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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 27727 | . . . 4 ⊢ (𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) → (𝑊 = 〈“𝐴(𝑊‘1)𝐶”〉 ∧ (𝑊‘1) ∈ 𝑉)) |
4 | anass 471 | . . . 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 585 | . . 3 ⊢ (𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) → ((𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ∧ 𝑊 = 〈“𝐴(𝑊‘1)𝐶”〉) ∧ (𝑊‘1) ∈ 𝑉)) |
6 | simpr 487 | . . . 4 ⊢ (((𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ∧ 𝑊 = 〈“𝐴(𝑊‘1)𝐶”〉) ∧ (𝑊‘1) ∈ 𝑉) → (𝑊‘1) ∈ 𝑉) | |
7 | s3eq2 14226 | . . . . . 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 632 | . . . . . 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 484 | . . . 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 487 | . . . . . 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 481 | . . . . . 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 514 | . . . . 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 483 | . . . 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 3626 | . . 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 479 | . . 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 211 | 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 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∃wrex 3139 ‘cfv 6350 (class class class)co 7150 1c1 10532 2c2 11686 〈“cs3 14198 Vtxcvtx 26775 WWalksNOn cwwlksnon 27599 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 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 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-map 8402 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-3 11695 df-n0 11892 df-z 11976 df-uz 12238 df-fz 12887 df-fzo 13028 df-hash 13685 df-word 13856 df-concat 13917 df-s1 13944 df-s2 14204 df-s3 14205 df-wwlks 27602 df-wwlksn 27603 df-wwlksnon 27604 |
This theorem is referenced by: elwwlks2on 27732 frgr2wwlk1 28102 |
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