑁 − 1)) → (𝑦 + 1) < ((𝑦 + 1) + 1))
216 fzdisj 13169 . . . . . . . . . . . . . . . 16 ((𝑦 + 1) < ((𝑦 + 1) + 1) → ((1...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁)) = ∅)
217215, 216syl 17 . . . . . . . . . . . . . . 15 (𝑦 ∈ (0...(𝑁 − 1)) → ((1...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁)) = ∅)
218217imaeq2d 5947 . . . . . . . . . . . . . 14 (𝑦 ∈ (0...(𝑁 − 1)) → (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ ((1...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁))) = (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ ∅))
219 ima0 5963 . . . . . . . . . . . . . 14 (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ ∅) = ∅
220218, 219eqtrdi 2796 . . . . . . . . . . . . 13 (𝑦 ∈ (0...(𝑁 − 1)) → (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ ((1...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁))) = ∅)
221213, 220sylan9req 2801 . . . . . . . . . . . 12 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) ∩ (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁))) = ∅)
222 fun 6603 . . . . . . . . . . . 12 (((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}):(((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1)))⟶{1} ∧ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}):(((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁))⟶{0}) ∧ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) ∩ (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁))) = ∅) → (((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0})):((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) ∪ (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)))⟶({1} ∪ {0}))
223109, 221, 222sylancr 590 . . . . . . . . . . 11 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0})):((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) ∪ (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)))⟶({1} ∪ {0}))
224 imaundi 6031 . . . . . . . . . . . . 13 (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ ((1...(𝑦 + 1)) ∪ (((𝑦 + 1) + 1)...𝑁))) = ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) ∪ (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)))
22561peano2nnd 11877 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑦 + 1) + 1) ∈ ℕ)
226225, 62eleqtrdi 2850 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑦 + 1) + 1) ∈ (ℤ‘1))
227226adantl 485 . . . . . . . . . . . . . . . 16 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((𝑦 + 1) + 1) ∈ (ℤ‘1))
228 eluzp1p1 12496 . . . . . . . . . . . . . . . . . . 19 ((𝑁 − 1) ∈ (ℤ𝑦) → ((𝑁 − 1) + 1) ∈ (ℤ‘(𝑦 + 1)))
22970, 228syl 17 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑁 − 1) + 1) ∈ (ℤ‘(𝑦 + 1)))
230229adantl 485 . . . . . . . . . . . . . . . . 17 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((𝑁 − 1) + 1) ∈ (ℤ‘(𝑦 + 1)))
23169, 230eqeltrrd 2841 . . . . . . . . . . . . . . . 16 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ (ℤ‘(𝑦 + 1)))
232 fzsplit2 13167 . . . . . . . . . . . . . . . 16 ((((𝑦 + 1) + 1) ∈ (ℤ‘1) ∧ 𝑁 ∈ (ℤ‘(𝑦 + 1))) → (1...𝑁) = ((1...(𝑦 + 1)) ∪ (((𝑦 + 1) + 1)...𝑁)))
233227, 231, 232syl2anc 587 . . . . . . . . . . . . . . 15 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (1...𝑁) = ((1...(𝑦 + 1)) ∪ (((𝑦 + 1) + 1)...𝑁)))
234233imaeq2d 5947 . . . . . . . . . . . . . 14 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...𝑁)) = (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ ((1...(𝑦 + 1)) ∪ (((𝑦 + 1) + 1)...𝑁))))
235 f1ofo 6690 . . . . . . . . . . . . . . . 16 (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))):(1...𝑁)–1-1-onto→(1...𝑁) → ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))):(1...𝑁)–onto→(1...𝑁))
236 foima 6660 . . . . . . . . . . . . . . . 16 (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))):(1...𝑁)–onto→(1...𝑁) → (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...𝑁)) = (1...𝑁))
237209, 235, 2363syl 18 . . . . . . . . . . . . . . 15 (𝜑 → (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...𝑁)) = (1...𝑁))
238237adantr 484 . . . . . . . . . . . . . 14 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...𝑁)) = (1...𝑁))
239234, 238eqtr3d 2781 . . . . . . . . . . . . 13 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ ((1...(𝑦 + 1)) ∪ (((𝑦 + 1) + 1)...𝑁))) = (1...𝑁))
240224, 239eqtr3id 2794 . . . . . . . . . . . 12 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) ∪ (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁))) = (1...𝑁))
241240feq2d 6553 . . . . . . . . . . 11 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0})):((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) ∪ (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)))⟶({1} ∪ {0}) ↔ (((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0})):(1...𝑁)⟶({1} ∪ {0})))
242223, 241mpbid 235 . . . . . . . . . 10 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0})):(1...𝑁)⟶({1} ∪ {0}))
243242ffvelrnda 6926 . . . . . . . . 9 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) → ((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) ∈ ({1} ∪ {0}))
244106, 243sseldi 3916 . . . . . . . 8 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) → ((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) ∈ ℂ)
24597, 101, 244subadd23d 11241 . . . . . . 7 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) → ((((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0)) + ((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛)) = (((1st ‘(1st𝑇))‘𝑛) + (((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))))
246 oveq2 7243 . . . . . . . . . 10 (1 = if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0) → (((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) − 1) = (((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0)))
247246eqeq1d 2741 . . . . . . . . 9 (1 = if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0) → ((((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) − 1) = (((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛) ↔ (((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0)) = (((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛)))
248 oveq2 7243 . . . . . . . . . 10 (0 = if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0) → (((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) − 0) = (((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0)))
249248eqeq1d 2741 . . . . . . . . 9 (0 = if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0) → ((((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) − 0) = (((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛) ↔ (((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0)) = (((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛)))
250 1m1e0 11932 . . . . . . . . . . . . 13 (1 − 1) = 0
251 f1ofn 6684 . . . . . . . . . . . . . . . . . . . 20 ((2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → (2nd ‘(1st𝑇)) Fn (1...𝑁))
25242, 251syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (2nd ‘(1st𝑇)) Fn (1...𝑁))
253252adantr 484 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (2nd ‘(1st𝑇)) Fn (1...𝑁))
254 imassrn 5958 . . . . . . . . . . . . . . . . . . . 20 ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ (1...(𝑦 + 1))) ⊆ ran (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))
255 f1of 6683 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))):(1...𝑁)–1-1-onto→(1...𝑁) → (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))):(1...𝑁)⟶(1...𝑁))
256207, 255syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))):(1...𝑁)⟶(1...𝑁))
257256frnd 6575 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ran (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) ⊆ (1...𝑁))
258254, 257sstrid 3929 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ (1...(𝑦 + 1))) ⊆ (1...𝑁))
259258adantr 484 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ (1...(𝑦 + 1))) ⊆ (1...𝑁))
260 eqidd 2740 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) = (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))
261 eluzfz1 13149 . . . . . . . . . . . . . . . . . . . . . 22 (𝑁 ∈ (ℤ‘1) → 1 ∈ (1...𝑁))
262184, 261syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → 1 ∈ (1...𝑁))
263260, 190, 262, 65fvmptd 6847 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))‘1) = 𝑁)
264263adantr 484 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))‘1) = 𝑁)
265 f1ofn 6684 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))):(1...𝑁)–1-1-onto→(1...𝑁) → (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) Fn (1...𝑁))
266207, 265syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) Fn (1...𝑁))
267266adantr 484 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) Fn (1...𝑁))
268 fzss2 13182 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ (ℤ‘(𝑦 + 1)) → (1...(𝑦 + 1)) ⊆ (1...𝑁))
269231, 268syl 17 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (1...(𝑦 + 1)) ⊆ (1...𝑁))
270 eluzfz1 13149 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑦 + 1) ∈ (ℤ‘1) → 1 ∈ (1...(𝑦 + 1)))
27163, 270syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 ∈ (0...(𝑁 − 1)) → 1 ∈ (1...(𝑦 + 1)))
272271adantl 485 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → 1 ∈ (1...(𝑦 + 1)))
273 fnfvima 7071 . . . . . . . . . . . . . . . . . . . 20 (((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) Fn (1...𝑁) ∧ (1...(𝑦 + 1)) ⊆ (1...𝑁) ∧ 1 ∈ (1...(𝑦 + 1))) → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))‘1) ∈ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ (1...(𝑦 + 1))))
274267, 269, 272, 273syl3anc 1373 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))‘1) ∈ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ (1...(𝑦 + 1))))
275264, 274eqeltrrd 2841 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ (1...(𝑦 + 1))))
276 fnfvima 7071 . . . . . . . . . . . . . . . . . 18 (((2nd ‘(1st𝑇)) Fn (1...𝑁) ∧ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ (1...(𝑦 + 1))) ⊆ (1...𝑁) ∧ 𝑁 ∈ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ (1...(𝑦 + 1)))) → ((2nd ‘(1st𝑇))‘𝑁) ∈ ((2nd ‘(1st𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ (1...(𝑦 + 1)))))
277253, 259, 275, 276syl3anc 1373 . . . . . . . . . . . . . . . . 17 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st𝑇))‘𝑁) ∈ ((2nd ‘(1st𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ (1...(𝑦 + 1)))))
278 imaco 6133 . . . . . . . . . . . . . . . . 17 (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) = ((2nd ‘(1st𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ (1...(𝑦 + 1))))
279277, 278eleqtrrdi 2851 . . . . . . . . . . . . . . . 16 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st𝑇))‘𝑁) ∈ (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))))
280 fnconstg 6629 . . . . . . . . . . . . . . . . . 18 (1 ∈ V → ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) Fn (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))))
28130, 280ax-mp 5 . . . . . . . . . . . . . . . . 17 ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) Fn (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1)))
282 fnconstg 6629 . . . . . . . . . . . . . . . . . 18 (0 ∈ V → ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}) Fn (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)))
28333, 282ax-mp 5 . . . . . . . . . . . . . . . . 17 ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}) Fn (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁))
284 fvun1 6824 . . . . . . . . . . . . . . . . 17 ((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) Fn (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) ∧ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}) Fn (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) ∧ (((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) ∩ (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁))) = ∅ ∧ ((2nd ‘(1st𝑇))‘𝑁) ∈ (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))))) → ((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘((2nd ‘(1st𝑇))‘𝑁)) = (((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1})‘((2nd ‘(1st𝑇))‘𝑁)))
285281, 283, 284mp3an12 1453 . . . . . . . . . . . . . . . 16 ((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) ∩ (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁))) = ∅ ∧ ((2nd ‘(1st𝑇))‘𝑁) ∈ (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1)))) → ((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘((2nd ‘(1st𝑇))‘𝑁)) = (((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1})‘((2nd ‘(1st𝑇))‘𝑁)))
286221, 279, 285syl2anc 587 . . . . . . . . . . . . . . 15 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘((2nd ‘(1st𝑇))‘𝑁)) = (((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1})‘((2nd ‘(1st𝑇))‘𝑁)))
28730fvconst2 7041 . . . . . . . . . . . . . . . 16 (((2nd ‘(1st𝑇))‘𝑁) ∈ (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) → (((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1})‘((2nd ‘(1st𝑇))‘𝑁)) = 1)
288279, 287syl 17 . . . . . . . . . . . . . . 15 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1})‘((2nd ‘(1st𝑇))‘𝑁)) = 1)
289286, 288eqtrd 2779 . . . . . . . . . . . . . 14 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘((2nd ‘(1st𝑇))‘𝑁)) = 1)
290289oveq1d 7250 . . . . . . . . . . . . 13 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘((2nd ‘(1st𝑇))‘𝑁)) − 1) = (1 − 1))
291 fzss1 13181 . . . . . . . . . . . . . . . . . 18 ((𝑦 + 1) ∈ (ℤ‘1) → ((𝑦 + 1)...𝑁) ⊆ (1...𝑁))
29263, 291syl 17 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑦 + 1)...𝑁) ⊆ (1...𝑁))
293292adantl 485 . . . . . . . . . . . . . . . 16 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((𝑦 + 1)...𝑁) ⊆ (1...𝑁))
294 eluzfz2 13150 . . . . . . . . . . . . . . . . 17 (𝑁 ∈ (ℤ‘(𝑦 + 1)) → 𝑁 ∈ ((𝑦 + 1)...𝑁))
295231, 294syl 17 . . . . . . . . . . . . . . . 16 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ ((𝑦 + 1)...𝑁))
296 fnfvima 7071 . . . . . . . . . . . . . . . 16 (((2nd ‘(1st𝑇)) Fn (1...𝑁) ∧ ((𝑦 + 1)...𝑁) ⊆ (1...𝑁) ∧ 𝑁 ∈ ((𝑦 + 1)...𝑁)) → ((2nd ‘(1st𝑇))‘𝑁) ∈ ((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)))
297253, 293, 295, 296syl3anc 1373 . . . . . . . . . . . . . . 15 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st𝑇))‘𝑁) ∈ ((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)))
298 fvun2 6825 . . . . . . . . . . . . . . . 16 (((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) Fn ((2nd ‘(1st𝑇)) “ (1...𝑦)) ∧ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}) Fn ((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) ∧ ((((2nd ‘(1st𝑇)) “ (1...𝑦)) ∩ ((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁))) = ∅ ∧ ((2nd ‘(1st𝑇))‘𝑁) ∈ ((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)))) → (((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘((2nd ‘(1st𝑇))‘𝑁)) = ((((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})‘((2nd ‘(1st𝑇))‘𝑁)))
29932, 35, 298mp3an12 1453 . . . . . . . . . . . . . . 15 (((((2nd ‘(1st𝑇)) “ (1...𝑦)) ∩ ((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁))) = ∅ ∧ ((2nd ‘(1st𝑇))‘𝑁) ∈ ((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁))) → (((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘((2nd ‘(1st𝑇))‘𝑁)) = ((((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})‘((2nd ‘(1st𝑇))‘𝑁)))
30056, 297, 299syl2anc 587 . . . . . . . . . . . . . 14 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘((2nd ‘(1st𝑇))‘𝑁)) = ((((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})‘((2nd ‘(1st𝑇))‘𝑁)))
30133fvconst2 7041 . . . . . . . . . . . . . . 15 (((2nd ‘(1st𝑇))‘𝑁) ∈ ((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) → ((((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})‘((2nd ‘(1st𝑇))‘𝑁)) = 0)
302297, 301syl 17 . . . . . . . . . . . . . 14 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})‘((2nd ‘(1st𝑇))‘𝑁)) = 0)
303300, 302eqtrd 2779 . . . . . . . . . . . . 13 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘((2nd ‘(1st𝑇))‘𝑁)) = 0)
304250, 290, 3033eqtr4a 2806 . . . . . . . . . . . 12 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘((2nd ‘(1st𝑇))‘𝑁)) − 1) = (((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘((2nd ‘(1st𝑇))‘𝑁)))
305 fveq2 6739 . . . . . . . . . . . . . 14 (𝑛 = ((2nd ‘(1st𝑇))‘𝑁) → ((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) = ((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘((2nd ‘(1st𝑇))‘𝑁)))
306305oveq1d 7250 . . . . . . . . . . . . 13 (𝑛 = ((2nd ‘(1st𝑇))‘𝑁) → (((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) − 1) = (((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘((2nd ‘(1st𝑇))‘𝑁)) − 1))
307 fveq2 6739 . . . . . . . . . . . . 13 (𝑛 = ((2nd ‘(1st𝑇))‘𝑁) → (((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛) = (((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘((2nd ‘(1st𝑇))‘𝑁)))
308306, 307eqeq12d 2755 . . . . . . . . . . . 12 (𝑛 = ((2nd ‘(1st𝑇))‘𝑁) → ((((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) − 1) = (((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛) ↔ (((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘((2nd ‘(1st𝑇))‘𝑁)) − 1) = (((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘((2nd ‘(1st𝑇))‘𝑁))))
309304, 308syl5ibrcom 250 . . . . . . . . . . 11 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (𝑛 = ((2nd ‘(1st𝑇))‘𝑁) → (((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) − 1) = (((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛)))
310309imp 410 . . . . . . . . . 10 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 = ((2nd ‘(1st𝑇))‘𝑁)) → (((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) − 1) = (((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛))
311310adantlr 715 . . . . . . . . 9 ((((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑛 = ((2nd ‘(1st𝑇))‘𝑁)) → (((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) − 1) = (((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛))
312244subid1d 11208 . . . . . . . . . . 11 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) → (((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) − 0) = ((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛))
313312adantr 484 . . . . . . . . . 10 ((((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) ∧ ¬ 𝑛 = ((2nd ‘(1st𝑇))‘𝑁)) → (((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) − 0) = ((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛))
314 eldifsn 4717 . . . . . . . . . . . . . 14 (𝑛 ∈ ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘𝑁)}) ↔ (𝑛 ∈ (1...𝑁) ∧ 𝑛 ≠ ((2nd ‘(1st𝑇))‘𝑁)))
315 df-ne 2944 . . . . . . . . . . . . . . 15 (𝑛 ≠ ((2nd ‘(1st𝑇))‘𝑁) ↔ ¬ 𝑛 = ((2nd ‘(1st𝑇))‘𝑁))
316315anbi2i 626 . . . . . . . . . . . . . 14 ((𝑛 ∈ (1...𝑁) ∧ 𝑛 ≠ ((2nd ‘(1st𝑇))‘𝑁)) ↔ (𝑛 ∈ (1...𝑁) ∧ ¬ 𝑛 = ((2nd ‘(1st𝑇))‘𝑁)))
317314, 316bitri 278 . . . . . . . . . . . . 13 (𝑛 ∈ ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘𝑁)}) ↔ (𝑛 ∈ (1...𝑁) ∧ ¬ 𝑛 = ((2nd ‘(1st𝑇))‘𝑁)))
318 fnconstg 6629 . . . . . . . . . . . . . . . . . 18 (0 ∈ V → (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0}) Fn ((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))))
31933, 318ax-mp 5 . . . . . . . . . . . . . . . . 17 (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0}) Fn ((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1)))
32032, 319pm3.2i 474 . . . . . . . . . . . . . . . 16 ((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) Fn ((2nd ‘(1st𝑇)) “ (1...𝑦)) ∧ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0}) Fn ((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))))
321 imain 6486 . . . . . . . . . . . . . . . . . 18 (Fun (2nd ‘(1st𝑇)) → ((2nd ‘(1st𝑇)) “ ((1...𝑦) ∩ ((𝑦 + 1)...(𝑁 − 1)))) = (((2nd ‘(1st𝑇)) “ (1...𝑦)) ∩ ((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1)))))
32245, 321syl 17 . . . . . . . . . . . . . . . . 17 (𝜑 → ((2nd ‘(1st𝑇)) “ ((1...𝑦) ∩ ((𝑦 + 1)...(𝑁 − 1)))) = (((2nd ‘(1st𝑇)) “ (1...𝑦)) ∩ ((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1)))))
323 fzdisj 13169 . . . . . . . . . . . . . . . . . . . 20 (𝑦 < (𝑦 + 1) → ((1...𝑦) ∩ ((𝑦 + 1)...(𝑁 − 1))) = ∅)
32450, 323syl 17 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ (0...(𝑁 − 1)) → ((1...𝑦) ∩ ((𝑦 + 1)...(𝑁 − 1))) = ∅)
325324imaeq2d 5947 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ (0...(𝑁 − 1)) → ((2nd ‘(1st𝑇)) “ ((1...𝑦) ∩ ((𝑦 + 1)...(𝑁 − 1)))) = ((2nd ‘(1st𝑇)) “ ∅))
326325, 54eqtrdi 2796 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ (0...(𝑁 − 1)) → ((2nd ‘(1st𝑇)) “ ((1...𝑦) ∩ ((𝑦 + 1)...(𝑁 − 1)))) = ∅)
327322, 326sylan9req 2801 . . . . . . . . . . . . . . . 16 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((2nd ‘(1st𝑇)) “ (1...𝑦)) ∩ ((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1)))) = ∅)
328 fnun 6512 . . . . . . . . . . . . . . . 16 ((((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) Fn ((2nd ‘(1st𝑇)) “ (1...𝑦)) ∧ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0}) Fn ((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1)))) ∧ (((2nd ‘(1st𝑇)) “ (1...𝑦)) ∩ ((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1)))) = ∅) → ((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) Fn (((2nd ‘(1st𝑇)) “ (1...𝑦)) ∪ ((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1)))))
329320, 327, 328sylancr 590 . . . . . . . . . . . . . . 15 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) Fn (((2nd ‘(1st𝑇)) “ (1...𝑦)) ∪ ((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1)))))
330 imaundi 6031 . . . . . . . . . . . . . . . . 17 ((2nd ‘(1st𝑇)) “ ((1...𝑦) ∪ ((𝑦 + 1)...(𝑁 − 1)))) = (((2nd ‘(1st𝑇)) “ (1...𝑦)) ∪ ((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))))
331199, 204eqtrd 2779 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → (1...𝑁) = ((1...(𝑁 − 1)) ∪ {𝑁}))
332331difeq1d 4053 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → ((1...𝑁) ∖ {𝑁}) = (((1...(𝑁 − 1)) ∪ {𝑁}) ∖ {𝑁}))
333 difun2 4412 . . . . . . . . . . . . . . . . . . . . . . 23 (((1...(𝑁 − 1)) ∪ {𝑁}) ∖ {𝑁}) = ((1...(𝑁 − 1)) ∖ {𝑁})
334332, 333eqtrdi 2796 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → ((1...𝑁) ∖ {𝑁}) = ((1...(𝑁 − 1)) ∖ {𝑁}))
335 difsn 4728 . . . . . . . . . . . . . . . . . . . . . . 23 𝑁 ∈ (1...(𝑁 − 1)) → ((1...(𝑁 − 1)) ∖ {𝑁}) = (1...(𝑁 − 1)))
336161, 335syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → ((1...(𝑁 − 1)) ∖ {𝑁}) = (1...(𝑁 − 1)))
337334, 336eqtrd 2779 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → ((1...𝑁) ∖ {𝑁}) = (1...(𝑁 − 1)))
338337adantr 484 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((1...𝑁) ∖ {𝑁}) = (1...(𝑁 − 1)))
33970adantl 485 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (𝑁 − 1) ∈ (ℤ𝑦))
340 fzsplit2 13167 . . . . . . . . . . . . . . . . . . . . 21 (((𝑦 + 1) ∈ (ℤ‘1) ∧ (𝑁 − 1) ∈ (ℤ𝑦)) → (1...(𝑁 − 1)) = ((1...𝑦) ∪ ((𝑦 + 1)...(𝑁 − 1))))
34164, 339, 340syl2anc 587 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (1...(𝑁 − 1)) = ((1...𝑦) ∪ ((𝑦 + 1)...(𝑁 − 1))))
342338, 341eqtrd 2779 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((1...𝑁) ∖ {𝑁}) = ((1...𝑦) ∪ ((𝑦 + 1)...(𝑁 − 1))))
343342imaeq2d 5947 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st𝑇)) “ ((1...𝑁) ∖ {𝑁})) = ((2nd ‘(1st𝑇)) “ ((1...𝑦) ∪ ((𝑦 + 1)...(𝑁 − 1)))))
344 imadif 6485 . . . . . . . . . . . . . . . . . . . . 21 (Fun (2nd ‘(1st𝑇)) → ((2nd ‘(1st𝑇)) “ ((1...𝑁) ∖ {𝑁})) = (((2nd ‘(1st𝑇)) “ (1...𝑁)) ∖ ((2nd ‘(1st𝑇)) “ {𝑁})))
34545, 344syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ((2nd ‘(1st𝑇)) “ ((1...𝑁) ∖ {𝑁})) = (((2nd ‘(1st𝑇)) “ (1...𝑁)) ∖ ((2nd ‘(1st𝑇)) “ {𝑁})))
346 elfz1end 13172 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑁 ∈ ℕ ↔ 𝑁 ∈ (1...𝑁))
34765, 346sylib 221 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑𝑁 ∈ (1...𝑁))
348 fnsnfv 6812 . . . . . . . . . . . . . . . . . . . . . . 23 (((2nd ‘(1st𝑇)) Fn (1...𝑁) ∧ 𝑁 ∈ (1...𝑁)) → {((2nd ‘(1st𝑇))‘𝑁)} = ((2nd ‘(1st𝑇)) “ {𝑁}))
349252, 347, 348syl2anc 587 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → {((2nd ‘(1st𝑇))‘𝑁)} = ((2nd ‘(1st𝑇)) “ {𝑁}))
350349eqcomd 2745 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → ((2nd ‘(1st𝑇)) “ {𝑁}) = {((2nd ‘(1st𝑇))‘𝑁)})
35180, 350difeq12d 4055 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (((2nd ‘(1st𝑇)) “ (1...𝑁)) ∖ ((2nd ‘(1st𝑇)) “ {𝑁})) = ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘𝑁)}))
352345, 351eqtrd 2779 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ((2nd ‘(1st𝑇)) “ ((1...𝑁) ∖ {𝑁})) = ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘𝑁)}))
353352adantr 484 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st𝑇)) “ ((1...𝑁) ∖ {𝑁})) = ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘𝑁)}))
354343, 353eqtr3d 2781 . . . . . . . . . . . . . . . . 17 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st𝑇)) “ ((1...𝑦) ∪ ((𝑦 + 1)...(𝑁 − 1)))) = ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘𝑁)}))
355330, 354eqtr3id 2794 . . . . . . . . . . . . . . . 16 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((2nd ‘(1st𝑇)) “ (1...𝑦)) ∪ ((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1)))) = ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘𝑁)}))
356355fneq2d 6494 . . . . . . . . . . . . . . 15 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) Fn (((2nd ‘(1st𝑇)) “ (1...𝑦)) ∪ ((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1)))) ↔ ((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) Fn ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘𝑁)})))
357329, 356mpbid 235 . . . . . . . . . . . . . 14 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) Fn ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘𝑁)}))
358 disjdifr 4404 . . . . . . . . . . . . . . 15 (((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘𝑁)}) ∩ {((2nd ‘(1st𝑇))‘𝑁)}) = ∅
359 fnconstg 6629 . . . . . . . . . . . . . . . . . 18 (1 ∈ V → ({((2nd ‘(1st𝑇))‘𝑁)} × {1}) Fn {((2nd ‘(1st𝑇))‘𝑁)})
36030, 359ax-mp 5 . . . . . . . . . . . . . . . . 17 ({((2nd ‘(1st𝑇))‘𝑁)} × {1}) Fn {((2nd ‘(1st𝑇))‘𝑁)}
361 fvun1 6824 . . . . . . . . . . . . . . . . 17 ((((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) Fn ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘𝑁)}) ∧ ({((2nd ‘(1st𝑇))‘𝑁)} × {1}) Fn {((2nd ‘(1st𝑇))‘𝑁)} ∧ ((((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘𝑁)}) ∩ {((2nd ‘(1st𝑇))‘𝑁)}) = ∅ ∧ 𝑛 ∈ ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘𝑁)}))) → ((((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) ∪ ({((2nd ‘(1st𝑇))‘𝑁)} × {1}))‘𝑛) = (((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0}))‘𝑛))
362360, 361mp3an2 1451 . . . . . . . . . . . . . . . 16 ((((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) Fn ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘𝑁)}) ∧ ((((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘𝑁)}) ∩ {((2nd ‘(1st𝑇))‘𝑁)}) = ∅ ∧ 𝑛 ∈ ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘𝑁)}))) → ((((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) ∪ ({((2nd ‘(1st𝑇))‘𝑁)} × {1}))‘𝑛) = (((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0}))‘𝑛))
363 fnconstg 6629 . . . . . . . . . . . . . . . . . 18 (0 ∈ V → ({((2nd ‘(1st𝑇))‘𝑁)} × {0}) Fn {((2nd ‘(1st𝑇))‘𝑁)})
36433, 363ax-mp 5 . . . . . . . . . . . . . . . . 17 ({((2nd ‘(1st𝑇))‘𝑁)} × {0}) Fn {((2nd ‘(1st𝑇))‘𝑁)}
365 fvun1 6824 . . . . . . . . . . . . . . . . 17 ((((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) Fn ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘𝑁)}) ∧ ({((2nd ‘(1st𝑇))‘𝑁)} × {0}) Fn {((2nd ‘(1st𝑇))‘𝑁)} ∧ ((((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘𝑁)}) ∩ {((2nd ‘(1st𝑇))‘𝑁)}) = ∅ ∧ 𝑛 ∈ ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘𝑁)}))) → ((((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) ∪ ({((2nd ‘(1st𝑇))‘𝑁)} × {0}))‘𝑛) = (((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0}))‘𝑛))
366364, 365mp3an2 1451 . . . . . . . . . . . . . . . 16 ((((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) Fn ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘𝑁)}) ∧ ((((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘𝑁)}) ∩ {((2nd ‘(1st𝑇))‘𝑁)}) = ∅ ∧ 𝑛 ∈ ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘𝑁)}))) → ((((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) ∪ ({((2nd ‘(1st𝑇))‘𝑁)} × {0}))‘𝑛) = (((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0}))‘𝑛))
367362, 366eqtr4d 2782 . . . . . . . . . . . . . . 15 ((((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) Fn ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘𝑁)}) ∧ ((((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘𝑁)}) ∩ {((2nd ‘(1st𝑇))‘𝑁)}) = ∅ ∧ 𝑛 ∈ ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘𝑁)}))) → ((((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) ∪ ({((2nd ‘(1st𝑇))‘𝑁)} × {1}))‘𝑛) = ((((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) ∪ ({((2nd ‘(1st𝑇))‘𝑁)} × {0}))‘𝑛))
368358, 367mpanr1 703 . . . . . . . . . . . . . 14 ((((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) Fn ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘𝑁)}) ∧ 𝑛 ∈ ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘𝑁)})) → ((((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) ∪ ({((2nd ‘(1st𝑇))‘𝑁)} × {1}))‘𝑛) = ((((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) ∪ ({((2nd ‘(1st𝑇))‘𝑁)} × {0}))‘𝑛))
369357, 368sylan 583 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘𝑁)})) → ((((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) ∪ ({((2nd ‘(1st𝑇))‘𝑁)} × {1}))‘𝑛) = ((((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) ∪ ({((2nd ‘(1st𝑇))‘𝑁)} × {0}))‘𝑛))
370317, 369sylan2br 598 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ (𝑛 ∈ (1...𝑁) ∧ ¬ 𝑛 = ((2nd ‘(1st𝑇))‘𝑁))) → ((((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) ∪ ({((2nd ‘(1st𝑇))‘𝑁)} × {1}))‘𝑛) = ((((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) ∪ ({((2nd ‘(1st𝑇))‘𝑁)} × {0}))‘𝑛))
371370anassrs 471 . . . . . . . . . . 11 ((((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) ∧ ¬ 𝑛 = ((2nd ‘(1st𝑇))‘𝑁)) → ((((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) ∪ ({((2nd ‘(1st𝑇))‘𝑁)} × {1}))‘𝑛) = ((((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) ∪ ({((2nd ‘(1st𝑇))‘𝑁)} × {0}))‘𝑛))
372 imaundi 6031 . . . . . . . . . . . . . . . . . . 19 (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((1 + 1)...(𝑦 + 1)) ∪ {1})) = ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ ((1 + 1)...(𝑦 + 1))) ∪ (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ {1}))
373 imaco 6133 . . . . . . . . . . . . . . . . . . . 20 (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ ((1 + 1)...(𝑦 + 1))) = ((2nd ‘(1st𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ ((1 + 1)...(𝑦 + 1))))
374 imaco 6133 . . . . . . . . . . . . . . . . . . . 20 (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ {1}) = ((2nd ‘(1st𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ {1}))
375373, 374uneq12i 4092 . . . . . . . . . . . . . . . . . . 19 ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ ((1 + 1)...(𝑦 + 1))) ∪ (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ {1})) = (((2nd ‘(1st𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ ((1 + 1)...(𝑦 + 1)))) ∪ ((2nd ‘(1st𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ {1})))
376372, 375eqtri 2767 . . . . . . . . . . . . . . . . . 18 (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((1 + 1)...(𝑦 + 1)) ∪ {1})) = (((2nd ‘(1st𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ ((1 + 1)...(𝑦 + 1)))) ∪ ((2nd ‘(1st𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ {1})))
377 fzpred 13190 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑦 + 1) ∈ (ℤ‘1) → (1...(𝑦 + 1)) = ({1} ∪ ((1 + 1)...(𝑦 + 1))))
37863, 377syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 ∈ (0...(𝑁 − 1)) → (1...(𝑦 + 1)) = ({1} ∪ ((1 + 1)...(𝑦 + 1))))
379 uncom 4084 . . . . . . . . . . . . . . . . . . . . 21 ({1} ∪ ((1 + 1)...(𝑦 + 1))) = (((1 + 1)...(𝑦 + 1)) ∪ {1})
380378, 379eqtrdi 2796 . . . . . . . . . . . . . . . . . . . 20 (𝑦 ∈ (0...(𝑁 − 1)) → (1...(𝑦 + 1)) = (((1 + 1)...(𝑦 + 1)) ∪ {1}))
381380imaeq2d 5947 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ (0...(𝑁 − 1)) → (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) = (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((1 + 1)...(𝑦 + 1)) ∪ {1})))
382381adantl 485 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) = (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((1 + 1)...(𝑦 + 1)) ∪ {1})))
383 elfzelz 13142 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 ∈ ℤ)
384121a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑦 ∈ ℤ → ((1 + 1) − 1) = 1)
385 zcn 12211 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 ∈ ℤ → 𝑦 ∈ ℂ)
386 pncan1 11286 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 ∈ ℂ → ((𝑦 + 1) − 1) = 𝑦)
387385, 386syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑦 ∈ ℤ → ((𝑦 + 1) − 1) = 𝑦)
388384, 387oveq12d 7253 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 ∈ ℤ → (((1 + 1) − 1)...((𝑦 + 1) − 1)) = (1...𝑦))
389 elfzelz 13142 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑗 ∈ (((1 + 1) − 1)...((𝑦 + 1) − 1)) → 𝑗 ∈ ℤ)
390389zcnd 12313 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑗 ∈ (((1 + 1) − 1)...((𝑦 + 1) − 1)) → 𝑗 ∈ ℂ)
391 pncan1 11286 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑗 ∈ ℂ → ((𝑗 + 1) − 1) = 𝑗)
392390, 391syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑗 ∈ (((1 + 1) − 1)...((𝑦 + 1) − 1)) → ((𝑗 + 1) − 1) = 𝑗)
393392eleq1d 2824 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑗 ∈ (((1 + 1) − 1)...((𝑦 + 1) − 1)) → (((𝑗 + 1) − 1) ∈ (((1 + 1) − 1)...((𝑦 + 1) − 1)) ↔ 𝑗 ∈ (((1 + 1) − 1)...((𝑦 + 1) − 1))))
394393ibir 271 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑗 ∈ (((1 + 1) − 1)...((𝑦 + 1) − 1)) → ((𝑗 + 1) − 1) ∈ (((1 + 1) − 1)...((𝑦 + 1) − 1)))
395394adantl 485 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑦 ∈ ℤ ∧ 𝑗 ∈ (((1 + 1) − 1)...((𝑦 + 1) − 1))) → ((𝑗 + 1) − 1) ∈ (((1 + 1) − 1)...((𝑦 + 1) − 1)))
396 peano2z 12248 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑦 ∈ ℤ → (𝑦 + 1) ∈ ℤ)
397396, 114jctil 523 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑦 ∈ ℤ → ((1 + 1) ∈ ℤ ∧ (𝑦 + 1) ∈ ℤ))
398389peano2zd 12315 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑗 ∈ (((1 + 1) − 1)...((𝑦 + 1) − 1)) → (𝑗 + 1) ∈ ℤ)
399398, 112jctir 524 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑗 ∈ (((1 + 1) − 1)...((𝑦 + 1) − 1)) → ((𝑗 + 1) ∈ ℤ ∧ 1 ∈ ℤ))
400 fzsubel 13178 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((1 + 1) ∈ ℤ ∧ (𝑦 + 1) ∈ ℤ) ∧ ((𝑗 + 1) ∈ ℤ ∧ 1 ∈ ℤ)) → ((𝑗 + 1) ∈ ((1 + 1)...(𝑦 + 1)) ↔ ((𝑗 + 1) − 1) ∈ (((1 + 1) − 1)...((𝑦 + 1) − 1))))
401397, 399, 400syl2an 599 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑦 ∈ ℤ ∧ 𝑗 ∈ (((1 + 1) − 1)...((𝑦 + 1) − 1))) → ((𝑗 + 1) ∈ ((1 + 1)...(𝑦 + 1)) ↔ ((𝑗 + 1) − 1) ∈ (((1 + 1) − 1)...((𝑦 + 1) − 1))))
402395, 401mpbird 260 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑦 ∈ ℤ ∧ 𝑗 ∈ (((1 + 1) − 1)...((𝑦 + 1) − 1))) → (𝑗 + 1) ∈ ((1 + 1)...(𝑦 + 1)))
403392eqcomd 2745 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑗 ∈ (((1 + 1) − 1)...((𝑦 + 1) − 1)) → 𝑗 = ((𝑗 + 1) − 1))
404403adantl 485 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑦 ∈ ℤ ∧ 𝑗 ∈ (((1 + 1) − 1)...((𝑦 + 1) − 1))) → 𝑗 = ((𝑗 + 1) − 1))
405 oveq1 7242 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑛 = (𝑗 + 1) → (𝑛 − 1) = ((𝑗 + 1) − 1))
406405rspceeqv 3567 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝑗 + 1) ∈ ((1 + 1)...(𝑦 + 1)) ∧ 𝑗 = ((𝑗 + 1) − 1)) → ∃𝑛 ∈ ((1 + 1)...(𝑦 + 1))𝑗 = (𝑛 − 1))
407402, 404, 406syl2anc 587 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑦 ∈ ℤ ∧ 𝑗 ∈ (((1 + 1) − 1)...((𝑦 + 1) − 1))) → ∃𝑛 ∈ ((1 + 1)...(𝑦 + 1))𝑗 = (𝑛 − 1))
408407ex 416 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑦 ∈ ℤ → (𝑗 ∈ (((1 + 1) − 1)...((𝑦 + 1) − 1)) → ∃𝑛 ∈ ((1 + 1)...(𝑦 + 1))𝑗 = (𝑛 − 1)))
409 simpr 488 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑦 ∈ ℤ ∧ 𝑛 ∈ ((1 + 1)...(𝑦 + 1))) → 𝑛 ∈ ((1 + 1)...(𝑦 + 1)))
410 elfzelz 13142 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑛 ∈ ((1 + 1)...(𝑦 + 1)) → 𝑛 ∈ ℤ)
411410, 112jctir 524 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑛 ∈ ((1 + 1)...(𝑦 + 1)) → (𝑛 ∈ ℤ ∧ 1 ∈ ℤ))
412 fzsubel 13178 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((1 + 1) ∈ ℤ ∧ (𝑦 + 1) ∈ ℤ) ∧ (𝑛 ∈ ℤ ∧ 1 ∈ ℤ)) → (𝑛 ∈ ((1 + 1)...(𝑦 + 1)) ↔ (𝑛 − 1) ∈ (((1 + 1) − 1)...((𝑦 + 1) − 1))))
413397, 411, 412syl2an 599 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑦 ∈ ℤ ∧ 𝑛 ∈ ((1 + 1)...(𝑦 + 1))) → (𝑛 ∈ ((1 + 1)...(𝑦 + 1)) ↔ (𝑛 − 1) ∈ (((1 + 1) − 1)...((𝑦 + 1) − 1))))
414409, 413mpbid 235 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑦 ∈ ℤ ∧ 𝑛 ∈ ((1 + 1)...(𝑦 + 1))) → (𝑛 − 1) ∈ (((1 + 1) − 1)...((𝑦 + 1) − 1)))
415 eleq1 2827 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑗 = (𝑛 − 1) → (𝑗 ∈ (((1 + 1) − 1)...((𝑦 + 1) − 1)) ↔ (𝑛 − 1) ∈ (((1 + 1) − 1)...((𝑦 + 1) − 1))))
416414, 415syl5ibrcom 250 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑦 ∈ ℤ ∧ 𝑛 ∈ ((1 + 1)...(𝑦 + 1))) → (𝑗 = (𝑛 − 1) → 𝑗 ∈ (((1 + 1) − 1)...((𝑦 + 1) − 1))))
417416rexlimdva 3213 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑦 ∈ ℤ → (∃𝑛 ∈ ((1 + 1)...(𝑦 + 1))𝑗 = (𝑛 − 1) → 𝑗 ∈ (((1 + 1) − 1)...((𝑦 + 1) − 1))))
418408, 417impbid 215 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 ∈ ℤ → (𝑗 ∈ (((1 + 1) − 1)...((𝑦 + 1) − 1)) ↔ ∃𝑛 ∈ ((1 + 1)...(𝑦 + 1))𝑗 = (𝑛 − 1)))
419 eqid 2739 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑛 ∈ ((1 + 1)...(𝑦 + 1)) ↦ (𝑛 − 1)) = (𝑛 ∈ ((1 + 1)...(𝑦 + 1)) ↦ (𝑛 − 1))
420419elrnmpt 5843 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑗 ∈ V → (𝑗 ∈ ran (𝑛 ∈ ((1 + 1)...(𝑦 + 1)) ↦ (𝑛 − 1)) ↔ ∃𝑛 ∈ ((1 + 1)...(𝑦 + 1))𝑗 = (𝑛 − 1)))
421420elv 3429 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑗 ∈ ran (𝑛 ∈ ((1 + 1)...(𝑦 + 1)) ↦ (𝑛 − 1)) ↔ ∃𝑛 ∈ ((1 + 1)...(𝑦 + 1))𝑗 = (𝑛 − 1))
422418, 421bitr4di 292 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑦 ∈ ℤ → (𝑗 ∈ (((1 + 1) − 1)...((𝑦 + 1) − 1)) ↔ 𝑗 ∈ ran (𝑛 ∈ ((1 + 1)...(𝑦 + 1)) ↦ (𝑛 − 1))))
423422eqrdv 2737 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 ∈ ℤ → (((1 + 1) − 1)...((𝑦 + 1) − 1)) = ran (𝑛 ∈ ((1 + 1)...(𝑦 + 1)) ↦ (𝑛 − 1)))
424388, 423eqtr3d 2781 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦 ∈ ℤ → (1...𝑦) = ran (𝑛 ∈ ((1 + 1)...(𝑦 + 1)) ↦ (𝑛 − 1)))
425383, 424syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 ∈ (0...(𝑁 − 1)) → (1...𝑦) = ran (𝑛 ∈ ((1 + 1)...(𝑦 + 1)) ↦ (𝑛 − 1)))
426425adantl 485 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (1...𝑦) = ran (𝑛 ∈ ((1 + 1)...(𝑦 + 1)) ↦ (𝑛 − 1)))
427 df-ima 5582 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ ((1 + 1)...(𝑦 + 1))) = ran ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) ↾ ((1 + 1)...(𝑦 + 1)))
428 uzid 12483 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (1 ∈ ℤ → 1 ∈ (ℤ‘1))
429 peano2uz 12527 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (1 ∈ (ℤ‘1) → (1 + 1) ∈ (ℤ‘1))
430112, 428, 429mp2b 10 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (1 + 1) ∈ (ℤ‘1)
431 fzss1 13181 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((1 + 1) ∈ (ℤ‘1) → ((1 + 1)...(𝑦 + 1)) ⊆ (1...(𝑦 + 1)))
432430, 431ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((1 + 1)...(𝑦 + 1)) ⊆ (1...(𝑦 + 1))
433432, 269sstrid 3929 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((1 + 1)...(𝑦 + 1)) ⊆ (1...𝑁))
434433resmptd 5926 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) ↾ ((1 + 1)...(𝑦 + 1))) = (𝑛 ∈ ((1 + 1)...(𝑦 + 1)) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))
435 elfzle1 13145 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (1 ∈ ((1 + 1)...(𝑦 + 1)) → (1 + 1) ≤ 1)
436168, 435mto 200 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ¬ 1 ∈ ((1 + 1)...(𝑦 + 1))
437 eleq1 2827 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑛 = 1 → (𝑛 ∈ ((1 + 1)...(𝑦 + 1)) ↔ 1 ∈ ((1 + 1)...(𝑦 + 1))))
438436, 437mtbiri 330 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑛 = 1 → ¬ 𝑛 ∈ ((1 + 1)...(𝑦 + 1)))
439438necon2ai 2973 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑛 ∈ ((1 + 1)...(𝑦 + 1)) → 𝑛 ≠ 1)
440439, 179syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑛 ∈ ((1 + 1)...(𝑦 + 1)) → if(𝑛 = 1, 𝑁, (𝑛 − 1)) = (𝑛 − 1))
441440mpteq2ia 5163 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑛 ∈ ((1 + 1)...(𝑦 + 1)) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) = (𝑛 ∈ ((1 + 1)...(𝑦 + 1)) ↦ (𝑛 − 1))
442434, 441eqtrdi 2796 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) ↾ ((1 + 1)...(𝑦 + 1))) = (𝑛 ∈ ((1 + 1)...(𝑦 + 1)) ↦ (𝑛 − 1)))
443442rneqd 5825 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ran ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) ↾ ((1 + 1)...(𝑦 + 1))) = ran (𝑛 ∈ ((1 + 1)...(𝑦 + 1)) ↦ (𝑛 − 1)))
444427, 443syl5eq 2792 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ ((1 + 1)...(𝑦 + 1))) = ran (𝑛 ∈ ((1 + 1)...(𝑦 + 1)) ↦ (𝑛 − 1)))
445426, 444eqtr4d 2782 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (1...𝑦) = ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ ((1 + 1)...(𝑦 + 1))))
446445imaeq2d 5947 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st𝑇)) “ (1...𝑦)) = ((2nd ‘(1st𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ ((1 + 1)...(𝑦 + 1)))))
447263sneqd 4570 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → {((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))‘1)} = {𝑁})
448 fnsnfv 6812 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) Fn (1...𝑁) ∧ 1 ∈ (1...𝑁)) → {((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))‘1)} = ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ {1}))
449266, 262, 448syl2anc 587 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → {((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))‘1)} = ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ {1}))
450447, 449eqtr3d 2781 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → {𝑁} = ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ {1}))
451450imaeq2d 5947 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → ((2nd ‘(1st𝑇)) “ {𝑁}) = ((2nd ‘(1st𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ {1})))
452349, 451eqtrd 2779 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → {((2nd ‘(1st𝑇))‘𝑁)} = ((2nd ‘(1st𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ {1})))
453452adantr 484 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → {((2nd ‘(1st𝑇))‘𝑁)} = ((2nd ‘(1st𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ {1})))
454446, 453uneq12d 4095 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((2nd ‘(1st𝑇)) “ (1...𝑦)) ∪ {((2nd ‘(1st𝑇))‘𝑁)}) = (((2nd ‘(1st𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ ((1 + 1)...(𝑦 + 1)))) ∪ ((2nd ‘(1st𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ {1}))))
455376, 382, 4543eqtr4a 2806 . . . . . . . . . . . . . . . . 17 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) = (((2nd ‘(1st𝑇)) “ (1...𝑦)) ∪ {((2nd ‘(1st𝑇))‘𝑁)}))
456455xpeq1d 5598 . . . . . . . . . . . . . . . 16 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) = ((((2nd ‘(1st𝑇)) “ (1...𝑦)) ∪ {((2nd ‘(1st𝑇))‘𝑁)}) × {1}))
457 xpundir 5636 . . . . . . . . . . . . . . . 16 ((((2nd ‘(1st𝑇)) “ (1...𝑦)) ∪ {((2nd ‘(1st𝑇))‘𝑁)}) × {1}) = ((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ ({((2nd ‘(1st𝑇))‘𝑁)} × {1}))
458456, 457eqtrdi 2796 . . . . . . . . . . . . . . 15 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) = ((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ ({((2nd ‘(1st𝑇))‘𝑁)} × {1})))
459 imaco 6133 . . . . . . . . . . . . . . . . 17 (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) = ((2nd ‘(1st𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ (((𝑦 + 1) + 1)...𝑁)))
460 df-ima 5582 . . . . . . . . . . . . . . . . . . . 20 ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ (((𝑦 + 1) + 1)...𝑁)) = ran ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) ↾ (((𝑦 + 1) + 1)...𝑁))
461 fzss1 13181 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑦 + 1) + 1) ∈ (ℤ‘1) → (((𝑦 + 1) + 1)...𝑁) ⊆ (1...𝑁))
462227, 461syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((𝑦 + 1) + 1)...𝑁) ⊆ (1...𝑁))
463462resmptd 5926 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) ↾ (((𝑦 + 1) + 1)...𝑁)) = (𝑛 ∈ (((𝑦 + 1) + 1)...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))
464 1red 10864 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑦 ∈ (0...(𝑁 − 1)) → 1 ∈ ℝ)
46561nnzd 12311 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑦 ∈ (0...(𝑁 − 1)) → (𝑦 + 1) ∈ ℤ)
466465peano2zd 12315 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑦 + 1) + 1) ∈ ℤ)
467466zred 12312 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑦 + 1) + 1) ∈ ℝ)
46861nnge1d 11908 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑦 ∈ (0...(𝑁 − 1)) → 1 ≤ (𝑦 + 1))
469464, 214, 467, 468, 215lelttrd 11020 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑦 ∈ (0...(𝑁 − 1)) → 1 < ((𝑦 + 1) + 1))
470464, 467ltnled 11009 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑦 ∈ (0...(𝑁 − 1)) → (1 < ((𝑦 + 1) + 1) ↔ ¬ ((𝑦 + 1) + 1) ≤ 1))
471469, 470mpbid 235 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑦 ∈ (0...(𝑁 − 1)) → ¬ ((𝑦 + 1) + 1) ≤ 1)
472 elfzle1 13145 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (1 ∈ (((𝑦 + 1) + 1)...𝑁) → ((𝑦 + 1) + 1) ≤ 1)
473471, 472nsyl 142 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑦 ∈ (0...(𝑁 − 1)) → ¬ 1 ∈ (((𝑦 + 1) + 1)...𝑁))
474 eleq1 2827 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑛 = 1 → (𝑛 ∈ (((𝑦 + 1) + 1)...𝑁) ↔ 1 ∈ (((𝑦 + 1) + 1)...𝑁)))
475474notbid 321 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑛 = 1 → (¬ 𝑛 ∈ (((𝑦 + 1) + 1)...𝑁) ↔ ¬ 1 ∈ (((𝑦 + 1) + 1)...𝑁)))
476473, 475syl5ibrcom 250 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑦 ∈ (0...(𝑁 − 1)) → (𝑛 = 1 → ¬ 𝑛 ∈ (((𝑦 + 1) + 1)...𝑁)))
477476necon2ad 2958 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 ∈ (0...(𝑁 − 1)) → (𝑛 ∈ (((𝑦 + 1) + 1)...𝑁) → 𝑛 ≠ 1))
478477imp 410 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑦 ∈ (0...(𝑁 − 1)) ∧ 𝑛 ∈ (((𝑦 + 1) + 1)...𝑁)) → 𝑛 ≠ 1)
479478, 179syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑦 ∈ (0...(𝑁 − 1)) ∧ 𝑛 ∈ (((𝑦 + 1) + 1)...𝑁)) → if(𝑛 = 1, 𝑁, (𝑛 − 1)) = (𝑛 − 1))
480479mpteq2dva 5167 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦 ∈ (0...(𝑁 − 1)) → (𝑛 ∈ (((𝑦 + 1) + 1)...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) = (𝑛 ∈ (((𝑦 + 1) + 1)...𝑁) ↦ (𝑛 − 1)))
481480adantl 485 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (𝑛 ∈ (((𝑦 + 1) + 1)...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) = (𝑛 ∈ (((𝑦 + 1) + 1)...𝑁) ↦ (𝑛 − 1)))
482463, 481eqtrd 2779 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) ↾ (((𝑦 + 1) + 1)...𝑁)) = (𝑛 ∈ (((𝑦 + 1) + 1)...𝑁) ↦ (𝑛 − 1)))
483482rneqd 5825 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ran ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) ↾ (((𝑦 + 1) + 1)...𝑁)) = ran (𝑛 ∈ (((𝑦 + 1) + 1)...𝑁) ↦ (𝑛 − 1)))
484460, 483syl5eq 2792 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ (((𝑦 + 1) + 1)...𝑁)) = ran (𝑛 ∈ (((𝑦 + 1) + 1)...𝑁) ↦ (𝑛 − 1)))
485 eqid 2739 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑛 ∈ (((𝑦 + 1) + 1)...𝑁) ↦ (𝑛 − 1)) = (𝑛 ∈ (((𝑦 + 1) + 1)...𝑁) ↦ (𝑛 − 1))
486485elrnmpt 5843 . . . . . . . . . . . . . . . . . . . . . 22 (𝑗 ∈ V → (𝑗 ∈ ran (𝑛 ∈ (((𝑦 + 1) + 1)...𝑁) ↦ (𝑛 − 1)) ↔ ∃𝑛 ∈ (((𝑦 + 1) + 1)...𝑁)𝑗 = (𝑛 − 1)))
487486elv 3429 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 ∈ ran (𝑛 ∈ (((𝑦 + 1) + 1)...𝑁) ↦ (𝑛 − 1)) ↔ ∃𝑛 ∈ (((𝑦 + 1) + 1)...𝑁)𝑗 = (𝑛 − 1))
488 simpr 488 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (((𝑦 + 1) + 1)...𝑁)) → 𝑛 ∈ (((𝑦 + 1) + 1)...𝑁))
489111, 466anim12ci 617 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((𝑦 + 1) + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ))
490 elfzelz 13142 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑛 ∈ (((𝑦 + 1) + 1)...𝑁) → 𝑛 ∈ ℤ)
491490, 112jctir 524 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑛 ∈ (((𝑦 + 1) + 1)...𝑁) → (𝑛 ∈ ℤ ∧ 1 ∈ ℤ))
492 fzsubel 13178 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝑦 + 1) + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑛 ∈ ℤ ∧ 1 ∈ ℤ)) → (𝑛 ∈ (((𝑦 + 1) + 1)...𝑁) ↔ (𝑛 − 1) ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1))))
493489, 491, 492syl2an 599 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (((𝑦 + 1) + 1)...𝑁)) → (𝑛 ∈ (((𝑦 + 1) + 1)...𝑁) ↔ (𝑛 − 1) ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1))))
494488, 493mpbid 235 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (((𝑦 + 1) + 1)...𝑁)) → (𝑛 − 1) ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1)))
495 eleq1 2827 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑗 = (𝑛 − 1) → (𝑗 ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1)) ↔ (𝑛 − 1) ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1))))
496494, 495syl5ibrcom 250 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (((𝑦 + 1) + 1)...𝑁)) → (𝑗 = (𝑛 − 1) → 𝑗 ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1))))
497496rexlimdva 3213 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (∃𝑛 ∈ (((𝑦 + 1) + 1)...𝑁)𝑗 = (𝑛 − 1) → 𝑗 ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1))))
498 elfzelz 13142 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑗 ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1)) → 𝑗 ∈ ℤ)
499498zcnd 12313 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑗 ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1)) → 𝑗 ∈ ℂ)
500499, 391syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑗 ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1)) → ((𝑗 + 1) − 1) = 𝑗)
501500eleq1d 2824 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑗 ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1)) → (((𝑗 + 1) − 1) ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1)) ↔ 𝑗 ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1))))
502501ibir 271 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑗 ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1)) → ((𝑗 + 1) − 1) ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1)))
503502adantl 485 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑗 ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1))) → ((𝑗 + 1) − 1) ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1)))
504498peano2zd 12315 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑗 ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1)) → (𝑗 + 1) ∈ ℤ)
505504, 112jctir 524 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑗 ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1)) → ((𝑗 + 1) ∈ ℤ ∧ 1 ∈ ℤ))
506 fzsubel 13178 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝑦 + 1) + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ((𝑗 + 1) ∈ ℤ ∧ 1 ∈ ℤ)) → ((𝑗 + 1) ∈ (((𝑦 + 1) + 1)...𝑁) ↔ ((𝑗 + 1) − 1) ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1))))
507489, 505, 506syl2an 599 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑗 ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1))) → ((𝑗 + 1) ∈ (((𝑦 + 1) + 1)...𝑁) ↔ ((𝑗 + 1) − 1) ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1))))
508503, 507mpbird 260 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑗 ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1))) → (𝑗 + 1) ∈ (((𝑦 + 1) + 1)...𝑁))
509500eqcomd 2745 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑗 ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1)) → 𝑗 = ((𝑗 + 1) − 1))
510509adantl 485 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑗 ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1))) → 𝑗 = ((𝑗 + 1) − 1))
511405rspceeqv 3567 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑗 + 1) ∈ (((𝑦 + 1) + 1)...𝑁) ∧ 𝑗 = ((𝑗 + 1) − 1)) → ∃𝑛 ∈ (((𝑦 + 1) + 1)...𝑁)𝑗 = (𝑛 − 1))
512508, 510, 511syl2anc 587 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑗 ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1))) → ∃𝑛 ∈ (((𝑦 + 1) + 1)...𝑁)𝑗 = (𝑛 − 1))
513512ex 416 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (𝑗 ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1)) → ∃𝑛 ∈ (((𝑦 + 1) + 1)...𝑁)𝑗 = (𝑛 − 1)))
514497, 513impbid 215 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (∃𝑛 ∈ (((𝑦 + 1) + 1)...𝑁)𝑗 = (𝑛 − 1) ↔ 𝑗 ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1))))
515487, 514syl5bb 286 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (𝑗 ∈ ran (𝑛 ∈ (((𝑦 + 1) + 1)...𝑁) ↦ (𝑛 − 1)) ↔ 𝑗 ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1))))
516515eqrdv 2737 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ran (𝑛 ∈ (((𝑦 + 1) + 1)...𝑁) ↦ (𝑛 − 1)) = ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1)))
51761nncnd 11876 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 ∈ (0...(𝑁 − 1)) → (𝑦 + 1) ∈ ℂ)
518 pncan1 11286 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑦 + 1) ∈ ℂ → (((𝑦 + 1) + 1) − 1) = (𝑦 + 1))
519517, 518syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 ∈ (0...(𝑁 − 1)) → (((𝑦 + 1) + 1) − 1) = (𝑦 + 1))
520519oveq1d 7250 . . . . . . . . . . . . . . . . . . . 20 (𝑦 ∈ (0...(𝑁 − 1)) → ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1)) = ((𝑦 + 1)...(𝑁 − 1)))
521520adantl 485 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1)) = ((𝑦 + 1)...(𝑁 − 1)))
522484, 516, 5213eqtrd 2783 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ (((𝑦 + 1) + 1)...𝑁)) = ((𝑦 + 1)...(𝑁 − 1)))
523522imaeq2d 5947 . . . . . . . . . . . . . . . . 17 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ (((𝑦 + 1) + 1)...𝑁))) = ((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))))
524459, 523syl5eq 2792 . . . . . . . . . . . . . . . 16 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) = ((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))))
525524xpeq1d 5598 . . . . . . . . . . . . . . 15 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}) = (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0}))
526458, 525uneq12d 4095 . . . . . . . . . . . . . 14 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) = (((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ ({((2nd ‘(1st𝑇))‘𝑁)} × {1})) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})))
527 un23 4099 . . . . . . . . . . . . . 14 (((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ ({((2nd ‘(1st𝑇))‘𝑁)} × {1})) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) = (((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) ∪ ({((2nd ‘(1st𝑇))‘𝑁)} × {1}))
528526, 527eqtrdi 2796 . . . . . . . . . . . . 13 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) = (((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) ∪ ({((2nd ‘(1st𝑇))‘𝑁)} × {1})))
529528fveq1d 6741 . . . . . . . . . . . 12 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) = ((((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) ∪ ({((2nd ‘(1st𝑇))‘𝑁)} × {1}))‘𝑛))
530529ad2antrr 726 . . . . . . . . . . 11 ((((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) ∧ ¬ 𝑛 = ((2nd ‘(1st𝑇))‘𝑁)) → ((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) = ((((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) ∪ ({((2nd ‘(1st𝑇))‘𝑁)} × {1}))‘𝑛))
531 imaundi 6031 . . . . . . . . . . . . . . . . . 18 ((2nd ‘(1st𝑇)) “ (((𝑦 + 1)...(𝑁 − 1)) ∪ {𝑁})) = (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) ∪ ((2nd ‘(1st𝑇)) “ {𝑁}))
532 fzsplit2 13167 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑁 − 1) + 1) ∈ (ℤ‘(𝑦 + 1)) ∧ 𝑁 ∈ (ℤ‘(𝑁 − 1))) → ((𝑦 + 1)...𝑁) = (((𝑦 + 1)...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)))
533229, 197, 532syl2anr 600 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((𝑦 + 1)...𝑁) = (((𝑦 + 1)...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)))
534203uneq2d 4094 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (((𝑦 + 1)...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)) = (((𝑦 + 1)...(𝑁 − 1)) ∪ {𝑁}))
535534adantr 484 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((𝑦 + 1)...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)) = (((𝑦 + 1)...(𝑁 − 1)) ∪ {𝑁}))
536533, 535eqtrd 2779 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((𝑦 + 1)...𝑁) = (((𝑦 + 1)...(𝑁 − 1)) ∪ {𝑁}))
537536imaeq2d 5947 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) = ((2nd ‘(1st𝑇)) “ (((𝑦 + 1)...(𝑁 − 1)) ∪ {𝑁})))
538349adantr 484 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → {((2nd ‘(1st𝑇))‘𝑁)} = ((2nd ‘(1st𝑇)) “ {𝑁}))
539538uneq2d 4094 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) ∪ {((2nd ‘(1st𝑇))‘𝑁)}) = (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) ∪ ((2nd ‘(1st𝑇)) “ {𝑁})))
540531, 537, 5393eqtr4a 2806 . . . . . . . . . . . . . . . . 17 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) = (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) ∪ {((2nd ‘(1st𝑇))‘𝑁)}))
541540xpeq1d 5598 . . . . . . . . . . . . . . . 16 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}) = ((((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) ∪ {((2nd ‘(1st𝑇))‘𝑁)}) × {0}))
542 xpundir 5636 . . . . . . . . . . . . . . . 16 ((((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) ∪ {((2nd ‘(1st𝑇))‘𝑁)}) × {0}) = ((((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0}) ∪ ({((2nd ‘(1st𝑇))‘𝑁)} × {0}))
543541, 542eqtrdi 2796 . . . . . . . . . . . . . . 15 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}) = ((((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0}) ∪ ({((2nd ‘(1st𝑇))‘𝑁)} × {0})))
544543uneq2d 4094 . . . . . . . . . . . . . 14 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})) = ((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0}) ∪ ({((2nd ‘(1st𝑇))‘𝑁)} × {0}))))
545 unass 4097 . . . . . . . . . . . . . 14 (((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) ∪ ({((2nd ‘(1st𝑇))‘𝑁)} × {0})) = ((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0}) ∪ ({((2nd ‘(1st𝑇))‘𝑁)} × {0})))
546544, 545eqtr4di 2798 . . . . . . . . . . . . 13 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})) = (((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) ∪ ({((2nd ‘(1st𝑇))‘𝑁)} × {0})))
547546fveq1d 6741 . . . . . . . . . . . 12 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛) = ((((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) ∪ ({((2nd ‘(1st𝑇))‘𝑁)} × {0}))‘𝑛))
548547ad2antrr 726 . . . . . . . . . . 11 ((((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) ∧ ¬ 𝑛 = ((2nd ‘(1st𝑇))‘𝑁)) → (((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛) = ((((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) ∪ ({((2nd ‘(1st𝑇))‘𝑁)} × {0}))‘𝑛))
549371, 530, 5483eqtr4d 2789 . . . . . . . . . 10 ((((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) ∧ ¬ 𝑛 = ((2nd ‘(1st𝑇))‘𝑁)) → ((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) = (((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛))
550313, 549eqtrd 2779 . . . . . . . . 9 ((((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) ∧ ¬ 𝑛 = ((2nd ‘(1st𝑇))‘𝑁)) → (((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) − 0) = (((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛))
551247, 249, 311, 550ifbothda 4494 . . . . . . . 8 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) → (((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0)) = (((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛))
552551oveq2d 7251 . . . . . . 7 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) → (((1st ‘(1st𝑇))‘𝑛) + (((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))) = (((1st ‘(1st𝑇))‘𝑛) + (((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛)))
553245, 552eqtr2d 2780 . . . . . 6 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) → (((1st ‘(1st𝑇))‘𝑛) + (((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛)) = ((((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0)) + ((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛)))
554553mpteq2dva 5167 . . . . 5 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) + (((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛))) = (𝑛 ∈ (1...𝑁) ↦ ((((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0)) + ((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛))))
55590, 554eqtrd 2779 . . . 4 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))) = (𝑛 ∈ (1...𝑁) ↦ ((((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0)) + ((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛))))
55649adantl 485 . . . . . . . . 9 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → 𝑦 ∈ ℝ)
557157adantr 484 . . . . . . . . 9 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (𝑁 − 1) ∈ ℝ)
558155adantr 484 . . . . . . . . 9 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ ℝ)
559 elfzle2 13146 . . . . . . . . . 10 (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 ≤ (𝑁 − 1))
560559adantl 485 . . . . . . . . 9 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → 𝑦 ≤ (𝑁 − 1))
561156adantr 484 . . . . . . . . 9 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (𝑁 − 1) < 𝑁)
562556, 557, 558, 560, 561lelttrd 11020 . . . . . . . 8 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → 𝑦 < 𝑁)
563 poimirlem21.4 . . . . . . . . 9 (𝜑 → (2nd𝑇) = 𝑁)
564563adantr 484 . . . . . . . 8 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (2nd𝑇) = 𝑁)
565562, 564breqtrrd 5098 . . . . . . 7 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → 𝑦 < (2nd𝑇))
566565iftrued 4464 . . . . . 6 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) = 𝑦)
567566csbeq1d 3832 . . . . 5 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = 𝑦 / 𝑗((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))
568 vex 3427 . . . . . 6 𝑦 ∈ V
569 oveq2 7243 . . . . . . . . . 10 (𝑗 = 𝑦 → (1...𝑗) = (1...𝑦))
570569imaeq2d 5947 . . . . . . . . 9 (𝑗 = 𝑦 → ((2nd ‘(1st𝑇)) “ (1...𝑗)) = ((2nd ‘(1st𝑇)) “ (1...𝑦)))
571570xpeq1d 5598 . . . . . . . 8 (𝑗 = 𝑦 → (((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}))
572 oveq1 7242 . . . . . . . . . . 11 (𝑗 = 𝑦 → (𝑗 + 1) = (𝑦 + 1))
573572oveq1d 7250 . . . . . . . . . 10 (𝑗 = 𝑦 → ((𝑗 + 1)...𝑁) = ((𝑦 + 1)...𝑁))
574573imaeq2d 5947 . . . . . . . . 9 (𝑗 = 𝑦 → ((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)))
575574xpeq1d 5598 . . . . . . . 8 (𝑗 = 𝑦 → (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))
576571, 575uneq12d 4095 . . . . . . 7 (𝑗 = 𝑦 → ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})))
577576oveq2d 7251 . . . . . 6 (𝑗 = 𝑦 → ((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))))
578568, 577csbie 3864 . . . . 5 𝑦 / 𝑗((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})))
579567, 578eqtrdi 2796 . . . 4 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))))
580 ovexd 7270 . . . . 5 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) → (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0)) ∈ V)
581 fvexd 6754 . . . . 5 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) → ((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) ∈ V)
582 eqidd 2740 . . . . 5 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))) = (𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))))
583242ffnd 6568 . . . . . 6 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) Fn (1...𝑁))
584 nfcv 2907 . . . . . . . . . . 11 𝑛(2nd ‘(1st𝑇))
585 nfmpt1 5170 . . . . . . . . . . 11 𝑛(𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))
586584, 585nfco 5752 . . . . . . . . . 10 𝑛((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))
587 nfcv 2907 . . . . . . . . . 10 𝑛(1...(𝑦 + 1))
588586, 587nfima 5955 . . . . . . . . 9 𝑛(((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1)))
589 nfcv 2907 . . . . . . . . 9 𝑛{1}
590588, 589nfxp 5602 . . . . . . . 8 𝑛((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1})
591 nfcv 2907 . . . . . . . . . 10 𝑛(((𝑦 + 1) + 1)...𝑁)
592586, 591nfima 5955 . . . . . . . . 9 𝑛(((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁))
593 nfcv 2907 . . . . . . . . 9 𝑛{0}
594592, 593nfxp 5602 . . . . . . . 8 𝑛((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0})
595590, 594nfun 4096 . . . . . . 7 𝑛(((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))
596595dffn5f 6805 . . . . . 6 ((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) Fn (1...𝑁) ↔ (((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) = (𝑛 ∈ (1...𝑁) ↦ ((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛)))
597583, 596sylib 221 . . . . 5 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) = (𝑛 ∈ (1...𝑁) ↦ ((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛)))
59886, 580, 581, 582, 597offval2 7510 . . . 4 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))) ∘f + (((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))) = (𝑛 ∈ (1...𝑁) ↦ ((((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0)) + ((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛))))
599555, 579, 5983eqtr4rd 2790 . . 3 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))) ∘f + (((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))) = if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))
600599mpteq2dva 5167 . 2 (𝜑 → (𝑦 ∈ (0...(𝑁 − 1)) ↦ ((𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))) ∘f + (((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0})))) = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))))
60119, 600eqtr4d 2782 1 (𝜑𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ((𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))) ∘f + (((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0})))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399   = wceq 1543  wcel 2112  {cab 2716  wne 2943  wral 3064  wrex 3065  ∃!wreu 3066  {crab 3068  Vcvv 3423  csb 3828  cdif 3880  cun 3881  cin 3882  wss 3883  c0 4254  ifcif 4456  {csn 4558  cop 4564   class class class wbr 5070  cmpt 5152   × cxp 5567  ccnv 5568  ran crn 5570  cres 5571  cima 5572  ccom 5573  Fun wfun 6395   Fn wfn 6396  wf 6397  ontowfo 6399  1-1-ontowf1o 6400  cfv 6401  (class class class)co 7235  f cof 7489  1st c1st 7781  2nd c2nd 7782  m cmap 8532  cc 10757  cr 10758  0cc0 10759  1c1 10760   + caddc 10762   < clt 10897  cle 10898  cmin 11092  cn 11860  0cn0 12120  cz 12206  cuz 12468  ...cfz 13125  ..^cfzo 13268
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2710  ax-rep 5196  ax-sep 5209  ax-nul 5216  ax-pow 5275  ax-pr 5339  ax-un 7545  ax-cnex 10815  ax-resscn 10816  ax-1cn 10817  ax-icn 10818  ax-addcl 10819  ax-addrcl 10820  ax-mulcl 10821  ax-mulrcl 10822  ax-mulcom 10823  ax-addass 10824  ax-mulass 10825  ax-distr 10826  ax-i2m1 10827  ax-1ne0 10828  ax-1rid 10829  ax-rnegex 10830  ax-rrecex 10831  ax-cnre 10832  ax-pre-lttri 10833  ax-pre-lttrn 10834  ax-pre-ltadd 10835  ax-pre-mulgt0 10836
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2818  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-reu 3071  df-rab 3073  df-v 3425  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4255  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5153  df-tr 5179  df-id 5472  df-eprel 5478  df-po 5486  df-so 5487  df-fr 5527  df-we 5529  df-xp 5575  df-rel 5576  df-cnv 5577  df-co 5578  df-dm 5579  df-rn 5580  df-res 5581  df-ima 5582  df-pred 6179  df-ord 6237  df-on 6238  df-lim 6239  df-suc 6240  df-iota 6359  df-fun 6403  df-fn 6404  df-f 6405  df-f1 6406  df-fo 6407  df-f1o 6408  df-fv 6409  df-riota 7192  df-ov 7238  df-oprab 7239  df-mpo 7240  df-of 7491  df-om 7667  df-1st 7783  df-2nd 7784  df-wrecs 8071  df-recs 8132  df-rdg 8170  df-er 8415  df-map 8534  df-en 8651  df-dom 8652  df-sdom 8653  df-pnf 10899  df-mnf 10900  df-xr 10901  df-ltxr 10902  df-le 10903  df-sub 11094  df-neg 11095  df-nn 11861  df-n0 12121  df-z 12207  df-uz 12469  df-fz 13126  df-fzo 13269
This theorem is referenced by:  poimirlem20  35571
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