uspgr2wlkeq2 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > uspgr2wlkeq2		Structured version Visualization version GIF version

Theorem uspgr2wlkeq2 29666

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 ∈ (Walks‘𝐺) ∧ (♯‘(1^st ‘𝐴)) = 𝑁) ∧ (𝐵 ∈ (Walks‘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 484	. . . . . 6 ⊢ ((𝐵 ∈ (Walks‘𝐺) ∧ (♯‘(1^st ‘𝐵)) = 𝑁) → (♯‘(1^st ‘𝐵)) = 𝑁)
2	1	eqcomd 2742	. . . . 5 ⊢ ((𝐵 ∈ (Walks‘𝐺) ∧ (♯‘(1^st ‘𝐵)) = 𝑁) → 𝑁 = (♯‘(1^st ‘𝐵)))
3	2	3ad2ant3 1135	. . . 4 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ₀) ∧ (𝐴 ∈ (Walks‘𝐺) ∧ (♯‘(1^st ‘𝐴)) = 𝑁) ∧ (𝐵 ∈ (Walks‘𝐺) ∧ (♯‘(1^st ‘𝐵)) = 𝑁)) → 𝑁 = (♯‘(1^st ‘𝐵)))
4	3	adantr 480	. . 3 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ₀) ∧ (𝐴 ∈ (Walks‘𝐺) ∧ (♯‘(1^st ‘𝐴)) = 𝑁) ∧ (𝐵 ∈ (Walks‘𝐺) ∧ (♯‘(1^st ‘𝐵)) = 𝑁)) ∧ (2^nd ‘𝐴) = (2^nd ‘𝐵)) → 𝑁 = (♯‘(1^st ‘𝐵)))
5		fveq1 6904	. . . . 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 ∈ (Walks‘𝐺) ∧ (♯‘(1^st ‘𝐴)) = 𝑁) ∧ (𝐵 ∈ (Walks‘𝐺) ∧ (♯‘(1^st ‘𝐵)) = 𝑁)) ∧ (2^nd ‘𝐴) = (2^nd ‘𝐵)) → ((2^nd ‘𝐴)‘𝑖) = ((2^nd ‘𝐵)‘𝑖))
7	6	ralrimivw 3149	. . 3 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ₀) ∧ (𝐴 ∈ (Walks‘𝐺) ∧ (♯‘(1^st ‘𝐴)) = 𝑁) ∧ (𝐵 ∈ (Walks‘𝐺) ∧ (♯‘(1^st ‘𝐵)) = 𝑁)) ∧ (2^nd ‘𝐴) = (2^nd ‘𝐵)) → ∀𝑖 ∈ (0...𝑁)((2^nd ‘𝐴)‘𝑖) = ((2^nd ‘𝐵)‘𝑖))
8		simpl1l 1224	. . . 4 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ₀) ∧ (𝐴 ∈ (Walks‘𝐺) ∧ (♯‘(1^st ‘𝐴)) = 𝑁) ∧ (𝐵 ∈ (Walks‘𝐺) ∧ (♯‘(1^st ‘𝐵)) = 𝑁)) ∧ (2^nd ‘𝐴) = (2^nd ‘𝐵)) → 𝐺 ∈ USPGraph)
9		simpl 482	. . . . . . 7 ⊢ ((𝐴 ∈ (Walks‘𝐺) ∧ (♯‘(1^st ‘𝐴)) = 𝑁) → 𝐴 ∈ (Walks‘𝐺))
10		simpl 482	. . . . . . 7 ⊢ ((𝐵 ∈ (Walks‘𝐺) ∧ (♯‘(1^st ‘𝐵)) = 𝑁) → 𝐵 ∈ (Walks‘𝐺))
11	9, 10	anim12i 613	. . . . . 6 ⊢ (((𝐴 ∈ (Walks‘𝐺) ∧ (♯‘(1^st ‘𝐴)) = 𝑁) ∧ (𝐵 ∈ (Walks‘𝐺) ∧ (♯‘(1^st ‘𝐵)) = 𝑁)) → (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)))
12	11	3adant1 1130	. . . . 5 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ₀) ∧ (𝐴 ∈ (Walks‘𝐺) ∧ (♯‘(1^st ‘𝐴)) = 𝑁) ∧ (𝐵 ∈ (Walks‘𝐺) ∧ (♯‘(1^st ‘𝐵)) = 𝑁)) → (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)))
13	12	adantr 480	. . . 4 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ₀) ∧ (𝐴 ∈ (Walks‘𝐺) ∧ (♯‘(1^st ‘𝐴)) = 𝑁) ∧ (𝐵 ∈ (Walks‘𝐺) ∧ (♯‘(1^st ‘𝐵)) = 𝑁)) ∧ (2^nd ‘𝐴) = (2^nd ‘𝐵)) → (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)))
14		simpr 484	. . . . . . 7 ⊢ ((𝐴 ∈ (Walks‘𝐺) ∧ (♯‘(1^st ‘𝐴)) = 𝑁) → (♯‘(1^st ‘𝐴)) = 𝑁)
15	14	eqcomd 2742	. . . . . 6 ⊢ ((𝐴 ∈ (Walks‘𝐺) ∧ (♯‘(1^st ‘𝐴)) = 𝑁) → 𝑁 = (♯‘(1^st ‘𝐴)))
16	15	3ad2ant2 1134	. . . . 5 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ₀) ∧ (𝐴 ∈ (Walks‘𝐺) ∧ (♯‘(1^st ‘𝐴)) = 𝑁) ∧ (𝐵 ∈ (Walks‘𝐺) ∧ (♯‘(1^st ‘𝐵)) = 𝑁)) → 𝑁 = (♯‘(1^st ‘𝐴)))
17	16	adantr 480	. . . 4 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ₀) ∧ (𝐴 ∈ (Walks‘𝐺) ∧ (♯‘(1^st ‘𝐴)) = 𝑁) ∧ (𝐵 ∈ (Walks‘𝐺) ∧ (♯‘(1^st ‘𝐵)) = 𝑁)) ∧ (2^nd ‘𝐴) = (2^nd ‘𝐵)) → 𝑁 = (♯‘(1^st ‘𝐴)))
18		uspgr2wlkeq 29665	. . . 4 ⊢ ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘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 1372	. . 3 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ₀) ∧ (𝐴 ∈ (Walks‘𝐺) ∧ (♯‘(1^st ‘𝐴)) = 𝑁) ∧ (𝐵 ∈ (Walks‘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 713	. 2 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ₀) ∧ (𝐴 ∈ (Walks‘𝐺) ∧ (♯‘(1^st ‘𝐴)) = 𝑁) ∧ (𝐵 ∈ (Walks‘𝐺) ∧ (♯‘(1^st ‘𝐵)) = 𝑁)) ∧ (2^nd ‘𝐴) = (2^nd ‘𝐵)) → 𝐴 = 𝐵)
21	20	ex 412	1 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ₀) ∧ (𝐴 ∈ (Walks‘𝐺) ∧ (♯‘(1^st ‘𝐴)) = 𝑁) ∧ (𝐵 ∈ (Walks‘𝐺) ∧ (♯‘(1^st ‘𝐵)) = 𝑁)) → ((2^nd ‘𝐴) = (2^nd ‘𝐵) → 𝐴 = 𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1539 ∈ wcel 2107 ∀wral 3060 ‘cfv 6560 (class class class)co 7432 1^st c1st 8013 2^nd c2nd 8014 0cc0 11156 ℕ₀cn0 12528 ...cfz 13548 ♯chash 14370 USPGraphcuspgr 29166 Walkscwlks 29615

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-ifp 1063 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-1st 8015 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-2o 8508 df-oadd 8511 df-er 8746 df-map 8869 df-pm 8870 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-dju 9942 df-card 9980 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-nn 12268 df-2 12330 df-n0 12529 df-xnn0 12602 df-z 12616 df-uz 12880 df-fz 13549 df-fzo 13696 df-hash 14371 df-word 14554 df-edg 29066 df-uhgr 29076 df-upgr 29100 df-uspgr 29168 df-wlks 29618

This theorem is referenced by: uspgr2wlkeqi 29667

W3C validator