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Mirrors > Home > MPE Home > Th. List > numclwwlk1lem2fv | Structured version Visualization version GIF version |
Description: Value of the function π. (Contributed by Alexander van der Vekens, 20-Sep-2018.) (Revised by AV, 29-May-2021.) (Revised by AV, 31-Oct-2022.) |
Ref | Expression |
---|---|
extwwlkfab.v | β’ π = (VtxβπΊ) |
extwwlkfab.c | β’ πΆ = (π£ β π, π β (β€β₯β2) β¦ {π€ β (π£(ClWWalksNOnβπΊ)π) β£ (π€β(π β 2)) = π£}) |
extwwlkfab.f | β’ πΉ = (π(ClWWalksNOnβπΊ)(π β 2)) |
numclwwlk.t | β’ π = (π’ β (ππΆπ) β¦ β¨(π’ prefix (π β 2)), (π’β(π β 1))β©) |
Ref | Expression |
---|---|
numclwwlk1lem2fv | β’ (π β (ππΆπ) β (πβπ) = β¨(π prefix (π β 2)), (πβ(π β 1))β©) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq1 7369 | . . 3 β’ (π’ = π β (π’ prefix (π β 2)) = (π prefix (π β 2))) | |
2 | fveq1 6846 | . . 3 β’ (π’ = π β (π’β(π β 1)) = (πβ(π β 1))) | |
3 | 1, 2 | opeq12d 4843 | . 2 β’ (π’ = π β β¨(π’ prefix (π β 2)), (π’β(π β 1))β© = β¨(π prefix (π β 2)), (πβ(π β 1))β©) |
4 | numclwwlk.t | . 2 β’ π = (π’ β (ππΆπ) β¦ β¨(π’ prefix (π β 2)), (π’β(π β 1))β©) | |
5 | opex 5426 | . 2 β’ β¨(π prefix (π β 2)), (πβ(π β 1))β© β V | |
6 | 3, 4, 5 | fvmpt 6953 | 1 β’ (π β (ππΆπ) β (πβπ) = β¨(π prefix (π β 2)), (πβ(π β 1))β©) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 {crab 3410 β¨cop 4597 β¦ cmpt 5193 βcfv 6501 (class class class)co 7362 β cmpo 7364 1c1 11059 β cmin 11392 2c2 12215 β€β₯cuz 12770 prefix cpfx 14565 Vtxcvtx 27989 ClWWalksNOncclwwlknon 29073 |
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-sep 5261 ax-nul 5268 ax-pr 5389 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 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-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-br 5111 df-opab 5173 df-mpt 5194 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-iota 6453 df-fun 6503 df-fv 6509 df-ov 7365 |
This theorem is referenced by: numclwwlk1lem2f1 29343 numclwwlk1lem2fo 29344 |
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