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Mirrors > Home > MPE Home > Th. List > elwwlks2ons3 | Structured version Visualization version GIF version |
Description: For each walk of length 2 between two vertices, there is a third vertex in the middle of the walk. (Contributed by Alexander van der Vekens, 15-Feb-2018.) (Revised by AV, 12-May-2021.) (Revised by AV, 14-Mar-2022.) |
Ref | Expression |
---|---|
wwlks2onv.v | ⊢ 𝑉 = (Vtx‘𝐺) |
Ref | Expression |
---|---|
elwwlks2ons3 | ⊢ (𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ↔ ∃𝑏 ∈ 𝑉 (𝑊 = 〈“𝐴𝑏𝐶”〉 ∧ 〈“𝐴𝑏𝐶”〉 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . . . 4 ⊢ (𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) → 𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)) | |
2 | wwlks2onv.v | . . . . 5 ⊢ 𝑉 = (Vtx‘𝐺) | |
3 | 2 | elwwlks2ons3im 27661 | . . . 4 ⊢ (𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) → (𝑊 = 〈“𝐴(𝑊‘1)𝐶”〉 ∧ (𝑊‘1) ∈ 𝑉)) |
4 | anass 469 | . . . 4 ⊢ (((𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ∧ 𝑊 = 〈“𝐴(𝑊‘1)𝐶”〉) ∧ (𝑊‘1) ∈ 𝑉) ↔ (𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ∧ (𝑊 = 〈“𝐴(𝑊‘1)𝐶”〉 ∧ (𝑊‘1) ∈ 𝑉))) | |
5 | 1, 3, 4 | sylanbrc 583 | . . 3 ⊢ (𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) → ((𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ∧ 𝑊 = 〈“𝐴(𝑊‘1)𝐶”〉) ∧ (𝑊‘1) ∈ 𝑉)) |
6 | simpr 485 | . . . 4 ⊢ (((𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ∧ 𝑊 = 〈“𝐴(𝑊‘1)𝐶”〉) ∧ (𝑊‘1) ∈ 𝑉) → (𝑊‘1) ∈ 𝑉) | |
7 | s3eq2 14222 | . . . . . 6 ⊢ (𝑏 = (𝑊‘1) → 〈“𝐴𝑏𝐶”〉 = 〈“𝐴(𝑊‘1)𝐶”〉) | |
8 | eqeq2 2833 | . . . . . . 7 ⊢ (〈“𝐴𝑏𝐶”〉 = 〈“𝐴(𝑊‘1)𝐶”〉 → (𝑊 = 〈“𝐴𝑏𝐶”〉 ↔ 𝑊 = 〈“𝐴(𝑊‘1)𝐶”〉)) | |
9 | eleq1 2900 | . . . . . . 7 ⊢ (〈“𝐴𝑏𝐶”〉 = 〈“𝐴(𝑊‘1)𝐶”〉 → (〈“𝐴𝑏𝐶”〉 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ↔ 〈“𝐴(𝑊‘1)𝐶”〉 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶))) | |
10 | 8, 9 | anbi12d 630 | . . . . . 6 ⊢ (〈“𝐴𝑏𝐶”〉 = 〈“𝐴(𝑊‘1)𝐶”〉 → ((𝑊 = 〈“𝐴𝑏𝐶”〉 ∧ 〈“𝐴𝑏𝐶”〉 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)) ↔ (𝑊 = 〈“𝐴(𝑊‘1)𝐶”〉 ∧ 〈“𝐴(𝑊‘1)𝐶”〉 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)))) |
11 | 7, 10 | syl 17 | . . . . 5 ⊢ (𝑏 = (𝑊‘1) → ((𝑊 = 〈“𝐴𝑏𝐶”〉 ∧ 〈“𝐴𝑏𝐶”〉 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)) ↔ (𝑊 = 〈“𝐴(𝑊‘1)𝐶”〉 ∧ 〈“𝐴(𝑊‘1)𝐶”〉 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)))) |
12 | 11 | adantl 482 | . . . 4 ⊢ ((((𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ∧ 𝑊 = 〈“𝐴(𝑊‘1)𝐶”〉) ∧ (𝑊‘1) ∈ 𝑉) ∧ 𝑏 = (𝑊‘1)) → ((𝑊 = 〈“𝐴𝑏𝐶”〉 ∧ 〈“𝐴𝑏𝐶”〉 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)) ↔ (𝑊 = 〈“𝐴(𝑊‘1)𝐶”〉 ∧ 〈“𝐴(𝑊‘1)𝐶”〉 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)))) |
13 | simpr 485 | . . . . . 6 ⊢ ((𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ∧ 𝑊 = 〈“𝐴(𝑊‘1)𝐶”〉) → 𝑊 = 〈“𝐴(𝑊‘1)𝐶”〉) | |
14 | eleq1 2900 | . . . . . . 7 ⊢ (𝑊 = 〈“𝐴(𝑊‘1)𝐶”〉 → (𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ↔ 〈“𝐴(𝑊‘1)𝐶”〉 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶))) | |
15 | 14 | biimpac 479 | . . . . . 6 ⊢ ((𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ∧ 𝑊 = 〈“𝐴(𝑊‘1)𝐶”〉) → 〈“𝐴(𝑊‘1)𝐶”〉 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)) |
16 | 13, 15 | jca 512 | . . . . 5 ⊢ ((𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ∧ 𝑊 = 〈“𝐴(𝑊‘1)𝐶”〉) → (𝑊 = 〈“𝐴(𝑊‘1)𝐶”〉 ∧ 〈“𝐴(𝑊‘1)𝐶”〉 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶))) |
17 | 16 | adantr 481 | . . . 4 ⊢ (((𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ∧ 𝑊 = 〈“𝐴(𝑊‘1)𝐶”〉) ∧ (𝑊‘1) ∈ 𝑉) → (𝑊 = 〈“𝐴(𝑊‘1)𝐶”〉 ∧ 〈“𝐴(𝑊‘1)𝐶”〉 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶))) |
18 | 6, 12, 17 | rspcedvd 3625 | . . 3 ⊢ (((𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ∧ 𝑊 = 〈“𝐴(𝑊‘1)𝐶”〉) ∧ (𝑊‘1) ∈ 𝑉) → ∃𝑏 ∈ 𝑉 (𝑊 = 〈“𝐴𝑏𝐶”〉 ∧ 〈“𝐴𝑏𝐶”〉 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶))) |
19 | 5, 18 | syl 17 | . 2 ⊢ (𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) → ∃𝑏 ∈ 𝑉 (𝑊 = 〈“𝐴𝑏𝐶”〉 ∧ 〈“𝐴𝑏𝐶”〉 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶))) |
20 | eleq1 2900 | . . . . 5 ⊢ (〈“𝐴𝑏𝐶”〉 = 𝑊 → (〈“𝐴𝑏𝐶”〉 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ↔ 𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶))) | |
21 | 20 | eqcoms 2829 | . . . 4 ⊢ (𝑊 = 〈“𝐴𝑏𝐶”〉 → (〈“𝐴𝑏𝐶”〉 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ↔ 𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶))) |
22 | 21 | biimpa 477 | . . 3 ⊢ ((𝑊 = 〈“𝐴𝑏𝐶”〉 ∧ 〈“𝐴𝑏𝐶”〉 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)) → 𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)) |
23 | 22 | rexlimivw 3282 | . 2 ⊢ (∃𝑏 ∈ 𝑉 (𝑊 = 〈“𝐴𝑏𝐶”〉 ∧ 〈“𝐴𝑏𝐶”〉 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)) → 𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)) |
24 | 19, 23 | impbii 210 | 1 ⊢ (𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ↔ ∃𝑏 ∈ 𝑉 (𝑊 = 〈“𝐴𝑏𝐶”〉 ∧ 〈“𝐴𝑏𝐶”〉 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∃wrex 3139 ‘cfv 6349 (class class class)co 7145 1c1 10527 2c2 11681 〈“cs3 14194 Vtxcvtx 26709 WWalksNOn cwwlksnon 27533 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7450 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-fal 1541 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4833 df-int 4870 df-iun 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7569 df-1st 7680 df-2nd 7681 df-wrecs 7938 df-recs 7999 df-rdg 8037 df-1o 8093 df-oadd 8097 df-er 8279 df-map 8398 df-en 8499 df-dom 8500 df-sdom 8501 df-fin 8502 df-card 9357 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11628 df-2 11689 df-3 11690 df-n0 11887 df-z 11971 df-uz 12233 df-fz 12883 df-fzo 13024 df-hash 13681 df-word 13852 df-concat 13913 df-s1 13940 df-s2 14200 df-s3 14201 df-wwlks 27536 df-wwlksn 27537 df-wwlksnon 27538 |
This theorem is referenced by: elwwlks2on 27666 frgr2wwlk1 28036 |
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