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Mirrors > Home > MPE Home > Th. List > wwlksonvtx | Structured version Visualization version GIF version |
Description: If a word ð represents a walk of length 2 on two classes ðŽ and ð¶, these classes are vertices. (Contributed by AV, 14-Mar-2022.) |
Ref | Expression |
---|---|
wwlksonvtx.v | ⢠ð = (Vtxâðº) |
Ref | Expression |
---|---|
wwlksonvtx | ⢠(ð â (ðŽ(ð WWalksNOn ðº)ð¶) â (ðŽ â ð ⧠ð¶ â ð)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvex 6888 | . . . . 5 ⢠(Vtxâð) â V | |
2 | 1, 1 | pm3.2i 471 | . . . 4 ⢠((Vtxâð) â V ⧠(Vtxâð) â V) |
3 | 2 | rgen2w 3065 | . . 3 ⢠âð â â0 âð â V ((Vtxâð) â V ⧠(Vtxâð) â V) |
4 | df-wwlksnon 28946 | . . . 4 ⢠WWalksNOn = (ð â â0, ð â V ⊠(ð â (Vtxâð), ð â (Vtxâð) ⊠{ð€ â (ð WWalksN ð) ⣠((ð€â0) = ð ⧠(ð€âð) = ð)})) | |
5 | fveq2 6875 | . . . . . 6 ⢠(ð = ðº â (Vtxâð) = (Vtxâðº)) | |
6 | 5, 5 | jca 512 | . . . . 5 ⢠(ð = ðº â ((Vtxâð) = (Vtxâðº) ⧠(Vtxâð) = (Vtxâðº))) |
7 | 6 | adantl 482 | . . . 4 ⢠((ð = ð ⧠ð = ðº) â ((Vtxâð) = (Vtxâðº) ⧠(Vtxâð) = (Vtxâðº))) |
8 | 4, 7 | el2mpocl 8051 | . . 3 ⢠(âð â â0 âð â V ((Vtxâð) â V ⧠(Vtxâð) â V) â (ð â (ðŽ(ð WWalksNOn ðº)ð¶) â ((ð â â0 ⧠ðº â V) ⧠(ðŽ â (Vtxâðº) ⧠ð¶ â (Vtxâðº))))) |
9 | 3, 8 | ax-mp 5 | . 2 ⢠(ð â (ðŽ(ð WWalksNOn ðº)ð¶) â ((ð â â0 ⧠ðº â V) ⧠(ðŽ â (Vtxâðº) ⧠ð¶ â (Vtxâðº)))) |
10 | wwlksonvtx.v | . . . . 5 ⢠ð = (Vtxâðº) | |
11 | 10 | eleq2i 2824 | . . . 4 ⢠(ðŽ â ð â ðŽ â (Vtxâðº)) |
12 | 10 | eleq2i 2824 | . . . 4 ⢠(ð¶ â ð â ð¶ â (Vtxâðº)) |
13 | 11, 12 | anbi12i 627 | . . 3 ⢠((ðŽ â ð ⧠ð¶ â ð) â (ðŽ â (Vtxâðº) ⧠ð¶ â (Vtxâðº))) |
14 | 13 | biimpri 227 | . 2 ⢠((ðŽ â (Vtxâðº) ⧠ð¶ â (Vtxâðº)) â (ðŽ â ð ⧠ð¶ â ð)) |
15 | 9, 14 | simpl2im 504 | 1 ⢠(ð â (ðŽ(ð WWalksNOn ðº)ð¶) â (ðŽ â ð ⧠ð¶ â ð)) |
Colors of variables: wff setvar class |
Syntax hints: â wi 4 ⧠wa 396 = wceq 1541 â wcel 2106 âwral 3060 {crab 3429 Vcvv 3470 âcfv 6529 (class class class)co 7390 0cc0 11089 â0cn0 12451 Vtxcvtx 28116 WWalksN cwwlksn 28940 WWalksNOn cwwlksnon 28941 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7705 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3430 df-v 3472 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4520 df-pw 4595 df-sn 4620 df-pr 4622 df-op 4626 df-uni 4899 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-id 5564 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-iota 6481 df-fun 6531 df-fn 6532 df-f 6533 df-f1 6534 df-fo 6535 df-f1o 6536 df-fv 6537 df-ov 7393 df-oprab 7394 df-mpo 7395 df-1st 7954 df-2nd 7955 df-wwlksnon 28946 |
This theorem is referenced by: iswspthsnon 28970 wwlks2onv 29067 elwwlks2ons3im 29068 |
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