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Mirrors > Home > MPE Home > Th. List > Mathboxes > cdleme42d | Structured version Visualization version GIF version |
Description: Part of proof of Lemma E in [Crawley] p. 113. Match (𝑠 ∨ (𝑥 ∧ 𝑊)) = 𝑥. (Contributed by NM, 6-Mar-2013.) |
Ref | Expression |
---|---|
cdleme42.b | ⊢ 𝐵 = (Base‘𝐾) |
cdleme42.l | ⊢ ≤ = (le‘𝐾) |
cdleme42.j | ⊢ ∨ = (join‘𝐾) |
cdleme42.m | ⊢ ∧ = (meet‘𝐾) |
cdleme42.a | ⊢ 𝐴 = (Atoms‘𝐾) |
cdleme42.h | ⊢ 𝐻 = (LHyp‘𝐾) |
cdleme42.v | ⊢ 𝑉 = ((𝑅 ∨ 𝑆) ∧ 𝑊) |
Ref | Expression |
---|---|
cdleme42d | ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) → (𝑅 ∨ ((𝑅 ∨ 𝑉) ∧ 𝑊)) = (𝑅 ∨ 𝑉)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cdleme42.v | . . 3 ⊢ 𝑉 = ((𝑅 ∨ 𝑆) ∧ 𝑊) | |
2 | 1 | oveq2i 7278 | . 2 ⊢ (𝑅 ∨ 𝑉) = (𝑅 ∨ ((𝑅 ∨ 𝑆) ∧ 𝑊)) |
3 | cdleme42.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐾) | |
4 | cdleme42.l | . . . . 5 ⊢ ≤ = (le‘𝐾) | |
5 | cdleme42.j | . . . . 5 ⊢ ∨ = (join‘𝐾) | |
6 | cdleme42.m | . . . . 5 ⊢ ∧ = (meet‘𝐾) | |
7 | cdleme42.a | . . . . 5 ⊢ 𝐴 = (Atoms‘𝐾) | |
8 | cdleme42.h | . . . . 5 ⊢ 𝐻 = (LHyp‘𝐾) | |
9 | 3, 4, 5, 6, 7, 8, 1 | cdleme42a 38493 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) → (𝑅 ∨ 𝑆) = (𝑅 ∨ 𝑉)) |
10 | 9 | oveq1d 7282 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) → ((𝑅 ∨ 𝑆) ∧ 𝑊) = ((𝑅 ∨ 𝑉) ∧ 𝑊)) |
11 | 10 | oveq2d 7283 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) → (𝑅 ∨ ((𝑅 ∨ 𝑆) ∧ 𝑊)) = (𝑅 ∨ ((𝑅 ∨ 𝑉) ∧ 𝑊))) |
12 | 2, 11 | eqtr2id 2791 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) → (𝑅 ∨ ((𝑅 ∨ 𝑉) ∧ 𝑊)) = (𝑅 ∨ 𝑉)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 class class class wbr 5073 ‘cfv 6426 (class class class)co 7267 Basecbs 16922 lecple 16979 joincjn 18039 meetcmee 18040 Atomscatm 37285 HLchlt 37372 LHypclh 38006 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5208 ax-sep 5221 ax-nul 5228 ax-pow 5286 ax-pr 5350 ax-un 7578 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-reu 3071 df-rab 3073 df-v 3431 df-sbc 3716 df-csb 3832 df-dif 3889 df-un 3891 df-in 3893 df-ss 3903 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-iin 4927 df-br 5074 df-opab 5136 df-mpt 5157 df-id 5484 df-xp 5590 df-rel 5591 df-cnv 5592 df-co 5593 df-dm 5594 df-rn 5595 df-res 5596 df-ima 5597 df-iota 6384 df-fun 6428 df-fn 6429 df-f 6430 df-f1 6431 df-fo 6432 df-f1o 6433 df-fv 6434 df-riota 7224 df-ov 7270 df-oprab 7271 df-mpo 7272 df-1st 7820 df-2nd 7821 df-proset 18023 df-poset 18041 df-plt 18058 df-lub 18074 df-glb 18075 df-join 18076 df-meet 18077 df-p0 18153 df-p1 18154 df-lat 18160 df-clat 18227 df-oposet 37198 df-ol 37200 df-oml 37201 df-covers 37288 df-ats 37289 df-atl 37320 df-cvlat 37344 df-hlat 37373 df-psubsp 37525 df-pmap 37526 df-padd 37818 df-lhyp 38010 |
This theorem is referenced by: cdleme42e 38501 |
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