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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > cdleme7a | Structured version Visualization version GIF version |
Description: Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme7ga 38714 and cdleme7 38715. (Contributed by NM, 7-Jun-2012.) |
Ref | Expression |
---|---|
cdleme4.l | β’ β€ = (leβπΎ) |
cdleme4.j | β’ β¨ = (joinβπΎ) |
cdleme4.m | β’ β§ = (meetβπΎ) |
cdleme4.a | β’ π΄ = (AtomsβπΎ) |
cdleme4.h | β’ π» = (LHypβπΎ) |
cdleme4.u | β’ π = ((π β¨ π) β§ π) |
cdleme4.f | β’ πΉ = ((π β¨ π) β§ (π β¨ ((π β¨ π) β§ π))) |
cdleme4.g | β’ πΊ = ((π β¨ π) β§ (πΉ β¨ ((π β¨ π) β§ π))) |
cdleme7.v | β’ π = ((π β¨ π) β§ π) |
Ref | Expression |
---|---|
cdleme7a | β’ πΊ = ((π β¨ π) β§ (πΉ β¨ π)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cdleme4.g | . 2 β’ πΊ = ((π β¨ π) β§ (πΉ β¨ ((π β¨ π) β§ π))) | |
2 | cdleme7.v | . . . 4 β’ π = ((π β¨ π) β§ π) | |
3 | 2 | oveq2i 7369 | . . 3 β’ (πΉ β¨ π) = (πΉ β¨ ((π β¨ π) β§ π)) |
4 | 3 | oveq2i 7369 | . 2 β’ ((π β¨ π) β§ (πΉ β¨ π)) = ((π β¨ π) β§ (πΉ β¨ ((π β¨ π) β§ π))) |
5 | 1, 4 | eqtr4i 2768 | 1 β’ πΊ = ((π β¨ π) β§ (πΉ β¨ π)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 βcfv 6497 (class class class)co 7358 lecple 17141 joincjn 18201 meetcmee 18202 Atomscatm 37728 LHypclh 38450 |
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-ext 2708 |
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-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 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-iota 6449 df-fv 6505 df-ov 7361 |
This theorem is referenced by: cdleme7d 38712 cdleme17a 38752 |
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