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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 39741 and cdleme3 39742. (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 7437 | . . 3 β’ (π β¨ π) = (π β¨ ((π β¨ π ) β§ π)) |
4 | 3 | oveq2i 7437 | . 2 β’ ((π β¨ π) β§ (π β¨ π)) = ((π β¨ π) β§ (π β¨ ((π β¨ π ) β§ π))) |
5 | 1, 4 | eqtr4i 2759 | 1 β’ πΉ = ((π β¨ π) β§ (π β¨ π)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 βcfv 6553 (class class class)co 7426 lecple 17247 joincjn 18310 meetcmee 18311 Atomscatm 38767 LHypclh 39489 |
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-ext 2699 |
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-sb 2060 df-clab 2706 df-cleq 2720 df-clel 2806 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-iota 6505 df-fv 6561 df-ov 7429 |
This theorem is referenced by: cdleme3g 39739 cdleme3h 39740 cdleme9 39758 |
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