uspgr2wlkeqi - Mathbox for Alexander van der Vekens

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Theorem uspgr2wlkeqi 40856

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 AV, 6-May-2021.)

**Assertion**
Ref	Expression
uspgr2wlkeqi	⊢ ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘𝐺)) ∧ (2^nd ‘𝐴) = (2^nd ‘𝐵)) → 𝐴 = 𝐵)

**Proof of Theorem uspgr2wlkeqi**
Step	Hyp	Ref	Expression
1		1wlkcpr 40833	. . . . 5 ⊢ (𝐴 ∈ (1Walks‘𝐺) ↔ (1^st ‘𝐴)(1Walks‘𝐺)(2^nd ‘𝐴))
2		1wlkcpr 40833	. . . . . 6 ⊢ (𝐵 ∈ (1Walks‘𝐺) ↔ (1^st ‘𝐵)(1Walks‘𝐺)(2^nd ‘𝐵))
3		1wlkcl 40820	. . . . . . 7 ⊢ ((1^st ‘𝐴)(1Walks‘𝐺)(2^nd ‘𝐴) → (#‘(1^st ‘𝐴)) ∈ ℕ₀)
4		fveq2 6103	. . . . . . . . . . . . 13 ⊢ ((2^nd ‘𝐴) = (2^nd ‘𝐵) → (#‘(2^nd ‘𝐴)) = (#‘(2^nd ‘𝐵)))
5	4	oveq1d 6564	. . . . . . . . . . . 12 ⊢ ((2^nd ‘𝐴) = (2^nd ‘𝐵) → ((#‘(2^nd ‘𝐴)) − 1) = ((#‘(2^nd ‘𝐵)) − 1))
6	5	eqcomd 2616	. . . . . . . . . . 11 ⊢ ((2^nd ‘𝐴) = (2^nd ‘𝐵) → ((#‘(2^nd ‘𝐵)) − 1) = ((#‘(2^nd ‘𝐴)) − 1))
7	6	adantl 481	. . . . . . . . . 10 ⊢ ((((1^st ‘𝐴)(1Walks‘𝐺)(2^nd ‘𝐴) ∧ (1^st ‘𝐵)(1Walks‘𝐺)(2^nd ‘𝐵)) ∧ (2^nd ‘𝐴) = (2^nd ‘𝐵)) → ((#‘(2^nd ‘𝐵)) − 1) = ((#‘(2^nd ‘𝐴)) − 1))
8		1wlklenvm1 40826	. . . . . . . . . . . 12 ⊢ ((1^st ‘𝐵)(1Walks‘𝐺)(2^nd ‘𝐵) → (#‘(1^st ‘𝐵)) = ((#‘(2^nd ‘𝐵)) − 1))
9		1wlklenvm1 40826	. . . . . . . . . . . 12 ⊢ ((1^st ‘𝐴)(1Walks‘𝐺)(2^nd ‘𝐴) → (#‘(1^st ‘𝐴)) = ((#‘(2^nd ‘𝐴)) − 1))
10	8, 9	eqeqan12rd 2628	. . . . . . . . . . 11 ⊢ (((1^st ‘𝐴)(1Walks‘𝐺)(2^nd ‘𝐴) ∧ (1^st ‘𝐵)(1Walks‘𝐺)(2^nd ‘𝐵)) → ((#‘(1^st ‘𝐵)) = (#‘(1^st ‘𝐴)) ↔ ((#‘(2^nd ‘𝐵)) − 1) = ((#‘(2^nd ‘𝐴)) − 1)))
11	10	adantr 480	. . . . . . . . . 10 ⊢ ((((1^st ‘𝐴)(1Walks‘𝐺)(2^nd ‘𝐴) ∧ (1^st ‘𝐵)(1Walks‘𝐺)(2^nd ‘𝐵)) ∧ (2^nd ‘𝐴) = (2^nd ‘𝐵)) → ((#‘(1^st ‘𝐵)) = (#‘(1^st ‘𝐴)) ↔ ((#‘(2^nd ‘𝐵)) − 1) = ((#‘(2^nd ‘𝐴)) − 1)))
12	7, 11	mpbird 246	. . . . . . . . 9 ⊢ ((((1^st ‘𝐴)(1Walks‘𝐺)(2^nd ‘𝐴) ∧ (1^st ‘𝐵)(1Walks‘𝐺)(2^nd ‘𝐵)) ∧ (2^nd ‘𝐴) = (2^nd ‘𝐵)) → (#‘(1^st ‘𝐵)) = (#‘(1^st ‘𝐴)))
13	12	anim2i 591	. . . . . . . 8 ⊢ (((#‘(1^st ‘𝐴)) ∈ ℕ₀ ∧ (((1^st ‘𝐴)(1Walks‘𝐺)(2^nd ‘𝐴) ∧ (1^st ‘𝐵)(1Walks‘𝐺)(2^nd ‘𝐵)) ∧ (2^nd ‘𝐴) = (2^nd ‘𝐵))) → ((#‘(1^st ‘𝐴)) ∈ ℕ₀ ∧ (#‘(1^st ‘𝐵)) = (#‘(1^st ‘𝐴))))
14	13	exp44 639	. . . . . . 7 ⊢ ((#‘(1^st ‘𝐴)) ∈ ℕ₀ → ((1^st ‘𝐴)(1Walks‘𝐺)(2^nd ‘𝐴) → ((1^st ‘𝐵)(1Walks‘𝐺)(2^nd ‘𝐵) → ((2^nd ‘𝐴) = (2^nd ‘𝐵) → ((#‘(1^st ‘𝐴)) ∈ ℕ₀ ∧ (#‘(1^st ‘𝐵)) = (#‘(1^st ‘𝐴)))))))
15	3, 14	mpcom 37	. . . . . 6 ⊢ ((1^st ‘𝐴)(1Walks‘𝐺)(2^nd ‘𝐴) → ((1^st ‘𝐵)(1Walks‘𝐺)(2^nd ‘𝐵) → ((2^nd ‘𝐴) = (2^nd ‘𝐵) → ((#‘(1^st ‘𝐴)) ∈ ℕ₀ ∧ (#‘(1^st ‘𝐵)) = (#‘(1^st ‘𝐴))))))
16	2, 15	syl5bi 231	. . . . 5 ⊢ ((1^st ‘𝐴)(1Walks‘𝐺)(2^nd ‘𝐴) → (𝐵 ∈ (1Walks‘𝐺) → ((2^nd ‘𝐴) = (2^nd ‘𝐵) → ((#‘(1^st ‘𝐴)) ∈ ℕ₀ ∧ (#‘(1^st ‘𝐵)) = (#‘(1^st ‘𝐴))))))
17	1, 16	sylbi 206	. . . 4 ⊢ (𝐴 ∈ (1Walks‘𝐺) → (𝐵 ∈ (1Walks‘𝐺) → ((2^nd ‘𝐴) = (2^nd ‘𝐵) → ((#‘(1^st ‘𝐴)) ∈ ℕ₀ ∧ (#‘(1^st ‘𝐵)) = (#‘(1^st ‘𝐴))))))
18	17	imp31 447	. . 3 ⊢ (((𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘𝐺)) ∧ (2^nd ‘𝐴) = (2^nd ‘𝐵)) → ((#‘(1^st ‘𝐴)) ∈ ℕ₀ ∧ (#‘(1^st ‘𝐵)) = (#‘(1^st ‘𝐴))))
19	18	3adant1 1072	. 2 ⊢ ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘𝐺)) ∧ (2^nd ‘𝐴) = (2^nd ‘𝐵)) → ((#‘(1^st ‘𝐴)) ∈ ℕ₀ ∧ (#‘(1^st ‘𝐵)) = (#‘(1^st ‘𝐴))))
20		simpl 472	. . . . . . 7 ⊢ ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘𝐺))) → 𝐺 ∈ USPGraph )
21		simpl 472	. . . . . . 7 ⊢ (((#‘(1^st ‘𝐴)) ∈ ℕ₀ ∧ (#‘(1^st ‘𝐵)) = (#‘(1^st ‘𝐴))) → (#‘(1^st ‘𝐴)) ∈ ℕ₀)
22	20, 21	anim12i 588	. . . . . 6 ⊢ (((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘𝐺))) ∧ ((#‘(1^st ‘𝐴)) ∈ ℕ₀ ∧ (#‘(1^st ‘𝐵)) = (#‘(1^st ‘𝐴)))) → (𝐺 ∈ USPGraph ∧ (#‘(1^st ‘𝐴)) ∈ ℕ₀))
23		simpl 472	. . . . . . . 8 ⊢ ((𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘𝐺)) → 𝐴 ∈ (1Walks‘𝐺))
24	23	adantl 481	. . . . . . 7 ⊢ ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘𝐺))) → 𝐴 ∈ (1Walks‘𝐺))
25		eqidd 2611	. . . . . . 7 ⊢ (((#‘(1^st ‘𝐴)) ∈ ℕ₀ ∧ (#‘(1^st ‘𝐵)) = (#‘(1^st ‘𝐴))) → (#‘(1^st ‘𝐴)) = (#‘(1^st ‘𝐴)))
26	24, 25	anim12i 588	. . . . . 6 ⊢ (((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘𝐺))) ∧ ((#‘(1^st ‘𝐴)) ∈ ℕ₀ ∧ (#‘(1^st ‘𝐵)) = (#‘(1^st ‘𝐴)))) → (𝐴 ∈ (1Walks‘𝐺) ∧ (#‘(1^st ‘𝐴)) = (#‘(1^st ‘𝐴))))
27		simpr 476	. . . . . . . 8 ⊢ ((𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘𝐺)) → 𝐵 ∈ (1Walks‘𝐺))
28	27	adantl 481	. . . . . . 7 ⊢ ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘𝐺))) → 𝐵 ∈ (1Walks‘𝐺))
29		simpr 476	. . . . . . 7 ⊢ (((#‘(1^st ‘𝐴)) ∈ ℕ₀ ∧ (#‘(1^st ‘𝐵)) = (#‘(1^st ‘𝐴))) → (#‘(1^st ‘𝐵)) = (#‘(1^st ‘𝐴)))
30	28, 29	anim12i 588	. . . . . 6 ⊢ (((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘𝐺))) ∧ ((#‘(1^st ‘𝐴)) ∈ ℕ₀ ∧ (#‘(1^st ‘𝐵)) = (#‘(1^st ‘𝐴)))) → (𝐵 ∈ (1Walks‘𝐺) ∧ (#‘(1^st ‘𝐵)) = (#‘(1^st ‘𝐴))))
31		uspgr2wlkeq2 40855	. . . . . 6 ⊢ (((𝐺 ∈ USPGraph ∧ (#‘(1^st ‘𝐴)) ∈ ℕ₀) ∧ (𝐴 ∈ (1Walks‘𝐺) ∧ (#‘(1^st ‘𝐴)) = (#‘(1^st ‘𝐴))) ∧ (𝐵 ∈ (1Walks‘𝐺) ∧ (#‘(1^st ‘𝐵)) = (#‘(1^st ‘𝐴)))) → ((2^nd ‘𝐴) = (2^nd ‘𝐵) → 𝐴 = 𝐵))
32	22, 26, 30, 31	syl3anc 1318	. . . . 5 ⊢ (((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘𝐺))) ∧ ((#‘(1^st ‘𝐴)) ∈ ℕ₀ ∧ (#‘(1^st ‘𝐵)) = (#‘(1^st ‘𝐴)))) → ((2^nd ‘𝐴) = (2^nd ‘𝐵) → 𝐴 = 𝐵))
33	32	ex 449	. . . 4 ⊢ ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘𝐺))) → (((#‘(1^st ‘𝐴)) ∈ ℕ₀ ∧ (#‘(1^st ‘𝐵)) = (#‘(1^st ‘𝐴))) → ((2^nd ‘𝐴) = (2^nd ‘𝐵) → 𝐴 = 𝐵)))
34	33	com23 84	. . 3 ⊢ ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘𝐺))) → ((2^nd ‘𝐴) = (2^nd ‘𝐵) → (((#‘(1^st ‘𝐴)) ∈ ℕ₀ ∧ (#‘(1^st ‘𝐵)) = (#‘(1^st ‘𝐴))) → 𝐴 = 𝐵)))
35	34	3impia 1253	. 2 ⊢ ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘𝐺)) ∧ (2^nd ‘𝐴) = (2^nd ‘𝐵)) → (((#‘(1^st ‘𝐴)) ∈ ℕ₀ ∧ (#‘(1^st ‘𝐵)) = (#‘(1^st ‘𝐴))) → 𝐴 = 𝐵))
36	19, 35	mpd 15	1 ⊢ ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘𝐺)) ∧ (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 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 1^st c1st 7057 2^nd c2nd 7058 1c1 9816 − cmin 10145 ℕ₀cn0 11169 #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: 1wlkpwwlkf1ouspgr 41076

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