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Theorem clwlkclwwlkf1lem3 27212
Description: Lemma 3 for clwlkclwwlkf1 27216. (Contributed by Alexander van der Vekens, 5-Jul-2018.) (Revised by AV, 3-May-2021.) (Revised by AV, 24-May-2022.)
Hypotheses
Ref Expression
clwlkclwwlkf.c 𝐶 = {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))}
clwlkclwwlkf.a 𝐴 = (1st𝑈)
clwlkclwwlkf.b 𝐵 = (2nd𝑈)
clwlkclwwlkf.d 𝐷 = (1st𝑊)
clwlkclwwlkf.e 𝐸 = (2nd𝑊)
Assertion
Ref Expression
clwlkclwwlkf1lem3 ((𝑈𝐶𝑊𝐶 ∧ (𝐵 substr ⟨0, (♯‘𝐴)⟩) = (𝐸 substr ⟨0, (♯‘𝐷)⟩)) → ∀𝑖 ∈ (0...(♯‘𝐴))(𝐵𝑖) = (𝐸𝑖))
Distinct variable groups:   𝑖,𝐺   𝑤,𝐺   𝑤,𝐴   𝑤,𝑈   𝐴,𝑖   𝐵,𝑖   𝐷,𝑖   𝑤,𝐷   𝑖,𝐸   𝑤,𝑊
Allowed substitution hints:   𝐵(𝑤)   𝐶(𝑤,𝑖)   𝑈(𝑖)   𝐸(𝑤)   𝑊(𝑖)

Proof of Theorem clwlkclwwlkf1lem3
StepHypRef Expression
1 clwlkclwwlkf.c . . . . 5 𝐶 = {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))}
2 clwlkclwwlkf.a . . . . 5 𝐴 = (1st𝑈)
3 clwlkclwwlkf.b . . . . 5 𝐵 = (2nd𝑈)
4 clwlkclwwlkf.d . . . . 5 𝐷 = (1st𝑊)
5 clwlkclwwlkf.e . . . . 5 𝐸 = (2nd𝑊)
61, 2, 3, 4, 5clwlkclwwlkf1lem2 27211 . . . 4 ((𝑈𝐶𝑊𝐶 ∧ (𝐵 substr ⟨0, (♯‘𝐴)⟩) = (𝐸 substr ⟨0, (♯‘𝐷)⟩)) → ((♯‘𝐴) = (♯‘𝐷) ∧ ∀𝑖 ∈ (0..^(♯‘𝐴))(𝐵𝑖) = (𝐸𝑖)))
7 simprr 789 . . . . 5 (((𝑈𝐶𝑊𝐶 ∧ (𝐵 substr ⟨0, (♯‘𝐴)⟩) = (𝐸 substr ⟨0, (♯‘𝐷)⟩)) ∧ ((♯‘𝐴) = (♯‘𝐷) ∧ ∀𝑖 ∈ (0..^(♯‘𝐴))(𝐵𝑖) = (𝐸𝑖))) → ∀𝑖 ∈ (0..^(♯‘𝐴))(𝐵𝑖) = (𝐸𝑖))
81, 2, 3clwlkclwwlkflem 27210 . . . . . . . . 9 (𝑈𝐶 → (𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ))
91, 4, 5clwlkclwwlkflem 27210 . . . . . . . . 9 (𝑊𝐶 → (𝐷(Walks‘𝐺)𝐸 ∧ (𝐸‘0) = (𝐸‘(♯‘𝐷)) ∧ (♯‘𝐷) ∈ ℕ))
10 lbfzo0 12716 . . . . . . . . . . . . . . . 16 (0 ∈ (0..^(♯‘𝐴)) ↔ (♯‘𝐴) ∈ ℕ)
1110biimpri 219 . . . . . . . . . . . . . . 15 ((♯‘𝐴) ∈ ℕ → 0 ∈ (0..^(♯‘𝐴)))
12113ad2ant3 1165 . . . . . . . . . . . . . 14 ((𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ) → 0 ∈ (0..^(♯‘𝐴)))
1312adantr 472 . . . . . . . . . . . . 13 (((𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ) ∧ (𝐷(Walks‘𝐺)𝐸 ∧ (𝐸‘0) = (𝐸‘(♯‘𝐷)) ∧ (♯‘𝐷) ∈ ℕ)) → 0 ∈ (0..^(♯‘𝐴)))
1413adantr 472 . . . . . . . . . . . 12 ((((𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ) ∧ (𝐷(Walks‘𝐺)𝐸 ∧ (𝐸‘0) = (𝐸‘(♯‘𝐷)) ∧ (♯‘𝐷) ∈ ℕ)) ∧ (♯‘𝐴) = (♯‘𝐷)) → 0 ∈ (0..^(♯‘𝐴)))
15 fveq2 6375 . . . . . . . . . . . . . 14 (𝑖 = 0 → (𝐵𝑖) = (𝐵‘0))
16 fveq2 6375 . . . . . . . . . . . . . 14 (𝑖 = 0 → (𝐸𝑖) = (𝐸‘0))
1715, 16eqeq12d 2780 . . . . . . . . . . . . 13 (𝑖 = 0 → ((𝐵𝑖) = (𝐸𝑖) ↔ (𝐵‘0) = (𝐸‘0)))
1817rspcv 3457 . . . . . . . . . . . 12 (0 ∈ (0..^(♯‘𝐴)) → (∀𝑖 ∈ (0..^(♯‘𝐴))(𝐵𝑖) = (𝐸𝑖) → (𝐵‘0) = (𝐸‘0)))
1914, 18syl 17 . . . . . . . . . . 11 ((((𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ) ∧ (𝐷(Walks‘𝐺)𝐸 ∧ (𝐸‘0) = (𝐸‘(♯‘𝐷)) ∧ (♯‘𝐷) ∈ ℕ)) ∧ (♯‘𝐴) = (♯‘𝐷)) → (∀𝑖 ∈ (0..^(♯‘𝐴))(𝐵𝑖) = (𝐸𝑖) → (𝐵‘0) = (𝐸‘0)))
20 simpl 474 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝐵‘(♯‘𝐴)) = (𝐵‘0) ∧ ((𝐵‘0) = (𝐸‘0) ∧ (𝐸‘0) = (𝐸‘(♯‘𝐷)))) → (𝐵‘(♯‘𝐴)) = (𝐵‘0))
21 eqtr 2784 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝐵‘0) = (𝐸‘0) ∧ (𝐸‘0) = (𝐸‘(♯‘𝐷))) → (𝐵‘0) = (𝐸‘(♯‘𝐷)))
2221adantl 473 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝐵‘(♯‘𝐴)) = (𝐵‘0) ∧ ((𝐵‘0) = (𝐸‘0) ∧ (𝐸‘0) = (𝐸‘(♯‘𝐷)))) → (𝐵‘0) = (𝐸‘(♯‘𝐷)))
2320, 22eqtrd 2799 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐵‘(♯‘𝐴)) = (𝐵‘0) ∧ ((𝐵‘0) = (𝐸‘0) ∧ (𝐸‘0) = (𝐸‘(♯‘𝐷)))) → (𝐵‘(♯‘𝐴)) = (𝐸‘(♯‘𝐷)))
2423exp32 411 . . . . . . . . . . . . . . . . . . . . 21 ((𝐵‘(♯‘𝐴)) = (𝐵‘0) → ((𝐵‘0) = (𝐸‘0) → ((𝐸‘0) = (𝐸‘(♯‘𝐷)) → (𝐵‘(♯‘𝐴)) = (𝐸‘(♯‘𝐷)))))
2524com23 86 . . . . . . . . . . . . . . . . . . . 20 ((𝐵‘(♯‘𝐴)) = (𝐵‘0) → ((𝐸‘0) = (𝐸‘(♯‘𝐷)) → ((𝐵‘0) = (𝐸‘0) → (𝐵‘(♯‘𝐴)) = (𝐸‘(♯‘𝐷)))))
2625eqcoms 2773 . . . . . . . . . . . . . . . . . . 19 ((𝐵‘0) = (𝐵‘(♯‘𝐴)) → ((𝐸‘0) = (𝐸‘(♯‘𝐷)) → ((𝐵‘0) = (𝐸‘0) → (𝐵‘(♯‘𝐴)) = (𝐸‘(♯‘𝐷)))))
27263ad2ant2 1164 . . . . . . . . . . . . . . . . . 18 ((𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ) → ((𝐸‘0) = (𝐸‘(♯‘𝐷)) → ((𝐵‘0) = (𝐸‘0) → (𝐵‘(♯‘𝐴)) = (𝐸‘(♯‘𝐷)))))
2827com12 32 . . . . . . . . . . . . . . . . 17 ((𝐸‘0) = (𝐸‘(♯‘𝐷)) → ((𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ) → ((𝐵‘0) = (𝐸‘0) → (𝐵‘(♯‘𝐴)) = (𝐸‘(♯‘𝐷)))))
29283ad2ant2 1164 . . . . . . . . . . . . . . . 16 ((𝐷(Walks‘𝐺)𝐸 ∧ (𝐸‘0) = (𝐸‘(♯‘𝐷)) ∧ (♯‘𝐷) ∈ ℕ) → ((𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ) → ((𝐵‘0) = (𝐸‘0) → (𝐵‘(♯‘𝐴)) = (𝐸‘(♯‘𝐷)))))
3029impcom 396 . . . . . . . . . . . . . . 15 (((𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ) ∧ (𝐷(Walks‘𝐺)𝐸 ∧ (𝐸‘0) = (𝐸‘(♯‘𝐷)) ∧ (♯‘𝐷) ∈ ℕ)) → ((𝐵‘0) = (𝐸‘0) → (𝐵‘(♯‘𝐴)) = (𝐸‘(♯‘𝐷))))
3130adantr 472 . . . . . . . . . . . . . 14 ((((𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ) ∧ (𝐷(Walks‘𝐺)𝐸 ∧ (𝐸‘0) = (𝐸‘(♯‘𝐷)) ∧ (♯‘𝐷) ∈ ℕ)) ∧ (♯‘𝐴) = (♯‘𝐷)) → ((𝐵‘0) = (𝐸‘0) → (𝐵‘(♯‘𝐴)) = (𝐸‘(♯‘𝐷))))
3231imp 395 . . . . . . . . . . . . 13 (((((𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ) ∧ (𝐷(Walks‘𝐺)𝐸 ∧ (𝐸‘0) = (𝐸‘(♯‘𝐷)) ∧ (♯‘𝐷) ∈ ℕ)) ∧ (♯‘𝐴) = (♯‘𝐷)) ∧ (𝐵‘0) = (𝐸‘0)) → (𝐵‘(♯‘𝐴)) = (𝐸‘(♯‘𝐷)))
33 fveq2 6375 . . . . . . . . . . . . . . . 16 ((♯‘𝐷) = (♯‘𝐴) → (𝐸‘(♯‘𝐷)) = (𝐸‘(♯‘𝐴)))
3433eqcoms 2773 . . . . . . . . . . . . . . 15 ((♯‘𝐴) = (♯‘𝐷) → (𝐸‘(♯‘𝐷)) = (𝐸‘(♯‘𝐴)))
3534adantl 473 . . . . . . . . . . . . . 14 ((((𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ) ∧ (𝐷(Walks‘𝐺)𝐸 ∧ (𝐸‘0) = (𝐸‘(♯‘𝐷)) ∧ (♯‘𝐷) ∈ ℕ)) ∧ (♯‘𝐴) = (♯‘𝐷)) → (𝐸‘(♯‘𝐷)) = (𝐸‘(♯‘𝐴)))
3635adantr 472 . . . . . . . . . . . . 13 (((((𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ) ∧ (𝐷(Walks‘𝐺)𝐸 ∧ (𝐸‘0) = (𝐸‘(♯‘𝐷)) ∧ (♯‘𝐷) ∈ ℕ)) ∧ (♯‘𝐴) = (♯‘𝐷)) ∧ (𝐵‘0) = (𝐸‘0)) → (𝐸‘(♯‘𝐷)) = (𝐸‘(♯‘𝐴)))
3732, 36eqtrd 2799 . . . . . . . . . . . 12 (((((𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ) ∧ (𝐷(Walks‘𝐺)𝐸 ∧ (𝐸‘0) = (𝐸‘(♯‘𝐷)) ∧ (♯‘𝐷) ∈ ℕ)) ∧ (♯‘𝐴) = (♯‘𝐷)) ∧ (𝐵‘0) = (𝐸‘0)) → (𝐵‘(♯‘𝐴)) = (𝐸‘(♯‘𝐴)))
3837ex 401 . . . . . . . . . . 11 ((((𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ) ∧ (𝐷(Walks‘𝐺)𝐸 ∧ (𝐸‘0) = (𝐸‘(♯‘𝐷)) ∧ (♯‘𝐷) ∈ ℕ)) ∧ (♯‘𝐴) = (♯‘𝐷)) → ((𝐵‘0) = (𝐸‘0) → (𝐵‘(♯‘𝐴)) = (𝐸‘(♯‘𝐴))))
3919, 38syld 47 . . . . . . . . . 10 ((((𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ) ∧ (𝐷(Walks‘𝐺)𝐸 ∧ (𝐸‘0) = (𝐸‘(♯‘𝐷)) ∧ (♯‘𝐷) ∈ ℕ)) ∧ (♯‘𝐴) = (♯‘𝐷)) → (∀𝑖 ∈ (0..^(♯‘𝐴))(𝐵𝑖) = (𝐸𝑖) → (𝐵‘(♯‘𝐴)) = (𝐸‘(♯‘𝐴))))
4039ex 401 . . . . . . . . 9 (((𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ) ∧ (𝐷(Walks‘𝐺)𝐸 ∧ (𝐸‘0) = (𝐸‘(♯‘𝐷)) ∧ (♯‘𝐷) ∈ ℕ)) → ((♯‘𝐴) = (♯‘𝐷) → (∀𝑖 ∈ (0..^(♯‘𝐴))(𝐵𝑖) = (𝐸𝑖) → (𝐵‘(♯‘𝐴)) = (𝐸‘(♯‘𝐴)))))
418, 9, 40syl2an 589 . . . . . . . 8 ((𝑈𝐶𝑊𝐶) → ((♯‘𝐴) = (♯‘𝐷) → (∀𝑖 ∈ (0..^(♯‘𝐴))(𝐵𝑖) = (𝐸𝑖) → (𝐵‘(♯‘𝐴)) = (𝐸‘(♯‘𝐴)))))
4241impd 398 . . . . . . 7 ((𝑈𝐶𝑊𝐶) → (((♯‘𝐴) = (♯‘𝐷) ∧ ∀𝑖 ∈ (0..^(♯‘𝐴))(𝐵𝑖) = (𝐸𝑖)) → (𝐵‘(♯‘𝐴)) = (𝐸‘(♯‘𝐴))))
43423adant3 1162 . . . . . 6 ((𝑈𝐶𝑊𝐶 ∧ (𝐵 substr ⟨0, (♯‘𝐴)⟩) = (𝐸 substr ⟨0, (♯‘𝐷)⟩)) → (((♯‘𝐴) = (♯‘𝐷) ∧ ∀𝑖 ∈ (0..^(♯‘𝐴))(𝐵𝑖) = (𝐸𝑖)) → (𝐵‘(♯‘𝐴)) = (𝐸‘(♯‘𝐴))))
4443imp 395 . . . . 5 (((𝑈𝐶𝑊𝐶 ∧ (𝐵 substr ⟨0, (♯‘𝐴)⟩) = (𝐸 substr ⟨0, (♯‘𝐷)⟩)) ∧ ((♯‘𝐴) = (♯‘𝐷) ∧ ∀𝑖 ∈ (0..^(♯‘𝐴))(𝐵𝑖) = (𝐸𝑖))) → (𝐵‘(♯‘𝐴)) = (𝐸‘(♯‘𝐴)))
457, 44jca 507 . . . 4 (((𝑈𝐶𝑊𝐶 ∧ (𝐵 substr ⟨0, (♯‘𝐴)⟩) = (𝐸 substr ⟨0, (♯‘𝐷)⟩)) ∧ ((♯‘𝐴) = (♯‘𝐷) ∧ ∀𝑖 ∈ (0..^(♯‘𝐴))(𝐵𝑖) = (𝐸𝑖))) → (∀𝑖 ∈ (0..^(♯‘𝐴))(𝐵𝑖) = (𝐸𝑖) ∧ (𝐵‘(♯‘𝐴)) = (𝐸‘(♯‘𝐴))))
466, 45mpdan 678 . . 3 ((𝑈𝐶𝑊𝐶 ∧ (𝐵 substr ⟨0, (♯‘𝐴)⟩) = (𝐸 substr ⟨0, (♯‘𝐷)⟩)) → (∀𝑖 ∈ (0..^(♯‘𝐴))(𝐵𝑖) = (𝐸𝑖) ∧ (𝐵‘(♯‘𝐴)) = (𝐸‘(♯‘𝐴))))
47 fvex 6388 . . . 4 (♯‘𝐴) ∈ V
48 fveq2 6375 . . . . . 6 (𝑖 = (♯‘𝐴) → (𝐵𝑖) = (𝐵‘(♯‘𝐴)))
49 fveq2 6375 . . . . . 6 (𝑖 = (♯‘𝐴) → (𝐸𝑖) = (𝐸‘(♯‘𝐴)))
5048, 49eqeq12d 2780 . . . . 5 (𝑖 = (♯‘𝐴) → ((𝐵𝑖) = (𝐸𝑖) ↔ (𝐵‘(♯‘𝐴)) = (𝐸‘(♯‘𝐴))))
5150ralunsn 4580 . . . 4 ((♯‘𝐴) ∈ V → (∀𝑖 ∈ ((0..^(♯‘𝐴)) ∪ {(♯‘𝐴)})(𝐵𝑖) = (𝐸𝑖) ↔ (∀𝑖 ∈ (0..^(♯‘𝐴))(𝐵𝑖) = (𝐸𝑖) ∧ (𝐵‘(♯‘𝐴)) = (𝐸‘(♯‘𝐴)))))
5247, 51ax-mp 5 . . 3 (∀𝑖 ∈ ((0..^(♯‘𝐴)) ∪ {(♯‘𝐴)})(𝐵𝑖) = (𝐸𝑖) ↔ (∀𝑖 ∈ (0..^(♯‘𝐴))(𝐵𝑖) = (𝐸𝑖) ∧ (𝐵‘(♯‘𝐴)) = (𝐸‘(♯‘𝐴))))
5346, 52sylibr 225 . 2 ((𝑈𝐶𝑊𝐶 ∧ (𝐵 substr ⟨0, (♯‘𝐴)⟩) = (𝐸 substr ⟨0, (♯‘𝐷)⟩)) → ∀𝑖 ∈ ((0..^(♯‘𝐴)) ∪ {(♯‘𝐴)})(𝐵𝑖) = (𝐸𝑖))
54 nnnn0 11546 . . . . . . . 8 ((♯‘𝐴) ∈ ℕ → (♯‘𝐴) ∈ ℕ0)
55 elnn0uz 11925 . . . . . . . 8 ((♯‘𝐴) ∈ ℕ0 ↔ (♯‘𝐴) ∈ (ℤ‘0))
5654, 55sylib 209 . . . . . . 7 ((♯‘𝐴) ∈ ℕ → (♯‘𝐴) ∈ (ℤ‘0))
57563ad2ant3 1165 . . . . . 6 ((𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ) → (♯‘𝐴) ∈ (ℤ‘0))
588, 57syl 17 . . . . 5 (𝑈𝐶 → (♯‘𝐴) ∈ (ℤ‘0))
59583ad2ant1 1163 . . . 4 ((𝑈𝐶𝑊𝐶 ∧ (𝐵 substr ⟨0, (♯‘𝐴)⟩) = (𝐸 substr ⟨0, (♯‘𝐷)⟩)) → (♯‘𝐴) ∈ (ℤ‘0))
60 fzisfzounsn 12788 . . . 4 ((♯‘𝐴) ∈ (ℤ‘0) → (0...(♯‘𝐴)) = ((0..^(♯‘𝐴)) ∪ {(♯‘𝐴)}))
6159, 60syl 17 . . 3 ((𝑈𝐶𝑊𝐶 ∧ (𝐵 substr ⟨0, (♯‘𝐴)⟩) = (𝐸 substr ⟨0, (♯‘𝐷)⟩)) → (0...(♯‘𝐴)) = ((0..^(♯‘𝐴)) ∪ {(♯‘𝐴)}))
6261raleqdv 3292 . 2 ((𝑈𝐶𝑊𝐶 ∧ (𝐵 substr ⟨0, (♯‘𝐴)⟩) = (𝐸 substr ⟨0, (♯‘𝐷)⟩)) → (∀𝑖 ∈ (0...(♯‘𝐴))(𝐵𝑖) = (𝐸𝑖) ↔ ∀𝑖 ∈ ((0..^(♯‘𝐴)) ∪ {(♯‘𝐴)})(𝐵𝑖) = (𝐸𝑖)))
6353, 62mpbird 248 1 ((𝑈𝐶𝑊𝐶 ∧ (𝐵 substr ⟨0, (♯‘𝐴)⟩) = (𝐸 substr ⟨0, (♯‘𝐷)⟩)) → ∀𝑖 ∈ (0...(♯‘𝐴))(𝐵𝑖) = (𝐸𝑖))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  w3a 1107   = wceq 1652  wcel 2155  wral 3055  {crab 3059  Vcvv 3350  cun 3730  {csn 4334  cop 4340   class class class wbr 4809  cfv 6068  (class class class)co 6842  1st c1st 7364  2nd c2nd 7365  0cc0 10189  1c1 10190  cle 10329  cn 11274  0cn0 11538  cuz 11886  ...cfz 12533  ..^cfzo 12673  chash 13321   substr csubstr 13616  Walkscwlks 26783  ClWalkscclwlks 26957
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 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-cnex 10245  ax-resscn 10246  ax-1cn 10247  ax-icn 10248  ax-addcl 10249  ax-addrcl 10250  ax-mulcl 10251  ax-mulrcl 10252  ax-mulcom 10253  ax-addass 10254  ax-mulass 10255  ax-distr 10256  ax-i2m1 10257  ax-1ne0 10258  ax-1rid 10259  ax-rnegex 10260  ax-rrecex 10261  ax-cnre 10262  ax-pre-lttri 10263  ax-pre-lttrn 10264  ax-pre-ltadd 10265  ax-pre-mulgt0 10266
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-ifp 1086  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-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-om 7264  df-1st 7366  df-2nd 7367  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-1o 7764  df-oadd 7768  df-er 7947  df-map 8062  df-pm 8063  df-en 8161  df-dom 8162  df-sdom 8163  df-fin 8164  df-card 9016  df-pnf 10330  df-mnf 10331  df-xr 10332  df-ltxr 10333  df-le 10334  df-sub 10522  df-neg 10523  df-nn 11275  df-n0 11539  df-z 11625  df-uz 11887  df-fz 12534  df-fzo 12674  df-hash 13322  df-word 13487  df-substr 13617  df-wlks 26786  df-clwlks 26958
This theorem is referenced by:  clwlkclwwlkf1  27216
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