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Theorem clwlkclwwlkf1lem3 27791
 Description: Lemma 3 for clwlkclwwlkf1 27795. (Contributed by Alexander van der Vekens, 5-Jul-2018.) (Revised by AV, 30-Oct-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 prefix (♯‘𝐴)) = (𝐸 prefix (♯‘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 27790 . . . 4 ((𝑈𝐶𝑊𝐶 ∧ (𝐵 prefix (♯‘𝐴)) = (𝐸 prefix (♯‘𝐷))) → ((♯‘𝐴) = (♯‘𝐷) ∧ ∀𝑖 ∈ (0..^(♯‘𝐴))(𝐵𝑖) = (𝐸𝑖)))
7 simprr 772 . . . . 5 (((𝑈𝐶𝑊𝐶 ∧ (𝐵 prefix (♯‘𝐴)) = (𝐸 prefix (♯‘𝐷))) ∧ ((♯‘𝐴) = (♯‘𝐷) ∧ ∀𝑖 ∈ (0..^(♯‘𝐴))(𝐵𝑖) = (𝐸𝑖))) → ∀𝑖 ∈ (0..^(♯‘𝐴))(𝐵𝑖) = (𝐸𝑖))
81, 2, 3clwlkclwwlkflem 27789 . . . . . . . . 9 (𝑈𝐶 → (𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ))
91, 4, 5clwlkclwwlkflem 27789 . . . . . . . . 9 (𝑊𝐶 → (𝐷(Walks‘𝐺)𝐸 ∧ (𝐸‘0) = (𝐸‘(♯‘𝐷)) ∧ (♯‘𝐷) ∈ ℕ))
10 lbfzo0 13072 . . . . . . . . . . . . . . . 16 (0 ∈ (0..^(♯‘𝐴)) ↔ (♯‘𝐴) ∈ ℕ)
1110biimpri 231 . . . . . . . . . . . . . . 15 ((♯‘𝐴) ∈ ℕ → 0 ∈ (0..^(♯‘𝐴)))
12113ad2ant3 1132 . . . . . . . . . . . . . 14 ((𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ) → 0 ∈ (0..^(♯‘𝐴)))
1312adantr 484 . . . . . . . . . . . . 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 484 . . . . . . . . . . . 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 6645 . . . . . . . . . . . . . 14 (𝑖 = 0 → (𝐵𝑖) = (𝐵‘0))
16 fveq2 6645 . . . . . . . . . . . . . 14 (𝑖 = 0 → (𝐸𝑖) = (𝐸‘0))
1715, 16eqeq12d 2814 . . . . . . . . . . . . 13 (𝑖 = 0 → ((𝐵𝑖) = (𝐸𝑖) ↔ (𝐵‘0) = (𝐸‘0)))
1817rspcv 3566 . . . . . . . . . . . 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 486 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝐵‘(♯‘𝐴)) = (𝐵‘0) ∧ ((𝐵‘0) = (𝐸‘0) ∧ (𝐸‘0) = (𝐸‘(♯‘𝐷)))) → (𝐵‘(♯‘𝐴)) = (𝐵‘0))
21 eqtr 2818 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝐵‘0) = (𝐸‘0) ∧ (𝐸‘0) = (𝐸‘(♯‘𝐷))) → (𝐵‘0) = (𝐸‘(♯‘𝐷)))
2221adantl 485 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝐵‘(♯‘𝐴)) = (𝐵‘0) ∧ ((𝐵‘0) = (𝐸‘0) ∧ (𝐸‘0) = (𝐸‘(♯‘𝐷)))) → (𝐵‘0) = (𝐸‘(♯‘𝐷)))
2320, 22eqtrd 2833 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐵‘(♯‘𝐴)) = (𝐵‘0) ∧ ((𝐵‘0) = (𝐸‘0) ∧ (𝐸‘0) = (𝐸‘(♯‘𝐷)))) → (𝐵‘(♯‘𝐴)) = (𝐸‘(♯‘𝐷)))
2423exp32 424 . . . . . . . . . . . . . . . . . . . . 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 2806 . . . . . . . . . . . . . . . . . . 19 ((𝐵‘0) = (𝐵‘(♯‘𝐴)) → ((𝐸‘0) = (𝐸‘(♯‘𝐷)) → ((𝐵‘0) = (𝐸‘0) → (𝐵‘(♯‘𝐴)) = (𝐸‘(♯‘𝐷)))))
27263ad2ant2 1131 . . . . . . . . . . . . . . . . . 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 1131 . . . . . . . . . . . . . . . 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 411 . . . . . . . . . . . . . . 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 484 . . . . . . . . . . . . . 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 410 . . . . . . . . . . . . 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 6645 . . . . . . . . . . . . . . . 16 ((♯‘𝐷) = (♯‘𝐴) → (𝐸‘(♯‘𝐷)) = (𝐸‘(♯‘𝐴)))
3433eqcoms 2806 . . . . . . . . . . . . . . 15 ((♯‘𝐴) = (♯‘𝐷) → (𝐸‘(♯‘𝐷)) = (𝐸‘(♯‘𝐴)))
3534adantl 485 . . . . . . . . . . . . . 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 484 . . . . . . . . . . . . 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 2833 . . . . . . . . . . . 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 416 . . . . . . . . . . 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 416 . . . . . . . . 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 598 . . . . . . . 8 ((𝑈𝐶𝑊𝐶) → ((♯‘𝐴) = (♯‘𝐷) → (∀𝑖 ∈ (0..^(♯‘𝐴))(𝐵𝑖) = (𝐸𝑖) → (𝐵‘(♯‘𝐴)) = (𝐸‘(♯‘𝐴)))))
4241impd 414 . . . . . . 7 ((𝑈𝐶𝑊𝐶) → (((♯‘𝐴) = (♯‘𝐷) ∧ ∀𝑖 ∈ (0..^(♯‘𝐴))(𝐵𝑖) = (𝐸𝑖)) → (𝐵‘(♯‘𝐴)) = (𝐸‘(♯‘𝐴))))
43423adant3 1129 . . . . . 6 ((𝑈𝐶𝑊𝐶 ∧ (𝐵 prefix (♯‘𝐴)) = (𝐸 prefix (♯‘𝐷))) → (((♯‘𝐴) = (♯‘𝐷) ∧ ∀𝑖 ∈ (0..^(♯‘𝐴))(𝐵𝑖) = (𝐸𝑖)) → (𝐵‘(♯‘𝐴)) = (𝐸‘(♯‘𝐴))))
4443imp 410 . . . . 5 (((𝑈𝐶𝑊𝐶 ∧ (𝐵 prefix (♯‘𝐴)) = (𝐸 prefix (♯‘𝐷))) ∧ ((♯‘𝐴) = (♯‘𝐷) ∧ ∀𝑖 ∈ (0..^(♯‘𝐴))(𝐵𝑖) = (𝐸𝑖))) → (𝐵‘(♯‘𝐴)) = (𝐸‘(♯‘𝐴)))
457, 44jca 515 . . . 4 (((𝑈𝐶𝑊𝐶 ∧ (𝐵 prefix (♯‘𝐴)) = (𝐸 prefix (♯‘𝐷))) ∧ ((♯‘𝐴) = (♯‘𝐷) ∧ ∀𝑖 ∈ (0..^(♯‘𝐴))(𝐵𝑖) = (𝐸𝑖))) → (∀𝑖 ∈ (0..^(♯‘𝐴))(𝐵𝑖) = (𝐸𝑖) ∧ (𝐵‘(♯‘𝐴)) = (𝐸‘(♯‘𝐴))))
466, 45mpdan 686 . . 3 ((𝑈𝐶𝑊𝐶 ∧ (𝐵 prefix (♯‘𝐴)) = (𝐸 prefix (♯‘𝐷))) → (∀𝑖 ∈ (0..^(♯‘𝐴))(𝐵𝑖) = (𝐸𝑖) ∧ (𝐵‘(♯‘𝐴)) = (𝐸‘(♯‘𝐴))))
47 fvex 6658 . . . 4 (♯‘𝐴) ∈ V
48 fveq2 6645 . . . . . 6 (𝑖 = (♯‘𝐴) → (𝐵𝑖) = (𝐵‘(♯‘𝐴)))
49 fveq2 6645 . . . . . 6 (𝑖 = (♯‘𝐴) → (𝐸𝑖) = (𝐸‘(♯‘𝐴)))
5048, 49eqeq12d 2814 . . . . 5 (𝑖 = (♯‘𝐴) → ((𝐵𝑖) = (𝐸𝑖) ↔ (𝐵‘(♯‘𝐴)) = (𝐸‘(♯‘𝐴))))
5150ralunsn 4786 . . . 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 237 . 2 ((𝑈𝐶𝑊𝐶 ∧ (𝐵 prefix (♯‘𝐴)) = (𝐸 prefix (♯‘𝐷))) → ∀𝑖 ∈ ((0..^(♯‘𝐴)) ∪ {(♯‘𝐴)})(𝐵𝑖) = (𝐸𝑖))
54 nnnn0 11892 . . . . . . . 8 ((♯‘𝐴) ∈ ℕ → (♯‘𝐴) ∈ ℕ0)
55 elnn0uz 12271 . . . . . . . 8 ((♯‘𝐴) ∈ ℕ0 ↔ (♯‘𝐴) ∈ (ℤ‘0))
5654, 55sylib 221 . . . . . . 7 ((♯‘𝐴) ∈ ℕ → (♯‘𝐴) ∈ (ℤ‘0))
57563ad2ant3 1132 . . . . . 6 ((𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ) → (♯‘𝐴) ∈ (ℤ‘0))
588, 57syl 17 . . . . 5 (𝑈𝐶 → (♯‘𝐴) ∈ (ℤ‘0))
59583ad2ant1 1130 . . . 4 ((𝑈𝐶𝑊𝐶 ∧ (𝐵 prefix (♯‘𝐴)) = (𝐸 prefix (♯‘𝐷))) → (♯‘𝐴) ∈ (ℤ‘0))
60 fzisfzounsn 13144 . . . 4 ((♯‘𝐴) ∈ (ℤ‘0) → (0...(♯‘𝐴)) = ((0..^(♯‘𝐴)) ∪ {(♯‘𝐴)}))
6159, 60syl 17 . . 3 ((𝑈𝐶𝑊𝐶 ∧ (𝐵 prefix (♯‘𝐴)) = (𝐸 prefix (♯‘𝐷))) → (0...(♯‘𝐴)) = ((0..^(♯‘𝐴)) ∪ {(♯‘𝐴)}))
6261raleqdv 3364 . 2 ((𝑈𝐶𝑊𝐶 ∧ (𝐵 prefix (♯‘𝐴)) = (𝐸 prefix (♯‘𝐷))) → (∀𝑖 ∈ (0...(♯‘𝐴))(𝐵𝑖) = (𝐸𝑖) ↔ ∀𝑖 ∈ ((0..^(♯‘𝐴)) ∪ {(♯‘𝐴)})(𝐵𝑖) = (𝐸𝑖)))
6353, 62mpbird 260 1 ((𝑈𝐶𝑊𝐶 ∧ (𝐵 prefix (♯‘𝐴)) = (𝐸 prefix (♯‘𝐷))) → ∀𝑖 ∈ (0...(♯‘𝐴))(𝐵𝑖) = (𝐸𝑖))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∀wral 3106  {crab 3110  Vcvv 3441   ∪ cun 3879  {csn 4525   class class class wbr 5030  ‘cfv 6324  (class class class)co 7135  1st c1st 7669  2nd c2nd 7670  0cc0 10526  1c1 10527   ≤ cle 10665  ℕcn 11625  ℕ0cn0 11885  ℤ≥cuz 12231  ...cfz 12885  ..^cfzo 13028  ♯chash 13686   prefix cpfx 14023  Walkscwlks 27386  ClWalkscclwlks 27559 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ifp 1059  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-1st 7671  df-2nd 7672  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-oadd 8089  df-er 8272  df-map 8391  df-en 8493  df-dom 8494  df-sdom 8495  df-fin 8496  df-card 9352  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11626  df-n0 11886  df-z 11970  df-uz 12232  df-fz 12886  df-fzo 13029  df-hash 13687  df-word 13858  df-substr 13994  df-pfx 14024  df-wlks 27389  df-clwlks 27560 This theorem is referenced by:  clwlkclwwlkf1  27795
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