uspgr2wlkeqi - Metamath Proof Explorer

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Theorem uspgr2wlkeqi 29173

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 29154	. . . . 5 ⊢ (𝐴 ∈ (Walks‘𝐺) ↔ (1^st ‘𝐴)(Walks‘𝐺)(2^nd ‘𝐴))
2		wlkcpr 29154	. . . . . 6 ⊢ (𝐵 ∈ (Walks‘𝐺) ↔ (1^st ‘𝐵)(Walks‘𝐺)(2^nd ‘𝐵))
3		wlkcl 29140	. . . . . . 7 ⊢ ((1^st ‘𝐴)(Walks‘𝐺)(2^nd ‘𝐴) → (♯‘(1^st ‘𝐴)) ∈ ℕ₀)
4		fveq2 6891	. . . . . . . . . . . . 13 ⊢ ((2^nd ‘𝐴) = (2^nd ‘𝐵) → (♯‘(2^nd ‘𝐴)) = (♯‘(2^nd ‘𝐵)))
5	4	oveq1d 7427	. . . . . . . . . . . 12 ⊢ ((2^nd ‘𝐴) = (2^nd ‘𝐵) → ((♯‘(2^nd ‘𝐴)) − 1) = ((♯‘(2^nd ‘𝐵)) − 1))
6	5	eqcomd 2737	. . . . . . . . . . 11 ⊢ ((2^nd ‘𝐴) = (2^nd ‘𝐵) → ((♯‘(2^nd ‘𝐵)) − 1) = ((♯‘(2^nd ‘𝐴)) − 1))
7	6	adantl 481	. . . . . . . . . 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 29147	. . . . . . . . . . . 12 ⊢ ((1^st ‘𝐵)(Walks‘𝐺)(2^nd ‘𝐵) → (♯‘(1^st ‘𝐵)) = ((♯‘(2^nd ‘𝐵)) − 1))
9		wlklenvm1 29147	. . . . . . . . . . . 12 ⊢ ((1^st ‘𝐴)(Walks‘𝐺)(2^nd ‘𝐴) → (♯‘(1^st ‘𝐴)) = ((♯‘(2^nd ‘𝐴)) − 1))
10	8, 9	eqeqan12rd 2746	. . . . . . . . . . 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 480	. . . . . . . . . 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 257	. . . . . . . . 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 616	. . . . . . . 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 437	. . . . . . 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 417	. . 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 1129	. 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 482	. . . . . . 7 ⊢ ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺))) → 𝐺 ∈ USPGraph)
21		simpl 482	. . . . . . 7 ⊢ (((♯‘(1^st ‘𝐴)) ∈ ℕ₀ ∧ (♯‘(1^st ‘𝐵)) = (♯‘(1^st ‘𝐴))) → (♯‘(1^st ‘𝐴)) ∈ ℕ₀)
22	20, 21	anim12i 612	. . . . . 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 482	. . . . . . . 8 ⊢ ((𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) → 𝐴 ∈ (Walks‘𝐺))
24	23	adantl 481	. . . . . . 7 ⊢ ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺))) → 𝐴 ∈ (Walks‘𝐺))
25		eqidd 2732	. . . . . . 7 ⊢ (((♯‘(1^st ‘𝐴)) ∈ ℕ₀ ∧ (♯‘(1^st ‘𝐵)) = (♯‘(1^st ‘𝐴))) → (♯‘(1^st ‘𝐴)) = (♯‘(1^st ‘𝐴)))
26	24, 25	anim12i 612	. . . . . 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 484	. . . . . . . 8 ⊢ ((𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) → 𝐵 ∈ (Walks‘𝐺))
28	27	adantl 481	. . . . . . 7 ⊢ ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺))) → 𝐵 ∈ (Walks‘𝐺))
29		simpr 484	. . . . . . 7 ⊢ (((♯‘(1^st ‘𝐴)) ∈ ℕ₀ ∧ (♯‘(1^st ‘𝐵)) = (♯‘(1^st ‘𝐴))) → (♯‘(1^st ‘𝐵)) = (♯‘(1^st ‘𝐴)))
30	28, 29	anim12i 612	. . . . . 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 29172	. . . . . 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 1370	. . . . 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 412	. . . 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 1116	. 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 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2105 class class class wbr 5148 ‘cfv 6543 (class class class)co 7412 1^st c1st 7977 2^nd c2nd 7978 1c1 11115 − cmin 11449 ℕ₀cn0 12477 ♯chash 14295 USPGraphcuspgr 28676 Walkscwlks 29121

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11170 ax-resscn 11171 ax-1cn 11172 ax-icn 11173 ax-addcl 11174 ax-addrcl 11175 ax-mulcl 11176 ax-mulrcl 11177 ax-mulcom 11178 ax-addass 11179 ax-mulass 11180 ax-distr 11181 ax-i2m1 11182 ax-1ne0 11183 ax-1rid 11184 ax-rnegex 11185 ax-rrecex 11186 ax-cnre 11187 ax-pre-lttri 11188 ax-pre-lttrn 11189 ax-pre-ltadd 11190 ax-pre-mulgt0 11191

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-ifp 1061 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8270 df-wrecs 8301 df-recs 8375 df-rdg 8414 df-1o 8470 df-2o 8471 df-oadd 8474 df-er 8707 df-map 8826 df-pm 8827 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-dju 9900 df-card 9938 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-nn 12218 df-2 12280 df-n0 12478 df-xnn0 12550 df-z 12564 df-uz 12828 df-fz 13490 df-fzo 13633 df-hash 14296 df-word 14470 df-edg 28576 df-uhgr 28586 df-upgr 28610 df-uspgr 28678 df-wlks 29124

This theorem is referenced by: wlkswwlksf1o 29401

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