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Mirrors > Home > MPE Home > Th. List > wlkp1lem4 | Structured version Visualization version GIF version |
Description: Lemma for wlkp1 28671. (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©}) |
wlkp1.q | β’ π = (π βͺ {β¨(π + 1), πΆβ©}) |
wlkp1.s | β’ (π β (Vtxβπ) = π) |
Ref | Expression |
---|---|
wlkp1lem4 | β’ (π β (π β V β§ π» β V β§ π β V)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wlkp1.w | . . 3 β’ (π β πΉ(WalksβπΊ)π) | |
2 | eqid 2737 | . . . . 5 β’ (iEdgβπΊ) = (iEdgβπΊ) | |
3 | 2 | wlkf 28604 | . . . 4 β’ (πΉ(WalksβπΊ)π β πΉ β Word dom (iEdgβπΊ)) |
4 | eqid 2737 | . . . . 5 β’ (VtxβπΊ) = (VtxβπΊ) | |
5 | 4 | wlkp 28606 | . . . 4 β’ (πΉ(WalksβπΊ)π β π:(0...(β―βπΉ))βΆ(VtxβπΊ)) |
6 | 3, 5 | jca 513 | . . 3 β’ (πΉ(WalksβπΊ)π β (πΉ β Word dom (iEdgβπΊ) β§ π:(0...(β―βπΉ))βΆ(VtxβπΊ))) |
7 | 1, 6 | syl 17 | . 2 β’ (π β (πΉ β Word dom (iEdgβπΊ) β§ π:(0...(β―βπΉ))βΆ(VtxβπΊ))) |
8 | wlkp1.c | . . . . . 6 β’ (π β πΆ β π) | |
9 | wlkp1.s | . . . . . 6 β’ (π β (Vtxβπ) = π) | |
10 | 8, 9 | eleqtrrd 2841 | . . . . 5 β’ (π β πΆ β (Vtxβπ)) |
11 | 10 | elfvexd 6886 | . . . 4 β’ (π β π β V) |
12 | 11 | adantr 482 | . . 3 β’ ((π β§ (πΉ β Word dom (iEdgβπΊ) β§ π:(0...(β―βπΉ))βΆ(VtxβπΊ))) β π β V) |
13 | wlkp1.h | . . . 4 β’ π» = (πΉ βͺ {β¨π, π΅β©}) | |
14 | simprl 770 | . . . . 5 β’ ((π β§ (πΉ β Word dom (iEdgβπΊ) β§ π:(0...(β―βπΉ))βΆ(VtxβπΊ))) β πΉ β Word dom (iEdgβπΊ)) | |
15 | snex 5393 | . . . . 5 β’ {β¨π, π΅β©} β V | |
16 | unexg 7688 | . . . . 5 β’ ((πΉ β Word dom (iEdgβπΊ) β§ {β¨π, π΅β©} β V) β (πΉ βͺ {β¨π, π΅β©}) β V) | |
17 | 14, 15, 16 | sylancl 587 | . . . 4 β’ ((π β§ (πΉ β Word dom (iEdgβπΊ) β§ π:(0...(β―βπΉ))βΆ(VtxβπΊ))) β (πΉ βͺ {β¨π, π΅β©}) β V) |
18 | 13, 17 | eqeltrid 2842 | . . 3 β’ ((π β§ (πΉ β Word dom (iEdgβπΊ) β§ π:(0...(β―βπΉ))βΆ(VtxβπΊ))) β π» β V) |
19 | wlkp1.q | . . . 4 β’ π = (π βͺ {β¨(π + 1), πΆβ©}) | |
20 | ovex 7395 | . . . . . . 7 β’ (0...(β―βπΉ)) β V | |
21 | fvex 6860 | . . . . . . 7 β’ (VtxβπΊ) β V | |
22 | 20, 21 | fpm 8820 | . . . . . 6 β’ (π:(0...(β―βπΉ))βΆ(VtxβπΊ) β π β ((VtxβπΊ) βpm (0...(β―βπΉ)))) |
23 | 22 | ad2antll 728 | . . . . 5 β’ ((π β§ (πΉ β Word dom (iEdgβπΊ) β§ π:(0...(β―βπΉ))βΆ(VtxβπΊ))) β π β ((VtxβπΊ) βpm (0...(β―βπΉ)))) |
24 | snex 5393 | . . . . 5 β’ {β¨(π + 1), πΆβ©} β V | |
25 | unexg 7688 | . . . . 5 β’ ((π β ((VtxβπΊ) βpm (0...(β―βπΉ))) β§ {β¨(π + 1), πΆβ©} β V) β (π βͺ {β¨(π + 1), πΆβ©}) β V) | |
26 | 23, 24, 25 | sylancl 587 | . . . 4 β’ ((π β§ (πΉ β Word dom (iEdgβπΊ) β§ π:(0...(β―βπΉ))βΆ(VtxβπΊ))) β (π βͺ {β¨(π + 1), πΆβ©}) β V) |
27 | 19, 26 | eqeltrid 2842 | . . 3 β’ ((π β§ (πΉ β Word dom (iEdgβπΊ) β§ π:(0...(β―βπΉ))βΆ(VtxβπΊ))) β π β V) |
28 | 12, 18, 27 | 3jca 1129 | . 2 β’ ((π β§ (πΉ β Word dom (iEdgβπΊ) β§ π:(0...(β―βπΉ))βΆ(VtxβπΊ))) β (π β V β§ π» β V β§ π β V)) |
29 | 7, 28 | mpdan 686 | 1 β’ (π β (π β V β§ π» β V β§ π β V)) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β§ wa 397 β§ w3a 1088 = wceq 1542 β wcel 2107 Vcvv 3448 βͺ cun 3913 β wss 3915 {csn 4591 {cpr 4593 β¨cop 4597 class class class wbr 5110 dom cdm 5638 Fun wfun 6495 βΆwf 6497 βcfv 6501 (class class class)co 7362 βpm cpm 8773 Fincfn 8890 0cc0 11058 1c1 11059 + caddc 11061 ...cfz 13431 β―chash 14237 Word cword 14409 Vtxcvtx 27989 iEdgciedg 27990 Edgcedg 28040 Walkscwlks 28586 |
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 2708 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-int 4913 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-1st 7926 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-er 8655 df-map 8774 df-pm 8775 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-card 9882 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-nn 12161 df-n0 12421 df-z 12507 df-uz 12771 df-fz 13432 df-fzo 13575 df-hash 14238 df-word 14410 df-wlks 28589 |
This theorem is referenced by: wlkp1 28671 |
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