uspgr2wlkeqi - Metamath Proof Explorer

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Theorem uspgr2wlkeqi 29172

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 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ (2^nd ‘𝐴) = (2^nd ‘𝐵)) → 𝐴 = 𝐵)

**Proof of Theorem uspgr2wlkeqi**
Step	Hyp	Ref	Expression
1		wlkcpr 29153	. . . . 5 ⊢ (𝐴 ∈ (Walks‘𝐺) ↔ (1^st ‘𝐴)(Walks‘𝐺)(2^nd ‘𝐴))
2		wlkcpr 29153	. . . . . 6 ⊢ (𝐵 ∈ (Walks‘𝐺) ↔ (1^st ‘𝐵)(Walks‘𝐺)(2^nd ‘𝐵))
3		wlkcl 29139	. . . . . . 7 ⊢ ((1^st ‘𝐴)(Walks‘𝐺)(2^nd ‘𝐴) → (♯‘(1^st ‘𝐴)) ∈ ℕ₀)
4		fveq2 6890	. . . . . . . . . . . . 13 ⊢ ((2^nd ‘𝐴) = (2^nd ‘𝐵) → (♯‘(2^nd ‘𝐴)) = (♯‘(2^nd ‘𝐵)))
5	4	oveq1d 7426	. . . . . . . . . . . 12 ⊢ ((2^nd ‘𝐴) = (2^nd ‘𝐵) → ((♯‘(2^nd ‘𝐴)) − 1) = ((♯‘(2^nd ‘𝐵)) − 1))
6	5	eqcomd 2736	. . . . . . . . . . 11 ⊢ ((2^nd ‘𝐴) = (2^nd ‘𝐵) → ((♯‘(2^nd ‘𝐵)) − 1) = ((♯‘(2^nd ‘𝐴)) − 1))
7	6	adantl 480	. . . . . . . . . 10 ⊢ ((((1^st ‘𝐴)(Walks‘𝐺)(2^nd ‘𝐴) ∧ (1^st ‘𝐵)(Walks‘𝐺)(2^nd ‘𝐵)) ∧ (2^nd ‘𝐴) = (2^nd ‘𝐵)) → ((♯‘(2^nd ‘𝐵)) − 1) = ((♯‘(2^nd ‘𝐴)) − 1))
8		wlklenvm1 29146	. . . . . . . . . . . 12 ⊢ ((1^st ‘𝐵)(Walks‘𝐺)(2^nd ‘𝐵) → (♯‘(1^st ‘𝐵)) = ((♯‘(2^nd ‘𝐵)) − 1))
9		wlklenvm1 29146	. . . . . . . . . . . 12 ⊢ ((1^st ‘𝐴)(Walks‘𝐺)(2^nd ‘𝐴) → (♯‘(1^st ‘𝐴)) = ((♯‘(2^nd ‘𝐴)) − 1))
10	8, 9	eqeqan12rd 2745	. . . . . . . . . . 11 ⊢ (((1^st ‘𝐴)(Walks‘𝐺)(2^nd ‘𝐴) ∧ (1^st ‘𝐵)(Walks‘𝐺)(2^nd ‘𝐵)) → ((♯‘(1^st ‘𝐵)) = (♯‘(1^st ‘𝐴)) ↔ ((♯‘(2^nd ‘𝐵)) − 1) = ((♯‘(2^nd ‘𝐴)) − 1)))
11	10	adantr 479	. . . . . . . . . 10 ⊢ ((((1^st ‘𝐴)(Walks‘𝐺)(2^nd ‘𝐴) ∧ (1^st ‘𝐵)(Walks‘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 256	. . . . . . . . 9 ⊢ ((((1^st ‘𝐴)(Walks‘𝐺)(2^nd ‘𝐴) ∧ (1^st ‘𝐵)(Walks‘𝐺)(2^nd ‘𝐵)) ∧ (2^nd ‘𝐴) = (2^nd ‘𝐵)) → (♯‘(1^st ‘𝐵)) = (♯‘(1^st ‘𝐴)))
13	12	anim2i 615	. . . . . . . 8 ⊢ (((♯‘(1^st ‘𝐴)) ∈ ℕ₀ ∧ (((1^st ‘𝐴)(Walks‘𝐺)(2^nd ‘𝐴) ∧ (1^st ‘𝐵)(Walks‘𝐺)(2^nd ‘𝐵)) ∧ (2^nd ‘𝐴) = (2^nd ‘𝐵))) → ((♯‘(1^st ‘𝐴)) ∈ ℕ₀ ∧ (♯‘(1^st ‘𝐵)) = (♯‘(1^st ‘𝐴))))
14	13	exp44 436	. . . . . . 7 ⊢ ((♯‘(1^st ‘𝐴)) ∈ ℕ₀ → ((1^st ‘𝐴)(Walks‘𝐺)(2^nd ‘𝐴) → ((1^st ‘𝐵)(Walks‘𝐺)(2^nd ‘𝐵) → ((2^nd ‘𝐴) = (2^nd ‘𝐵) → ((♯‘(1^st ‘𝐴)) ∈ ℕ₀ ∧ (♯‘(1^st ‘𝐵)) = (♯‘(1^st ‘𝐴)))))))
15	3, 14	mpcom 38	. . . . . 6 ⊢ ((1^st ‘𝐴)(Walks‘𝐺)(2^nd ‘𝐴) → ((1^st ‘𝐵)(Walks‘𝐺)(2^nd ‘𝐵) → ((2^nd ‘𝐴) = (2^nd ‘𝐵) → ((♯‘(1^st ‘𝐴)) ∈ ℕ₀ ∧ (♯‘(1^st ‘𝐵)) = (♯‘(1^st ‘𝐴))))))
16	2, 15	biimtrid 241	. . . . 5 ⊢ ((1^st ‘𝐴)(Walks‘𝐺)(2^nd ‘𝐴) → (𝐵 ∈ (Walks‘𝐺) → ((2^nd ‘𝐴) = (2^nd ‘𝐵) → ((♯‘(1^st ‘𝐴)) ∈ ℕ₀ ∧ (♯‘(1^st ‘𝐵)) = (♯‘(1^st ‘𝐴))))))
17	1, 16	sylbi 216	. . . 4 ⊢ (𝐴 ∈ (Walks‘𝐺) → (𝐵 ∈ (Walks‘𝐺) → ((2^nd ‘𝐴) = (2^nd ‘𝐵) → ((♯‘(1^st ‘𝐴)) ∈ ℕ₀ ∧ (♯‘(1^st ‘𝐵)) = (♯‘(1^st ‘𝐴))))))
18	17	imp31 416	. . 3 ⊢ (((𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ (2^nd ‘𝐴) = (2^nd ‘𝐵)) → ((♯‘(1^st ‘𝐴)) ∈ ℕ₀ ∧ (♯‘(1^st ‘𝐵)) = (♯‘(1^st ‘𝐴))))
19	18	3adant1 1128	. 2 ⊢ ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ (2^nd ‘𝐴) = (2^nd ‘𝐵)) → ((♯‘(1^st ‘𝐴)) ∈ ℕ₀ ∧ (♯‘(1^st ‘𝐵)) = (♯‘(1^st ‘𝐴))))
20		simpl 481	. . . . . . 7 ⊢ ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺))) → 𝐺 ∈ USPGraph)
21		simpl 481	. . . . . . 7 ⊢ (((♯‘(1^st ‘𝐴)) ∈ ℕ₀ ∧ (♯‘(1^st ‘𝐵)) = (♯‘(1^st ‘𝐴))) → (♯‘(1^st ‘𝐴)) ∈ ℕ₀)
22	20, 21	anim12i 611	. . . . . 6 ⊢ (((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺))) ∧ ((♯‘(1^st ‘𝐴)) ∈ ℕ₀ ∧ (♯‘(1^st ‘𝐵)) = (♯‘(1^st ‘𝐴)))) → (𝐺 ∈ USPGraph ∧ (♯‘(1^st ‘𝐴)) ∈ ℕ₀))
23		simpl 481	. . . . . . . 8 ⊢ ((𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) → 𝐴 ∈ (Walks‘𝐺))
24	23	adantl 480	. . . . . . 7 ⊢ ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺))) → 𝐴 ∈ (Walks‘𝐺))
25		eqidd 2731	. . . . . . 7 ⊢ (((♯‘(1^st ‘𝐴)) ∈ ℕ₀ ∧ (♯‘(1^st ‘𝐵)) = (♯‘(1^st ‘𝐴))) → (♯‘(1^st ‘𝐴)) = (♯‘(1^st ‘𝐴)))
26	24, 25	anim12i 611	. . . . . 6 ⊢ (((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺))) ∧ ((♯‘(1^st ‘𝐴)) ∈ ℕ₀ ∧ (♯‘(1^st ‘𝐵)) = (♯‘(1^st ‘𝐴)))) → (𝐴 ∈ (Walks‘𝐺) ∧ (♯‘(1^st ‘𝐴)) = (♯‘(1^st ‘𝐴))))
27		simpr 483	. . . . . . . 8 ⊢ ((𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) → 𝐵 ∈ (Walks‘𝐺))
28	27	adantl 480	. . . . . . 7 ⊢ ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺))) → 𝐵 ∈ (Walks‘𝐺))
29		simpr 483	. . . . . . 7 ⊢ (((♯‘(1^st ‘𝐴)) ∈ ℕ₀ ∧ (♯‘(1^st ‘𝐵)) = (♯‘(1^st ‘𝐴))) → (♯‘(1^st ‘𝐵)) = (♯‘(1^st ‘𝐴)))
30	28, 29	anim12i 611	. . . . . 6 ⊢ (((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺))) ∧ ((♯‘(1^st ‘𝐴)) ∈ ℕ₀ ∧ (♯‘(1^st ‘𝐵)) = (♯‘(1^st ‘𝐴)))) → (𝐵 ∈ (Walks‘𝐺) ∧ (♯‘(1^st ‘𝐵)) = (♯‘(1^st ‘𝐴))))
31		uspgr2wlkeq2 29171	. . . . . 6 ⊢ (((𝐺 ∈ USPGraph ∧ (♯‘(1^st ‘𝐴)) ∈ ℕ₀) ∧ (𝐴 ∈ (Walks‘𝐺) ∧ (♯‘(1^st ‘𝐴)) = (♯‘(1^st ‘𝐴))) ∧ (𝐵 ∈ (Walks‘𝐺) ∧ (♯‘(1^st ‘𝐵)) = (♯‘(1^st ‘𝐴)))) → ((2^nd ‘𝐴) = (2^nd ‘𝐵) → 𝐴 = 𝐵))
32	22, 26, 30, 31	syl3anc 1369	. . . . 5 ⊢ (((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺))) ∧ ((♯‘(1^st ‘𝐴)) ∈ ℕ₀ ∧ (♯‘(1^st ‘𝐵)) = (♯‘(1^st ‘𝐴)))) → ((2^nd ‘𝐴) = (2^nd ‘𝐵) → 𝐴 = 𝐵))
33	32	ex 411	. . . 4 ⊢ ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺))) → (((♯‘(1^st ‘𝐴)) ∈ ℕ₀ ∧ (♯‘(1^st ‘𝐵)) = (♯‘(1^st ‘𝐴))) → ((2^nd ‘𝐴) = (2^nd ‘𝐵) → 𝐴 = 𝐵)))
34	33	com23 86	. . 3 ⊢ ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺))) → ((2^nd ‘𝐴) = (2^nd ‘𝐵) → (((♯‘(1^st ‘𝐴)) ∈ ℕ₀ ∧ (♯‘(1^st ‘𝐵)) = (♯‘(1^st ‘𝐴))) → 𝐴 = 𝐵)))
35	34	3impia 1115	. 2 ⊢ ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘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 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ (2^nd ‘𝐴) = (2^nd ‘𝐵)) → 𝐴 = 𝐵)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1085 = wceq 1539 ∈ wcel 2104 class class class wbr 5147 ‘cfv 6542 (class class class)co 7411 1^st c1st 7975 2^nd c2nd 7976 1c1 11113 − cmin 11448 ℕ₀cn0 12476 ♯chash 14294 USPGraphcuspgr 28675 Walkscwlks 29120

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189

This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-ifp 1060 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-2o 8469 df-oadd 8472 df-er 8705 df-map 8824 df-pm 8825 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-dju 9898 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-n0 12477 df-xnn0 12549 df-z 12563 df-uz 12827 df-fz 13489 df-fzo 13632 df-hash 14295 df-word 14469 df-edg 28575 df-uhgr 28585 df-upgr 28609 df-uspgr 28677 df-wlks 29123

This theorem is referenced by: wlkswwlksf1o 29400

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