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Mirrors > Home > MPE Home > Th. List > wlkp1lem3 | Structured version Visualization version GIF version |
Description: Lemma for wlkp1 28678. (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 6848 | . . 3 β’ (π β (π»βπ) = ((πΉ βͺ {β¨π, π΅β©})βπ)) |
5 | wlkp1.n | . . . . 5 β’ π = (β―βπΉ) | |
6 | 5 | fvexi 6860 | . . . 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 28611 | . . . . 5 β’ (πΉ(WalksβπΊ)π β πΉ β Word dom πΌ) |
11 | lencl 14430 | . . . . . 6 β’ (πΉ β Word dom πΌ β (β―βπΉ) β β0) | |
12 | wrddm 14418 | . . . . . 6 β’ (πΉ β Word dom πΌ β dom πΉ = (0..^(β―βπΉ))) | |
13 | fzonel 13595 | . . . . . . 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 486 | . . . . . . . 8 β’ (((β―βπΉ) β β0 β§ dom πΉ = (0..^(β―βπΉ))) β dom πΉ = (0..^(β―βπΉ))) | |
16 | 14, 15 | eleq12d 2828 | . . . . . . 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 585 | . . . . 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 7137 | . . . 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 1467 | . . 3 β’ (π β ((πΉ βͺ {β¨π, π΅β©})βπ) = π΅) |
22 | 4, 21 | eqtrd 2773 | . 2 β’ (π β (π»βπ) = π΅) |
23 | 1, 22 | fveq12d 6853 | 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 397 = wceq 1542 β wcel 2107 Vcvv 3447 βͺ cun 3912 β wss 3914 {csn 4590 {cpr 4592 β¨cop 4596 class class class wbr 5109 dom cdm 5637 Fun wfun 6494 βcfv 6500 (class class class)co 7361 Fincfn 8889 0cc0 11059 β0cn0 12421 ..^cfzo 13576 β―chash 14239 Word cword 14411 Vtxcvtx 27996 iEdgciedg 27997 Edgcedg 28047 Walkscwlks 28593 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-ifp 1063 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-int 4912 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7807 df-1st 7925 df-2nd 7926 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-1o 8416 df-er 8654 df-map 8773 df-en 8890 df-dom 8891 df-sdom 8892 df-fin 8893 df-card 9883 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-nn 12162 df-n0 12422 df-z 12508 df-uz 12772 df-fz 13434 df-fzo 13577 df-hash 14240 df-word 14412 df-wlks 28596 |
This theorem is referenced by: wlkp1lem7 28676 wlkp1lem8 28677 |
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