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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > cdleme3d | Structured version Visualization version GIF version |
Description: Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme3fa 39107 and cdleme3 39108. (Contributed by NM, 6-Jun-2012.) |
Ref | Expression |
---|---|
cdleme1.l | β’ β€ = (leβπΎ) |
cdleme1.j | β’ β¨ = (joinβπΎ) |
cdleme1.m | β’ β§ = (meetβπΎ) |
cdleme1.a | β’ π΄ = (AtomsβπΎ) |
cdleme1.h | β’ π» = (LHypβπΎ) |
cdleme1.u | β’ π = ((π β¨ π) β§ π) |
cdleme1.f | β’ πΉ = ((π β¨ π) β§ (π β¨ ((π β¨ π ) β§ π))) |
cdleme3.3 | β’ π = ((π β¨ π ) β§ π) |
Ref | Expression |
---|---|
cdleme3d | β’ πΉ = ((π β¨ π) β§ (π β¨ π)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cdleme1.f | . 2 β’ πΉ = ((π β¨ π) β§ (π β¨ ((π β¨ π ) β§ π))) | |
2 | cdleme3.3 | . . . 4 β’ π = ((π β¨ π ) β§ π) | |
3 | 2 | oveq2i 7420 | . . 3 β’ (π β¨ π) = (π β¨ ((π β¨ π ) β§ π)) |
4 | 3 | oveq2i 7420 | . 2 β’ ((π β¨ π) β§ (π β¨ π)) = ((π β¨ π) β§ (π β¨ ((π β¨ π ) β§ π))) |
5 | 1, 4 | eqtr4i 2764 | 1 β’ πΉ = ((π β¨ π) β§ (π β¨ π)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 βcfv 6544 (class class class)co 7409 lecple 17204 joincjn 18264 meetcmee 18265 Atomscatm 38133 LHypclh 38855 |
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 2704 |
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 2711 df-cleq 2725 df-clel 2811 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-iota 6496 df-fv 6552 df-ov 7412 |
This theorem is referenced by: cdleme3g 39105 cdleme3h 39106 cdleme9 39124 |
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