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Theorem wlkdvspthlem 26137
 Description: Lemma for wlkdvspth 26138. (Contributed by Alexander van der Vekens, 27-Oct-2017.)
Assertion
Ref Expression
wlkdvspthlem ((𝐹 ∈ Word dom 𝐸𝑃:(0...(#‘𝐹))–1-1𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}) → Fun 𝐹)
Distinct variable groups:   𝑘,𝐹   𝑘,𝐸   𝑃,𝑘
Allowed substitution hint:   𝑉(𝑘)

Proof of Theorem wlkdvspthlem
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 wrdf 13165 . . . 4 (𝐹 ∈ Word dom 𝐸𝐹:(0..^(#‘𝐹))⟶dom 𝐸)
213ad2ant1 1075 . . 3 ((𝐹 ∈ Word dom 𝐸𝑃:(0...(#‘𝐹))–1-1𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}) → 𝐹:(0..^(#‘𝐹))⟶dom 𝐸)
3 fveq2 6103 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 𝑥 → (𝐹𝑘) = (𝐹𝑥))
43fveq2d 6107 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑥 → (𝐸‘(𝐹𝑘)) = (𝐸‘(𝐹𝑥)))
5 fveq2 6103 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 𝑥 → (𝑃𝑘) = (𝑃𝑥))
6 oveq1 6556 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = 𝑥 → (𝑘 + 1) = (𝑥 + 1))
76fveq2d 6107 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 𝑥 → (𝑃‘(𝑘 + 1)) = (𝑃‘(𝑥 + 1)))
85, 7preq12d 4220 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑥 → {(𝑃𝑘), (𝑃‘(𝑘 + 1))} = {(𝑃𝑥), (𝑃‘(𝑥 + 1))})
94, 8eqeq12d 2625 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑥 → ((𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ↔ (𝐸‘(𝐹𝑥)) = {(𝑃𝑥), (𝑃‘(𝑥 + 1))}))
109rspcva 3280 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ (0..^(#‘𝐹)) ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}) → (𝐸‘(𝐹𝑥)) = {(𝑃𝑥), (𝑃‘(𝑥 + 1))})
11 fveq2 6103 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 = 𝑦 → (𝐹𝑘) = (𝐹𝑦))
1211fveq2d 6107 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 = 𝑦 → (𝐸‘(𝐹𝑘)) = (𝐸‘(𝐹𝑦)))
13 fveq2 6103 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 = 𝑦 → (𝑃𝑘) = (𝑃𝑦))
14 oveq1 6556 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 = 𝑦 → (𝑘 + 1) = (𝑦 + 1))
1514fveq2d 6107 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 = 𝑦 → (𝑃‘(𝑘 + 1)) = (𝑃‘(𝑦 + 1)))
1613, 15preq12d 4220 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 = 𝑦 → {(𝑃𝑘), (𝑃‘(𝑘 + 1))} = {(𝑃𝑦), (𝑃‘(𝑦 + 1))})
1712, 16eqeq12d 2625 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = 𝑦 → ((𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ↔ (𝐸‘(𝐹𝑦)) = {(𝑃𝑦), (𝑃‘(𝑦 + 1))}))
1817rspcva 3280 . . . . . . . . . . . . . . . . . . 19 ((𝑦 ∈ (0..^(#‘𝐹)) ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}) → (𝐸‘(𝐹𝑦)) = {(𝑃𝑦), (𝑃‘(𝑦 + 1))})
19 pm3.2 462 . . . . . . . . . . . . . . . . . . 19 ((𝐸‘(𝐹𝑦)) = {(𝑃𝑦), (𝑃‘(𝑦 + 1))} → ((𝐸‘(𝐹𝑥)) = {(𝑃𝑥), (𝑃‘(𝑥 + 1))} → ((𝐸‘(𝐹𝑦)) = {(𝑃𝑦), (𝑃‘(𝑦 + 1))} ∧ (𝐸‘(𝐹𝑥)) = {(𝑃𝑥), (𝑃‘(𝑥 + 1))})))
2018, 19syl 17 . . . . . . . . . . . . . . . . . 18 ((𝑦 ∈ (0..^(#‘𝐹)) ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}) → ((𝐸‘(𝐹𝑥)) = {(𝑃𝑥), (𝑃‘(𝑥 + 1))} → ((𝐸‘(𝐹𝑦)) = {(𝑃𝑦), (𝑃‘(𝑦 + 1))} ∧ (𝐸‘(𝐹𝑥)) = {(𝑃𝑥), (𝑃‘(𝑥 + 1))})))
2120ex 449 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ (0..^(#‘𝐹)) → (∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))} → ((𝐸‘(𝐹𝑥)) = {(𝑃𝑥), (𝑃‘(𝑥 + 1))} → ((𝐸‘(𝐹𝑦)) = {(𝑃𝑦), (𝑃‘(𝑦 + 1))} ∧ (𝐸‘(𝐹𝑥)) = {(𝑃𝑥), (𝑃‘(𝑥 + 1))}))))
2221com3r 85 . . . . . . . . . . . . . . . 16 ((𝐸‘(𝐹𝑥)) = {(𝑃𝑥), (𝑃‘(𝑥 + 1))} → (𝑦 ∈ (0..^(#‘𝐹)) → (∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))} → ((𝐸‘(𝐹𝑦)) = {(𝑃𝑦), (𝑃‘(𝑦 + 1))} ∧ (𝐸‘(𝐹𝑥)) = {(𝑃𝑥), (𝑃‘(𝑥 + 1))}))))
2310, 22syl 17 . . . . . . . . . . . . . . 15 ((𝑥 ∈ (0..^(#‘𝐹)) ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}) → (𝑦 ∈ (0..^(#‘𝐹)) → (∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))} → ((𝐸‘(𝐹𝑦)) = {(𝑃𝑦), (𝑃‘(𝑦 + 1))} ∧ (𝐸‘(𝐹𝑥)) = {(𝑃𝑥), (𝑃‘(𝑥 + 1))}))))
2423impancom 455 . . . . . . . . . . . . . 14 ((𝑥 ∈ (0..^(#‘𝐹)) ∧ 𝑦 ∈ (0..^(#‘𝐹))) → (∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))} → (∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))} → ((𝐸‘(𝐹𝑦)) = {(𝑃𝑦), (𝑃‘(𝑦 + 1))} ∧ (𝐸‘(𝐹𝑥)) = {(𝑃𝑥), (𝑃‘(𝑥 + 1))}))))
2524com3r 85 . . . . . . . . . . . . 13 (∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))} → ((𝑥 ∈ (0..^(#‘𝐹)) ∧ 𝑦 ∈ (0..^(#‘𝐹))) → (∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))} → ((𝐸‘(𝐹𝑦)) = {(𝑃𝑦), (𝑃‘(𝑦 + 1))} ∧ (𝐸‘(𝐹𝑥)) = {(𝑃𝑥), (𝑃‘(𝑥 + 1))}))))
2625pm2.43a 52 . . . . . . . . . . . 12 (∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))} → ((𝑥 ∈ (0..^(#‘𝐹)) ∧ 𝑦 ∈ (0..^(#‘𝐹))) → ((𝐸‘(𝐹𝑦)) = {(𝑃𝑦), (𝑃‘(𝑦 + 1))} ∧ (𝐸‘(𝐹𝑥)) = {(𝑃𝑥), (𝑃‘(𝑥 + 1))})))
2726impcom 445 . . . . . . . . . . 11 (((𝑥 ∈ (0..^(#‘𝐹)) ∧ 𝑦 ∈ (0..^(#‘𝐹))) ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}) → ((𝐸‘(𝐹𝑦)) = {(𝑃𝑦), (𝑃‘(𝑦 + 1))} ∧ (𝐸‘(𝐹𝑥)) = {(𝑃𝑥), (𝑃‘(𝑥 + 1))}))
28 fveq2 6103 . . . . . . . . . . . . . 14 ((𝐹𝑥) = (𝐹𝑦) → (𝐸‘(𝐹𝑥)) = (𝐸‘(𝐹𝑦)))
29 eqtr2 2630 . . . . . . . . . . . . . . . . . . . 20 (((𝐸‘(𝐹𝑦)) = (𝐸‘(𝐹𝑥)) ∧ (𝐸‘(𝐹𝑦)) = {(𝑃𝑦), (𝑃‘(𝑦 + 1))}) → (𝐸‘(𝐹𝑥)) = {(𝑃𝑦), (𝑃‘(𝑦 + 1))})
3029ex 449 . . . . . . . . . . . . . . . . . . 19 ((𝐸‘(𝐹𝑦)) = (𝐸‘(𝐹𝑥)) → ((𝐸‘(𝐹𝑦)) = {(𝑃𝑦), (𝑃‘(𝑦 + 1))} → (𝐸‘(𝐹𝑥)) = {(𝑃𝑦), (𝑃‘(𝑦 + 1))}))
3130eqcoms 2618 . . . . . . . . . . . . . . . . . 18 ((𝐸‘(𝐹𝑥)) = (𝐸‘(𝐹𝑦)) → ((𝐸‘(𝐹𝑦)) = {(𝑃𝑦), (𝑃‘(𝑦 + 1))} → (𝐸‘(𝐹𝑥)) = {(𝑃𝑦), (𝑃‘(𝑦 + 1))}))
32 eqtr2 2630 . . . . . . . . . . . . . . . . . . 19 (((𝐸‘(𝐹𝑥)) = {(𝑃𝑦), (𝑃‘(𝑦 + 1))} ∧ (𝐸‘(𝐹𝑥)) = {(𝑃𝑥), (𝑃‘(𝑥 + 1))}) → {(𝑃𝑦), (𝑃‘(𝑦 + 1))} = {(𝑃𝑥), (𝑃‘(𝑥 + 1))})
3332ex 449 . . . . . . . . . . . . . . . . . 18 ((𝐸‘(𝐹𝑥)) = {(𝑃𝑦), (𝑃‘(𝑦 + 1))} → ((𝐸‘(𝐹𝑥)) = {(𝑃𝑥), (𝑃‘(𝑥 + 1))} → {(𝑃𝑦), (𝑃‘(𝑦 + 1))} = {(𝑃𝑥), (𝑃‘(𝑥 + 1))}))
3431, 33syl6com 36 . . . . . . . . . . . . . . . . 17 ((𝐸‘(𝐹𝑦)) = {(𝑃𝑦), (𝑃‘(𝑦 + 1))} → ((𝐸‘(𝐹𝑥)) = (𝐸‘(𝐹𝑦)) → ((𝐸‘(𝐹𝑥)) = {(𝑃𝑥), (𝑃‘(𝑥 + 1))} → {(𝑃𝑦), (𝑃‘(𝑦 + 1))} = {(𝑃𝑥), (𝑃‘(𝑥 + 1))})))
3534com23 84 . . . . . . . . . . . . . . . 16 ((𝐸‘(𝐹𝑦)) = {(𝑃𝑦), (𝑃‘(𝑦 + 1))} → ((𝐸‘(𝐹𝑥)) = {(𝑃𝑥), (𝑃‘(𝑥 + 1))} → ((𝐸‘(𝐹𝑥)) = (𝐸‘(𝐹𝑦)) → {(𝑃𝑦), (𝑃‘(𝑦 + 1))} = {(𝑃𝑥), (𝑃‘(𝑥 + 1))})))
3635imp 444 . . . . . . . . . . . . . . 15 (((𝐸‘(𝐹𝑦)) = {(𝑃𝑦), (𝑃‘(𝑦 + 1))} ∧ (𝐸‘(𝐹𝑥)) = {(𝑃𝑥), (𝑃‘(𝑥 + 1))}) → ((𝐸‘(𝐹𝑥)) = (𝐸‘(𝐹𝑦)) → {(𝑃𝑦), (𝑃‘(𝑦 + 1))} = {(𝑃𝑥), (𝑃‘(𝑥 + 1))}))
37 elfzofz 12354 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑥 ∈ (0..^(#‘𝐹)) → 𝑥 ∈ (0...(#‘𝐹)))
38 elfzofz 12354 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑦 ∈ (0..^(#‘𝐹)) → 𝑦 ∈ (0...(#‘𝐹)))
3937, 38anim12i 588 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑥 ∈ (0..^(#‘𝐹)) ∧ 𝑦 ∈ (0..^(#‘𝐹))) → (𝑥 ∈ (0...(#‘𝐹)) ∧ 𝑦 ∈ (0...(#‘𝐹))))
4039anim2i 591 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑃:(0...(#‘𝐹))–1-1𝑉 ∧ (𝑥 ∈ (0..^(#‘𝐹)) ∧ 𝑦 ∈ (0..^(#‘𝐹)))) → (𝑃:(0...(#‘𝐹))–1-1𝑉 ∧ (𝑥 ∈ (0...(#‘𝐹)) ∧ 𝑦 ∈ (0...(#‘𝐹)))))
4140ancoms 468 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑥 ∈ (0..^(#‘𝐹)) ∧ 𝑦 ∈ (0..^(#‘𝐹))) ∧ 𝑃:(0...(#‘𝐹))–1-1𝑉) → (𝑃:(0...(#‘𝐹))–1-1𝑉 ∧ (𝑥 ∈ (0...(#‘𝐹)) ∧ 𝑦 ∈ (0...(#‘𝐹)))))
42 f1fveq 6420 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑃:(0...(#‘𝐹))–1-1𝑉 ∧ (𝑥 ∈ (0...(#‘𝐹)) ∧ 𝑦 ∈ (0...(#‘𝐹)))) → ((𝑃𝑥) = (𝑃𝑦) ↔ 𝑥 = 𝑦))
4341, 42syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑥 ∈ (0..^(#‘𝐹)) ∧ 𝑦 ∈ (0..^(#‘𝐹))) ∧ 𝑃:(0...(#‘𝐹))–1-1𝑉) → ((𝑃𝑥) = (𝑃𝑦) ↔ 𝑥 = 𝑦))
4443notbid 307 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑥 ∈ (0..^(#‘𝐹)) ∧ 𝑦 ∈ (0..^(#‘𝐹))) ∧ 𝑃:(0...(#‘𝐹))–1-1𝑉) → (¬ (𝑃𝑥) = (𝑃𝑦) ↔ ¬ 𝑥 = 𝑦))
4544biimparc 503 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((¬ 𝑥 = 𝑦 ∧ ((𝑥 ∈ (0..^(#‘𝐹)) ∧ 𝑦 ∈ (0..^(#‘𝐹))) ∧ 𝑃:(0...(#‘𝐹))–1-1𝑉)) → ¬ (𝑃𝑥) = (𝑃𝑦))
46 simpr 476 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝑥 ∈ (0..^(#‘𝐹)) ∧ 𝑦 ∈ (0..^(#‘𝐹))) ∧ 𝑃:(0...(#‘𝐹))–1-1𝑉) → 𝑃:(0...(#‘𝐹))–1-1𝑉)
47 fzofzp1 12431 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑥 ∈ (0..^(#‘𝐹)) → (𝑥 + 1) ∈ (0...(#‘𝐹)))
48 fzofzp1 12431 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑦 ∈ (0..^(#‘𝐹)) → (𝑦 + 1) ∈ (0...(#‘𝐹)))
4947, 48anim12i 588 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑥 ∈ (0..^(#‘𝐹)) ∧ 𝑦 ∈ (0..^(#‘𝐹))) → ((𝑥 + 1) ∈ (0...(#‘𝐹)) ∧ (𝑦 + 1) ∈ (0...(#‘𝐹))))
5049adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝑥 ∈ (0..^(#‘𝐹)) ∧ 𝑦 ∈ (0..^(#‘𝐹))) ∧ 𝑃:(0...(#‘𝐹))–1-1𝑉) → ((𝑥 + 1) ∈ (0...(#‘𝐹)) ∧ (𝑦 + 1) ∈ (0...(#‘𝐹))))
51 f1fveq 6420 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑃:(0...(#‘𝐹))–1-1𝑉 ∧ ((𝑥 + 1) ∈ (0...(#‘𝐹)) ∧ (𝑦 + 1) ∈ (0...(#‘𝐹)))) → ((𝑃‘(𝑥 + 1)) = (𝑃‘(𝑦 + 1)) ↔ (𝑥 + 1) = (𝑦 + 1)))
5246, 50, 51syl2anc 691 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝑥 ∈ (0..^(#‘𝐹)) ∧ 𝑦 ∈ (0..^(#‘𝐹))) ∧ 𝑃:(0...(#‘𝐹))–1-1𝑉) → ((𝑃‘(𝑥 + 1)) = (𝑃‘(𝑦 + 1)) ↔ (𝑥 + 1) = (𝑦 + 1)))
53 elfzoelz 12339 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑥 ∈ (0..^(#‘𝐹)) → 𝑥 ∈ ℤ)
5453zcnd 11359 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑥 ∈ (0..^(#‘𝐹)) → 𝑥 ∈ ℂ)
5554ad2antrr 758 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝑥 ∈ (0..^(#‘𝐹)) ∧ 𝑦 ∈ (0..^(#‘𝐹))) ∧ 𝑃:(0...(#‘𝐹))–1-1𝑉) → 𝑥 ∈ ℂ)
56 elfzoelz 12339 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑦 ∈ (0..^(#‘𝐹)) → 𝑦 ∈ ℤ)
5756zcnd 11359 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑦 ∈ (0..^(#‘𝐹)) → 𝑦 ∈ ℂ)
5857adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑥 ∈ (0..^(#‘𝐹)) ∧ 𝑦 ∈ (0..^(#‘𝐹))) → 𝑦 ∈ ℂ)
5958adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝑥 ∈ (0..^(#‘𝐹)) ∧ 𝑦 ∈ (0..^(#‘𝐹))) ∧ 𝑃:(0...(#‘𝐹))–1-1𝑉) → 𝑦 ∈ ℂ)
60 1cnd 9935 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝑥 ∈ (0..^(#‘𝐹)) ∧ 𝑦 ∈ (0..^(#‘𝐹))) ∧ 𝑃:(0...(#‘𝐹))–1-1𝑉) → 1 ∈ ℂ)
6155, 59, 60addcan2d 10119 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝑥 ∈ (0..^(#‘𝐹)) ∧ 𝑦 ∈ (0..^(#‘𝐹))) ∧ 𝑃:(0...(#‘𝐹))–1-1𝑉) → ((𝑥 + 1) = (𝑦 + 1) ↔ 𝑥 = 𝑦))
6252, 61bitrd 267 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑥 ∈ (0..^(#‘𝐹)) ∧ 𝑦 ∈ (0..^(#‘𝐹))) ∧ 𝑃:(0...(#‘𝐹))–1-1𝑉) → ((𝑃‘(𝑥 + 1)) = (𝑃‘(𝑦 + 1)) ↔ 𝑥 = 𝑦))
6362notbid 307 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑥 ∈ (0..^(#‘𝐹)) ∧ 𝑦 ∈ (0..^(#‘𝐹))) ∧ 𝑃:(0...(#‘𝐹))–1-1𝑉) → (¬ (𝑃‘(𝑥 + 1)) = (𝑃‘(𝑦 + 1)) ↔ ¬ 𝑥 = 𝑦))
64 pm3.2 462 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (¬ (𝑃‘(𝑥 + 1)) = (𝑃‘(𝑦 + 1)) → (¬ (𝑃𝑥) = (𝑃𝑦) → (¬ (𝑃‘(𝑥 + 1)) = (𝑃‘(𝑦 + 1)) ∧ ¬ (𝑃𝑥) = (𝑃𝑦))))
6563, 64syl6bir 243 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑥 ∈ (0..^(#‘𝐹)) ∧ 𝑦 ∈ (0..^(#‘𝐹))) ∧ 𝑃:(0...(#‘𝐹))–1-1𝑉) → (¬ 𝑥 = 𝑦 → (¬ (𝑃𝑥) = (𝑃𝑦) → (¬ (𝑃‘(𝑥 + 1)) = (𝑃‘(𝑦 + 1)) ∧ ¬ (𝑃𝑥) = (𝑃𝑦)))))
6665com13 86 . . . . . . . . . . . . . . . . . . . . . . . . 25 (¬ (𝑃𝑥) = (𝑃𝑦) → (¬ 𝑥 = 𝑦 → (((𝑥 ∈ (0..^(#‘𝐹)) ∧ 𝑦 ∈ (0..^(#‘𝐹))) ∧ 𝑃:(0...(#‘𝐹))–1-1𝑉) → (¬ (𝑃‘(𝑥 + 1)) = (𝑃‘(𝑦 + 1)) ∧ ¬ (𝑃𝑥) = (𝑃𝑦)))))
6745, 66syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 ((¬ 𝑥 = 𝑦 ∧ ((𝑥 ∈ (0..^(#‘𝐹)) ∧ 𝑦 ∈ (0..^(#‘𝐹))) ∧ 𝑃:(0...(#‘𝐹))–1-1𝑉)) → (¬ 𝑥 = 𝑦 → (((𝑥 ∈ (0..^(#‘𝐹)) ∧ 𝑦 ∈ (0..^(#‘𝐹))) ∧ 𝑃:(0...(#‘𝐹))–1-1𝑉) → (¬ (𝑃‘(𝑥 + 1)) = (𝑃‘(𝑦 + 1)) ∧ ¬ (𝑃𝑥) = (𝑃𝑦)))))
68 fvex 6113 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑃𝑦) ∈ V
69 fvex 6113 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑃‘(𝑦 + 1)) ∈ V
70 fvex 6113 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑃𝑥) ∈ V
71 fvex 6113 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑃‘(𝑥 + 1)) ∈ V
7268, 69, 70, 71preq12b 4322 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ({(𝑃𝑦), (𝑃‘(𝑦 + 1))} = {(𝑃𝑥), (𝑃‘(𝑥 + 1))} ↔ (((𝑃𝑦) = (𝑃𝑥) ∧ (𝑃‘(𝑦 + 1)) = (𝑃‘(𝑥 + 1))) ∨ ((𝑃𝑦) = (𝑃‘(𝑥 + 1)) ∧ (𝑃‘(𝑦 + 1)) = (𝑃𝑥))))
73 pm2.24 120 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑃‘(𝑥 + 1)) = (𝑃‘(𝑦 + 1)) → (¬ (𝑃‘(𝑥 + 1)) = (𝑃‘(𝑦 + 1)) → (𝑃𝑦) = (𝑃‘(𝑥 + 1))))
7473eqcoms 2618 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑃‘(𝑦 + 1)) = (𝑃‘(𝑥 + 1)) → (¬ (𝑃‘(𝑥 + 1)) = (𝑃‘(𝑦 + 1)) → (𝑃𝑦) = (𝑃‘(𝑥 + 1))))
75 pm2.24 120 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑃𝑥) = (𝑃𝑦) → (¬ (𝑃𝑥) = (𝑃𝑦) → (𝑃‘(𝑦 + 1)) = (𝑃𝑥)))
7675eqcoms 2618 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑃𝑦) = (𝑃𝑥) → (¬ (𝑃𝑥) = (𝑃𝑦) → (𝑃‘(𝑦 + 1)) = (𝑃𝑥)))
7774, 76im2anan9r 877 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑃𝑦) = (𝑃𝑥) ∧ (𝑃‘(𝑦 + 1)) = (𝑃‘(𝑥 + 1))) → ((¬ (𝑃‘(𝑥 + 1)) = (𝑃‘(𝑦 + 1)) ∧ ¬ (𝑃𝑥) = (𝑃𝑦)) → ((𝑃𝑦) = (𝑃‘(𝑥 + 1)) ∧ (𝑃‘(𝑦 + 1)) = (𝑃𝑥))))
78 ax-1 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑃𝑦) = (𝑃‘(𝑥 + 1)) ∧ (𝑃‘(𝑦 + 1)) = (𝑃𝑥)) → ((¬ (𝑃‘(𝑥 + 1)) = (𝑃‘(𝑦 + 1)) ∧ ¬ (𝑃𝑥) = (𝑃𝑦)) → ((𝑃𝑦) = (𝑃‘(𝑥 + 1)) ∧ (𝑃‘(𝑦 + 1)) = (𝑃𝑥))))
7977, 78jaoi 393 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((𝑃𝑦) = (𝑃𝑥) ∧ (𝑃‘(𝑦 + 1)) = (𝑃‘(𝑥 + 1))) ∨ ((𝑃𝑦) = (𝑃‘(𝑥 + 1)) ∧ (𝑃‘(𝑦 + 1)) = (𝑃𝑥))) → ((¬ (𝑃‘(𝑥 + 1)) = (𝑃‘(𝑦 + 1)) ∧ ¬ (𝑃𝑥) = (𝑃𝑦)) → ((𝑃𝑦) = (𝑃‘(𝑥 + 1)) ∧ (𝑃‘(𝑦 + 1)) = (𝑃𝑥))))
8072, 79sylbi 206 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ({(𝑃𝑦), (𝑃‘(𝑦 + 1))} = {(𝑃𝑥), (𝑃‘(𝑥 + 1))} → ((¬ (𝑃‘(𝑥 + 1)) = (𝑃‘(𝑦 + 1)) ∧ ¬ (𝑃𝑥) = (𝑃𝑦)) → ((𝑃𝑦) = (𝑃‘(𝑥 + 1)) ∧ (𝑃‘(𝑦 + 1)) = (𝑃𝑥))))
8137, 48anim12ci 589 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑥 ∈ (0..^(#‘𝐹)) ∧ 𝑦 ∈ (0..^(#‘𝐹))) → ((𝑦 + 1) ∈ (0...(#‘𝐹)) ∧ 𝑥 ∈ (0...(#‘𝐹))))
82 f1fveq 6420 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑃:(0...(#‘𝐹))–1-1𝑉 ∧ ((𝑦 + 1) ∈ (0...(#‘𝐹)) ∧ 𝑥 ∈ (0...(#‘𝐹)))) → ((𝑃‘(𝑦 + 1)) = (𝑃𝑥) ↔ (𝑦 + 1) = 𝑥))
8381, 82sylan2 490 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑃:(0...(#‘𝐹))–1-1𝑉 ∧ (𝑥 ∈ (0..^(#‘𝐹)) ∧ 𝑦 ∈ (0..^(#‘𝐹)))) → ((𝑃‘(𝑦 + 1)) = (𝑃𝑥) ↔ (𝑦 + 1) = 𝑥))
8483biimpd 218 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑃:(0...(#‘𝐹))–1-1𝑉 ∧ (𝑥 ∈ (0..^(#‘𝐹)) ∧ 𝑦 ∈ (0..^(#‘𝐹)))) → ((𝑃‘(𝑦 + 1)) = (𝑃𝑥) → (𝑦 + 1) = 𝑥))
8584ancoms 468 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝑥 ∈ (0..^(#‘𝐹)) ∧ 𝑦 ∈ (0..^(#‘𝐹))) ∧ 𝑃:(0...(#‘𝐹))–1-1𝑉) → ((𝑃‘(𝑦 + 1)) = (𝑃𝑥) → (𝑦 + 1) = 𝑥))
8647, 38anim12ci 589 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝑥 ∈ (0..^(#‘𝐹)) ∧ 𝑦 ∈ (0..^(#‘𝐹))) → (𝑦 ∈ (0...(#‘𝐹)) ∧ (𝑥 + 1) ∈ (0...(#‘𝐹))))
87 f1fveq 6420 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝑃:(0...(#‘𝐹))–1-1𝑉 ∧ (𝑦 ∈ (0...(#‘𝐹)) ∧ (𝑥 + 1) ∈ (0...(#‘𝐹)))) → ((𝑃𝑦) = (𝑃‘(𝑥 + 1)) ↔ 𝑦 = (𝑥 + 1)))
8886, 87sylan2 490 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝑃:(0...(#‘𝐹))–1-1𝑉 ∧ (𝑥 ∈ (0..^(#‘𝐹)) ∧ 𝑦 ∈ (0..^(#‘𝐹)))) → ((𝑃𝑦) = (𝑃‘(𝑥 + 1)) ↔ 𝑦 = (𝑥 + 1)))
8988ancoms 468 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((𝑥 ∈ (0..^(#‘𝐹)) ∧ 𝑦 ∈ (0..^(#‘𝐹))) ∧ 𝑃:(0...(#‘𝐹))–1-1𝑉) → ((𝑃𝑦) = (𝑃‘(𝑥 + 1)) ↔ 𝑦 = (𝑥 + 1)))
9089biimpa 500 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((((𝑥 ∈ (0..^(#‘𝐹)) ∧ 𝑦 ∈ (0..^(#‘𝐹))) ∧ 𝑃:(0...(#‘𝐹))–1-1𝑉) ∧ (𝑃𝑦) = (𝑃‘(𝑥 + 1))) → 𝑦 = (𝑥 + 1))
91 oveq1 6556 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (𝑦 = (𝑥 + 1) → (𝑦 + 1) = ((𝑥 + 1) + 1))
9291eqeq1d 2612 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (𝑦 = (𝑥 + 1) → ((𝑦 + 1) = 𝑥 ↔ ((𝑥 + 1) + 1) = 𝑥))
9392adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (((𝑥 ∈ (0..^(#‘𝐹)) ∧ 𝑦 ∈ (0..^(#‘𝐹))) ∧ 𝑦 = (𝑥 + 1)) → ((𝑦 + 1) = 𝑥 ↔ ((𝑥 + 1) + 1) = 𝑥))
94 1cnd 9935 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (𝑥 ∈ (0..^(#‘𝐹)) → 1 ∈ ℂ)
9554, 94, 943jca 1235 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (𝑥 ∈ (0..^(#‘𝐹)) → (𝑥 ∈ ℂ ∧ 1 ∈ ℂ ∧ 1 ∈ ℂ))
9695ad2antrr 758 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (((𝑥 ∈ (0..^(#‘𝐹)) ∧ 𝑦 ∈ (0..^(#‘𝐹))) ∧ 𝑦 = (𝑥 + 1)) → (𝑥 ∈ ℂ ∧ 1 ∈ ℂ ∧ 1 ∈ ℂ))
97 addass 9902 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((𝑥 ∈ ℂ ∧ 1 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑥 + 1) + 1) = (𝑥 + (1 + 1)))
9897eqeq1d 2612 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((𝑥 ∈ ℂ ∧ 1 ∈ ℂ ∧ 1 ∈ ℂ) → (((𝑥 + 1) + 1) = 𝑥 ↔ (𝑥 + (1 + 1)) = 𝑥))
9996, 98syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (((𝑥 ∈ (0..^(#‘𝐹)) ∧ 𝑦 ∈ (0..^(#‘𝐹))) ∧ 𝑦 = (𝑥 + 1)) → (((𝑥 + 1) + 1) = 𝑥 ↔ (𝑥 + (1 + 1)) = 𝑥))
100 1p1e2 11011 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 (1 + 1) = 2
101100oveq2i 6560 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (𝑥 + (1 + 1)) = (𝑥 + 2)
102101eqeq1i 2615 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((𝑥 + (1 + 1)) = 𝑥 ↔ (𝑥 + 2) = 𝑥)
103 zcn 11259 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 (𝑥 ∈ ℤ → 𝑥 ∈ ℂ)
104 2cn 10968 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 2 ∈ ℂ
105103, 104jctir 559 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 (𝑥 ∈ ℤ → (𝑥 ∈ ℂ ∧ 2 ∈ ℂ))
106 addcl 9897 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ((𝑥 ∈ ℂ ∧ 2 ∈ ℂ) → (𝑥 + 2) ∈ ℂ)
107105, 106syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 (𝑥 ∈ ℤ → (𝑥 + 2) ∈ ℂ)
108107, 103, 1033jca 1235 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 (𝑥 ∈ ℤ → ((𝑥 + 2) ∈ ℂ ∧ 𝑥 ∈ ℂ ∧ 𝑥 ∈ ℂ))
109 subcan2 10185 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 (((𝑥 + 2) ∈ ℂ ∧ 𝑥 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (((𝑥 + 2) − 𝑥) = (𝑥𝑥) ↔ (𝑥 + 2) = 𝑥))
110109bicomd 212 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 (((𝑥 + 2) ∈ ℂ ∧ 𝑥 ∈ ℂ ∧ 𝑥 ∈ ℂ) → ((𝑥 + 2) = 𝑥 ↔ ((𝑥 + 2) − 𝑥) = (𝑥𝑥)))
11153, 108, 1103syl 18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (𝑥 ∈ (0..^(#‘𝐹)) → ((𝑥 + 2) = 𝑥 ↔ ((𝑥 + 2) − 𝑥) = (𝑥𝑥)))
112 pncan2 10167 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ((𝑥 ∈ ℂ ∧ 2 ∈ ℂ) → ((𝑥 + 2) − 𝑥) = 2)
11353, 105, 1123syl 18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 (𝑥 ∈ (0..^(#‘𝐹)) → ((𝑥 + 2) − 𝑥) = 2)
11454subidd 10259 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 (𝑥 ∈ (0..^(#‘𝐹)) → (𝑥𝑥) = 0)
115113, 114eqeq12d 2625 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (𝑥 ∈ (0..^(#‘𝐹)) → (((𝑥 + 2) − 𝑥) = (𝑥𝑥) ↔ 2 = 0))
116111, 115bitrd 267 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (𝑥 ∈ (0..^(#‘𝐹)) → ((𝑥 + 2) = 𝑥 ↔ 2 = 0))
117102, 116syl5bb 271 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (𝑥 ∈ (0..^(#‘𝐹)) → ((𝑥 + (1 + 1)) = 𝑥 ↔ 2 = 0))
118117ad2antrr 758 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (((𝑥 ∈ (0..^(#‘𝐹)) ∧ 𝑦 ∈ (0..^(#‘𝐹))) ∧ 𝑦 = (𝑥 + 1)) → ((𝑥 + (1 + 1)) = 𝑥 ↔ 2 = 0))
11993, 99, 1183bitrd 293 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (((𝑥 ∈ (0..^(#‘𝐹)) ∧ 𝑦 ∈ (0..^(#‘𝐹))) ∧ 𝑦 = (𝑥 + 1)) → ((𝑦 + 1) = 𝑥 ↔ 2 = 0))
120 2ne0 10990 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2 ≠ 0
121 df-ne 2782 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (2 ≠ 0 ↔ ¬ 2 = 0)
122 pm2.21 119 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (¬ 2 = 0 → (2 = 0 → 𝑥 = 𝑦))
123121, 122sylbi 206 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (2 ≠ 0 → (2 = 0 → 𝑥 = 𝑦))
124120, 123ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (2 = 0 → 𝑥 = 𝑦)
125119, 124syl6bi 242 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (((𝑥 ∈ (0..^(#‘𝐹)) ∧ 𝑦 ∈ (0..^(#‘𝐹))) ∧ 𝑦 = (𝑥 + 1)) → ((𝑦 + 1) = 𝑥𝑥 = 𝑦))
126125ex 449 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑥 ∈ (0..^(#‘𝐹)) ∧ 𝑦 ∈ (0..^(#‘𝐹))) → (𝑦 = (𝑥 + 1) → ((𝑦 + 1) = 𝑥𝑥 = 𝑦)))
127126ad2antrr 758 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((((𝑥 ∈ (0..^(#‘𝐹)) ∧ 𝑦 ∈ (0..^(#‘𝐹))) ∧ 𝑃:(0...(#‘𝐹))–1-1𝑉) ∧ (𝑃𝑦) = (𝑃‘(𝑥 + 1))) → (𝑦 = (𝑥 + 1) → ((𝑦 + 1) = 𝑥𝑥 = 𝑦)))
12890, 127mpd 15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((𝑥 ∈ (0..^(#‘𝐹)) ∧ 𝑦 ∈ (0..^(#‘𝐹))) ∧ 𝑃:(0...(#‘𝐹))–1-1𝑉) ∧ (𝑃𝑦) = (𝑃‘(𝑥 + 1))) → ((𝑦 + 1) = 𝑥𝑥 = 𝑦))
129128expcom 450 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑃𝑦) = (𝑃‘(𝑥 + 1)) → (((𝑥 ∈ (0..^(#‘𝐹)) ∧ 𝑦 ∈ (0..^(#‘𝐹))) ∧ 𝑃:(0...(#‘𝐹))–1-1𝑉) → ((𝑦 + 1) = 𝑥𝑥 = 𝑦)))
130129com13 86 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑦 + 1) = 𝑥 → (((𝑥 ∈ (0..^(#‘𝐹)) ∧ 𝑦 ∈ (0..^(#‘𝐹))) ∧ 𝑃:(0...(#‘𝐹))–1-1𝑉) → ((𝑃𝑦) = (𝑃‘(𝑥 + 1)) → 𝑥 = 𝑦)))
13185, 130syl6 34 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝑥 ∈ (0..^(#‘𝐹)) ∧ 𝑦 ∈ (0..^(#‘𝐹))) ∧ 𝑃:(0...(#‘𝐹))–1-1𝑉) → ((𝑃‘(𝑦 + 1)) = (𝑃𝑥) → (((𝑥 ∈ (0..^(#‘𝐹)) ∧ 𝑦 ∈ (0..^(#‘𝐹))) ∧ 𝑃:(0...(#‘𝐹))–1-1𝑉) → ((𝑃𝑦) = (𝑃‘(𝑥 + 1)) → 𝑥 = 𝑦))))
132131pm2.43a 52 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑥 ∈ (0..^(#‘𝐹)) ∧ 𝑦 ∈ (0..^(#‘𝐹))) ∧ 𝑃:(0...(#‘𝐹))–1-1𝑉) → ((𝑃‘(𝑦 + 1)) = (𝑃𝑥) → ((𝑃𝑦) = (𝑃‘(𝑥 + 1)) → 𝑥 = 𝑦)))
133132com13 86 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑃𝑦) = (𝑃‘(𝑥 + 1)) → ((𝑃‘(𝑦 + 1)) = (𝑃𝑥) → (((𝑥 ∈ (0..^(#‘𝐹)) ∧ 𝑦 ∈ (0..^(#‘𝐹))) ∧ 𝑃:(0...(#‘𝐹))–1-1𝑉) → 𝑥 = 𝑦)))
134133imp 444 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑃𝑦) = (𝑃‘(𝑥 + 1)) ∧ (𝑃‘(𝑦 + 1)) = (𝑃𝑥)) → (((𝑥 ∈ (0..^(#‘𝐹)) ∧ 𝑦 ∈ (0..^(#‘𝐹))) ∧ 𝑃:(0...(#‘𝐹))–1-1𝑉) → 𝑥 = 𝑦))
13580, 134syl6com 36 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((¬ (𝑃‘(𝑥 + 1)) = (𝑃‘(𝑦 + 1)) ∧ ¬ (𝑃𝑥) = (𝑃𝑦)) → ({(𝑃𝑦), (𝑃‘(𝑦 + 1))} = {(𝑃𝑥), (𝑃‘(𝑥 + 1))} → (((𝑥 ∈ (0..^(#‘𝐹)) ∧ 𝑦 ∈ (0..^(#‘𝐹))) ∧ 𝑃:(0...(#‘𝐹))–1-1𝑉) → 𝑥 = 𝑦)))
136135com23 84 . . . . . . . . . . . . . . . . . . . . . . . 24 ((¬ (𝑃‘(𝑥 + 1)) = (𝑃‘(𝑦 + 1)) ∧ ¬ (𝑃𝑥) = (𝑃𝑦)) → (((𝑥 ∈ (0..^(#‘𝐹)) ∧ 𝑦 ∈ (0..^(#‘𝐹))) ∧ 𝑃:(0...(#‘𝐹))–1-1𝑉) → ({(𝑃𝑦), (𝑃‘(𝑦 + 1))} = {(𝑃𝑥), (𝑃‘(𝑥 + 1))} → 𝑥 = 𝑦)))
13767, 136syl8 74 . . . . . . . . . . . . . . . . . . . . . . 23 ((¬ 𝑥 = 𝑦 ∧ ((𝑥 ∈ (0..^(#‘𝐹)) ∧ 𝑦 ∈ (0..^(#‘𝐹))) ∧ 𝑃:(0...(#‘𝐹))–1-1𝑉)) → (¬ 𝑥 = 𝑦 → (((𝑥 ∈ (0..^(#‘𝐹)) ∧ 𝑦 ∈ (0..^(#‘𝐹))) ∧ 𝑃:(0...(#‘𝐹))–1-1𝑉) → (((𝑥 ∈ (0..^(#‘𝐹)) ∧ 𝑦 ∈ (0..^(#‘𝐹))) ∧ 𝑃:(0...(#‘𝐹))–1-1𝑉) → ({(𝑃𝑦), (𝑃‘(𝑦 + 1))} = {(𝑃𝑥), (𝑃‘(𝑥 + 1))} → 𝑥 = 𝑦)))))
138137ex 449 . . . . . . . . . . . . . . . . . . . . . 22 𝑥 = 𝑦 → (((𝑥 ∈ (0..^(#‘𝐹)) ∧ 𝑦 ∈ (0..^(#‘𝐹))) ∧ 𝑃:(0...(#‘𝐹))–1-1𝑉) → (¬ 𝑥 = 𝑦 → (((𝑥 ∈ (0..^(#‘𝐹)) ∧ 𝑦 ∈ (0..^(#‘𝐹))) ∧ 𝑃:(0...(#‘𝐹))–1-1𝑉) → (((𝑥 ∈ (0..^(#‘𝐹)) ∧ 𝑦 ∈ (0..^(#‘𝐹))) ∧ 𝑃:(0...(#‘𝐹))–1-1𝑉) → ({(𝑃𝑦), (𝑃‘(𝑦 + 1))} = {(𝑃𝑥), (𝑃‘(𝑥 + 1))} → 𝑥 = 𝑦))))))
139138pm2.43a 52 . . . . . . . . . . . . . . . . . . . . 21 𝑥 = 𝑦 → (((𝑥 ∈ (0..^(#‘𝐹)) ∧ 𝑦 ∈ (0..^(#‘𝐹))) ∧ 𝑃:(0...(#‘𝐹))–1-1𝑉) → (((𝑥 ∈ (0..^(#‘𝐹)) ∧ 𝑦 ∈ (0..^(#‘𝐹))) ∧ 𝑃:(0...(#‘𝐹))–1-1𝑉) → (((𝑥 ∈ (0..^(#‘𝐹)) ∧ 𝑦 ∈ (0..^(#‘𝐹))) ∧ 𝑃:(0...(#‘𝐹))–1-1𝑉) → ({(𝑃𝑦), (𝑃‘(𝑦 + 1))} = {(𝑃𝑥), (𝑃‘(𝑥 + 1))} → 𝑥 = 𝑦)))))
140139com14 94 . . . . . . . . . . . . . . . . . . . 20 (((𝑥 ∈ (0..^(#‘𝐹)) ∧ 𝑦 ∈ (0..^(#‘𝐹))) ∧ 𝑃:(0...(#‘𝐹))–1-1𝑉) → (((𝑥 ∈ (0..^(#‘𝐹)) ∧ 𝑦 ∈ (0..^(#‘𝐹))) ∧ 𝑃:(0...(#‘𝐹))–1-1𝑉) → (((𝑥 ∈ (0..^(#‘𝐹)) ∧ 𝑦 ∈ (0..^(#‘𝐹))) ∧ 𝑃:(0...(#‘𝐹))–1-1𝑉) → (¬ 𝑥 = 𝑦 → ({(𝑃𝑦), (𝑃‘(𝑦 + 1))} = {(𝑃𝑥), (𝑃‘(𝑥 + 1))} → 𝑥 = 𝑦)))))
141140pm2.43a 52 . . . . . . . . . . . . . . . . . . 19 (((𝑥 ∈ (0..^(#‘𝐹)) ∧ 𝑦 ∈ (0..^(#‘𝐹))) ∧ 𝑃:(0...(#‘𝐹))–1-1𝑉) → (((𝑥 ∈ (0..^(#‘𝐹)) ∧ 𝑦 ∈ (0..^(#‘𝐹))) ∧ 𝑃:(0...(#‘𝐹))–1-1𝑉) → (¬ 𝑥 = 𝑦 → ({(𝑃𝑦), (𝑃‘(𝑦 + 1))} = {(𝑃𝑥), (𝑃‘(𝑥 + 1))} → 𝑥 = 𝑦))))
142141pm2.43i 50 . . . . . . . . . . . . . . . . . 18 (((𝑥 ∈ (0..^(#‘𝐹)) ∧ 𝑦 ∈ (0..^(#‘𝐹))) ∧ 𝑃:(0...(#‘𝐹))–1-1𝑉) → (¬ 𝑥 = 𝑦 → ({(𝑃𝑦), (𝑃‘(𝑦 + 1))} = {(𝑃𝑥), (𝑃‘(𝑥 + 1))} → 𝑥 = 𝑦)))
143142ex 449 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ (0..^(#‘𝐹)) ∧ 𝑦 ∈ (0..^(#‘𝐹))) → (𝑃:(0...(#‘𝐹))–1-1𝑉 → (¬ 𝑥 = 𝑦 → ({(𝑃𝑦), (𝑃‘(𝑦 + 1))} = {(𝑃𝑥), (𝑃‘(𝑥 + 1))} → 𝑥 = 𝑦))))
144143com23 84 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ (0..^(#‘𝐹)) ∧ 𝑦 ∈ (0..^(#‘𝐹))) → (¬ 𝑥 = 𝑦 → (𝑃:(0...(#‘𝐹))–1-1𝑉 → ({(𝑃𝑦), (𝑃‘(𝑦 + 1))} = {(𝑃𝑥), (𝑃‘(𝑥 + 1))} → 𝑥 = 𝑦))))
145144com14 94 . . . . . . . . . . . . . . 15 ({(𝑃𝑦), (𝑃‘(𝑦 + 1))} = {(𝑃𝑥), (𝑃‘(𝑥 + 1))} → (¬ 𝑥 = 𝑦 → (𝑃:(0...(#‘𝐹))–1-1𝑉 → ((𝑥 ∈ (0..^(#‘𝐹)) ∧ 𝑦 ∈ (0..^(#‘𝐹))) → 𝑥 = 𝑦))))
14636, 145syl6com 36 . . . . . . . . . . . . . 14 ((𝐸‘(𝐹𝑥)) = (𝐸‘(𝐹𝑦)) → (((𝐸‘(𝐹𝑦)) = {(𝑃𝑦), (𝑃‘(𝑦 + 1))} ∧ (𝐸‘(𝐹𝑥)) = {(𝑃𝑥), (𝑃‘(𝑥 + 1))}) → (¬ 𝑥 = 𝑦 → (𝑃:(0...(#‘𝐹))–1-1𝑉 → ((𝑥 ∈ (0..^(#‘𝐹)) ∧ 𝑦 ∈ (0..^(#‘𝐹))) → 𝑥 = 𝑦)))))
14728, 146syl 17 . . . . . . . . . . . . 13 ((𝐹𝑥) = (𝐹𝑦) → (((𝐸‘(𝐹𝑦)) = {(𝑃𝑦), (𝑃‘(𝑦 + 1))} ∧ (𝐸‘(𝐹𝑥)) = {(𝑃𝑥), (𝑃‘(𝑥 + 1))}) → (¬ 𝑥 = 𝑦 → (𝑃:(0...(#‘𝐹))–1-1𝑉 → ((𝑥 ∈ (0..^(#‘𝐹)) ∧ 𝑦 ∈ (0..^(#‘𝐹))) → 𝑥 = 𝑦)))))
148147com15 99 . . . . . . . . . . . 12 ((𝑥 ∈ (0..^(#‘𝐹)) ∧ 𝑦 ∈ (0..^(#‘𝐹))) → (((𝐸‘(𝐹𝑦)) = {(𝑃𝑦), (𝑃‘(𝑦 + 1))} ∧ (𝐸‘(𝐹𝑥)) = {(𝑃𝑥), (𝑃‘(𝑥 + 1))}) → (¬ 𝑥 = 𝑦 → (𝑃:(0...(#‘𝐹))–1-1𝑉 → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))))
149148adantr 480 . . . . . . . . . . 11 (((𝑥 ∈ (0..^(#‘𝐹)) ∧ 𝑦 ∈ (0..^(#‘𝐹))) ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}) → (((𝐸‘(𝐹𝑦)) = {(𝑃𝑦), (𝑃‘(𝑦 + 1))} ∧ (𝐸‘(𝐹𝑥)) = {(𝑃𝑥), (𝑃‘(𝑥 + 1))}) → (¬ 𝑥 = 𝑦 → (𝑃:(0...(#‘𝐹))–1-1𝑉 → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))))
15027, 149mpd 15 . . . . . . . . . 10 (((𝑥 ∈ (0..^(#‘𝐹)) ∧ 𝑦 ∈ (0..^(#‘𝐹))) ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}) → (¬ 𝑥 = 𝑦 → (𝑃:(0...(#‘𝐹))–1-1𝑉 → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))))
151150ex 449 . . . . . . . . 9 ((𝑥 ∈ (0..^(#‘𝐹)) ∧ 𝑦 ∈ (0..^(#‘𝐹))) → (∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))} → (¬ 𝑥 = 𝑦 → (𝑃:(0...(#‘𝐹))–1-1𝑉 → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))))
152151com14 94 . . . . . . . 8 (𝑃:(0...(#‘𝐹))–1-1𝑉 → (∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))} → (¬ 𝑥 = 𝑦 → ((𝑥 ∈ (0..^(#‘𝐹)) ∧ 𝑦 ∈ (0..^(#‘𝐹))) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))))
153152a1i 11 . . . . . . 7 (𝐹:(0..^(#‘𝐹))⟶dom 𝐸 → (𝑃:(0...(#‘𝐹))–1-1𝑉 → (∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))} → (¬ 𝑥 = 𝑦 → ((𝑥 ∈ (0..^(#‘𝐹)) ∧ 𝑦 ∈ (0..^(#‘𝐹))) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))))))
1541533imp 1249 . . . . . 6 ((𝐹:(0..^(#‘𝐹))⟶dom 𝐸𝑃:(0...(#‘𝐹))–1-1𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}) → (¬ 𝑥 = 𝑦 → ((𝑥 ∈ (0..^(#‘𝐹)) ∧ 𝑦 ∈ (0..^(#‘𝐹))) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))))
155 2a1 28 . . . . . 6 (𝑥 = 𝑦 → ((𝑥 ∈ (0..^(#‘𝐹)) ∧ 𝑦 ∈ (0..^(#‘𝐹))) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
156154, 155pm2.61d2 171 . . . . 5 ((𝐹:(0..^(#‘𝐹))⟶dom 𝐸𝑃:(0...(#‘𝐹))–1-1𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}) → ((𝑥 ∈ (0..^(#‘𝐹)) ∧ 𝑦 ∈ (0..^(#‘𝐹))) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
157156ralrimivv 2953 . . . 4 ((𝐹:(0..^(#‘𝐹))⟶dom 𝐸𝑃:(0...(#‘𝐹))–1-1𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}) → ∀𝑥 ∈ (0..^(#‘𝐹))∀𝑦 ∈ (0..^(#‘𝐹))((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
1581, 157syl3an1 1351 . . 3 ((𝐹 ∈ Word dom 𝐸𝑃:(0...(#‘𝐹))–1-1𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}) → ∀𝑥 ∈ (0..^(#‘𝐹))∀𝑦 ∈ (0..^(#‘𝐹))((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
159 dff13 6416 . . 3 (𝐹:(0..^(#‘𝐹))–1-1→dom 𝐸 ↔ (𝐹:(0..^(#‘𝐹))⟶dom 𝐸 ∧ ∀𝑥 ∈ (0..^(#‘𝐹))∀𝑦 ∈ (0..^(#‘𝐹))((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
1602, 158, 159sylanbrc 695 . 2 ((𝐹 ∈ Word dom 𝐸𝑃:(0...(#‘𝐹))–1-1𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}) → 𝐹:(0..^(#‘𝐹))–1-1→dom 𝐸)
1612biantrurd 528 . . 3 ((𝐹 ∈ Word dom 𝐸𝑃:(0...(#‘𝐹))–1-1𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}) → (Fun 𝐹 ↔ (𝐹:(0..^(#‘𝐹))⟶dom 𝐸 ∧ Fun 𝐹)))
162 df-f1 5809 . . 3 (𝐹:(0..^(#‘𝐹))–1-1→dom 𝐸 ↔ (𝐹:(0..^(#‘𝐹))⟶dom 𝐸 ∧ Fun 𝐹))
163161, 162syl6bbr 277 . 2 ((𝐹 ∈ Word dom 𝐸𝑃:(0...(#‘𝐹))–1-1𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}) → (Fun 𝐹𝐹:(0..^(#‘𝐹))–1-1→dom 𝐸))
164160, 163mpbird 246 1 ((𝐹 ∈ Word dom 𝐸𝑃:(0...(#‘𝐹))–1-1𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}) → Fun 𝐹)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  {cpr 4127  ◡ccnv 5037  dom cdm 5038  Fun wfun 5798  ⟶wf 5800  –1-1→wf1 5801  ‘cfv 5804  (class class class)co 6549  ℂcc 9813  0cc0 9815  1c1 9816   + caddc 9818   − cmin 10145  2c2 10947  ℤcz 11254  ...cfz 12197  ..^cfzo 12334  #chash 12979  Word cword 13146 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154 This theorem is referenced by:  wlkdvspth  26138
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