![]() |
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > cdlemksv | Structured version Visualization version GIF version |
Description: Part of proof of Lemma K of [Crawley] p. 118. Value of the sigma(p) function. (Contributed by NM, 26-Jun-2013.) |
Ref | Expression |
---|---|
cdlemk.b | β’ π΅ = (BaseβπΎ) |
cdlemk.l | β’ β€ = (leβπΎ) |
cdlemk.j | β’ β¨ = (joinβπΎ) |
cdlemk.a | β’ π΄ = (AtomsβπΎ) |
cdlemk.h | β’ π» = (LHypβπΎ) |
cdlemk.t | β’ π = ((LTrnβπΎ)βπ) |
cdlemk.r | β’ π = ((trLβπΎ)βπ) |
cdlemk.m | β’ β§ = (meetβπΎ) |
cdlemk.s | β’ π = (π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘πΉ)))))) |
Ref | Expression |
---|---|
cdlemksv | β’ (πΊ β π β (πβπΊ) = (β©π β π (πβπ) = ((π β¨ (π βπΊ)) β§ ((πβπ) β¨ (π β(πΊ β β‘πΉ)))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6902 | . . . . . 6 β’ (π = πΊ β (π βπ) = (π βπΊ)) | |
2 | 1 | oveq2d 7442 | . . . . 5 β’ (π = πΊ β (π β¨ (π βπ)) = (π β¨ (π βπΊ))) |
3 | coeq1 5864 | . . . . . . 7 β’ (π = πΊ β (π β β‘πΉ) = (πΊ β β‘πΉ)) | |
4 | 3 | fveq2d 6906 | . . . . . 6 β’ (π = πΊ β (π β(π β β‘πΉ)) = (π β(πΊ β β‘πΉ))) |
5 | 4 | oveq2d 7442 | . . . . 5 β’ (π = πΊ β ((πβπ) β¨ (π β(π β β‘πΉ))) = ((πβπ) β¨ (π β(πΊ β β‘πΉ)))) |
6 | 2, 5 | oveq12d 7444 | . . . 4 β’ (π = πΊ β ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘πΉ)))) = ((π β¨ (π βπΊ)) β§ ((πβπ) β¨ (π β(πΊ β β‘πΉ))))) |
7 | 6 | eqeq2d 2739 | . . 3 β’ (π = πΊ β ((πβπ) = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘πΉ)))) β (πβπ) = ((π β¨ (π βπΊ)) β§ ((πβπ) β¨ (π β(πΊ β β‘πΉ)))))) |
8 | 7 | riotabidv 7384 | . 2 β’ (π = πΊ β (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘πΉ))))) = (β©π β π (πβπ) = ((π β¨ (π βπΊ)) β§ ((πβπ) β¨ (π β(πΊ β β‘πΉ)))))) |
9 | cdlemk.s | . 2 β’ π = (π β π β¦ (β©π β π (πβπ) = ((π β¨ (π βπ)) β§ ((πβπ) β¨ (π β(π β β‘πΉ)))))) | |
10 | riotaex 7386 | . 2 β’ (β©π β π (πβπ) = ((π β¨ (π βπΊ)) β§ ((πβπ) β¨ (π β(πΊ β β‘πΉ))))) β V | |
11 | 8, 9, 10 | fvmpt 7010 | 1 β’ (πΊ β π β (πβπΊ) = (β©π β π (πβπ) = ((π β¨ (π βπΊ)) β§ ((πβπ) β¨ (π β(πΊ β β‘πΉ)))))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 β¦ cmpt 5235 β‘ccnv 5681 β ccom 5686 βcfv 6553 β©crio 7381 (class class class)co 7426 Basecbs 17189 lecple 17249 joincjn 18312 meetcmee 18313 Atomscatm 38775 LHypclh 39497 LTrncltrn 39614 trLctrl 39671 |
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 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-iota 6505 df-fun 6555 df-fv 6561 df-riota 7382 df-ov 7429 |
This theorem is referenced by: cdlemksel 40358 cdlemksv2 40360 cdlemkuvN 40377 cdlemkuel 40378 cdlemkuv2 40380 cdlemkuv-2N 40396 cdlemkuu 40408 |
Copyright terms: Public domain | W3C validator |