uspgr2wlkeqi - Metamath Proof Explorer

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Theorem uspgr2wlkeqi 30255

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 30236	. . . . 5 ⊢ (𝐴 ∈ (Walks‘𝐺) ↔ (1^st ‘𝐴)(Walks‘𝐺)(2^nd ‘𝐴))
2		wlkcpr 30236	. . . . . 6 ⊢ (𝐵 ∈ (Walks‘𝐺) ↔ (1^st ‘𝐵)(Walks‘𝐺)(2^nd ‘𝐵))
3		wlkcl 30223	. . . . . . 7 ⊢ ((1^st ‘𝐴)(Walks‘𝐺)(2^nd ‘𝐴) → (♯‘(1^st ‘𝐴)) ∈ ℕ₀)
4		fveq2 6905	. . . . . . . . . . . . 13 ⊢ ((2^nd ‘𝐴) = (2^nd ‘𝐵) → (♯‘(2^nd ‘𝐴)) = (♯‘(2^nd ‘𝐵)))
5	4	oveq1d 7455	. . . . . . . . . . . 12 ⊢ ((2^nd ‘𝐴) = (2^nd ‘𝐵) → ((♯‘(2^nd ‘𝐴)) − 1) = ((♯‘(2^nd ‘𝐵)) − 1))
6	5	eqcomd 2769	. . . . . . . . . . 11 ⊢ ((2^nd ‘𝐴) = (2^nd ‘𝐵) → ((♯‘(2^nd ‘𝐵)) − 1) = ((♯‘(2^nd ‘𝐴)) − 1))
7	6	adantl 487	. . . . . . . . . 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 30229	. . . . . . . . . . . 12 ⊢ ((1^st ‘𝐵)(Walks‘𝐺)(2^nd ‘𝐵) → (♯‘(1^st ‘𝐵)) = ((♯‘(2^nd ‘𝐵)) − 1))
9		wlklenvm1 30229	. . . . . . . . . . . 12 ⊢ ((1^st ‘𝐴)(Walks‘𝐺)(2^nd ‘𝐴) → (♯‘(1^st ‘𝐴)) = ((♯‘(2^nd ‘𝐴)) − 1))
10	8, 9	eqeqan12rd 2778	. . . . . . . . . . 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 486	. . . . . . . . . 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 260	. . . . . . . . 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 629	. . . . . . . 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 443	. . . . . . 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 39	. . . . . 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 245	. . . . 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 220	. . . 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 423	. . 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 1148	. 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 488	. . . . . . 7 ⊢ ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺))) → 𝐺 ∈ USPGraph)
21		simpl 488	. . . . . . 7 ⊢ (((♯‘(1^st ‘𝐴)) ∈ ℕ₀ ∧ (♯‘(1^st ‘𝐵)) = (♯‘(1^st ‘𝐴))) → (♯‘(1^st ‘𝐴)) ∈ ℕ₀)
22	20, 21	anim12i 625	. . . . . 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 488	. . . . . . . 8 ⊢ ((𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) → 𝐴 ∈ (Walks‘𝐺))
24	23	adantl 487	. . . . . . 7 ⊢ ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺))) → 𝐴 ∈ (Walks‘𝐺))
25		eqidd 2764	. . . . . . 7 ⊢ (((♯‘(1^st ‘𝐴)) ∈ ℕ₀ ∧ (♯‘(1^st ‘𝐵)) = (♯‘(1^st ‘𝐴))) → (♯‘(1^st ‘𝐴)) = (♯‘(1^st ‘𝐴)))
26	24, 25	anim12i 625	. . . . . 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 490	. . . . . . . 8 ⊢ ((𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) → 𝐵 ∈ (Walks‘𝐺))
28	27	adantl 487	. . . . . . 7 ⊢ ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺))) → 𝐵 ∈ (Walks‘𝐺))
29		simpr 490	. . . . . . 7 ⊢ (((♯‘(1^st ‘𝐴)) ∈ ℕ₀ ∧ (♯‘(1^st ‘𝐵)) = (♯‘(1^st ‘𝐴))) → (♯‘(1^st ‘𝐵)) = (♯‘(1^st ‘𝐴)))
30	28, 29	anim12i 625	. . . . . 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 30254	. . . . . 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 1399	. . . . 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 418	. . . 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 87	. . 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 1135	. 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 16	1 ⊢ ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ (2^nd ‘𝐴) = (2^nd ‘𝐵)) → 𝐴 = 𝐵)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2147 class class class wbr 5106 ‘cfv 6555 (class class class)co 7440 1^st c1st 8020 2^nd c2nd 8021 1c1 11225 − cmin 11565 ℕ₀cn0 12630 ♯chash 14498 USPGraphcuspgr 29756 Walkscwlks 30204

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5235 ax-sep 5252 ax-nul 5263 ax-pow 5331 ax-pr 5395 ax-un 7772 ax-cnex 11280 ax-resscn 11281 ax-1cn 11282 ax-icn 11283 ax-addcl 11284 ax-addrcl 11285 ax-mulcl 11286 ax-mulrcl 11287 ax-mulcom 11288 ax-addass 11289 ax-mulass 11290 ax-distr 11291 ax-i2m1 11292 ax-1ne0 11293 ax-1rid 11294 ax-rnegex 11295 ax-rrecex 11296 ax-cnre 11297 ax-pre-lttri 11298 ax-pre-lttrn 11299 ax-pre-ltadd 11300 ax-pre-mulgt0 11301

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-ifp 1079 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4839 df-uni 4871 df-int 4911 df-iun 4956 df-br 5107 df-opab 5171 df-mpt 5190 df-tr 5216 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6319 df-ord 6380 df-on 6381 df-lim 6382 df-suc 6383 df-iota 6509 df-fun 6557 df-fn 6558 df-f 6559 df-f1 6560 df-fo 6561 df-f1o 6562 df-fv 6563 df-riota 7397 df-ov 7443 df-oprab 7444 df-mpo 7445 df-om 7899 df-1st 8022 df-2nd 8023 df-frecs 8324 df-wrecs 8355 df-recs 8404 df-rdg 8443 df-1o 8501 df-2o 8502 df-oadd 8505 df-er 8742 df-map 8874 df-pm 8875 df-en 8999 df-dom 9000 df-sdom 9001 df-fin 9002 df-dju 10006 df-card 10044 df-pnf 11369 df-mnf 11370 df-xr 11371 df-ltxr 11372 df-le 11373 df-sub 11567 df-neg 11568 df-nn 12360 df-2 12429 df-n0 12631 df-xnn0 12704 df-z 12718 df-uz 12990 df-fz 13664 df-fzo 13813 df-hash 14499 df-word 14683 df-edg 29653 df-uhgr 29663 df-upgr 29687 df-uspgr 29758 df-wlks 30207

This theorem is used by: wlkswwlksf1o 30495

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