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 29669

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 2740	. . . . 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 6831	. . . . 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 3130	. . 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 1225	. . . 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 2740	. . . . . 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 29668	. . . 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 1373	. . 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 1541 ∈ wcel 2113 ∀wral 3049 ‘cfv 6490 (class class class)co 7356 1^st c1st 7929 2^nd c2nd 7930 0cc0 11024 ℕ₀cn0 12399 ...cfz 13421 ♯chash 14251 USPGraphcuspgr 29170 Walkscwlks 29619

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 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2706 ax-rep 5222 ax-sep 5239 ax-nul 5249 ax-pow 5308 ax-pr 5375 ax-un 7678 ax-cnex 11080 ax-resscn 11081 ax-1cn 11082 ax-icn 11083 ax-addcl 11084 ax-addrcl 11085 ax-mulcl 11086 ax-mulrcl 11087 ax-mulcom 11088 ax-addass 11089 ax-mulass 11090 ax-distr 11091 ax-i2m1 11092 ax-1ne0 11093 ax-1rid 11094 ax-rnegex 11095 ax-rrecex 11096 ax-cnre 11097 ax-pre-lttri 11098 ax-pre-lttrn 11099 ax-pre-ltadd 11100 ax-pre-mulgt0 11101

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 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2809 df-nfc 2883 df-ne 2931 df-nel 3035 df-ral 3050 df-rex 3059 df-reu 3349 df-rab 3398 df-v 3440 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4579 df-pr 4581 df-op 4585 df-uni 4862 df-int 4901 df-iun 4946 df-br 5097 df-opab 5159 df-mpt 5178 df-tr 5204 df-id 5517 df-eprel 5522 df-po 5530 df-so 5531 df-fr 5575 df-we 5577 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-pred 6257 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7807 df-1st 7931 df-2nd 7932 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-2o 8396 df-oadd 8399 df-er 8633 df-map 8763 df-pm 8764 df-en 8882 df-dom 8883 df-sdom 8884 df-fin 8885 df-dju 9811 df-card 9849 df-pnf 11166 df-mnf 11167 df-xr 11168 df-ltxr 11169 df-le 11170 df-sub 11364 df-neg 11365 df-nn 12144 df-2 12206 df-n0 12400 df-xnn0 12473 df-z 12487 df-uz 12750 df-fz 13422 df-fzo 13569 df-hash 14252 df-word 14435 df-edg 29070 df-uhgr 29080 df-upgr 29104 df-uspgr 29172 df-wlks 29622

This theorem is referenced by: uspgr2wlkeqi 29670

W3C validator