uspgr2wlkeq2 - Metamath Proof Explorer

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Theorem uspgr2wlkeq2 28014

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 485	. . . . . 6 ⊢ ((𝐵 ∈ (Walks‘𝐺) ∧ (♯‘(1^st ‘𝐵)) = 𝑁) → (♯‘(1^st ‘𝐵)) = 𝑁)
2	1	eqcomd 2744	. . . . 5 ⊢ ((𝐵 ∈ (Walks‘𝐺) ∧ (♯‘(1^st ‘𝐵)) = 𝑁) → 𝑁 = (♯‘(1^st ‘𝐵)))
3	2	3ad2ant3 1134	. . . 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 481	. . 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 6773	. . . . 5 ⊢ ((2^nd ‘𝐴) = (2^nd ‘𝐵) → ((2^nd ‘𝐴)‘𝑖) = ((2^nd ‘𝐵)‘𝑖))
6	5	adantl 482	. . . 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 3104	. . 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 1223	. . . 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 483	. . . . . . 7 ⊢ ((𝐴 ∈ (Walks‘𝐺) ∧ (♯‘(1^st ‘𝐴)) = 𝑁) → 𝐴 ∈ (Walks‘𝐺))
10		simpl 483	. . . . . . 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 1129	. . . . 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 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)) → (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)))
14		simpr 485	. . . . . . 7 ⊢ ((𝐴 ∈ (Walks‘𝐺) ∧ (♯‘(1^st ‘𝐴)) = 𝑁) → (♯‘(1^st ‘𝐴)) = 𝑁)
15	14	eqcomd 2744	. . . . . 6 ⊢ ((𝐴 ∈ (Walks‘𝐺) ∧ (♯‘(1^st ‘𝐴)) = 𝑁) → 𝑁 = (♯‘(1^st ‘𝐴)))
16	15	3ad2ant2 1133	. . . . 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 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)) → 𝑁 = (♯‘(1^st ‘𝐴)))
18		uspgr2wlkeq 28013	. . . 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 1370	. . 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 710	. 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 413	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 205 ∧ wa 396 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 ∀wral 3064 ‘cfv 6433 (class class class)co 7275 1^st c1st 7829 2^nd c2nd 7830 0cc0 10871 ℕ₀cn0 12233 ...cfz 13239 ♯chash 14044 USPGraphcuspgr 27518 Walkscwlks 27963

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-ifp 1061 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-2o 8298 df-oadd 8301 df-er 8498 df-map 8617 df-pm 8618 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-dju 9659 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-n0 12234 df-xnn0 12306 df-z 12320 df-uz 12583 df-fz 13240 df-fzo 13383 df-hash 14045 df-word 14218 df-edg 27418 df-uhgr 27428 df-upgr 27452 df-uspgr 27520 df-wlks 27966

This theorem is referenced by: uspgr2wlkeqi 28015

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