elwwlks2ons3im - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > elwwlks2ons3im		Structured version Visualization version GIF version

Theorem elwwlks2ons3im 29752

Description: A walk as word of length 2 between two vertices is a length 3 string and its second symbol is a vertex. (Contributed by AV, 14-Mar-2022.)

**Hypothesis**
Ref	Expression
wwlks2onv.v	⊢ 𝑉 = (Vtx‘𝐺)

**Assertion**
Ref	Expression
elwwlks2ons3im	⊢ (𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) → (𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩ ∧ (𝑊‘1) ∈ 𝑉))

**Proof of Theorem elwwlks2ons3im**
Step	Hyp	Ref	Expression
1		wwlks2onv.v	. . 3 ⊢ 𝑉 = (Vtx‘𝐺)
2	1	wwlksonvtx 29653	. 2 ⊢ (𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) → (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉))
3		wwlknon 29655	. . 3 ⊢ (𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ↔ (𝑊 ∈ (2 WWalksN 𝐺) ∧ (𝑊‘0) = 𝐴 ∧ (𝑊‘2) = 𝐶))
4		wwlknbp1 29642	. . . . 5 ⊢ (𝑊 ∈ (2 WWalksN 𝐺) → (2 ∈ ℕ₀ ∧ 𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (2 + 1)))
5		2p1e3 12376	. . . . . . . 8 ⊢ (2 + 1) = 3
6	5	eqeq2i 2740	. . . . . . 7 ⊢ ((♯‘𝑊) = (2 + 1) ↔ (♯‘𝑊) = 3)
7		1ex 11232	. . . . . . . . . . . . . 14 ⊢ 1 ∈ V
8	7	tpid2 4770	. . . . . . . . . . . . 13 ⊢ 1 ∈ {0, 1, 2}
9		fzo0to3tp 13742	. . . . . . . . . . . . 13 ⊢ (0..^3) = {0, 1, 2}
10	8, 9	eleqtrri 2827	. . . . . . . . . . . 12 ⊢ 1 ∈ (0..^3)
11		oveq2 7422	. . . . . . . . . . . 12 ⊢ ((♯‘𝑊) = 3 → (0..^(♯‘𝑊)) = (0..^3))
12	10, 11	eleqtrrid 2835	. . . . . . . . . . 11 ⊢ ((♯‘𝑊) = 3 → 1 ∈ (0..^(♯‘𝑊)))
13		wrdsymbcl 14501	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ 1 ∈ (0..^(♯‘𝑊))) → (𝑊‘1) ∈ (Vtx‘𝐺))
14	12, 13	sylan2 592	. . . . . . . . . 10 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = 3) → (𝑊‘1) ∈ (Vtx‘𝐺))
15	14	3ad2ant1 1131	. . . . . . . . 9 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = 3) ∧ ((𝑊‘0) = 𝐴 ∧ (𝑊‘2) = 𝐶) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (𝑊‘1) ∈ (Vtx‘𝐺))
16		simpl1r 1223	. . . . . . . . . . 11 ⊢ ((((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = 3) ∧ ((𝑊‘0) = 𝐴 ∧ (𝑊‘2) = 𝐶) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ (𝑊‘1) ∈ (Vtx‘𝐺)) → (♯‘𝑊) = 3)
17		simpl 482	. . . . . . . . . . . . . 14 ⊢ (((𝑊‘0) = 𝐴 ∧ (𝑊‘2) = 𝐶) → (𝑊‘0) = 𝐴)
18		eqidd 2728	. . . . . . . . . . . . . 14 ⊢ (((𝑊‘0) = 𝐴 ∧ (𝑊‘2) = 𝐶) → (𝑊‘1) = (𝑊‘1))
19		simpr 484	. . . . . . . . . . . . . 14 ⊢ (((𝑊‘0) = 𝐴 ∧ (𝑊‘2) = 𝐶) → (𝑊‘2) = 𝐶)
20	17, 18, 19	3jca 1126	. . . . . . . . . . . . 13 ⊢ (((𝑊‘0) = 𝐴 ∧ (𝑊‘2) = 𝐶) → ((𝑊‘0) = 𝐴 ∧ (𝑊‘1) = (𝑊‘1) ∧ (𝑊‘2) = 𝐶))
21	20	3ad2ant2 1132	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = 3) ∧ ((𝑊‘0) = 𝐴 ∧ (𝑊‘2) = 𝐶) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → ((𝑊‘0) = 𝐴 ∧ (𝑊‘1) = (𝑊‘1) ∧ (𝑊‘2) = 𝐶))
22	21	adantr 480	. . . . . . . . . . 11 ⊢ ((((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = 3) ∧ ((𝑊‘0) = 𝐴 ∧ (𝑊‘2) = 𝐶) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ (𝑊‘1) ∈ (Vtx‘𝐺)) → ((𝑊‘0) = 𝐴 ∧ (𝑊‘1) = (𝑊‘1) ∧ (𝑊‘2) = 𝐶))
23	1	eqcomi 2736	. . . . . . . . . . . . . . . . . 18 ⊢ (Vtx‘𝐺) = 𝑉
24	23	wrdeqi 14511	. . . . . . . . . . . . . . . . 17 ⊢ Word (Vtx‘𝐺) = Word 𝑉
25	24	eleq2i 2820	. . . . . . . . . . . . . . . 16 ⊢ (𝑊 ∈ Word (Vtx‘𝐺) ↔ 𝑊 ∈ Word 𝑉)
26	25	biimpi 215	. . . . . . . . . . . . . . 15 ⊢ (𝑊 ∈ Word (Vtx‘𝐺) → 𝑊 ∈ Word 𝑉)
27	26	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = 3) → 𝑊 ∈ Word 𝑉)
28	27	3ad2ant1 1131	. . . . . . . . . . . . 13 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = 3) ∧ ((𝑊‘0) = 𝐴 ∧ (𝑊‘2) = 𝐶) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → 𝑊 ∈ Word 𝑉)
29	28	adantr 480	. . . . . . . . . . . 12 ⊢ ((((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = 3) ∧ ((𝑊‘0) = 𝐴 ∧ (𝑊‘2) = 𝐶) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ (𝑊‘1) ∈ (Vtx‘𝐺)) → 𝑊 ∈ Word 𝑉)
30		simpl3l 1226	. . . . . . . . . . . 12 ⊢ ((((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = 3) ∧ ((𝑊‘0) = 𝐴 ∧ (𝑊‘2) = 𝐶) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ (𝑊‘1) ∈ (Vtx‘𝐺)) → 𝐴 ∈ 𝑉)
31	23	eleq2i 2820	. . . . . . . . . . . . . 14 ⊢ ((𝑊‘1) ∈ (Vtx‘𝐺) ↔ (𝑊‘1) ∈ 𝑉)
32	31	biimpi 215	. . . . . . . . . . . . 13 ⊢ ((𝑊‘1) ∈ (Vtx‘𝐺) → (𝑊‘1) ∈ 𝑉)
33	32	adantl 481	. . . . . . . . . . . 12 ⊢ ((((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = 3) ∧ ((𝑊‘0) = 𝐴 ∧ (𝑊‘2) = 𝐶) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ (𝑊‘1) ∈ (Vtx‘𝐺)) → (𝑊‘1) ∈ 𝑉)
34		simpl3r 1227	. . . . . . . . . . . 12 ⊢ ((((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = 3) ∧ ((𝑊‘0) = 𝐴 ∧ (𝑊‘2) = 𝐶) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ (𝑊‘1) ∈ (Vtx‘𝐺)) → 𝐶 ∈ 𝑉)
35		eqwrds3 14936	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (𝐴 ∈ 𝑉 ∧ (𝑊‘1) ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩ ↔ ((♯‘𝑊) = 3 ∧ ((𝑊‘0) = 𝐴 ∧ (𝑊‘1) = (𝑊‘1) ∧ (𝑊‘2) = 𝐶))))
36	29, 30, 33, 34, 35	syl13anc 1370	. . . . . . . . . . 11 ⊢ ((((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = 3) ∧ ((𝑊‘0) = 𝐴 ∧ (𝑊‘2) = 𝐶) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ (𝑊‘1) ∈ (Vtx‘𝐺)) → (𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩ ↔ ((♯‘𝑊) = 3 ∧ ((𝑊‘0) = 𝐴 ∧ (𝑊‘1) = (𝑊‘1) ∧ (𝑊‘2) = 𝐶))))
37	16, 22, 36	mpbir2and 712	. . . . . . . . . 10 ⊢ ((((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = 3) ∧ ((𝑊‘0) = 𝐴 ∧ (𝑊‘2) = 𝐶) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ (𝑊‘1) ∈ (Vtx‘𝐺)) → 𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩)
38	37, 33	jca 511	. . . . . . . . 9 ⊢ ((((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = 3) ∧ ((𝑊‘0) = 𝐴 ∧ (𝑊‘2) = 𝐶) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ (𝑊‘1) ∈ (Vtx‘𝐺)) → (𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩ ∧ (𝑊‘1) ∈ 𝑉))
39	15, 38	mpdan 686	. . . . . . . 8 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = 3) ∧ ((𝑊‘0) = 𝐴 ∧ (𝑊‘2) = 𝐶) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩ ∧ (𝑊‘1) ∈ 𝑉))
40	39	3exp 1117	. . . . . . 7 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = 3) → (((𝑊‘0) = 𝐴 ∧ (𝑊‘2) = 𝐶) → ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩ ∧ (𝑊‘1) ∈ 𝑉))))
41	6, 40	sylan2b 593	. . . . . 6 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (2 + 1)) → (((𝑊‘0) = 𝐴 ∧ (𝑊‘2) = 𝐶) → ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩ ∧ (𝑊‘1) ∈ 𝑉))))
42	41	3adant1 1128	. . . . 5 ⊢ ((2 ∈ ℕ₀ ∧ 𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (2 + 1)) → (((𝑊‘0) = 𝐴 ∧ (𝑊‘2) = 𝐶) → ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩ ∧ (𝑊‘1) ∈ 𝑉))))
43	4, 42	syl 17	. . . 4 ⊢ (𝑊 ∈ (2 WWalksN 𝐺) → (((𝑊‘0) = 𝐴 ∧ (𝑊‘2) = 𝐶) → ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩ ∧ (𝑊‘1) ∈ 𝑉))))
44	43	3impib 1114	. . 3 ⊢ ((𝑊 ∈ (2 WWalksN 𝐺) ∧ (𝑊‘0) = 𝐴 ∧ (𝑊‘2) = 𝐶) → ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩ ∧ (𝑊‘1) ∈ 𝑉)))
45	3, 44	sylbi 216	. 2 ⊢ (𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) → ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩ ∧ (𝑊‘1) ∈ 𝑉)))
46	2, 45	mpd 15	1 ⊢ (𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) → (𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩ ∧ (𝑊‘1) ∈ 𝑉))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 {ctp 4628 ‘cfv 6542 (class class class)co 7414 0cc0 11130 1c1 11131 + caddc 11133 2c2 12289 3c3 12290 ℕ₀cn0 12494 ..^cfzo 13651 ♯chash 14313 Word cword 14488 ⟨“cs3 14817 Vtxcvtx 28796 WWalksN cwwlksn 29624 WWalksNOn cwwlksnon 29625

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11186 ax-resscn 11187 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-addrcl 11191 ax-mulcl 11192 ax-mulrcl 11193 ax-mulcom 11194 ax-addass 11195 ax-mulass 11196 ax-distr 11197 ax-i2m1 11198 ax-1ne0 11199 ax-1rid 11200 ax-rnegex 11201 ax-rrecex 11202 ax-cnre 11203 ax-pre-lttri 11204 ax-pre-lttrn 11205 ax-pre-ltadd 11206 ax-pre-mulgt0 11207

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-map 8838 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-card 9954 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11468 df-neg 11469 df-nn 12235 df-2 12297 df-3 12298 df-n0 12495 df-z 12581 df-uz 12845 df-fz 13509 df-fzo 13652 df-hash 14314 df-word 14489 df-concat 14545 df-s1 14570 df-s2 14823 df-s3 14824 df-wwlks 29628 df-wwlksn 29629 df-wwlksnon 29630

This theorem is referenced by: elwwlks2ons3 29753

W3C validator