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Mirrors > Home > MPE Home > Th. List > numclwlk2lem2fv | Structured version Visualization version GIF version |
Description: Value of the function π . (Contributed by Alexander van der Vekens, 6-Oct-2018.) (Revised by AV, 31-May-2021.) (Revised by AV, 1-Nov-2022.) |
Ref | Expression |
---|---|
numclwwlk.v | β’ π = (VtxβπΊ) |
numclwwlk.q | β’ π = (π£ β π, π β β β¦ {π€ β (π WWalksN πΊ) β£ ((π€β0) = π£ β§ (lastSβπ€) β π£)}) |
numclwwlk.h | β’ π» = (π£ β π, π β (β€β₯β2) β¦ {π€ β (π£(ClWWalksNOnβπΊ)π) β£ (π€β(π β 2)) β π£}) |
numclwwlk.r | β’ π = (π₯ β (ππ»(π + 2)) β¦ (π₯ prefix (π + 1))) |
Ref | Expression |
---|---|
numclwlk2lem2fv | β’ ((π β π β§ π β β) β (π β (ππ»(π + 2)) β (π βπ) = (π prefix (π + 1)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | numclwwlk.r | . . 3 β’ π = (π₯ β (ππ»(π + 2)) β¦ (π₯ prefix (π + 1))) | |
2 | oveq1 7421 | . . 3 β’ (π₯ = π β (π₯ prefix (π + 1)) = (π prefix (π + 1))) | |
3 | simpr 484 | . . 3 β’ (((π β π β§ π β β) β§ π β (ππ»(π + 2))) β π β (ππ»(π + 2))) | |
4 | ovexd 7449 | . . 3 β’ (((π β π β§ π β β) β§ π β (ππ»(π + 2))) β (π prefix (π + 1)) β V) | |
5 | 1, 2, 3, 4 | fvmptd3 7022 | . 2 β’ (((π β π β§ π β β) β§ π β (ππ»(π + 2))) β (π βπ) = (π prefix (π + 1))) |
6 | 5 | ex 412 | 1 β’ ((π β π β§ π β β) β (π β (ππ»(π + 2)) β (π βπ) = (π prefix (π + 1)))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1534 β wcel 2099 β wne 2935 {crab 3427 Vcvv 3469 β¦ cmpt 5225 βcfv 6542 (class class class)co 7414 β cmpo 7416 0cc0 11130 1c1 11131 + caddc 11133 β cmin 11466 βcn 12234 2c2 12289 β€β₯cuz 12844 lastSclsw 14536 prefix cpfx 14644 Vtxcvtx 28796 WWalksN cwwlksn 29624 ClWWalksNOncclwwlknon 29884 |
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-sep 5293 ax-nul 5300 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 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-ral 3057 df-rex 3066 df-rab 3428 df-v 3471 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-iota 6494 df-fun 6544 df-fv 6550 df-ov 7417 |
This theorem is referenced by: numclwlk2lem2f1o 30176 |
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