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Mirrors > Home > MPE Home > Th. List > eupth2lem3lem2 | Structured version Visualization version GIF version |
Description: Lemma for eupth2lem3 30033. (Contributed by AV, 21-Feb-2021.) |
Ref | Expression |
---|---|
trlsegvdeg.v | β’ π = (VtxβπΊ) |
trlsegvdeg.i | β’ πΌ = (iEdgβπΊ) |
trlsegvdeg.f | β’ (π β Fun πΌ) |
trlsegvdeg.n | β’ (π β π β (0..^(β―βπΉ))) |
trlsegvdeg.u | β’ (π β π β π) |
trlsegvdeg.w | β’ (π β πΉ(TrailsβπΊ)π) |
trlsegvdeg.vx | β’ (π β (Vtxβπ) = π) |
trlsegvdeg.vy | β’ (π β (Vtxβπ) = π) |
trlsegvdeg.vz | β’ (π β (Vtxβπ) = π) |
trlsegvdeg.ix | β’ (π β (iEdgβπ) = (πΌ βΎ (πΉ β (0..^π)))) |
trlsegvdeg.iy | β’ (π β (iEdgβπ) = {β¨(πΉβπ), (πΌβ(πΉβπ))β©}) |
trlsegvdeg.iz | β’ (π β (iEdgβπ) = (πΌ βΎ (πΉ β (0...π)))) |
Ref | Expression |
---|---|
eupth2lem3lem2 | β’ (π β ((VtxDegβπ)βπ) β β0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | trlsegvdeg.u | . . . . 5 β’ (π β π β π) | |
2 | trlsegvdeg.vy | . . . . 5 β’ (π β (Vtxβπ) = π) | |
3 | 1, 2 | eleqtrrd 2831 | . . . 4 β’ (π β π β (Vtxβπ)) |
4 | 3 | elfvexd 6930 | . . 3 β’ (π β π β V) |
5 | trlsegvdeg.v | . . . 4 β’ π = (VtxβπΊ) | |
6 | trlsegvdeg.i | . . . 4 β’ πΌ = (iEdgβπΊ) | |
7 | trlsegvdeg.f | . . . 4 β’ (π β Fun πΌ) | |
8 | trlsegvdeg.n | . . . 4 β’ (π β π β (0..^(β―βπΉ))) | |
9 | trlsegvdeg.w | . . . 4 β’ (π β πΉ(TrailsβπΊ)π) | |
10 | trlsegvdeg.vx | . . . 4 β’ (π β (Vtxβπ) = π) | |
11 | trlsegvdeg.vz | . . . 4 β’ (π β (Vtxβπ) = π) | |
12 | trlsegvdeg.ix | . . . 4 β’ (π β (iEdgβπ) = (πΌ βΎ (πΉ β (0..^π)))) | |
13 | trlsegvdeg.iy | . . . 4 β’ (π β (iEdgβπ) = {β¨(πΉβπ), (πΌβ(πΉβπ))β©}) | |
14 | trlsegvdeg.iz | . . . 4 β’ (π β (iEdgβπ) = (πΌ βΎ (πΉ β (0...π)))) | |
15 | 5, 6, 7, 8, 1, 9, 10, 2, 11, 12, 13, 14 | trlsegvdeglem7 30023 | . . 3 β’ (π β dom (iEdgβπ) β Fin) |
16 | eqid 2727 | . . . 4 β’ (Vtxβπ) = (Vtxβπ) | |
17 | eqid 2727 | . . . 4 β’ (iEdgβπ) = (iEdgβπ) | |
18 | eqid 2727 | . . . 4 β’ dom (iEdgβπ) = dom (iEdgβπ) | |
19 | 16, 17, 18 | vtxdgfisf 29277 | . . 3 β’ ((π β V β§ dom (iEdgβπ) β Fin) β (VtxDegβπ):(Vtxβπ)βΆβ0) |
20 | 4, 15, 19 | syl2anc 583 | . 2 β’ (π β (VtxDegβπ):(Vtxβπ)βΆβ0) |
21 | 20, 3 | ffvelcdmd 7089 | 1 β’ (π β ((VtxDegβπ)βπ) β β0) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1534 β wcel 2099 Vcvv 3469 {csn 4624 β¨cop 4630 class class class wbr 5142 dom cdm 5672 βΎ cres 5674 β cima 5675 Fun wfun 6536 βΆwf 6538 βcfv 6542 (class class class)co 7414 Fincfn 8955 0cc0 11130 β0cn0 12494 ...cfz 13508 ..^cfzo 13651 β―chash 14313 Vtxcvtx 28796 iEdgciedg 28797 VtxDegcvtxdg 29266 Trailsctrls 29491 |
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 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11186 ax-resscn 11187 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-addrcl 11191 ax-mulcl 11192 ax-mulrcl 11193 ax-mulcom 11194 ax-addass 11195 ax-mulass 11196 ax-distr 11197 ax-i2m1 11198 ax-1ne0 11199 ax-1rid 11200 ax-rnegex 11201 ax-rrecex 11202 ax-cnre 11203 ax-pre-lttri 11204 ax-pre-lttrn 11205 ax-pre-ltadd 11206 ax-pre-mulgt0 11207 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-card 9954 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11468 df-neg 11469 df-nn 12235 df-n0 12495 df-xnn0 12567 df-z 12581 df-uz 12845 df-xadd 13117 df-hash 14314 df-vtxdg 29267 |
This theorem is referenced by: eupth2lem3lem3 30027 |
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