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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 6843 | . . . . . 6 β’ (π = πΊ β (π βπ) = (π βπΊ)) | |
2 | 1 | oveq2d 7374 | . . . . 5 β’ (π = πΊ β (π β¨ (π βπ)) = (π β¨ (π βπΊ))) |
3 | coeq1 5814 | . . . . . . 7 β’ (π = πΊ β (π β β‘πΉ) = (πΊ β β‘πΉ)) | |
4 | 3 | fveq2d 6847 | . . . . . 6 β’ (π = πΊ β (π β(π β β‘πΉ)) = (π β(πΊ β β‘πΉ))) |
5 | 4 | oveq2d 7374 | . . . . 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 7376 | . . . 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 2748 | . . 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 7316 | . 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 7318 | . 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 6949 | 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 1542 β wcel 2107 β¦ cmpt 5189 β‘ccnv 5633 β ccom 5638 βcfv 6497 β©crio 7313 (class class class)co 7358 Basecbs 17084 lecple 17141 joincjn 18201 meetcmee 18202 Atomscatm 37728 LHypclh 38450 LTrncltrn 38567 trLctrl 38624 |
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 5257 ax-nul 5264 ax-pr 5385 |
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-ne 2945 df-ral 3066 df-rex 3075 df-rab 3409 df-v 3448 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-iota 6449 df-fun 6499 df-fv 6505 df-riota 7314 df-ov 7361 |
This theorem is referenced by: cdlemksel 39311 cdlemksv2 39313 cdlemkuvN 39330 cdlemkuel 39331 cdlemkuv2 39333 cdlemkuv-2N 39349 cdlemkuu 39361 |
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