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Theorem poimirlem1 33855
Description: Lemma for poimir 33887- the vertices on either side of a skipped vertex differ in at least two dimensions. (Contributed by Brendan Leahy, 21-Aug-2020.)
Hypotheses
Ref Expression
poimir.0 (𝜑𝑁 ∈ ℕ)
poimirlem2.1 (𝜑𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < 𝑀, 𝑦, (𝑦 + 1)) / 𝑗(𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))))
poimirlem2.2 (𝜑𝑇:(1...𝑁)⟶ℤ)
poimirlem2.3 (𝜑𝑈:(1...𝑁)–1-1-onto→(1...𝑁))
poimirlem1.4 (𝜑𝑀 ∈ (1...(𝑁 − 1)))
Assertion
Ref Expression
poimirlem1 (𝜑 → ¬ ∃*𝑛 ∈ (1...𝑁)((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹𝑀)‘𝑛))
Distinct variable groups:   𝑗,𝑛,𝑦,𝜑   𝑗,𝐹,𝑛,𝑦   𝑗,𝑀,𝑛,𝑦   𝑗,𝑁,𝑛,𝑦   𝑇,𝑗,𝑛,𝑦   𝑈,𝑗,𝑛,𝑦

Proof of Theorem poimirlem1
Dummy variable 𝑚 is distinct from all other variables.
StepHypRef Expression
1 poimirlem2.3 . . . . 5 (𝜑𝑈:(1...𝑁)–1-1-onto→(1...𝑁))
2 f1of 6324 . . . . 5 (𝑈:(1...𝑁)–1-1-onto→(1...𝑁) → 𝑈:(1...𝑁)⟶(1...𝑁))
31, 2syl 17 . . . 4 (𝜑𝑈:(1...𝑁)⟶(1...𝑁))
4 poimir.0 . . . . . . . . 9 (𝜑𝑁 ∈ ℕ)
54nncnd 11296 . . . . . . . 8 (𝜑𝑁 ∈ ℂ)
6 npcan1 10713 . . . . . . . 8 (𝑁 ∈ ℂ → ((𝑁 − 1) + 1) = 𝑁)
75, 6syl 17 . . . . . . 7 (𝜑 → ((𝑁 − 1) + 1) = 𝑁)
84nnzd 11733 . . . . . . . 8 (𝜑𝑁 ∈ ℤ)
9 peano2zm 11672 . . . . . . . 8 (𝑁 ∈ ℤ → (𝑁 − 1) ∈ ℤ)
10 uzid 11906 . . . . . . . 8 ((𝑁 − 1) ∈ ℤ → (𝑁 − 1) ∈ (ℤ‘(𝑁 − 1)))
11 peano2uz 11946 . . . . . . . 8 ((𝑁 − 1) ∈ (ℤ‘(𝑁 − 1)) → ((𝑁 − 1) + 1) ∈ (ℤ‘(𝑁 − 1)))
128, 9, 10, 114syl 19 . . . . . . 7 (𝜑 → ((𝑁 − 1) + 1) ∈ (ℤ‘(𝑁 − 1)))
137, 12eqeltrrd 2845 . . . . . 6 (𝜑𝑁 ∈ (ℤ‘(𝑁 − 1)))
14 fzss2 12593 . . . . . 6 (𝑁 ∈ (ℤ‘(𝑁 − 1)) → (1...(𝑁 − 1)) ⊆ (1...𝑁))
1513, 14syl 17 . . . . 5 (𝜑 → (1...(𝑁 − 1)) ⊆ (1...𝑁))
16 poimirlem1.4 . . . . 5 (𝜑𝑀 ∈ (1...(𝑁 − 1)))
1715, 16sseldd 3764 . . . 4 (𝜑𝑀 ∈ (1...𝑁))
183, 17ffvelrnd 6554 . . 3 (𝜑 → (𝑈𝑀) ∈ (1...𝑁))
19 fzp1elp1 12606 . . . . . 6 (𝑀 ∈ (1...(𝑁 − 1)) → (𝑀 + 1) ∈ (1...((𝑁 − 1) + 1)))
2016, 19syl 17 . . . . 5 (𝜑 → (𝑀 + 1) ∈ (1...((𝑁 − 1) + 1)))
217oveq2d 6862 . . . . 5 (𝜑 → (1...((𝑁 − 1) + 1)) = (1...𝑁))
2220, 21eleqtrd 2846 . . . 4 (𝜑 → (𝑀 + 1) ∈ (1...𝑁))
233, 22ffvelrnd 6554 . . 3 (𝜑 → (𝑈‘(𝑀 + 1)) ∈ (1...𝑁))
24 imassrn 5661 . . . . . . . . . 10 (𝑈 “ (𝑀...(𝑀 + 1))) ⊆ ran 𝑈
25 frn 6231 . . . . . . . . . . 11 (𝑈:(1...𝑁)⟶(1...𝑁) → ran 𝑈 ⊆ (1...𝑁))
261, 2, 253syl 18 . . . . . . . . . 10 (𝜑 → ran 𝑈 ⊆ (1...𝑁))
2724, 26syl5ss 3774 . . . . . . . . 9 (𝜑 → (𝑈 “ (𝑀...(𝑀 + 1))) ⊆ (1...𝑁))
2827sselda 3763 . . . . . . . 8 ((𝜑𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))) → 𝑛 ∈ (1...𝑁))
29 poimirlem2.2 . . . . . . . . . . 11 (𝜑𝑇:(1...𝑁)⟶ℤ)
3029ffvelrnda 6553 . . . . . . . . . 10 ((𝜑𝑛 ∈ (1...𝑁)) → (𝑇𝑛) ∈ ℤ)
3130zred 11734 . . . . . . . . 9 ((𝜑𝑛 ∈ (1...𝑁)) → (𝑇𝑛) ∈ ℝ)
3231ltp1d 11212 . . . . . . . . 9 ((𝜑𝑛 ∈ (1...𝑁)) → (𝑇𝑛) < ((𝑇𝑛) + 1))
3331, 32ltned 10431 . . . . . . . 8 ((𝜑𝑛 ∈ (1...𝑁)) → (𝑇𝑛) ≠ ((𝑇𝑛) + 1))
3428, 33syldan 585 . . . . . . 7 ((𝜑𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))) → (𝑇𝑛) ≠ ((𝑇𝑛) + 1))
35 poimirlem2.1 . . . . . . . . . . 11 (𝜑𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < 𝑀, 𝑦, (𝑦 + 1)) / 𝑗(𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))))
36 breq1 4814 . . . . . . . . . . . . . . 15 (𝑦 = (𝑀 − 1) → (𝑦 < 𝑀 ↔ (𝑀 − 1) < 𝑀))
37 id 22 . . . . . . . . . . . . . . 15 (𝑦 = (𝑀 − 1) → 𝑦 = (𝑀 − 1))
3836, 37ifbieq1d 4268 . . . . . . . . . . . . . 14 (𝑦 = (𝑀 − 1) → if(𝑦 < 𝑀, 𝑦, (𝑦 + 1)) = if((𝑀 − 1) < 𝑀, (𝑀 − 1), (𝑦 + 1)))
39 elfzelz 12554 . . . . . . . . . . . . . . . . . 18 (𝑀 ∈ (1...(𝑁 − 1)) → 𝑀 ∈ ℤ)
4016, 39syl 17 . . . . . . . . . . . . . . . . 17 (𝜑𝑀 ∈ ℤ)
4140zred 11734 . . . . . . . . . . . . . . . 16 (𝜑𝑀 ∈ ℝ)
4241ltm1d 11214 . . . . . . . . . . . . . . 15 (𝜑 → (𝑀 − 1) < 𝑀)
4342iftrued 4253 . . . . . . . . . . . . . 14 (𝜑 → if((𝑀 − 1) < 𝑀, (𝑀 − 1), (𝑦 + 1)) = (𝑀 − 1))
4438, 43sylan9eqr 2821 . . . . . . . . . . . . 13 ((𝜑𝑦 = (𝑀 − 1)) → if(𝑦 < 𝑀, 𝑦, (𝑦 + 1)) = (𝑀 − 1))
4544csbeq1d 3700 . . . . . . . . . . . 12 ((𝜑𝑦 = (𝑀 − 1)) → if(𝑦 < 𝑀, 𝑦, (𝑦 + 1)) / 𝑗(𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) = (𝑀 − 1) / 𝑗(𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))))
468, 9syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑁 − 1) ∈ ℤ)
47 elfzm1b 12630 . . . . . . . . . . . . . . . 16 ((𝑀 ∈ ℤ ∧ (𝑁 − 1) ∈ ℤ) → (𝑀 ∈ (1...(𝑁 − 1)) ↔ (𝑀 − 1) ∈ (0...((𝑁 − 1) − 1))))
4840, 46, 47syl2anc 579 . . . . . . . . . . . . . . 15 (𝜑 → (𝑀 ∈ (1...(𝑁 − 1)) ↔ (𝑀 − 1) ∈ (0...((𝑁 − 1) − 1))))
4916, 48mpbid 223 . . . . . . . . . . . . . 14 (𝜑 → (𝑀 − 1) ∈ (0...((𝑁 − 1) − 1)))
50 oveq2 6854 . . . . . . . . . . . . . . . . . . 19 (𝑗 = (𝑀 − 1) → (1...𝑗) = (1...(𝑀 − 1)))
5150imaeq2d 5650 . . . . . . . . . . . . . . . . . 18 (𝑗 = (𝑀 − 1) → (𝑈 “ (1...𝑗)) = (𝑈 “ (1...(𝑀 − 1))))
5251xpeq1d 5308 . . . . . . . . . . . . . . . . 17 (𝑗 = (𝑀 − 1) → ((𝑈 “ (1...𝑗)) × {1}) = ((𝑈 “ (1...(𝑀 − 1))) × {1}))
5352adantl 473 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 = (𝑀 − 1)) → ((𝑈 “ (1...𝑗)) × {1}) = ((𝑈 “ (1...(𝑀 − 1))) × {1}))
54 oveq1 6853 . . . . . . . . . . . . . . . . . . . 20 (𝑗 = (𝑀 − 1) → (𝑗 + 1) = ((𝑀 − 1) + 1))
5540zcnd 11735 . . . . . . . . . . . . . . . . . . . . 21 (𝜑𝑀 ∈ ℂ)
56 npcan1 10713 . . . . . . . . . . . . . . . . . . . . 21 (𝑀 ∈ ℂ → ((𝑀 − 1) + 1) = 𝑀)
5755, 56syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ((𝑀 − 1) + 1) = 𝑀)
5854, 57sylan9eqr 2821 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 = (𝑀 − 1)) → (𝑗 + 1) = 𝑀)
5958oveq1d 6861 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗 = (𝑀 − 1)) → ((𝑗 + 1)...𝑁) = (𝑀...𝑁))
6059imaeq2d 5650 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 = (𝑀 − 1)) → (𝑈 “ ((𝑗 + 1)...𝑁)) = (𝑈 “ (𝑀...𝑁)))
6160xpeq1d 5308 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 = (𝑀 − 1)) → ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}) = ((𝑈 “ (𝑀...𝑁)) × {0}))
6253, 61uneq12d 3932 . . . . . . . . . . . . . . 15 ((𝜑𝑗 = (𝑀 − 1)) → (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})) = (((𝑈 “ (1...(𝑀 − 1))) × {1}) ∪ ((𝑈 “ (𝑀...𝑁)) × {0})))
6362oveq2d 6862 . . . . . . . . . . . . . 14 ((𝜑𝑗 = (𝑀 − 1)) → (𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) = (𝑇𝑓 + (((𝑈 “ (1...(𝑀 − 1))) × {1}) ∪ ((𝑈 “ (𝑀...𝑁)) × {0}))))
6449, 63csbied 3720 . . . . . . . . . . . . 13 (𝜑(𝑀 − 1) / 𝑗(𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) = (𝑇𝑓 + (((𝑈 “ (1...(𝑀 − 1))) × {1}) ∪ ((𝑈 “ (𝑀...𝑁)) × {0}))))
6564adantr 472 . . . . . . . . . . . 12 ((𝜑𝑦 = (𝑀 − 1)) → (𝑀 − 1) / 𝑗(𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) = (𝑇𝑓 + (((𝑈 “ (1...(𝑀 − 1))) × {1}) ∪ ((𝑈 “ (𝑀...𝑁)) × {0}))))
6645, 65eqtrd 2799 . . . . . . . . . . 11 ((𝜑𝑦 = (𝑀 − 1)) → if(𝑦 < 𝑀, 𝑦, (𝑦 + 1)) / 𝑗(𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) = (𝑇𝑓 + (((𝑈 “ (1...(𝑀 − 1))) × {1}) ∪ ((𝑈 “ (𝑀...𝑁)) × {0}))))
6746zcnd 11735 . . . . . . . . . . . . . . 15 (𝜑 → (𝑁 − 1) ∈ ℂ)
68 npcan1 10713 . . . . . . . . . . . . . . 15 ((𝑁 − 1) ∈ ℂ → (((𝑁 − 1) − 1) + 1) = (𝑁 − 1))
6967, 68syl 17 . . . . . . . . . . . . . 14 (𝜑 → (((𝑁 − 1) − 1) + 1) = (𝑁 − 1))
70 peano2zm 11672 . . . . . . . . . . . . . . 15 ((𝑁 − 1) ∈ ℤ → ((𝑁 − 1) − 1) ∈ ℤ)
71 uzid 11906 . . . . . . . . . . . . . . 15 (((𝑁 − 1) − 1) ∈ ℤ → ((𝑁 − 1) − 1) ∈ (ℤ‘((𝑁 − 1) − 1)))
72 peano2uz 11946 . . . . . . . . . . . . . . 15 (((𝑁 − 1) − 1) ∈ (ℤ‘((𝑁 − 1) − 1)) → (((𝑁 − 1) − 1) + 1) ∈ (ℤ‘((𝑁 − 1) − 1)))
7346, 70, 71, 724syl 19 . . . . . . . . . . . . . 14 (𝜑 → (((𝑁 − 1) − 1) + 1) ∈ (ℤ‘((𝑁 − 1) − 1)))
7469, 73eqeltrrd 2845 . . . . . . . . . . . . 13 (𝜑 → (𝑁 − 1) ∈ (ℤ‘((𝑁 − 1) − 1)))
75 fzss2 12593 . . . . . . . . . . . . 13 ((𝑁 − 1) ∈ (ℤ‘((𝑁 − 1) − 1)) → (0...((𝑁 − 1) − 1)) ⊆ (0...(𝑁 − 1)))
7674, 75syl 17 . . . . . . . . . . . 12 (𝜑 → (0...((𝑁 − 1) − 1)) ⊆ (0...(𝑁 − 1)))
7776, 49sseldd 3764 . . . . . . . . . . 11 (𝜑 → (𝑀 − 1) ∈ (0...(𝑁 − 1)))
78 ovexd 6880 . . . . . . . . . . 11 (𝜑 → (𝑇𝑓 + (((𝑈 “ (1...(𝑀 − 1))) × {1}) ∪ ((𝑈 “ (𝑀...𝑁)) × {0}))) ∈ V)
7935, 66, 77, 78fvmptd 6481 . . . . . . . . . 10 (𝜑 → (𝐹‘(𝑀 − 1)) = (𝑇𝑓 + (((𝑈 “ (1...(𝑀 − 1))) × {1}) ∪ ((𝑈 “ (𝑀...𝑁)) × {0}))))
8079fveq1d 6381 . . . . . . . . 9 (𝜑 → ((𝐹‘(𝑀 − 1))‘𝑛) = ((𝑇𝑓 + (((𝑈 “ (1...(𝑀 − 1))) × {1}) ∪ ((𝑈 “ (𝑀...𝑁)) × {0})))‘𝑛))
8180adantr 472 . . . . . . . 8 ((𝜑𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))) → ((𝐹‘(𝑀 − 1))‘𝑛) = ((𝑇𝑓 + (((𝑈 “ (1...(𝑀 − 1))) × {1}) ∪ ((𝑈 “ (𝑀...𝑁)) × {0})))‘𝑛))
8229ffnd 6226 . . . . . . . . . . 11 (𝜑𝑇 Fn (1...𝑁))
8382adantr 472 . . . . . . . . . 10 ((𝜑𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))) → 𝑇 Fn (1...𝑁))
84 1ex 10293 . . . . . . . . . . . . . . 15 1 ∈ V
85 fnconstg 6277 . . . . . . . . . . . . . . 15 (1 ∈ V → ((𝑈 “ (1...(𝑀 − 1))) × {1}) Fn (𝑈 “ (1...(𝑀 − 1))))
8684, 85ax-mp 5 . . . . . . . . . . . . . 14 ((𝑈 “ (1...(𝑀 − 1))) × {1}) Fn (𝑈 “ (1...(𝑀 − 1)))
87 c0ex 10291 . . . . . . . . . . . . . . 15 0 ∈ V
88 fnconstg 6277 . . . . . . . . . . . . . . 15 (0 ∈ V → ((𝑈 “ (𝑀...𝑁)) × {0}) Fn (𝑈 “ (𝑀...𝑁)))
8987, 88ax-mp 5 . . . . . . . . . . . . . 14 ((𝑈 “ (𝑀...𝑁)) × {0}) Fn (𝑈 “ (𝑀...𝑁))
9086, 89pm3.2i 462 . . . . . . . . . . . . 13 (((𝑈 “ (1...(𝑀 − 1))) × {1}) Fn (𝑈 “ (1...(𝑀 − 1))) ∧ ((𝑈 “ (𝑀...𝑁)) × {0}) Fn (𝑈 “ (𝑀...𝑁)))
91 dff1o3 6330 . . . . . . . . . . . . . . . 16 (𝑈:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (𝑈:(1...𝑁)–onto→(1...𝑁) ∧ Fun 𝑈))
9291simprbi 490 . . . . . . . . . . . . . . 15 (𝑈:(1...𝑁)–1-1-onto→(1...𝑁) → Fun 𝑈)
93 imain 6154 . . . . . . . . . . . . . . 15 (Fun 𝑈 → (𝑈 “ ((1...(𝑀 − 1)) ∩ (𝑀...𝑁))) = ((𝑈 “ (1...(𝑀 − 1))) ∩ (𝑈 “ (𝑀...𝑁))))
941, 92, 933syl 18 . . . . . . . . . . . . . 14 (𝜑 → (𝑈 “ ((1...(𝑀 − 1)) ∩ (𝑀...𝑁))) = ((𝑈 “ (1...(𝑀 − 1))) ∩ (𝑈 “ (𝑀...𝑁))))
95 fzdisj 12580 . . . . . . . . . . . . . . . . 17 ((𝑀 − 1) < 𝑀 → ((1...(𝑀 − 1)) ∩ (𝑀...𝑁)) = ∅)
9642, 95syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → ((1...(𝑀 − 1)) ∩ (𝑀...𝑁)) = ∅)
9796imaeq2d 5650 . . . . . . . . . . . . . . 15 (𝜑 → (𝑈 “ ((1...(𝑀 − 1)) ∩ (𝑀...𝑁))) = (𝑈 “ ∅))
98 ima0 5665 . . . . . . . . . . . . . . 15 (𝑈 “ ∅) = ∅
9997, 98syl6eq 2815 . . . . . . . . . . . . . 14 (𝜑 → (𝑈 “ ((1...(𝑀 − 1)) ∩ (𝑀...𝑁))) = ∅)
10094, 99eqtr3d 2801 . . . . . . . . . . . . 13 (𝜑 → ((𝑈 “ (1...(𝑀 − 1))) ∩ (𝑈 “ (𝑀...𝑁))) = ∅)
101 fnun 6177 . . . . . . . . . . . . 13 (((((𝑈 “ (1...(𝑀 − 1))) × {1}) Fn (𝑈 “ (1...(𝑀 − 1))) ∧ ((𝑈 “ (𝑀...𝑁)) × {0}) Fn (𝑈 “ (𝑀...𝑁))) ∧ ((𝑈 “ (1...(𝑀 − 1))) ∩ (𝑈 “ (𝑀...𝑁))) = ∅) → (((𝑈 “ (1...(𝑀 − 1))) × {1}) ∪ ((𝑈 “ (𝑀...𝑁)) × {0})) Fn ((𝑈 “ (1...(𝑀 − 1))) ∪ (𝑈 “ (𝑀...𝑁))))
10290, 100, 101sylancr 581 . . . . . . . . . . . 12 (𝜑 → (((𝑈 “ (1...(𝑀 − 1))) × {1}) ∪ ((𝑈 “ (𝑀...𝑁)) × {0})) Fn ((𝑈 “ (1...(𝑀 − 1))) ∪ (𝑈 “ (𝑀...𝑁))))
103 elfzuz 12550 . . . . . . . . . . . . . . . . . . . 20 (𝑀 ∈ (1...(𝑁 − 1)) → 𝑀 ∈ (ℤ‘1))
10416, 103syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜑𝑀 ∈ (ℤ‘1))
10557, 104eqeltrd 2844 . . . . . . . . . . . . . . . . . 18 (𝜑 → ((𝑀 − 1) + 1) ∈ (ℤ‘1))
106 peano2zm 11672 . . . . . . . . . . . . . . . . . . . . . 22 (𝑀 ∈ ℤ → (𝑀 − 1) ∈ ℤ)
107 uzid 11906 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑀 − 1) ∈ ℤ → (𝑀 − 1) ∈ (ℤ‘(𝑀 − 1)))
108 peano2uz 11946 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑀 − 1) ∈ (ℤ‘(𝑀 − 1)) → ((𝑀 − 1) + 1) ∈ (ℤ‘(𝑀 − 1)))
10940, 106, 107, 1084syl 19 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → ((𝑀 − 1) + 1) ∈ (ℤ‘(𝑀 − 1)))
11057, 109eqeltrrd 2845 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝑀 ∈ (ℤ‘(𝑀 − 1)))
111 peano2uz 11946 . . . . . . . . . . . . . . . . . . . 20 (𝑀 ∈ (ℤ‘(𝑀 − 1)) → (𝑀 + 1) ∈ (ℤ‘(𝑀 − 1)))
112 uzss 11912 . . . . . . . . . . . . . . . . . . . 20 ((𝑀 + 1) ∈ (ℤ‘(𝑀 − 1)) → (ℤ‘(𝑀 + 1)) ⊆ (ℤ‘(𝑀 − 1)))
113110, 111, 1123syl 18 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (ℤ‘(𝑀 + 1)) ⊆ (ℤ‘(𝑀 − 1)))
114 elfzuz3 12551 . . . . . . . . . . . . . . . . . . . . 21 (𝑀 ∈ (1...(𝑁 − 1)) → (𝑁 − 1) ∈ (ℤ𝑀))
115 eluzp1p1 11917 . . . . . . . . . . . . . . . . . . . . 21 ((𝑁 − 1) ∈ (ℤ𝑀) → ((𝑁 − 1) + 1) ∈ (ℤ‘(𝑀 + 1)))
11616, 114, 1153syl 18 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ((𝑁 − 1) + 1) ∈ (ℤ‘(𝑀 + 1)))
1177, 116eqeltrrd 2845 . . . . . . . . . . . . . . . . . . 19 (𝜑𝑁 ∈ (ℤ‘(𝑀 + 1)))
118113, 117sseldd 3764 . . . . . . . . . . . . . . . . . 18 (𝜑𝑁 ∈ (ℤ‘(𝑀 − 1)))
119 fzsplit2 12578 . . . . . . . . . . . . . . . . . 18 ((((𝑀 − 1) + 1) ∈ (ℤ‘1) ∧ 𝑁 ∈ (ℤ‘(𝑀 − 1))) → (1...𝑁) = ((1...(𝑀 − 1)) ∪ (((𝑀 − 1) + 1)...𝑁)))
120105, 118, 119syl2anc 579 . . . . . . . . . . . . . . . . 17 (𝜑 → (1...𝑁) = ((1...(𝑀 − 1)) ∪ (((𝑀 − 1) + 1)...𝑁)))
12157oveq1d 6861 . . . . . . . . . . . . . . . . . 18 (𝜑 → (((𝑀 − 1) + 1)...𝑁) = (𝑀...𝑁))
122121uneq2d 3931 . . . . . . . . . . . . . . . . 17 (𝜑 → ((1...(𝑀 − 1)) ∪ (((𝑀 − 1) + 1)...𝑁)) = ((1...(𝑀 − 1)) ∪ (𝑀...𝑁)))
123120, 122eqtrd 2799 . . . . . . . . . . . . . . . 16 (𝜑 → (1...𝑁) = ((1...(𝑀 − 1)) ∪ (𝑀...𝑁)))
124123imaeq2d 5650 . . . . . . . . . . . . . . 15 (𝜑 → (𝑈 “ (1...𝑁)) = (𝑈 “ ((1...(𝑀 − 1)) ∪ (𝑀...𝑁))))
125 imaundi 5730 . . . . . . . . . . . . . . 15 (𝑈 “ ((1...(𝑀 − 1)) ∪ (𝑀...𝑁))) = ((𝑈 “ (1...(𝑀 − 1))) ∪ (𝑈 “ (𝑀...𝑁)))
126124, 125syl6eq 2815 . . . . . . . . . . . . . 14 (𝜑 → (𝑈 “ (1...𝑁)) = ((𝑈 “ (1...(𝑀 − 1))) ∪ (𝑈 “ (𝑀...𝑁))))
127 f1ofo 6331 . . . . . . . . . . . . . . 15 (𝑈:(1...𝑁)–1-1-onto→(1...𝑁) → 𝑈:(1...𝑁)–onto→(1...𝑁))
128 foima 6305 . . . . . . . . . . . . . . 15 (𝑈:(1...𝑁)–onto→(1...𝑁) → (𝑈 “ (1...𝑁)) = (1...𝑁))
1291, 127, 1283syl 18 . . . . . . . . . . . . . 14 (𝜑 → (𝑈 “ (1...𝑁)) = (1...𝑁))
130126, 129eqtr3d 2801 . . . . . . . . . . . . 13 (𝜑 → ((𝑈 “ (1...(𝑀 − 1))) ∪ (𝑈 “ (𝑀...𝑁))) = (1...𝑁))
131130fneq2d 6162 . . . . . . . . . . . 12 (𝜑 → ((((𝑈 “ (1...(𝑀 − 1))) × {1}) ∪ ((𝑈 “ (𝑀...𝑁)) × {0})) Fn ((𝑈 “ (1...(𝑀 − 1))) ∪ (𝑈 “ (𝑀...𝑁))) ↔ (((𝑈 “ (1...(𝑀 − 1))) × {1}) ∪ ((𝑈 “ (𝑀...𝑁)) × {0})) Fn (1...𝑁)))
132102, 131mpbid 223 . . . . . . . . . . 11 (𝜑 → (((𝑈 “ (1...(𝑀 − 1))) × {1}) ∪ ((𝑈 “ (𝑀...𝑁)) × {0})) Fn (1...𝑁))
133132adantr 472 . . . . . . . . . 10 ((𝜑𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))) → (((𝑈 “ (1...(𝑀 − 1))) × {1}) ∪ ((𝑈 “ (𝑀...𝑁)) × {0})) Fn (1...𝑁))
134 ovexd 6880 . . . . . . . . . 10 ((𝜑𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))) → (1...𝑁) ∈ V)
135 inidm 3984 . . . . . . . . . 10 ((1...𝑁) ∩ (1...𝑁)) = (1...𝑁)
136 eqidd 2766 . . . . . . . . . 10 (((𝜑𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))) ∧ 𝑛 ∈ (1...𝑁)) → (𝑇𝑛) = (𝑇𝑛))
137100adantr 472 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))) → ((𝑈 “ (1...(𝑀 − 1))) ∩ (𝑈 “ (𝑀...𝑁))) = ∅)
138 fzss2 12593 . . . . . . . . . . . . . . 15 (𝑁 ∈ (ℤ‘(𝑀 + 1)) → (𝑀...(𝑀 + 1)) ⊆ (𝑀...𝑁))
139 imass2 5685 . . . . . . . . . . . . . . 15 ((𝑀...(𝑀 + 1)) ⊆ (𝑀...𝑁) → (𝑈 “ (𝑀...(𝑀 + 1))) ⊆ (𝑈 “ (𝑀...𝑁)))
140117, 138, 1393syl 18 . . . . . . . . . . . . . 14 (𝜑 → (𝑈 “ (𝑀...(𝑀 + 1))) ⊆ (𝑈 “ (𝑀...𝑁)))
141140sselda 3763 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))) → 𝑛 ∈ (𝑈 “ (𝑀...𝑁)))
142 fvun2 6463 . . . . . . . . . . . . . 14 ((((𝑈 “ (1...(𝑀 − 1))) × {1}) Fn (𝑈 “ (1...(𝑀 − 1))) ∧ ((𝑈 “ (𝑀...𝑁)) × {0}) Fn (𝑈 “ (𝑀...𝑁)) ∧ (((𝑈 “ (1...(𝑀 − 1))) ∩ (𝑈 “ (𝑀...𝑁))) = ∅ ∧ 𝑛 ∈ (𝑈 “ (𝑀...𝑁)))) → ((((𝑈 “ (1...(𝑀 − 1))) × {1}) ∪ ((𝑈 “ (𝑀...𝑁)) × {0}))‘𝑛) = (((𝑈 “ (𝑀...𝑁)) × {0})‘𝑛))
14386, 89, 142mp3an12 1575 . . . . . . . . . . . . 13 ((((𝑈 “ (1...(𝑀 − 1))) ∩ (𝑈 “ (𝑀...𝑁))) = ∅ ∧ 𝑛 ∈ (𝑈 “ (𝑀...𝑁))) → ((((𝑈 “ (1...(𝑀 − 1))) × {1}) ∪ ((𝑈 “ (𝑀...𝑁)) × {0}))‘𝑛) = (((𝑈 “ (𝑀...𝑁)) × {0})‘𝑛))
144137, 141, 143syl2anc 579 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))) → ((((𝑈 “ (1...(𝑀 − 1))) × {1}) ∪ ((𝑈 “ (𝑀...𝑁)) × {0}))‘𝑛) = (((𝑈 “ (𝑀...𝑁)) × {0})‘𝑛))
14587fvconst2 6666 . . . . . . . . . . . . 13 (𝑛 ∈ (𝑈 “ (𝑀...𝑁)) → (((𝑈 “ (𝑀...𝑁)) × {0})‘𝑛) = 0)
146141, 145syl 17 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))) → (((𝑈 “ (𝑀...𝑁)) × {0})‘𝑛) = 0)
147144, 146eqtrd 2799 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))) → ((((𝑈 “ (1...(𝑀 − 1))) × {1}) ∪ ((𝑈 “ (𝑀...𝑁)) × {0}))‘𝑛) = 0)
148147adantr 472 . . . . . . . . . 10 (((𝜑𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))) ∧ 𝑛 ∈ (1...𝑁)) → ((((𝑈 “ (1...(𝑀 − 1))) × {1}) ∪ ((𝑈 “ (𝑀...𝑁)) × {0}))‘𝑛) = 0)
14983, 133, 134, 134, 135, 136, 148ofval 7108 . . . . . . . . 9 (((𝜑𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))) ∧ 𝑛 ∈ (1...𝑁)) → ((𝑇𝑓 + (((𝑈 “ (1...(𝑀 − 1))) × {1}) ∪ ((𝑈 “ (𝑀...𝑁)) × {0})))‘𝑛) = ((𝑇𝑛) + 0))
15028, 149mpdan 678 . . . . . . . 8 ((𝜑𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))) → ((𝑇𝑓 + (((𝑈 “ (1...(𝑀 − 1))) × {1}) ∪ ((𝑈 “ (𝑀...𝑁)) × {0})))‘𝑛) = ((𝑇𝑛) + 0))
15130zcnd 11735 . . . . . . . . . 10 ((𝜑𝑛 ∈ (1...𝑁)) → (𝑇𝑛) ∈ ℂ)
152151addid1d 10494 . . . . . . . . 9 ((𝜑𝑛 ∈ (1...𝑁)) → ((𝑇𝑛) + 0) = (𝑇𝑛))
15328, 152syldan 585 . . . . . . . 8 ((𝜑𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))) → ((𝑇𝑛) + 0) = (𝑇𝑛))
15481, 150, 1533eqtrd 2803 . . . . . . 7 ((𝜑𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))) → ((𝐹‘(𝑀 − 1))‘𝑛) = (𝑇𝑛))
155 breq1 4814 . . . . . . . . . . . . . . 15 (𝑦 = 𝑀 → (𝑦 < 𝑀𝑀 < 𝑀))
156 oveq1 6853 . . . . . . . . . . . . . . 15 (𝑦 = 𝑀 → (𝑦 + 1) = (𝑀 + 1))
157155, 156ifbieq2d 4270 . . . . . . . . . . . . . 14 (𝑦 = 𝑀 → if(𝑦 < 𝑀, 𝑦, (𝑦 + 1)) = if(𝑀 < 𝑀, 𝑦, (𝑀 + 1)))
15841ltnrd 10429 . . . . . . . . . . . . . . 15 (𝜑 → ¬ 𝑀 < 𝑀)
159158iffalsed 4256 . . . . . . . . . . . . . 14 (𝜑 → if(𝑀 < 𝑀, 𝑦, (𝑀 + 1)) = (𝑀 + 1))
160157, 159sylan9eqr 2821 . . . . . . . . . . . . 13 ((𝜑𝑦 = 𝑀) → if(𝑦 < 𝑀, 𝑦, (𝑦 + 1)) = (𝑀 + 1))
161160csbeq1d 3700 . . . . . . . . . . . 12 ((𝜑𝑦 = 𝑀) → if(𝑦 < 𝑀, 𝑦, (𝑦 + 1)) / 𝑗(𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) = (𝑀 + 1) / 𝑗(𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))))
162 ovex 6878 . . . . . . . . . . . . 13 (𝑀 + 1) ∈ V
163 oveq2 6854 . . . . . . . . . . . . . . . . 17 (𝑗 = (𝑀 + 1) → (1...𝑗) = (1...(𝑀 + 1)))
164163imaeq2d 5650 . . . . . . . . . . . . . . . 16 (𝑗 = (𝑀 + 1) → (𝑈 “ (1...𝑗)) = (𝑈 “ (1...(𝑀 + 1))))
165164xpeq1d 5308 . . . . . . . . . . . . . . 15 (𝑗 = (𝑀 + 1) → ((𝑈 “ (1...𝑗)) × {1}) = ((𝑈 “ (1...(𝑀 + 1))) × {1}))
166 oveq1 6853 . . . . . . . . . . . . . . . . . 18 (𝑗 = (𝑀 + 1) → (𝑗 + 1) = ((𝑀 + 1) + 1))
167166oveq1d 6861 . . . . . . . . . . . . . . . . 17 (𝑗 = (𝑀 + 1) → ((𝑗 + 1)...𝑁) = (((𝑀 + 1) + 1)...𝑁))
168167imaeq2d 5650 . . . . . . . . . . . . . . . 16 (𝑗 = (𝑀 + 1) → (𝑈 “ ((𝑗 + 1)...𝑁)) = (𝑈 “ (((𝑀 + 1) + 1)...𝑁)))
169168xpeq1d 5308 . . . . . . . . . . . . . . 15 (𝑗 = (𝑀 + 1) → ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}) = ((𝑈 “ (((𝑀 + 1) + 1)...𝑁)) × {0}))
170165, 169uneq12d 3932 . . . . . . . . . . . . . 14 (𝑗 = (𝑀 + 1) → (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})) = (((𝑈 “ (1...(𝑀 + 1))) × {1}) ∪ ((𝑈 “ (((𝑀 + 1) + 1)...𝑁)) × {0})))
171170oveq2d 6862 . . . . . . . . . . . . 13 (𝑗 = (𝑀 + 1) → (𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) = (𝑇𝑓 + (((𝑈 “ (1...(𝑀 + 1))) × {1}) ∪ ((𝑈 “ (((𝑀 + 1) + 1)...𝑁)) × {0}))))
172162, 171csbie 3719 . . . . . . . . . . . 12 (𝑀 + 1) / 𝑗(𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) = (𝑇𝑓 + (((𝑈 “ (1...(𝑀 + 1))) × {1}) ∪ ((𝑈 “ (((𝑀 + 1) + 1)...𝑁)) × {0})))
173161, 172syl6eq 2815 . . . . . . . . . . 11 ((𝜑𝑦 = 𝑀) → if(𝑦 < 𝑀, 𝑦, (𝑦 + 1)) / 𝑗(𝑇𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) = (𝑇𝑓 + (((𝑈 “ (1...(𝑀 + 1))) × {1}) ∪ ((𝑈 “ (((𝑀 + 1) + 1)...𝑁)) × {0}))))
174 fz1ssfz0 12648 . . . . . . . . . . . 12 (1...(𝑁 − 1)) ⊆ (0...(𝑁 − 1))
175174, 16sseldi 3761 . . . . . . . . . . 11 (𝜑𝑀 ∈ (0...(𝑁 − 1)))
176 ovexd 6880 . . . . . . . . . . 11 (𝜑 → (𝑇𝑓 + (((𝑈 “ (1...(𝑀 + 1))) × {1}) ∪ ((𝑈 “ (((𝑀 + 1) + 1)...𝑁)) × {0}))) ∈ V)
17735, 173, 175, 176fvmptd 6481 . . . . . . . . . 10 (𝜑 → (𝐹𝑀) = (𝑇𝑓 + (((𝑈 “ (1...(𝑀 + 1))) × {1}) ∪ ((𝑈 “ (((𝑀 + 1) + 1)...𝑁)) × {0}))))
178177fveq1d 6381 . . . . . . . . 9 (𝜑 → ((𝐹𝑀)‘𝑛) = ((𝑇𝑓 + (((𝑈 “ (1...(𝑀 + 1))) × {1}) ∪ ((𝑈 “ (((𝑀 + 1) + 1)...𝑁)) × {0})))‘𝑛))
179178adantr 472 . . . . . . . 8 ((𝜑𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))) → ((𝐹𝑀)‘𝑛) = ((𝑇𝑓 + (((𝑈 “ (1...(𝑀 + 1))) × {1}) ∪ ((𝑈 “ (((𝑀 + 1) + 1)...𝑁)) × {0})))‘𝑛))
180 fnconstg 6277 . . . . . . . . . . . . . . 15 (1 ∈ V → ((𝑈 “ (1...(𝑀 + 1))) × {1}) Fn (𝑈 “ (1...(𝑀 + 1))))
18184, 180ax-mp 5 . . . . . . . . . . . . . 14 ((𝑈 “ (1...(𝑀 + 1))) × {1}) Fn (𝑈 “ (1...(𝑀 + 1)))
182 fnconstg 6277 . . . . . . . . . . . . . . 15 (0 ∈ V → ((𝑈 “ (((𝑀 + 1) + 1)...𝑁)) × {0}) Fn (𝑈 “ (((𝑀 + 1) + 1)...𝑁)))
18387, 182ax-mp 5 . . . . . . . . . . . . . 14 ((𝑈 “ (((𝑀 + 1) + 1)...𝑁)) × {0}) Fn (𝑈 “ (((𝑀 + 1) + 1)...𝑁))
184181, 183pm3.2i 462 . . . . . . . . . . . . 13 (((𝑈 “ (1...(𝑀 + 1))) × {1}) Fn (𝑈 “ (1...(𝑀 + 1))) ∧ ((𝑈 “ (((𝑀 + 1) + 1)...𝑁)) × {0}) Fn (𝑈 “ (((𝑀 + 1) + 1)...𝑁)))
185 imain 6154 . . . . . . . . . . . . . . . 16 (Fun 𝑈 → (𝑈 “ ((1...(𝑀 + 1)) ∩ (((𝑀 + 1) + 1)...𝑁))) = ((𝑈 “ (1...(𝑀 + 1))) ∩ (𝑈 “ (((𝑀 + 1) + 1)...𝑁))))
1861, 92, 1853syl 18 . . . . . . . . . . . . . . 15 (𝜑 → (𝑈 “ ((1...(𝑀 + 1)) ∩ (((𝑀 + 1) + 1)...𝑁))) = ((𝑈 “ (1...(𝑀 + 1))) ∩ (𝑈 “ (((𝑀 + 1) + 1)...𝑁))))
187 peano2re 10467 . . . . . . . . . . . . . . . . . . 19 (𝑀 ∈ ℝ → (𝑀 + 1) ∈ ℝ)
18841, 187syl 17 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝑀 + 1) ∈ ℝ)
189188ltp1d 11212 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝑀 + 1) < ((𝑀 + 1) + 1))
190 fzdisj 12580 . . . . . . . . . . . . . . . . 17 ((𝑀 + 1) < ((𝑀 + 1) + 1) → ((1...(𝑀 + 1)) ∩ (((𝑀 + 1) + 1)...𝑁)) = ∅)
191189, 190syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → ((1...(𝑀 + 1)) ∩ (((𝑀 + 1) + 1)...𝑁)) = ∅)
192191imaeq2d 5650 . . . . . . . . . . . . . . 15 (𝜑 → (𝑈 “ ((1...(𝑀 + 1)) ∩ (((𝑀 + 1) + 1)...𝑁))) = (𝑈 “ ∅))
193186, 192eqtr3d 2801 . . . . . . . . . . . . . 14 (𝜑 → ((𝑈 “ (1...(𝑀 + 1))) ∩ (𝑈 “ (((𝑀 + 1) + 1)...𝑁))) = (𝑈 “ ∅))
194193, 98syl6eq 2815 . . . . . . . . . . . . 13 (𝜑 → ((𝑈 “ (1...(𝑀 + 1))) ∩ (𝑈 “ (((𝑀 + 1) + 1)...𝑁))) = ∅)
195 fnun 6177 . . . . . . . . . . . . 13 (((((𝑈 “ (1...(𝑀 + 1))) × {1}) Fn (𝑈 “ (1...(𝑀 + 1))) ∧ ((𝑈 “ (((𝑀 + 1) + 1)...𝑁)) × {0}) Fn (𝑈 “ (((𝑀 + 1) + 1)...𝑁))) ∧ ((𝑈 “ (1...(𝑀 + 1))) ∩ (𝑈 “ (((𝑀 + 1) + 1)...𝑁))) = ∅) → (((𝑈 “ (1...(𝑀 + 1))) × {1}) ∪ ((𝑈 “ (((𝑀 + 1) + 1)...𝑁)) × {0})) Fn ((𝑈 “ (1...(𝑀 + 1))) ∪ (𝑈 “ (((𝑀 + 1) + 1)...𝑁))))
196184, 194, 195sylancr 581 . . . . . . . . . . . 12 (𝜑 → (((𝑈 “ (1...(𝑀 + 1))) × {1}) ∪ ((𝑈 “ (((𝑀 + 1) + 1)...𝑁)) × {0})) Fn ((𝑈 “ (1...(𝑀 + 1))) ∪ (𝑈 “ (((𝑀 + 1) + 1)...𝑁))))
197 fzsplit 12579 . . . . . . . . . . . . . . . . 17 ((𝑀 + 1) ∈ (1...𝑁) → (1...𝑁) = ((1...(𝑀 + 1)) ∪ (((𝑀 + 1) + 1)...𝑁)))
19822, 197syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → (1...𝑁) = ((1...(𝑀 + 1)) ∪ (((𝑀 + 1) + 1)...𝑁)))
199198imaeq2d 5650 . . . . . . . . . . . . . . 15 (𝜑 → (𝑈 “ (1...𝑁)) = (𝑈 “ ((1...(𝑀 + 1)) ∪ (((𝑀 + 1) + 1)...𝑁))))
200 imaundi 5730 . . . . . . . . . . . . . . 15 (𝑈 “ ((1...(𝑀 + 1)) ∪ (((𝑀 + 1) + 1)...𝑁))) = ((𝑈 “ (1...(𝑀 + 1))) ∪ (𝑈 “ (((𝑀 + 1) + 1)...𝑁)))
201199, 200syl6eq 2815 . . . . . . . . . . . . . 14 (𝜑 → (𝑈 “ (1...𝑁)) = ((𝑈 “ (1...(𝑀 + 1))) ∪ (𝑈 “ (((𝑀 + 1) + 1)...𝑁))))
202201, 129eqtr3d 2801 . . . . . . . . . . . . 13 (𝜑 → ((𝑈 “ (1...(𝑀 + 1))) ∪ (𝑈 “ (((𝑀 + 1) + 1)...𝑁))) = (1...𝑁))
203202fneq2d 6162 . . . . . . . . . . . 12 (𝜑 → ((((𝑈 “ (1...(𝑀 + 1))) × {1}) ∪ ((𝑈 “ (((𝑀 + 1) + 1)...𝑁)) × {0})) Fn ((𝑈 “ (1...(𝑀 + 1))) ∪ (𝑈 “ (((𝑀 + 1) + 1)...𝑁))) ↔ (((𝑈 “ (1...(𝑀 + 1))) × {1}) ∪ ((𝑈 “ (((𝑀 + 1) + 1)...𝑁)) × {0})) Fn (1...𝑁)))
204196, 203mpbid 223 . . . . . . . . . . 11 (𝜑 → (((𝑈 “ (1...(𝑀 + 1))) × {1}) ∪ ((𝑈 “ (((𝑀 + 1) + 1)...𝑁)) × {0})) Fn (1...𝑁))
205204adantr 472 . . . . . . . . . 10 ((𝜑𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))) → (((𝑈 “ (1...(𝑀 + 1))) × {1}) ∪ ((𝑈 “ (((𝑀 + 1) + 1)...𝑁)) × {0})) Fn (1...𝑁))
206194adantr 472 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))) → ((𝑈 “ (1...(𝑀 + 1))) ∩ (𝑈 “ (((𝑀 + 1) + 1)...𝑁))) = ∅)
207 fzss1 12592 . . . . . . . . . . . . . . 15 (𝑀 ∈ (ℤ‘1) → (𝑀...(𝑀 + 1)) ⊆ (1...(𝑀 + 1)))
208 imass2 5685 . . . . . . . . . . . . . . 15 ((𝑀...(𝑀 + 1)) ⊆ (1...(𝑀 + 1)) → (𝑈 “ (𝑀...(𝑀 + 1))) ⊆ (𝑈 “ (1...(𝑀 + 1))))
209104, 207, 2083syl 18 . . . . . . . . . . . . . 14 (𝜑 → (𝑈 “ (𝑀...(𝑀 + 1))) ⊆ (𝑈 “ (1...(𝑀 + 1))))
210209sselda 3763 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))) → 𝑛 ∈ (𝑈 “ (1...(𝑀 + 1))))
211 fvun1 6462 . . . . . . . . . . . . . 14 ((((𝑈 “ (1...(𝑀 + 1))) × {1}) Fn (𝑈 “ (1...(𝑀 + 1))) ∧ ((𝑈 “ (((𝑀 + 1) + 1)...𝑁)) × {0}) Fn (𝑈 “ (((𝑀 + 1) + 1)...𝑁)) ∧ (((𝑈 “ (1...(𝑀 + 1))) ∩ (𝑈 “ (((𝑀 + 1) + 1)...𝑁))) = ∅ ∧ 𝑛 ∈ (𝑈 “ (1...(𝑀 + 1))))) → ((((𝑈 “ (1...(𝑀 + 1))) × {1}) ∪ ((𝑈 “ (((𝑀 + 1) + 1)...𝑁)) × {0}))‘𝑛) = (((𝑈 “ (1...(𝑀 + 1))) × {1})‘𝑛))
212181, 183, 211mp3an12 1575 . . . . . . . . . . . . 13 ((((𝑈 “ (1...(𝑀 + 1))) ∩ (𝑈 “ (((𝑀 + 1) + 1)...𝑁))) = ∅ ∧ 𝑛 ∈ (𝑈 “ (1...(𝑀 + 1)))) → ((((𝑈 “ (1...(𝑀 + 1))) × {1}) ∪ ((𝑈 “ (((𝑀 + 1) + 1)...𝑁)) × {0}))‘𝑛) = (((𝑈 “ (1...(𝑀 + 1))) × {1})‘𝑛))
213206, 210, 212syl2anc 579 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))) → ((((𝑈 “ (1...(𝑀 + 1))) × {1}) ∪ ((𝑈 “ (((𝑀 + 1) + 1)...𝑁)) × {0}))‘𝑛) = (((𝑈 “ (1...(𝑀 + 1))) × {1})‘𝑛))
21484fvconst2 6666 . . . . . . . . . . . . 13 (𝑛 ∈ (𝑈 “ (1...(𝑀 + 1))) → (((𝑈 “ (1...(𝑀 + 1))) × {1})‘𝑛) = 1)
215210, 214syl 17 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))) → (((𝑈 “ (1...(𝑀 + 1))) × {1})‘𝑛) = 1)
216213, 215eqtrd 2799 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))) → ((((𝑈 “ (1...(𝑀 + 1))) × {1}) ∪ ((𝑈 “ (((𝑀 + 1) + 1)...𝑁)) × {0}))‘𝑛) = 1)
217216adantr 472 . . . . . . . . . 10 (((𝜑𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))) ∧ 𝑛 ∈ (1...𝑁)) → ((((𝑈 “ (1...(𝑀 + 1))) × {1}) ∪ ((𝑈 “ (((𝑀 + 1) + 1)...𝑁)) × {0}))‘𝑛) = 1)
21883, 205, 134, 134, 135, 136, 217ofval 7108 . . . . . . . . 9 (((𝜑𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))) ∧ 𝑛 ∈ (1...𝑁)) → ((𝑇𝑓 + (((𝑈 “ (1...(𝑀 + 1))) × {1}) ∪ ((𝑈 “ (((𝑀 + 1) + 1)...𝑁)) × {0})))‘𝑛) = ((𝑇𝑛) + 1))
21928, 218mpdan 678 . . . . . . . 8 ((𝜑𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))) → ((𝑇𝑓 + (((𝑈 “ (1...(𝑀 + 1))) × {1}) ∪ ((𝑈 “ (((𝑀 + 1) + 1)...𝑁)) × {0})))‘𝑛) = ((𝑇𝑛) + 1))
220179, 219eqtrd 2799 . . . . . . 7 ((𝜑𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))) → ((𝐹𝑀)‘𝑛) = ((𝑇𝑛) + 1))
22134, 154, 2203netr4d 3014 . . . . . 6 ((𝜑𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))) → ((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹𝑀)‘𝑛))
222221ralrimiva 3113 . . . . 5 (𝜑 → ∀𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹𝑀)‘𝑛))
223 fzpr 12608 . . . . . . . . 9 (𝑀 ∈ ℤ → (𝑀...(𝑀 + 1)) = {𝑀, (𝑀 + 1)})
22416, 39, 2233syl 18 . . . . . . . 8 (𝜑 → (𝑀...(𝑀 + 1)) = {𝑀, (𝑀 + 1)})
225224imaeq2d 5650 . . . . . . 7 (𝜑 → (𝑈 “ (𝑀...(𝑀 + 1))) = (𝑈 “ {𝑀, (𝑀 + 1)}))
226 f1ofn 6325 . . . . . . . . 9 (𝑈:(1...𝑁)–1-1-onto→(1...𝑁) → 𝑈 Fn (1...𝑁))
2271, 226syl 17 . . . . . . . 8 (𝜑𝑈 Fn (1...𝑁))
228 fnimapr 6455 . . . . . . . 8 ((𝑈 Fn (1...𝑁) ∧ 𝑀 ∈ (1...𝑁) ∧ (𝑀 + 1) ∈ (1...𝑁)) → (𝑈 “ {𝑀, (𝑀 + 1)}) = {(𝑈𝑀), (𝑈‘(𝑀 + 1))})
229227, 17, 22, 228syl3anc 1490 . . . . . . 7 (𝜑 → (𝑈 “ {𝑀, (𝑀 + 1)}) = {(𝑈𝑀), (𝑈‘(𝑀 + 1))})
230225, 229eqtrd 2799 . . . . . 6 (𝜑 → (𝑈 “ (𝑀...(𝑀 + 1))) = {(𝑈𝑀), (𝑈‘(𝑀 + 1))})
231230raleqdv 3292 . . . . 5 (𝜑 → (∀𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹𝑀)‘𝑛) ↔ ∀𝑛 ∈ {(𝑈𝑀), (𝑈‘(𝑀 + 1))} ((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹𝑀)‘𝑛)))
232222, 231mpbid 223 . . . 4 (𝜑 → ∀𝑛 ∈ {(𝑈𝑀), (𝑈‘(𝑀 + 1))} ((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹𝑀)‘𝑛))
233 fvex 6392 . . . . 5 (𝑈𝑀) ∈ V
234 fvex 6392 . . . . 5 (𝑈‘(𝑀 + 1)) ∈ V
235 fveq2 6379 . . . . . 6 (𝑛 = (𝑈𝑀) → ((𝐹‘(𝑀 − 1))‘𝑛) = ((𝐹‘(𝑀 − 1))‘(𝑈𝑀)))
236 fveq2 6379 . . . . . 6 (𝑛 = (𝑈𝑀) → ((𝐹𝑀)‘𝑛) = ((𝐹𝑀)‘(𝑈𝑀)))
237235, 236neeq12d 2998 . . . . 5 (𝑛 = (𝑈𝑀) → (((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹𝑀)‘𝑛) ↔ ((𝐹‘(𝑀 − 1))‘(𝑈𝑀)) ≠ ((𝐹𝑀)‘(𝑈𝑀))))
238 fveq2 6379 . . . . . 6 (𝑛 = (𝑈‘(𝑀 + 1)) → ((𝐹‘(𝑀 − 1))‘𝑛) = ((𝐹‘(𝑀 − 1))‘(𝑈‘(𝑀 + 1))))
239 fveq2 6379 . . . . . 6 (𝑛 = (𝑈‘(𝑀 + 1)) → ((𝐹𝑀)‘𝑛) = ((𝐹𝑀)‘(𝑈‘(𝑀 + 1))))
240238, 239neeq12d 2998 . . . . 5 (𝑛 = (𝑈‘(𝑀 + 1)) → (((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹𝑀)‘𝑛) ↔ ((𝐹‘(𝑀 − 1))‘(𝑈‘(𝑀 + 1))) ≠ ((𝐹𝑀)‘(𝑈‘(𝑀 + 1)))))
241233, 234, 237, 240ralpr 4396 . . . 4 (∀𝑛 ∈ {(𝑈𝑀), (𝑈‘(𝑀 + 1))} ((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹𝑀)‘𝑛) ↔ (((𝐹‘(𝑀 − 1))‘(𝑈𝑀)) ≠ ((𝐹𝑀)‘(𝑈𝑀)) ∧ ((𝐹‘(𝑀 − 1))‘(𝑈‘(𝑀 + 1))) ≠ ((𝐹𝑀)‘(𝑈‘(𝑀 + 1)))))
242232, 241sylib 209 . . 3 (𝜑 → (((𝐹‘(𝑀 − 1))‘(𝑈𝑀)) ≠ ((𝐹𝑀)‘(𝑈𝑀)) ∧ ((𝐹‘(𝑀 − 1))‘(𝑈‘(𝑀 + 1))) ≠ ((𝐹𝑀)‘(𝑈‘(𝑀 + 1)))))
24341ltp1d 11212 . . . . 5 (𝜑𝑀 < (𝑀 + 1))
24441, 243ltned 10431 . . . 4 (𝜑𝑀 ≠ (𝑀 + 1))
245 f1of1 6323 . . . . . . 7 (𝑈:(1...𝑁)–1-1-onto→(1...𝑁) → 𝑈:(1...𝑁)–1-1→(1...𝑁))
2461, 245syl 17 . . . . . 6 (𝜑𝑈:(1...𝑁)–1-1→(1...𝑁))
247 f1veqaeq 6710 . . . . . 6 ((𝑈:(1...𝑁)–1-1→(1...𝑁) ∧ (𝑀 ∈ (1...𝑁) ∧ (𝑀 + 1) ∈ (1...𝑁))) → ((𝑈𝑀) = (𝑈‘(𝑀 + 1)) → 𝑀 = (𝑀 + 1)))
248246, 17, 22, 247syl12anc 865 . . . . 5 (𝜑 → ((𝑈𝑀) = (𝑈‘(𝑀 + 1)) → 𝑀 = (𝑀 + 1)))
249248necon3d 2958 . . . 4 (𝜑 → (𝑀 ≠ (𝑀 + 1) → (𝑈𝑀) ≠ (𝑈‘(𝑀 + 1))))
250244, 249mpd 15 . . 3 (𝜑 → (𝑈𝑀) ≠ (𝑈‘(𝑀 + 1)))
251237anbi1d 623 . . . . 5 (𝑛 = (𝑈𝑀) → ((((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹𝑀)‘𝑛) ∧ ((𝐹‘(𝑀 − 1))‘𝑚) ≠ ((𝐹𝑀)‘𝑚)) ↔ (((𝐹‘(𝑀 − 1))‘(𝑈𝑀)) ≠ ((𝐹𝑀)‘(𝑈𝑀)) ∧ ((𝐹‘(𝑀 − 1))‘𝑚) ≠ ((𝐹𝑀)‘𝑚))))
252 neeq1 2999 . . . . 5 (𝑛 = (𝑈𝑀) → (𝑛𝑚 ↔ (𝑈𝑀) ≠ 𝑚))
253251, 252anbi12d 624 . . . 4 (𝑛 = (𝑈𝑀) → (((((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹𝑀)‘𝑛) ∧ ((𝐹‘(𝑀 − 1))‘𝑚) ≠ ((𝐹𝑀)‘𝑚)) ∧ 𝑛𝑚) ↔ ((((𝐹‘(𝑀 − 1))‘(𝑈𝑀)) ≠ ((𝐹𝑀)‘(𝑈𝑀)) ∧ ((𝐹‘(𝑀 − 1))‘𝑚) ≠ ((𝐹𝑀)‘𝑚)) ∧ (𝑈𝑀) ≠ 𝑚)))
254 fveq2 6379 . . . . . . 7 (𝑚 = (𝑈‘(𝑀 + 1)) → ((𝐹‘(𝑀 − 1))‘𝑚) = ((𝐹‘(𝑀 − 1))‘(𝑈‘(𝑀 + 1))))
255 fveq2 6379 . . . . . . 7 (𝑚 = (𝑈‘(𝑀 + 1)) → ((𝐹𝑀)‘𝑚) = ((𝐹𝑀)‘(𝑈‘(𝑀 + 1))))
256254, 255neeq12d 2998 . . . . . 6 (𝑚 = (𝑈‘(𝑀 + 1)) → (((𝐹‘(𝑀 − 1))‘𝑚) ≠ ((𝐹𝑀)‘𝑚) ↔ ((𝐹‘(𝑀 − 1))‘(𝑈‘(𝑀 + 1))) ≠ ((𝐹𝑀)‘(𝑈‘(𝑀 + 1)))))
257256anbi2d 622 . . . . 5 (𝑚 = (𝑈‘(𝑀 + 1)) → ((((𝐹‘(𝑀 − 1))‘(𝑈𝑀)) ≠ ((𝐹𝑀)‘(𝑈𝑀)) ∧ ((𝐹‘(𝑀 − 1))‘𝑚) ≠ ((𝐹𝑀)‘𝑚)) ↔ (((𝐹‘(𝑀 − 1))‘(𝑈𝑀)) ≠ ((𝐹𝑀)‘(𝑈𝑀)) ∧ ((𝐹‘(𝑀 − 1))‘(𝑈‘(𝑀 + 1))) ≠ ((𝐹𝑀)‘(𝑈‘(𝑀 + 1))))))
258 neeq2 3000 . . . . 5 (𝑚 = (𝑈‘(𝑀 + 1)) → ((𝑈𝑀) ≠ 𝑚 ↔ (𝑈𝑀) ≠ (𝑈‘(𝑀 + 1))))
259257, 258anbi12d 624 . . . 4 (𝑚 = (𝑈‘(𝑀 + 1)) → (((((𝐹‘(𝑀 − 1))‘(𝑈𝑀)) ≠ ((𝐹𝑀)‘(𝑈𝑀)) ∧ ((𝐹‘(𝑀 − 1))‘𝑚) ≠ ((𝐹𝑀)‘𝑚)) ∧ (𝑈𝑀) ≠ 𝑚) ↔ ((((𝐹‘(𝑀 − 1))‘(𝑈𝑀)) ≠ ((𝐹𝑀)‘(𝑈𝑀)) ∧ ((𝐹‘(𝑀 − 1))‘(𝑈‘(𝑀 + 1))) ≠ ((𝐹𝑀)‘(𝑈‘(𝑀 + 1)))) ∧ (𝑈𝑀) ≠ (𝑈‘(𝑀 + 1)))))
260253, 259rspc2ev 3477 . . 3 (((𝑈𝑀) ∈ (1...𝑁) ∧ (𝑈‘(𝑀 + 1)) ∈ (1...𝑁) ∧ ((((𝐹‘(𝑀 − 1))‘(𝑈𝑀)) ≠ ((𝐹𝑀)‘(𝑈𝑀)) ∧ ((𝐹‘(𝑀 − 1))‘(𝑈‘(𝑀 + 1))) ≠ ((𝐹𝑀)‘(𝑈‘(𝑀 + 1)))) ∧ (𝑈𝑀) ≠ (𝑈‘(𝑀 + 1)))) → ∃𝑛 ∈ (1...𝑁)∃𝑚 ∈ (1...𝑁)((((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹𝑀)‘𝑛) ∧ ((𝐹‘(𝑀 − 1))‘𝑚) ≠ ((𝐹𝑀)‘𝑚)) ∧ 𝑛𝑚))
26118, 23, 242, 250, 260syl112anc 1493 . 2 (𝜑 → ∃𝑛 ∈ (1...𝑁)∃𝑚 ∈ (1...𝑁)((((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹𝑀)‘𝑛) ∧ ((𝐹‘(𝑀 − 1))‘𝑚) ≠ ((𝐹𝑀)‘𝑚)) ∧ 𝑛𝑚))
262 dfrex2 3142 . . 3 (∃𝑛 ∈ (1...𝑁)∃𝑚 ∈ (1...𝑁)((((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹𝑀)‘𝑛) ∧ ((𝐹‘(𝑀 − 1))‘𝑚) ≠ ((𝐹𝑀)‘𝑚)) ∧ 𝑛𝑚) ↔ ¬ ∀𝑛 ∈ (1...𝑁) ¬ ∃𝑚 ∈ (1...𝑁)((((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹𝑀)‘𝑛) ∧ ((𝐹‘(𝑀 − 1))‘𝑚) ≠ ((𝐹𝑀)‘𝑚)) ∧ 𝑛𝑚))
263 fveq2 6379 . . . . . 6 (𝑛 = 𝑚 → ((𝐹‘(𝑀 − 1))‘𝑛) = ((𝐹‘(𝑀 − 1))‘𝑚))
264 fveq2 6379 . . . . . 6 (𝑛 = 𝑚 → ((𝐹𝑀)‘𝑛) = ((𝐹𝑀)‘𝑚))
265263, 264neeq12d 2998 . . . . 5 (𝑛 = 𝑚 → (((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹𝑀)‘𝑛) ↔ ((𝐹‘(𝑀 − 1))‘𝑚) ≠ ((𝐹𝑀)‘𝑚)))
266265rmo4 3560 . . . 4 (∃*𝑛 ∈ (1...𝑁)((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹𝑀)‘𝑛) ↔ ∀𝑛 ∈ (1...𝑁)∀𝑚 ∈ (1...𝑁)((((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹𝑀)‘𝑛) ∧ ((𝐹‘(𝑀 − 1))‘𝑚) ≠ ((𝐹𝑀)‘𝑚)) → 𝑛 = 𝑚))
267 dfral2 3140 . . . . . 6 (∀𝑚 ∈ (1...𝑁)((((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹𝑀)‘𝑛) ∧ ((𝐹‘(𝑀 − 1))‘𝑚) ≠ ((𝐹𝑀)‘𝑚)) → 𝑛 = 𝑚) ↔ ¬ ∃𝑚 ∈ (1...𝑁) ¬ ((((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹𝑀)‘𝑛) ∧ ((𝐹‘(𝑀 − 1))‘𝑚) ≠ ((𝐹𝑀)‘𝑚)) → 𝑛 = 𝑚))
268 df-ne 2938 . . . . . . . . 9 (𝑛𝑚 ↔ ¬ 𝑛 = 𝑚)
269268anbi2i 616 . . . . . . . 8 (((((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹𝑀)‘𝑛) ∧ ((𝐹‘(𝑀 − 1))‘𝑚) ≠ ((𝐹𝑀)‘𝑚)) ∧ 𝑛𝑚) ↔ ((((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹𝑀)‘𝑛) ∧ ((𝐹‘(𝑀 − 1))‘𝑚) ≠ ((𝐹𝑀)‘𝑚)) ∧ ¬ 𝑛 = 𝑚))
270 annim 392 . . . . . . . 8 (((((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹𝑀)‘𝑛) ∧ ((𝐹‘(𝑀 − 1))‘𝑚) ≠ ((𝐹𝑀)‘𝑚)) ∧ ¬ 𝑛 = 𝑚) ↔ ¬ ((((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹𝑀)‘𝑛) ∧ ((𝐹‘(𝑀 − 1))‘𝑚) ≠ ((𝐹𝑀)‘𝑚)) → 𝑛 = 𝑚))
271269, 270bitri 266 . . . . . . 7 (((((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹𝑀)‘𝑛) ∧ ((𝐹‘(𝑀 − 1))‘𝑚) ≠ ((𝐹𝑀)‘𝑚)) ∧ 𝑛𝑚) ↔ ¬ ((((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹𝑀)‘𝑛) ∧ ((𝐹‘(𝑀 − 1))‘𝑚) ≠ ((𝐹𝑀)‘𝑚)) → 𝑛 = 𝑚))
272271rexbii 3188 . . . . . 6 (∃𝑚 ∈ (1...𝑁)((((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹𝑀)‘𝑛) ∧ ((𝐹‘(𝑀 − 1))‘𝑚) ≠ ((𝐹𝑀)‘𝑚)) ∧ 𝑛𝑚) ↔ ∃𝑚 ∈ (1...𝑁) ¬ ((((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹𝑀)‘𝑛) ∧ ((𝐹‘(𝑀 − 1))‘𝑚) ≠ ((𝐹𝑀)‘𝑚)) → 𝑛 = 𝑚))
273267, 272xchbinxr 326 . . . . 5 (∀𝑚 ∈ (1...𝑁)((((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹𝑀)‘𝑛) ∧ ((𝐹‘(𝑀 − 1))‘𝑚) ≠ ((𝐹𝑀)‘𝑚)) → 𝑛 = 𝑚) ↔ ¬ ∃𝑚 ∈ (1...𝑁)((((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹𝑀)‘𝑛) ∧ ((𝐹‘(𝑀 − 1))‘𝑚) ≠ ((𝐹𝑀)‘𝑚)) ∧ 𝑛𝑚))
274273ralbii 3127 . . . 4 (∀𝑛 ∈ (1...𝑁)∀𝑚 ∈ (1...𝑁)((((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹𝑀)‘𝑛) ∧ ((𝐹‘(𝑀 − 1))‘𝑚) ≠ ((𝐹𝑀)‘𝑚)) → 𝑛 = 𝑚) ↔ ∀𝑛 ∈ (1...𝑁) ¬ ∃𝑚 ∈ (1...𝑁)((((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹𝑀)‘𝑛) ∧ ((𝐹‘(𝑀 − 1))‘𝑚) ≠ ((𝐹𝑀)‘𝑚)) ∧ 𝑛𝑚))
275266, 274bitri 266 . . 3 (∃*𝑛 ∈ (1...𝑁)((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹𝑀)‘𝑛) ↔ ∀𝑛 ∈ (1...𝑁) ¬ ∃𝑚 ∈ (1...𝑁)((((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹𝑀)‘𝑛) ∧ ((𝐹‘(𝑀 − 1))‘𝑚) ≠ ((𝐹𝑀)‘𝑚)) ∧ 𝑛𝑚))
276262, 275xchbinxr 326 . 2 (∃𝑛 ∈ (1...𝑁)∃𝑚 ∈ (1...𝑁)((((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹𝑀)‘𝑛) ∧ ((𝐹‘(𝑀 − 1))‘𝑚) ≠ ((𝐹𝑀)‘𝑚)) ∧ 𝑛𝑚) ↔ ¬ ∃*𝑛 ∈ (1...𝑁)((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹𝑀)‘𝑛))
277261, 276sylib 209 1 (𝜑 → ¬ ∃*𝑛 ∈ (1...𝑁)((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹𝑀)‘𝑛))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384   = wceq 1652  wcel 2155  wne 2937  wral 3055  wrex 3056  ∃*wrmo 3058  Vcvv 3350  csb 3693  cun 3732  cin 3733  wss 3734  c0 4081  ifcif 4245  {csn 4336  {cpr 4338   class class class wbr 4811  cmpt 4890   × cxp 5277  ccnv 5278  ran crn 5280  cima 5282  Fun wfun 6064   Fn wfn 6065  wf 6066  1-1wf1 6067  ontowfo 6068  1-1-ontowf1o 6069  cfv 6070  (class class class)co 6846  𝑓 cof 7097  cc 10191  cr 10192  0cc0 10193  1c1 10194   + caddc 10196   < clt 10332  cmin 10524  cn 11278  cz 11628  cuz 11891  ...cfz 12538
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4932  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7151  ax-cnex 10249  ax-resscn 10250  ax-1cn 10251  ax-icn 10252  ax-addcl 10253  ax-addrcl 10254  ax-mulcl 10255  ax-mulrcl 10256  ax-mulcom 10257  ax-addass 10258  ax-mulass 10259  ax-distr 10260  ax-i2m1 10261  ax-1ne0 10262  ax-1rid 10263  ax-rnegex 10264  ax-rrecex 10265  ax-cnre 10266  ax-pre-lttri 10267  ax-pre-lttrn 10268  ax-pre-ltadd 10269  ax-pre-mulgt0 10270
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-pss 3750  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-tp 4341  df-op 4343  df-uni 4597  df-iun 4680  df-br 4812  df-opab 4874  df-mpt 4891  df-tr 4914  df-id 5187  df-eprel 5192  df-po 5200  df-so 5201  df-fr 5238  df-we 5240  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-pred 5867  df-ord 5913  df-on 5914  df-lim 5915  df-suc 5916  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-f1 6075  df-fo 6076  df-f1o 6077  df-fv 6078  df-riota 6807  df-ov 6849  df-oprab 6850  df-mpt2 6851  df-of 7099  df-om 7268  df-1st 7370  df-2nd 7371  df-wrecs 7614  df-recs 7676  df-rdg 7714  df-er 7951  df-en 8165  df-dom 8166  df-sdom 8167  df-pnf 10334  df-mnf 10335  df-xr 10336  df-ltxr 10337  df-le 10338  df-sub 10526  df-neg 10527  df-nn 11279  df-n0 11543  df-z 11629  df-uz 11892  df-fz 12539
This theorem is referenced by:  poimirlem8  33862  poimirlem18  33872  poimirlem21  33875  poimirlem22  33876
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