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Theorem clwlkclwwlkf1lem3OLD 27516
 Description: Obsolete version of clwlkclwwlkf1lem3 27515 as of 12-Oct-2022. (Contributed by Alexander van der Vekens, 5-Jul-2018.) (Revised by AV, 3-May-2021.) (Revised by AV, 24-May-2022.) (New usage is discouraged.) (Proof modification is discouraged.)
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
clwlkclwwlkf1lem3OLD ((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 clwlkclwwlkf1lem3OLD
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, 5clwlkclwwlkf1lem2OLD 27514 . . . 4 ((𝑈𝐶𝑊𝐶 ∧ (𝐵 substr ⟨0, (♯‘𝐴)⟩) = (𝐸 substr ⟨0, (♯‘𝐷)⟩)) → ((♯‘𝐴) = (♯‘𝐷) ∧ ∀𝑖 ∈ (0..^(♯‘𝐴))(𝐵𝑖) = (𝐸𝑖)))
7 simprr 760 . . . . 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 27512 . . . . . . . . 9 (𝑈𝐶 → (𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ))
91, 4, 5clwlkclwwlkflem 27512 . . . . . . . . 9 (𝑊𝐶 → (𝐷(Walks‘𝐺)𝐸 ∧ (𝐸‘0) = (𝐸‘(♯‘𝐷)) ∧ (♯‘𝐷) ∈ ℕ))
10 lbfzo0 12892 . . . . . . . . . . . . . . . 16 (0 ∈ (0..^(♯‘𝐴)) ↔ (♯‘𝐴) ∈ ℕ)
1110biimpri 220 . . . . . . . . . . . . . . 15 ((♯‘𝐴) ∈ ℕ → 0 ∈ (0..^(♯‘𝐴)))
12113ad2ant3 1115 . . . . . . . . . . . . . 14 ((𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ) → 0 ∈ (0..^(♯‘𝐴)))
1312adantr 473 . . . . . . . . . . . . 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 473 . . . . . . . . . . . 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 6499 . . . . . . . . . . . . . 14 (𝑖 = 0 → (𝐵𝑖) = (𝐵‘0))
16 fveq2 6499 . . . . . . . . . . . . . 14 (𝑖 = 0 → (𝐸𝑖) = (𝐸‘0))
1715, 16eqeq12d 2794 . . . . . . . . . . . . 13 (𝑖 = 0 → ((𝐵𝑖) = (𝐸𝑖) ↔ (𝐵‘0) = (𝐸‘0)))
1817rspcv 3532 . . . . . . . . . . . 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 475 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝐵‘(♯‘𝐴)) = (𝐵‘0) ∧ ((𝐵‘0) = (𝐸‘0) ∧ (𝐸‘0) = (𝐸‘(♯‘𝐷)))) → (𝐵‘(♯‘𝐴)) = (𝐵‘0))
21 eqtr 2800 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝐵‘0) = (𝐸‘0) ∧ (𝐸‘0) = (𝐸‘(♯‘𝐷))) → (𝐵‘0) = (𝐸‘(♯‘𝐷)))
2221adantl 474 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝐵‘(♯‘𝐴)) = (𝐵‘0) ∧ ((𝐵‘0) = (𝐸‘0) ∧ (𝐸‘0) = (𝐸‘(♯‘𝐷)))) → (𝐵‘0) = (𝐸‘(♯‘𝐷)))
2320, 22eqtrd 2815 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐵‘(♯‘𝐴)) = (𝐵‘0) ∧ ((𝐵‘0) = (𝐸‘0) ∧ (𝐸‘0) = (𝐸‘(♯‘𝐷)))) → (𝐵‘(♯‘𝐴)) = (𝐸‘(♯‘𝐷)))
2423exp32 413 . . . . . . . . . . . . . . . . . . . . 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 2787 . . . . . . . . . . . . . . . . . . 19 ((𝐵‘0) = (𝐵‘(♯‘𝐴)) → ((𝐸‘0) = (𝐸‘(♯‘𝐷)) → ((𝐵‘0) = (𝐸‘0) → (𝐵‘(♯‘𝐴)) = (𝐸‘(♯‘𝐷)))))
27263ad2ant2 1114 . . . . . . . . . . . . . . . . . 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 1114 . . . . . . . . . . . . . . . 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 399 . . . . . . . . . . . . . . 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 473 . . . . . . . . . . . . . 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 398 . . . . . . . . . . . . 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 6499 . . . . . . . . . . . . . . . 16 ((♯‘𝐷) = (♯‘𝐴) → (𝐸‘(♯‘𝐷)) = (𝐸‘(♯‘𝐴)))
3433eqcoms 2787 . . . . . . . . . . . . . . 15 ((♯‘𝐴) = (♯‘𝐷) → (𝐸‘(♯‘𝐷)) = (𝐸‘(♯‘𝐴)))
3534adantl 474 . . . . . . . . . . . . . 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 473 . . . . . . . . . . . . 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 2815 . . . . . . . . . . . 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 405 . . . . . . . . . . 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 405 . . . . . . . . 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 586 . . . . . . . 8 ((𝑈𝐶𝑊𝐶) → ((♯‘𝐴) = (♯‘𝐷) → (∀𝑖 ∈ (0..^(♯‘𝐴))(𝐵𝑖) = (𝐸𝑖) → (𝐵‘(♯‘𝐴)) = (𝐸‘(♯‘𝐴)))))
4241impd 402 . . . . . . 7 ((𝑈𝐶𝑊𝐶) → (((♯‘𝐴) = (♯‘𝐷) ∧ ∀𝑖 ∈ (0..^(♯‘𝐴))(𝐵𝑖) = (𝐸𝑖)) → (𝐵‘(♯‘𝐴)) = (𝐸‘(♯‘𝐴))))
43423adant3 1112 . . . . . 6 ((𝑈𝐶𝑊𝐶 ∧ (𝐵 substr ⟨0, (♯‘𝐴)⟩) = (𝐸 substr ⟨0, (♯‘𝐷)⟩)) → (((♯‘𝐴) = (♯‘𝐷) ∧ ∀𝑖 ∈ (0..^(♯‘𝐴))(𝐵𝑖) = (𝐸𝑖)) → (𝐵‘(♯‘𝐴)) = (𝐸‘(♯‘𝐴))))
4443imp 398 . . . . 5 (((𝑈𝐶𝑊𝐶 ∧ (𝐵 substr ⟨0, (♯‘𝐴)⟩) = (𝐸 substr ⟨0, (♯‘𝐷)⟩)) ∧ ((♯‘𝐴) = (♯‘𝐷) ∧ ∀𝑖 ∈ (0..^(♯‘𝐴))(𝐵𝑖) = (𝐸𝑖))) → (𝐵‘(♯‘𝐴)) = (𝐸‘(♯‘𝐴)))
457, 44jca 504 . . . 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 674 . . 3 ((𝑈𝐶𝑊𝐶 ∧ (𝐵 substr ⟨0, (♯‘𝐴)⟩) = (𝐸 substr ⟨0, (♯‘𝐷)⟩)) → (∀𝑖 ∈ (0..^(♯‘𝐴))(𝐵𝑖) = (𝐸𝑖) ∧ (𝐵‘(♯‘𝐴)) = (𝐸‘(♯‘𝐴))))
47 fvex 6512 . . . 4 (♯‘𝐴) ∈ V
48 fveq2 6499 . . . . . 6 (𝑖 = (♯‘𝐴) → (𝐵𝑖) = (𝐵‘(♯‘𝐴)))
49 fveq2 6499 . . . . . 6 (𝑖 = (♯‘𝐴) → (𝐸𝑖) = (𝐸‘(♯‘𝐴)))
5048, 49eqeq12d 2794 . . . . 5 (𝑖 = (♯‘𝐴) → ((𝐵𝑖) = (𝐸𝑖) ↔ (𝐵‘(♯‘𝐴)) = (𝐸‘(♯‘𝐴))))
5150ralunsn 4698 . . . 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 226 . 2 ((𝑈𝐶𝑊𝐶 ∧ (𝐵 substr ⟨0, (♯‘𝐴)⟩) = (𝐸 substr ⟨0, (♯‘𝐷)⟩)) → ∀𝑖 ∈ ((0..^(♯‘𝐴)) ∪ {(♯‘𝐴)})(𝐵𝑖) = (𝐸𝑖))
54 nnnn0 11715 . . . . . . . 8 ((♯‘𝐴) ∈ ℕ → (♯‘𝐴) ∈ ℕ0)
55 elnn0uz 12097 . . . . . . . 8 ((♯‘𝐴) ∈ ℕ0 ↔ (♯‘𝐴) ∈ (ℤ‘0))
5654, 55sylib 210 . . . . . . 7 ((♯‘𝐴) ∈ ℕ → (♯‘𝐴) ∈ (ℤ‘0))
57563ad2ant3 1115 . . . . . 6 ((𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ) → (♯‘𝐴) ∈ (ℤ‘0))
588, 57syl 17 . . . . 5 (𝑈𝐶 → (♯‘𝐴) ∈ (ℤ‘0))
59583ad2ant1 1113 . . . 4 ((𝑈𝐶𝑊𝐶 ∧ (𝐵 substr ⟨0, (♯‘𝐴)⟩) = (𝐸 substr ⟨0, (♯‘𝐷)⟩)) → (♯‘𝐴) ∈ (ℤ‘0))
60 fzisfzounsn 12964 . . . 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 3356 . 2 ((𝑈𝐶𝑊𝐶 ∧ (𝐵 substr ⟨0, (♯‘𝐴)⟩) = (𝐸 substr ⟨0, (♯‘𝐷)⟩)) → (∀𝑖 ∈ (0...(♯‘𝐴))(𝐵𝑖) = (𝐸𝑖) ↔ ∀𝑖 ∈ ((0..^(♯‘𝐴)) ∪ {(♯‘𝐴)})(𝐵𝑖) = (𝐸𝑖)))
6353, 62mpbird 249 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 198   ∧ wa 387   ∧ w3a 1068   = wceq 1507   ∈ wcel 2050  ∀wral 3089  {crab 3093  Vcvv 3416   ∪ cun 3828  {csn 4441  ⟨cop 4447   class class class wbr 4929  ‘cfv 6188  (class class class)co 6976  1st c1st 7499  2nd c2nd 7500  0cc0 10335  1c1 10336   ≤ cle 10475  ℕcn 11439  ℕ0cn0 11707  ℤ≥cuz 12058  ...cfz 12708  ..^cfzo 12849  ♯chash 13505   substr csubstr 13803  Walkscwlks 27081  ClWalkscclwlks 27259 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2751  ax-rep 5049  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186  ax-un 7279  ax-cnex 10391  ax-resscn 10392  ax-1cn 10393  ax-icn 10394  ax-addcl 10395  ax-addrcl 10396  ax-mulcl 10397  ax-mulrcl 10398  ax-mulcom 10399  ax-addass 10400  ax-mulass 10401  ax-distr 10402  ax-i2m1 10403  ax-1ne0 10404  ax-1rid 10405  ax-rnegex 10406  ax-rrecex 10407  ax-cnre 10408  ax-pre-lttri 10409  ax-pre-lttrn 10410  ax-pre-ltadd 10411  ax-pre-mulgt0 10412 This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-ifp 1044  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2760  df-cleq 2772  df-clel 2847  df-nfc 2919  df-ne 2969  df-nel 3075  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3418  df-sbc 3683  df-csb 3788  df-dif 3833  df-un 3835  df-in 3837  df-ss 3844  df-pss 3846  df-nul 4180  df-if 4351  df-pw 4424  df-sn 4442  df-pr 4444  df-tp 4446  df-op 4448  df-uni 4713  df-int 4750  df-iun 4794  df-br 4930  df-opab 4992  df-mpt 5009  df-tr 5031  df-id 5312  df-eprel 5317  df-po 5326  df-so 5327  df-fr 5366  df-we 5368  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-ima 5420  df-pred 5986  df-ord 6032  df-on 6033  df-lim 6034  df-suc 6035  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-f1 6193  df-fo 6194  df-f1o 6195  df-fv 6196  df-riota 6937  df-ov 6979  df-oprab 6980  df-mpo 6981  df-om 7397  df-1st 7501  df-2nd 7502  df-wrecs 7750  df-recs 7812  df-rdg 7850  df-1o 7905  df-oadd 7909  df-er 8089  df-map 8208  df-en 8307  df-dom 8308  df-sdom 8309  df-fin 8310  df-card 9162  df-pnf 10476  df-mnf 10477  df-xr 10478  df-ltxr 10479  df-le 10480  df-sub 10672  df-neg 10673  df-nn 11440  df-n0 11708  df-z 11794  df-uz 12059  df-fz 12709  df-fzo 12850  df-hash 13506  df-word 13673  df-substr 13804  df-wlks 27084  df-clwlks 27260 This theorem is referenced by:  clwlkclwwlkf1OLD  27520
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