![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > rusgrnumwwlklem | Structured version Visualization version GIF version |
Description: Lemma for rusgrnumwwlk 28920 etc. (Contributed by Alexander van der Vekens, 21-Jul-2018.) (Revised by AV, 7-May-2021.) |
Ref | Expression |
---|---|
rusgrnumwwlk.v | ⢠ð = (Vtxâðº) |
rusgrnumwwlk.l | ⢠ð¿ = (ð£ â ð, ð â â0 ⊠(â¯â{ð€ â (ð WWalksN ðº) ⣠(ð€â0) = ð£})) |
Ref | Expression |
---|---|
rusgrnumwwlklem | ⢠((ð â ð ⧠ð â â0) â (ðð¿ð) = (â¯â{ð€ â (ð WWalksN ðº) ⣠(ð€â0) = ð})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq1 7364 | . . . . 5 ⢠(ð = ð â (ð WWalksN ðº) = (ð WWalksN ðº)) | |
2 | 1 | adantl 482 | . . . 4 ⢠((ð£ = ð ⧠ð = ð) â (ð WWalksN ðº) = (ð WWalksN ðº)) |
3 | eqeq2 2748 | . . . . 5 ⢠(ð£ = ð â ((ð€â0) = ð£ â (ð€â0) = ð)) | |
4 | 3 | adantr 481 | . . . 4 ⢠((ð£ = ð ⧠ð = ð) â ((ð€â0) = ð£ â (ð€â0) = ð)) |
5 | 2, 4 | rabeqbidv 3424 | . . 3 ⢠((ð£ = ð ⧠ð = ð) â {ð€ â (ð WWalksN ðº) ⣠(ð€â0) = ð£} = {ð€ â (ð WWalksN ðº) ⣠(ð€â0) = ð}) |
6 | 5 | fveq2d 6846 | . 2 ⢠((ð£ = ð ⧠ð = ð) â (â¯â{ð€ â (ð WWalksN ðº) ⣠(ð€â0) = ð£}) = (â¯â{ð€ â (ð WWalksN ðº) ⣠(ð€â0) = ð})) |
7 | rusgrnumwwlk.l | . 2 ⢠ð¿ = (ð£ â ð, ð â â0 ⊠(â¯â{ð€ â (ð WWalksN ðº) ⣠(ð€â0) = ð£})) | |
8 | fvex 6855 | . 2 ⢠(â¯â{ð€ â (ð WWalksN ðº) ⣠(ð€â0) = ð}) â V | |
9 | 6, 7, 8 | ovmpoa 7510 | 1 ⢠((ð â ð ⧠ð â â0) â (ðð¿ð) = (â¯â{ð€ â (ð WWalksN ðº) ⣠(ð€â0) = ð})) |
Colors of variables: wff setvar class |
Syntax hints: â wi 4 â wb 205 ⧠wa 396 = wceq 1541 â wcel 2106 {crab 3407 âcfv 6496 (class class class)co 7357 â cmpo 7359 0cc0 11051 â0cn0 12413 â¯chash 14230 Vtxcvtx 27947 WWalksN cwwlksn 28771 |
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 2707 ax-sep 5256 ax-nul 5263 ax-pr 5384 |
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 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-ral 3065 df-rex 3074 df-rab 3408 df-v 3447 df-sbc 3740 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-br 5106 df-opab 5168 df-id 5531 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-iota 6448 df-fun 6498 df-fv 6504 df-ov 7360 df-oprab 7361 df-mpo 7362 |
This theorem is referenced by: rusgrnumwwlkb0 28916 rusgrnumwwlkb1 28917 rusgr0edg 28918 rusgrnumwwlks 28919 rusgrnumwwlkg 28921 |
Copyright terms: Public domain | W3C validator |