uspgr2wlkeqi - Metamath Proof Explorer

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Theorem uspgr2wlkeqi 28015

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 27996	. . . . 5 ⊢ (𝐴 ∈ (Walks‘𝐺) ↔ (1^st ‘𝐴)(Walks‘𝐺)(2^nd ‘𝐴))
2		wlkcpr 27996	. . . . . 6 ⊢ (𝐵 ∈ (Walks‘𝐺) ↔ (1^st ‘𝐵)(Walks‘𝐺)(2^nd ‘𝐵))
3		wlkcl 27982	. . . . . . 7 ⊢ ((1^st ‘𝐴)(Walks‘𝐺)(2^nd ‘𝐴) → (♯‘(1^st ‘𝐴)) ∈ ℕ₀)
4		fveq2 6774	. . . . . . . . . . . . 13 ⊢ ((2^nd ‘𝐴) = (2^nd ‘𝐵) → (♯‘(2^nd ‘𝐴)) = (♯‘(2^nd ‘𝐵)))
5	4	oveq1d 7290	. . . . . . . . . . . 12 ⊢ ((2^nd ‘𝐴) = (2^nd ‘𝐵) → ((♯‘(2^nd ‘𝐴)) − 1) = ((♯‘(2^nd ‘𝐵)) − 1))
6	5	eqcomd 2744	. . . . . . . . . . 11 ⊢ ((2^nd ‘𝐴) = (2^nd ‘𝐵) → ((♯‘(2^nd ‘𝐵)) − 1) = ((♯‘(2^nd ‘𝐴)) − 1))
7	6	adantl 482	. . . . . . . . . 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 27989	. . . . . . . . . . . 12 ⊢ ((1^st ‘𝐵)(Walks‘𝐺)(2^nd ‘𝐵) → (♯‘(1^st ‘𝐵)) = ((♯‘(2^nd ‘𝐵)) − 1))
9		wlklenvm1 27989	. . . . . . . . . . . 12 ⊢ ((1^st ‘𝐴)(Walks‘𝐺)(2^nd ‘𝐴) → (♯‘(1^st ‘𝐴)) = ((♯‘(2^nd ‘𝐴)) − 1))
10	8, 9	eqeqan12rd 2753	. . . . . . . . . . 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 481	. . . . . . . . . 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 256	. . . . . . . . 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 617	. . . . . . . 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 438	. . . . . . 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	syl5bi 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 418	. . 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 483	. . . . . . 7 ⊢ ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺))) → 𝐺 ∈ USPGraph)
21		simpl 483	. . . . . . 7 ⊢ (((♯‘(1^st ‘𝐴)) ∈ ℕ₀ ∧ (♯‘(1^st ‘𝐵)) = (♯‘(1^st ‘𝐴))) → (♯‘(1^st ‘𝐴)) ∈ ℕ₀)
22	20, 21	anim12i 613	. . . . . 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 483	. . . . . . . 8 ⊢ ((𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) → 𝐴 ∈ (Walks‘𝐺))
24	23	adantl 482	. . . . . . 7 ⊢ ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺))) → 𝐴 ∈ (Walks‘𝐺))
25		eqidd 2739	. . . . . . 7 ⊢ (((♯‘(1^st ‘𝐴)) ∈ ℕ₀ ∧ (♯‘(1^st ‘𝐵)) = (♯‘(1^st ‘𝐴))) → (♯‘(1^st ‘𝐴)) = (♯‘(1^st ‘𝐴)))
26	24, 25	anim12i 613	. . . . . 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 485	. . . . . . . 8 ⊢ ((𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) → 𝐵 ∈ (Walks‘𝐺))
28	27	adantl 482	. . . . . . 7 ⊢ ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺))) → 𝐵 ∈ (Walks‘𝐺))
29		simpr 485	. . . . . . 7 ⊢ (((♯‘(1^st ‘𝐴)) ∈ ℕ₀ ∧ (♯‘(1^st ‘𝐵)) = (♯‘(1^st ‘𝐴))) → (♯‘(1^st ‘𝐵)) = (♯‘(1^st ‘𝐴)))
30	28, 29	anim12i 613	. . . . . 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 28014	. . . . . 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 413	. . . 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 396 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 class class class wbr 5074 ‘cfv 6433 (class class class)co 7275 1^st c1st 7829 2^nd c2nd 7830 1c1 10872 − cmin 11205 ℕ₀cn0 12233 ♯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: wlkswwlksf1o 28244

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