![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > wlkp1lem3 | Structured version Visualization version GIF version |
Description: Lemma for wlkp1 29508. (Contributed by AV, 6-Mar-2021.) |
Ref | Expression |
---|---|
wlkp1.v | β’ π = (VtxβπΊ) |
wlkp1.i | β’ πΌ = (iEdgβπΊ) |
wlkp1.f | β’ (π β Fun πΌ) |
wlkp1.a | β’ (π β πΌ β Fin) |
wlkp1.b | β’ (π β π΅ β π) |
wlkp1.c | β’ (π β πΆ β π) |
wlkp1.d | β’ (π β Β¬ π΅ β dom πΌ) |
wlkp1.w | β’ (π β πΉ(WalksβπΊ)π) |
wlkp1.n | β’ π = (β―βπΉ) |
wlkp1.e | β’ (π β πΈ β (EdgβπΊ)) |
wlkp1.x | β’ (π β {(πβπ), πΆ} β πΈ) |
wlkp1.u | β’ (π β (iEdgβπ) = (πΌ βͺ {β¨π΅, πΈβ©})) |
wlkp1.h | β’ π» = (πΉ βͺ {β¨π, π΅β©}) |
Ref | Expression |
---|---|
wlkp1lem3 | β’ (π β ((iEdgβπ)β(π»βπ)) = ((πΌ βͺ {β¨π΅, πΈβ©})βπ΅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wlkp1.u | . 2 β’ (π β (iEdgβπ) = (πΌ βͺ {β¨π΅, πΈβ©})) | |
2 | wlkp1.h | . . . . 5 β’ π» = (πΉ βͺ {β¨π, π΅β©}) | |
3 | 2 | a1i 11 | . . . 4 β’ (π β π» = (πΉ βͺ {β¨π, π΅β©})) |
4 | 3 | fveq1d 6899 | . . 3 β’ (π β (π»βπ) = ((πΉ βͺ {β¨π, π΅β©})βπ)) |
5 | wlkp1.n | . . . . 5 β’ π = (β―βπΉ) | |
6 | 5 | fvexi 6911 | . . . 4 β’ π β V |
7 | wlkp1.b | . . . 4 β’ (π β π΅ β π) | |
8 | wlkp1.w | . . . . 5 β’ (π β πΉ(WalksβπΊ)π) | |
9 | wlkp1.i | . . . . . 6 β’ πΌ = (iEdgβπΊ) | |
10 | 9 | wlkf 29441 | . . . . 5 β’ (πΉ(WalksβπΊ)π β πΉ β Word dom πΌ) |
11 | lencl 14516 | . . . . . 6 β’ (πΉ β Word dom πΌ β (β―βπΉ) β β0) | |
12 | wrddm 14504 | . . . . . 6 β’ (πΉ β Word dom πΌ β dom πΉ = (0..^(β―βπΉ))) | |
13 | fzonel 13679 | . . . . . . 7 β’ Β¬ (β―βπΉ) β (0..^(β―βπΉ)) | |
14 | 5 | a1i 11 | . . . . . . . 8 β’ (((β―βπΉ) β β0 β§ dom πΉ = (0..^(β―βπΉ))) β π = (β―βπΉ)) |
15 | simpr 484 | . . . . . . . 8 β’ (((β―βπΉ) β β0 β§ dom πΉ = (0..^(β―βπΉ))) β dom πΉ = (0..^(β―βπΉ))) | |
16 | 14, 15 | eleq12d 2823 | . . . . . . 7 β’ (((β―βπΉ) β β0 β§ dom πΉ = (0..^(β―βπΉ))) β (π β dom πΉ β (β―βπΉ) β (0..^(β―βπΉ)))) |
17 | 13, 16 | mtbiri 327 | . . . . . 6 β’ (((β―βπΉ) β β0 β§ dom πΉ = (0..^(β―βπΉ))) β Β¬ π β dom πΉ) |
18 | 11, 12, 17 | syl2anc 583 | . . . . 5 β’ (πΉ β Word dom πΌ β Β¬ π β dom πΉ) |
19 | 8, 10, 18 | 3syl 18 | . . . 4 β’ (π β Β¬ π β dom πΉ) |
20 | fsnunfv 7196 | . . . 4 β’ ((π β V β§ π΅ β π β§ Β¬ π β dom πΉ) β ((πΉ βͺ {β¨π, π΅β©})βπ) = π΅) | |
21 | 6, 7, 19, 20 | mp3an2i 1463 | . . 3 β’ (π β ((πΉ βͺ {β¨π, π΅β©})βπ) = π΅) |
22 | 4, 21 | eqtrd 2768 | . 2 β’ (π β (π»βπ) = π΅) |
23 | 1, 22 | fveq12d 6904 | 1 β’ (π β ((iEdgβπ)β(π»βπ)) = ((πΌ βͺ {β¨π΅, πΈβ©})βπ΅)) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β§ wa 395 = wceq 1534 β wcel 2099 Vcvv 3471 βͺ cun 3945 β wss 3947 {csn 4629 {cpr 4631 β¨cop 4635 class class class wbr 5148 dom cdm 5678 Fun wfun 6542 βcfv 6548 (class class class)co 7420 Fincfn 8964 0cc0 11139 β0cn0 12503 ..^cfzo 13660 β―chash 14322 Word cword 14497 Vtxcvtx 28822 iEdgciedg 28823 Edgcedg 28873 Walkscwlks 29423 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-ifp 1062 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-card 9963 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-nn 12244 df-n0 12504 df-z 12590 df-uz 12854 df-fz 13518 df-fzo 13661 df-hash 14323 df-word 14498 df-wlks 29426 |
This theorem is referenced by: wlkp1lem7 29506 wlkp1lem8 29507 |
Copyright terms: Public domain | W3C validator |