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Mathbox for Norm Megill |
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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 6885 | . . . . . 6 β’ (π = πΊ β (π βπ) = (π βπΊ)) | |
2 | 1 | oveq2d 7421 | . . . . 5 β’ (π = πΊ β (π β¨ (π βπ)) = (π β¨ (π βπΊ))) |
3 | coeq1 5851 | . . . . . . 7 β’ (π = πΊ β (π β β‘πΉ) = (πΊ β β‘πΉ)) | |
4 | 3 | fveq2d 6889 | . . . . . 6 β’ (π = πΊ β (π β(π β β‘πΉ)) = (π β(πΊ β β‘πΉ))) |
5 | 4 | oveq2d 7421 | . . . . 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 7423 | . . . 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 2737 | . . 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 7363 | . 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 7365 | . 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 6992 | 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 5224 β‘ccnv 5668 β ccom 5673 βcfv 6537 β©crio 7360 (class class class)co 7405 Basecbs 17153 lecple 17213 joincjn 18276 meetcmee 18277 Atomscatm 38646 LHypclh 39368 LTrncltrn 39485 trLctrl 39542 |
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 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-iota 6489 df-fun 6539 df-fv 6545 df-riota 7361 df-ov 7408 |
This theorem is referenced by: cdlemksel 40229 cdlemksv2 40231 cdlemkuvN 40248 cdlemkuel 40249 cdlemkuv2 40251 cdlemkuv-2N 40267 cdlemkuu 40279 |
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