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Mirrors > Home > MPE Home > Th. List > wlkcompim | Structured version Visualization version GIF version |
Description: Implications for the properties of the components of a walk. (Contributed by Alexander van der Vekens, 23-Jun-2018.) (Revised by AV, 2-Jan-2021.) |
Ref | Expression |
---|---|
wlkcomp.v | β’ π = (VtxβπΊ) |
wlkcomp.i | β’ πΌ = (iEdgβπΊ) |
wlkcomp.1 | β’ πΉ = (1st βπ) |
wlkcomp.2 | β’ π = (2nd βπ) |
Ref | Expression |
---|---|
wlkcompim | β’ (π β (WalksβπΊ) β (πΉ β Word dom πΌ β§ π:(0...(β―βπΉ))βΆπ β§ βπ β (0..^(β―βπΉ))if-((πβπ) = (πβ(π + 1)), (πΌβ(πΉβπ)) = {(πβπ)}, {(πβπ), (πβ(π + 1))} β (πΌβ(πΉβπ))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfvex 6930 | . 2 β’ (π β (WalksβπΊ) β πΊ β V) | |
2 | wlkcpr 29487 | . . 3 β’ (π β (WalksβπΊ) β (1st βπ)(WalksβπΊ)(2nd βπ)) | |
3 | wlkvv 29485 | . . 3 β’ ((1st βπ)(WalksβπΊ)(2nd βπ) β π β (V Γ V)) | |
4 | 2, 3 | sylbi 216 | . 2 β’ (π β (WalksβπΊ) β π β (V Γ V)) |
5 | wlkcomp.v | . . . 4 β’ π = (VtxβπΊ) | |
6 | wlkcomp.i | . . . 4 β’ πΌ = (iEdgβπΊ) | |
7 | wlkcomp.1 | . . . 4 β’ πΉ = (1st βπ) | |
8 | wlkcomp.2 | . . . 4 β’ π = (2nd βπ) | |
9 | 5, 6, 7, 8 | wlkcomp 29489 | . . 3 β’ ((πΊ β V β§ π β (V Γ V)) β (π β (WalksβπΊ) β (πΉ β Word dom πΌ β§ π:(0...(β―βπΉ))βΆπ β§ βπ β (0..^(β―βπΉ))if-((πβπ) = (πβ(π + 1)), (πΌβ(πΉβπ)) = {(πβπ)}, {(πβπ), (πβ(π + 1))} β (πΌβ(πΉβπ)))))) |
10 | 9 | biimpcd 248 | . 2 β’ (π β (WalksβπΊ) β ((πΊ β V β§ π β (V Γ V)) β (πΉ β Word dom πΌ β§ π:(0...(β―βπΉ))βΆπ β§ βπ β (0..^(β―βπΉ))if-((πβπ) = (πβ(π + 1)), (πΌβ(πΉβπ)) = {(πβπ)}, {(πβπ), (πβ(π + 1))} β (πΌβ(πΉβπ)))))) |
11 | 1, 4, 10 | mp2and 697 | 1 β’ (π β (WalksβπΊ) β (πΉ β Word dom πΌ β§ π:(0...(β―βπΉ))βΆπ β§ βπ β (0..^(β―βπΉ))if-((πβπ) = (πβ(π + 1)), (πΌβ(πΉβπ)) = {(πβπ)}, {(πβπ), (πβ(π + 1))} β (πΌβ(πΉβπ))))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 if-wif 1060 β§ w3a 1084 = wceq 1533 β wcel 2098 βwral 3051 Vcvv 3463 β wss 3939 {csn 4624 {cpr 4626 class class class wbr 5143 Γ cxp 5670 dom cdm 5672 βΆwf 6539 βcfv 6543 (class class class)co 7416 1st c1st 7989 2nd c2nd 7990 0cc0 11138 1c1 11139 + caddc 11141 ...cfz 13516 ..^cfzo 13659 β―chash 14321 Word cword 14496 Vtxcvtx 28853 iEdgciedg 28854 Walkscwlks 29454 |
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 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-ifp 1061 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 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 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 5144 df-opab 5206 df-mpt 5227 df-tr 5261 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 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7991 df-2nd 7992 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-er 8723 df-map 8845 df-pm 8846 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 df-card 9962 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-n0 12503 df-z 12589 df-uz 12853 df-fz 13517 df-fzo 13660 df-hash 14322 df-word 14497 df-wlks 29457 |
This theorem is referenced by: wlkelwrd 29491 |
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