uspgr2wlkeq2 - Mathbox for Alexander van der Vekens

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Theorem uspgr2wlkeq2 40855

Description: Conditions for two walks within the same simple pseudograph to be identical. It is sufficient that the vertices (in the same order) are identical. (Contributed by Alexander van der Vekens, 25-Aug-2018.) (Revised by AV, 14-Apr-2021.)

**Assertion**
Ref	Expression
uspgr2wlkeq2	⊢ (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ₀) ∧ (𝐴 ∈ (1Walks‘𝐺) ∧ (#‘(1^st ‘𝐴)) = 𝑁) ∧ (𝐵 ∈ (1Walks‘𝐺) ∧ (#‘(1^st ‘𝐵)) = 𝑁)) → ((2^nd ‘𝐴) = (2^nd ‘𝐵) → 𝐴 = 𝐵))

Proof of Theorem uspgr2wlkeq2 Dummy variable 𝑖 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 476	. . . . . 6 ⊢ ((𝐵 ∈ (1Walks‘𝐺) ∧ (#‘(1^st ‘𝐵)) = 𝑁) → (#‘(1^st ‘𝐵)) = 𝑁)
2	1	eqcomd 2616	. . . . 5 ⊢ ((𝐵 ∈ (1Walks‘𝐺) ∧ (#‘(1^st ‘𝐵)) = 𝑁) → 𝑁 = (#‘(1^st ‘𝐵)))
3	2	3ad2ant3 1077	. . . 4 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ₀) ∧ (𝐴 ∈ (1Walks‘𝐺) ∧ (#‘(1^st ‘𝐴)) = 𝑁) ∧ (𝐵 ∈ (1Walks‘𝐺) ∧ (#‘(1^st ‘𝐵)) = 𝑁)) → 𝑁 = (#‘(1^st ‘𝐵)))
4	3	adantr 480	. . 3 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ₀) ∧ (𝐴 ∈ (1Walks‘𝐺) ∧ (#‘(1^st ‘𝐴)) = 𝑁) ∧ (𝐵 ∈ (1Walks‘𝐺) ∧ (#‘(1^st ‘𝐵)) = 𝑁)) ∧ (2^nd ‘𝐴) = (2^nd ‘𝐵)) → 𝑁 = (#‘(1^st ‘𝐵)))
5		fveq1 6102	. . . . 5 ⊢ ((2^nd ‘𝐴) = (2^nd ‘𝐵) → ((2^nd ‘𝐴)‘𝑖) = ((2^nd ‘𝐵)‘𝑖))
6	5	adantl 481	. . . 4 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ₀) ∧ (𝐴 ∈ (1Walks‘𝐺) ∧ (#‘(1^st ‘𝐴)) = 𝑁) ∧ (𝐵 ∈ (1Walks‘𝐺) ∧ (#‘(1^st ‘𝐵)) = 𝑁)) ∧ (2^nd ‘𝐴) = (2^nd ‘𝐵)) → ((2^nd ‘𝐴)‘𝑖) = ((2^nd ‘𝐵)‘𝑖))
7	6	ralrimivw 2950	. . 3 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ₀) ∧ (𝐴 ∈ (1Walks‘𝐺) ∧ (#‘(1^st ‘𝐴)) = 𝑁) ∧ (𝐵 ∈ (1Walks‘𝐺) ∧ (#‘(1^st ‘𝐵)) = 𝑁)) ∧ (2^nd ‘𝐴) = (2^nd ‘𝐵)) → ∀𝑖 ∈ (0...𝑁)((2^nd ‘𝐴)‘𝑖) = ((2^nd ‘𝐵)‘𝑖))
8		simpl1l 1105	. . . 4 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ₀) ∧ (𝐴 ∈ (1Walks‘𝐺) ∧ (#‘(1^st ‘𝐴)) = 𝑁) ∧ (𝐵 ∈ (1Walks‘𝐺) ∧ (#‘(1^st ‘𝐵)) = 𝑁)) ∧ (2^nd ‘𝐴) = (2^nd ‘𝐵)) → 𝐺 ∈ USPGraph )
9		simpl 472	. . . . . . 7 ⊢ ((𝐴 ∈ (1Walks‘𝐺) ∧ (#‘(1^st ‘𝐴)) = 𝑁) → 𝐴 ∈ (1Walks‘𝐺))
10		simpl 472	. . . . . . 7 ⊢ ((𝐵 ∈ (1Walks‘𝐺) ∧ (#‘(1^st ‘𝐵)) = 𝑁) → 𝐵 ∈ (1Walks‘𝐺))
11	9, 10	anim12i 588	. . . . . 6 ⊢ (((𝐴 ∈ (1Walks‘𝐺) ∧ (#‘(1^st ‘𝐴)) = 𝑁) ∧ (𝐵 ∈ (1Walks‘𝐺) ∧ (#‘(1^st ‘𝐵)) = 𝑁)) → (𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘𝐺)))
12	11	3adant1 1072	. . . . 5 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ₀) ∧ (𝐴 ∈ (1Walks‘𝐺) ∧ (#‘(1^st ‘𝐴)) = 𝑁) ∧ (𝐵 ∈ (1Walks‘𝐺) ∧ (#‘(1^st ‘𝐵)) = 𝑁)) → (𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘𝐺)))
13	12	adantr 480	. . . 4 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ₀) ∧ (𝐴 ∈ (1Walks‘𝐺) ∧ (#‘(1^st ‘𝐴)) = 𝑁) ∧ (𝐵 ∈ (1Walks‘𝐺) ∧ (#‘(1^st ‘𝐵)) = 𝑁)) ∧ (2^nd ‘𝐴) = (2^nd ‘𝐵)) → (𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘𝐺)))
14		simpr 476	. . . . . . 7 ⊢ ((𝐴 ∈ (1Walks‘𝐺) ∧ (#‘(1^st ‘𝐴)) = 𝑁) → (#‘(1^st ‘𝐴)) = 𝑁)
15	14	eqcomd 2616	. . . . . 6 ⊢ ((𝐴 ∈ (1Walks‘𝐺) ∧ (#‘(1^st ‘𝐴)) = 𝑁) → 𝑁 = (#‘(1^st ‘𝐴)))
16	15	3ad2ant2 1076	. . . . 5 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ₀) ∧ (𝐴 ∈ (1Walks‘𝐺) ∧ (#‘(1^st ‘𝐴)) = 𝑁) ∧ (𝐵 ∈ (1Walks‘𝐺) ∧ (#‘(1^st ‘𝐵)) = 𝑁)) → 𝑁 = (#‘(1^st ‘𝐴)))
17	16	adantr 480	. . . 4 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ₀) ∧ (𝐴 ∈ (1Walks‘𝐺) ∧ (#‘(1^st ‘𝐴)) = 𝑁) ∧ (𝐵 ∈ (1Walks‘𝐺) ∧ (#‘(1^st ‘𝐵)) = 𝑁)) ∧ (2^nd ‘𝐴) = (2^nd ‘𝐵)) → 𝑁 = (#‘(1^st ‘𝐴)))
18		uspgr2wlkeq 40854	. . . 4 ⊢ ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘𝐺)) ∧ 𝑁 = (#‘(1^st ‘𝐴))) → (𝐴 = 𝐵 ↔ (𝑁 = (#‘(1^st ‘𝐵)) ∧ ∀𝑖 ∈ (0...𝑁)((2^nd ‘𝐴)‘𝑖) = ((2^nd ‘𝐵)‘𝑖))))
19	8, 13, 17, 18	syl3anc 1318	. . 3 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ₀) ∧ (𝐴 ∈ (1Walks‘𝐺) ∧ (#‘(1^st ‘𝐴)) = 𝑁) ∧ (𝐵 ∈ (1Walks‘𝐺) ∧ (#‘(1^st ‘𝐵)) = 𝑁)) ∧ (2^nd ‘𝐴) = (2^nd ‘𝐵)) → (𝐴 = 𝐵 ↔ (𝑁 = (#‘(1^st ‘𝐵)) ∧ ∀𝑖 ∈ (0...𝑁)((2^nd ‘𝐴)‘𝑖) = ((2^nd ‘𝐵)‘𝑖))))
20	4, 7, 19	mpbir2and 959	. 2 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ₀) ∧ (𝐴 ∈ (1Walks‘𝐺) ∧ (#‘(1^st ‘𝐴)) = 𝑁) ∧ (𝐵 ∈ (1Walks‘𝐺) ∧ (#‘(1^st ‘𝐵)) = 𝑁)) ∧ (2^nd ‘𝐴) = (2^nd ‘𝐵)) → 𝐴 = 𝐵)
21	20	ex 449	1 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ₀) ∧ (𝐴 ∈ (1Walks‘𝐺) ∧ (#‘(1^st ‘𝐴)) = 𝑁) ∧ (𝐵 ∈ (1Walks‘𝐺) ∧ (#‘(1^st ‘𝐵)) = 𝑁)) → ((2^nd ‘𝐴) = (2^nd ‘𝐵) → 𝐴 = 𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ‘cfv 5804 (class class class)co 6549 1^st c1st 7057 2^nd c2nd 7058 0cc0 9815 ℕ₀cn0 11169 ...cfz 12197 #chash 12979 USPGraph cuspgr 40378 1Walksc1wlks 40796

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-ifp 1007 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-rmo 2904 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-2o 7448 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-cda 8873 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 df-uhgr 25724 df-upgr 25749 df-edga 25793 df-uspgr 40380 df-1wlks 40800 df-wlks 40801

This theorem is referenced by: uspgr2wlkeqi 40856 wlknwwlksninj 41086 wlkwwlkinj 41093

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