uspgr2wlkeqi - Metamath Proof Explorer

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Theorem uspgr2wlkeqi 29702

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 29683	. . . . 5 ⊢ (𝐴 ∈ (Walks‘𝐺) ↔ (1^st ‘𝐴)(Walks‘𝐺)(2^nd ‘𝐴))
2		wlkcpr 29683	. . . . . 6 ⊢ (𝐵 ∈ (Walks‘𝐺) ↔ (1^st ‘𝐵)(Walks‘𝐺)(2^nd ‘𝐵))
3		wlkcl 29670	. . . . . . 7 ⊢ ((1^st ‘𝐴)(Walks‘𝐺)(2^nd ‘𝐴) → (♯‘(1^st ‘𝐴)) ∈ ℕ₀)
4		fveq2 6833	. . . . . . . . . . . . 13 ⊢ ((2^nd ‘𝐴) = (2^nd ‘𝐵) → (♯‘(2^nd ‘𝐴)) = (♯‘(2^nd ‘𝐵)))
5	4	oveq1d 7373	. . . . . . . . . . . 12 ⊢ ((2^nd ‘𝐴) = (2^nd ‘𝐵) → ((♯‘(2^nd ‘𝐴)) − 1) = ((♯‘(2^nd ‘𝐵)) − 1))
6	5	eqcomd 2741	. . . . . . . . . . 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 29676	. . . . . . . . . . . 12 ⊢ ((1^st ‘𝐵)(Walks‘𝐺)(2^nd ‘𝐵) → (♯‘(1^st ‘𝐵)) = ((♯‘(2^nd ‘𝐵)) − 1))
9		wlklenvm1 29676	. . . . . . . . . . . 12 ⊢ ((1^st ‘𝐴)(Walks‘𝐺)(2^nd ‘𝐴) → (♯‘(1^st ‘𝐴)) = ((♯‘(2^nd ‘𝐴)) − 1))
10	8, 9	eqeqan12rd 2750	. . . . . . . . . . 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 618	. . . . . . . 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 242	. . . . 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 217	. . . 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 1131	. 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 614	. . . . . 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 2736	. . . . . . 7 ⊢ (((♯‘(1^st ‘𝐴)) ∈ ℕ₀ ∧ (♯‘(1^st ‘𝐵)) = (♯‘(1^st ‘𝐴))) → (♯‘(1^st ‘𝐴)) = (♯‘(1^st ‘𝐴)))
26	24, 25	anim12i 614	. . . . . 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 614	. . . . . 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 29701	. . . . . 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 1374	. . . . 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 1118	. 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 206 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 class class class wbr 5097 ‘cfv 6491 (class class class)co 7358 1^st c1st 7931 2^nd c2nd 7932 1c1 11029 − cmin 11366 ℕ₀cn0 12403 ♯chash 14255 USPGraphcuspgr 29202 Walkscwlks 29651

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2183 ax-ext 2707 ax-rep 5223 ax-sep 5240 ax-nul 5250 ax-pow 5309 ax-pr 5376 ax-un 7680 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-ifp 1064 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2538 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2810 df-nfc 2884 df-ne 2932 df-nel 3036 df-ral 3051 df-rex 3060 df-reu 3350 df-rab 3399 df-v 3441 df-sbc 3740 df-csb 3849 df-dif 3903 df-un 3905 df-in 3907 df-ss 3917 df-pss 3920 df-nul 4285 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4863 df-int 4902 df-iun 4947 df-br 5098 df-opab 5160 df-mpt 5179 df-tr 5205 df-id 5518 df-eprel 5523 df-po 5531 df-so 5532 df-fr 5576 df-we 5578 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-pred 6258 df-ord 6319 df-on 6320 df-lim 6321 df-suc 6322 df-iota 6447 df-fun 6493 df-fn 6494 df-f 6495 df-f1 6496 df-fo 6497 df-f1o 6498 df-fv 6499 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7809 df-1st 7933 df-2nd 7934 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-1o 8397 df-2o 8398 df-oadd 8401 df-er 8635 df-map 8767 df-pm 8768 df-en 8886 df-dom 8887 df-sdom 8888 df-fin 8889 df-dju 9815 df-card 9853 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11368 df-neg 11369 df-nn 12148 df-2 12210 df-n0 12404 df-xnn0 12477 df-z 12491 df-uz 12754 df-fz 13426 df-fzo 13573 df-hash 14256 df-word 14439 df-edg 29102 df-uhgr 29112 df-upgr 29136 df-uspgr 29204 df-wlks 29654

This theorem is referenced by: wlkswwlksf1o 29933

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