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Mirrors > Home > MPE Home > Th. List > wlkp1lem4 | Structured version Visualization version GIF version |
Description: Lemma for wlkp1 29447. (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 2726 | . . . . 5 β’ (iEdgβπΊ) = (iEdgβπΊ) | |
3 | 2 | wlkf 29380 | . . . 4 β’ (πΉ(WalksβπΊ)π β πΉ β Word dom (iEdgβπΊ)) |
4 | eqid 2726 | . . . . 5 β’ (VtxβπΊ) = (VtxβπΊ) | |
5 | 4 | wlkp 29382 | . . . 4 β’ (πΉ(WalksβπΊ)π β π:(0...(β―βπΉ))βΆ(VtxβπΊ)) |
6 | 3, 5 | jca 511 | . . 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 2830 | . . . . 5 β’ (π β πΆ β (Vtxβπ)) |
11 | 10 | elfvexd 6924 | . . . 4 β’ (π β π β V) |
12 | 11 | adantr 480 | . . 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 768 | . . . . 5 β’ ((π β§ (πΉ β Word dom (iEdgβπΊ) β§ π:(0...(β―βπΉ))βΆ(VtxβπΊ))) β πΉ β Word dom (iEdgβπΊ)) | |
15 | snex 5424 | . . . . 5 β’ {β¨π, π΅β©} β V | |
16 | unexg 7733 | . . . . 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 585 | . . . 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 2831 | . . 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 7438 | . . . . . . 7 β’ (0...(β―βπΉ)) β V | |
21 | fvex 6898 | . . . . . . 7 β’ (VtxβπΊ) β V | |
22 | 20, 21 | fpm 8871 | . . . . . 6 β’ (π:(0...(β―βπΉ))βΆ(VtxβπΊ) β π β ((VtxβπΊ) βpm (0...(β―βπΉ)))) |
23 | 22 | ad2antll 726 | . . . . 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 5424 | . . . . 5 β’ {β¨(π + 1), πΆβ©} β V | |
25 | unexg 7733 | . . . . 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 585 | . . . 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 2831 | . . 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 1125 | . 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 684 | 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 395 β§ w3a 1084 = wceq 1533 β wcel 2098 Vcvv 3468 βͺ cun 3941 β wss 3943 {csn 4623 {cpr 4625 β¨cop 4629 class class class wbr 5141 dom cdm 5669 Fun wfun 6531 βΆwf 6533 βcfv 6537 (class class class)co 7405 βpm cpm 8823 Fincfn 8941 0cc0 11112 1c1 11113 + caddc 11115 ...cfz 13490 β―chash 14295 Word cword 14470 Vtxcvtx 28764 iEdgciedg 28765 Edgcedg 28815 Walkscwlks 29362 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-ifp 1060 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-er 8705 df-map 8824 df-pm 8825 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-n0 12477 df-z 12563 df-uz 12827 df-fz 13491 df-fzo 13634 df-hash 14296 df-word 14471 df-wlks 29365 |
This theorem is referenced by: wlkp1 29447 |
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