Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > trlsegvdeglem1 | Structured version Visualization version GIF version |
Description: Lemma for trlsegvdeg 28636. (Contributed by AV, 20-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) |
Ref | Expression |
---|---|
trlsegvdeglem1 | β’ (π β ((πβπ) β π β§ (πβ(π + 1)) β π)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | trlsegvdeg.n | . 2 β’ (π β π β (0..^(β―βπΉ))) | |
2 | trlsegvdeg.w | . . 3 β’ (π β πΉ(TrailsβπΊ)π) | |
3 | trliswlk 28110 | . . 3 β’ (πΉ(TrailsβπΊ)π β πΉ(WalksβπΊ)π) | |
4 | trlsegvdeg.v | . . . . . . 7 β’ π = (VtxβπΊ) | |
5 | 4 | wlkpvtx 28072 | . . . . . 6 β’ (πΉ(WalksβπΊ)π β (π β (0...(β―βπΉ)) β (πβπ) β π)) |
6 | elfzofz 13449 | . . . . . 6 β’ (π β (0..^(β―βπΉ)) β π β (0...(β―βπΉ))) | |
7 | 5, 6 | impel 507 | . . . . 5 β’ ((πΉ(WalksβπΊ)π β§ π β (0..^(β―βπΉ))) β (πβπ) β π) |
8 | 4 | wlkpvtx 28072 | . . . . . 6 β’ (πΉ(WalksβπΊ)π β ((π + 1) β (0...(β―βπΉ)) β (πβ(π + 1)) β π)) |
9 | fzofzp1 13530 | . . . . . 6 β’ (π β (0..^(β―βπΉ)) β (π + 1) β (0...(β―βπΉ))) | |
10 | 8, 9 | impel 507 | . . . . 5 β’ ((πΉ(WalksβπΊ)π β§ π β (0..^(β―βπΉ))) β (πβ(π + 1)) β π) |
11 | 7, 10 | jca 513 | . . . 4 β’ ((πΉ(WalksβπΊ)π β§ π β (0..^(β―βπΉ))) β ((πβπ) β π β§ (πβ(π + 1)) β π)) |
12 | 11 | ex 414 | . . 3 β’ (πΉ(WalksβπΊ)π β (π β (0..^(β―βπΉ)) β ((πβπ) β π β§ (πβ(π + 1)) β π))) |
13 | 2, 3, 12 | 3syl 18 | . 2 β’ (π β (π β (0..^(β―βπΉ)) β ((πβπ) β π β§ (πβ(π + 1)) β π))) |
14 | 1, 13 | mpd 15 | 1 β’ (π β ((πβπ) β π β§ (πβ(π + 1)) β π)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1539 β wcel 2104 class class class wbr 5081 Fun wfun 6452 βcfv 6458 (class class class)co 7307 0cc0 10917 1c1 10918 + caddc 10920 ...cfz 13285 ..^cfzo 13428 β―chash 14090 Vtxcvtx 27411 iEdgciedg 27412 Walkscwlks 28008 Trailsctrls 28103 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2707 ax-rep 5218 ax-sep 5232 ax-nul 5239 ax-pow 5297 ax-pr 5361 ax-un 7620 ax-cnex 10973 ax-resscn 10974 ax-1cn 10975 ax-icn 10976 ax-addcl 10977 ax-addrcl 10978 ax-mulcl 10979 ax-mulrcl 10980 ax-mulcom 10981 ax-addass 10982 ax-mulass 10983 ax-distr 10984 ax-i2m1 10985 ax-1ne0 10986 ax-1rid 10987 ax-rnegex 10988 ax-rrecex 10989 ax-cnre 10990 ax-pre-lttri 10991 ax-pre-lttrn 10992 ax-pre-ltadd 10993 ax-pre-mulgt0 10994 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-ifp 1062 df-3or 1088 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3286 df-rab 3287 df-v 3439 df-sbc 3722 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4566 df-pr 4568 df-op 4572 df-uni 4845 df-int 4887 df-iun 4933 df-br 5082 df-opab 5144 df-mpt 5165 df-tr 5199 df-id 5500 df-eprel 5506 df-po 5514 df-so 5515 df-fr 5555 df-we 5557 df-xp 5606 df-rel 5607 df-cnv 5608 df-co 5609 df-dm 5610 df-rn 5611 df-res 5612 df-ima 5613 df-pred 6217 df-ord 6284 df-on 6285 df-lim 6286 df-suc 6287 df-iota 6410 df-fun 6460 df-fn 6461 df-f 6462 df-f1 6463 df-fo 6464 df-f1o 6465 df-fv 6466 df-riota 7264 df-ov 7310 df-oprab 7311 df-mpo 7312 df-om 7745 df-1st 7863 df-2nd 7864 df-frecs 8128 df-wrecs 8159 df-recs 8233 df-rdg 8272 df-1o 8328 df-er 8529 df-map 8648 df-en 8765 df-dom 8766 df-sdom 8767 df-fin 8768 df-card 9741 df-pnf 11057 df-mnf 11058 df-xr 11059 df-ltxr 11060 df-le 11061 df-sub 11253 df-neg 11254 df-nn 12020 df-n0 12280 df-z 12366 df-uz 12629 df-fz 13286 df-fzo 13429 df-hash 14091 df-word 14263 df-wlks 28011 df-trls 28105 |
This theorem is referenced by: eupth2lem3lem3 28639 eupth2lem3lem4 28640 eupth2lem3lem5 28641 |
Copyright terms: Public domain | W3C validator |