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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 27735 | . . . 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 14234 | . . . . . 6 ⊢ (𝑏 = (𝑊‘1) → 〈“𝐴𝑏𝐶”〉 = 〈“𝐴(𝑊‘1)𝐶”〉) | |
8 | eqeq2 2835 | . . . . . . 7 ⊢ (〈“𝐴𝑏𝐶”〉 = 〈“𝐴(𝑊‘1)𝐶”〉 → (𝑊 = 〈“𝐴𝑏𝐶”〉 ↔ 𝑊 = 〈“𝐴(𝑊‘1)𝐶”〉)) | |
9 | eleq1 2902 | . . . . . . 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 2902 | . . . . . . 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 3628 | . . 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 2902 | . . . . 5 ⊢ (〈“𝐴𝑏𝐶”〉 = 𝑊 → (〈“𝐴𝑏𝐶”〉 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ↔ 𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶))) | |
21 | 20 | eqcoms 2831 | . . . 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 3284 | . 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 1537 ∈ wcel 2114 ∃wrex 3141 ‘cfv 6357 (class class class)co 7158 1c1 10540 2c2 11695 〈“cs3 14206 Vtxcvtx 26783 WWalksNOn cwwlksnon 27607 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-3 11704 df-n0 11901 df-z 11985 df-uz 12247 df-fz 12896 df-fzo 13037 df-hash 13694 df-word 13865 df-concat 13925 df-s1 13952 df-s2 14212 df-s3 14213 df-wwlks 27610 df-wwlksn 27611 df-wwlksnon 27612 |
This theorem is referenced by: elwwlks2on 27740 frgr2wwlk1 28110 |
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