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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 29984 | . . . 4 ⊢ (𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) → (𝑊 = 〈“𝐴(𝑊‘1)𝐶”〉 ∧ (𝑊‘1) ∈ 𝑉)) |
4 | anass 468 | . . . 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 484 | . . . 4 ⊢ (((𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ∧ 𝑊 = 〈“𝐴(𝑊‘1)𝐶”〉) ∧ (𝑊‘1) ∈ 𝑉) → (𝑊‘1) ∈ 𝑉) | |
7 | s3eq2 14906 | . . . . . 6 ⊢ (𝑏 = (𝑊‘1) → 〈“𝐴𝑏𝐶”〉 = 〈“𝐴(𝑊‘1)𝐶”〉) | |
8 | eqeq2 2747 | . . . . . . 7 ⊢ (〈“𝐴𝑏𝐶”〉 = 〈“𝐴(𝑊‘1)𝐶”〉 → (𝑊 = 〈“𝐴𝑏𝐶”〉 ↔ 𝑊 = 〈“𝐴(𝑊‘1)𝐶”〉)) | |
9 | eleq1 2827 | . . . . . . 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 481 | . . . 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 484 | . . . . . 6 ⊢ ((𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ∧ 𝑊 = 〈“𝐴(𝑊‘1)𝐶”〉) → 𝑊 = 〈“𝐴(𝑊‘1)𝐶”〉) | |
14 | eleq1 2827 | . . . . . . 7 ⊢ (𝑊 = 〈“𝐴(𝑊‘1)𝐶”〉 → (𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ↔ 〈“𝐴(𝑊‘1)𝐶”〉 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶))) | |
15 | 14 | biimpac 478 | . . . . . 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 511 | . . . . 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 480 | . . . 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 3624 | . . 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 2827 | . . . . 5 ⊢ (〈“𝐴𝑏𝐶”〉 = 𝑊 → (〈“𝐴𝑏𝐶”〉 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ↔ 𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶))) | |
21 | 20 | eqcoms 2743 | . . . 4 ⊢ (𝑊 = 〈“𝐴𝑏𝐶”〉 → (〈“𝐴𝑏𝐶”〉 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ↔ 𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶))) |
22 | 21 | biimpa 476 | . . 3 ⊢ ((𝑊 = 〈“𝐴𝑏𝐶”〉 ∧ 〈“𝐴𝑏𝐶”〉 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)) → 𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)) |
23 | 22 | rexlimivw 3149 | . 2 ⊢ (∃𝑏 ∈ 𝑉 (𝑊 = 〈“𝐴𝑏𝐶”〉 ∧ 〈“𝐴𝑏𝐶”〉 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)) → 𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)) |
24 | 19, 23 | impbii 209 | 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 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∃wrex 3068 ‘cfv 6563 (class class class)co 7431 1c1 11154 2c2 12319 〈“cs3 14878 Vtxcvtx 29028 WWalksNOn cwwlksnon 29857 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-n0 12525 df-z 12612 df-uz 12877 df-fz 13545 df-fzo 13692 df-hash 14367 df-word 14550 df-concat 14606 df-s1 14631 df-s2 14884 df-s3 14885 df-wwlks 29860 df-wwlksn 29861 df-wwlksnon 29862 |
This theorem is referenced by: elwwlks2on 29989 frgr2wwlk1 30358 |
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