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Mirrors > Home > MPE Home > Th. List > Mathboxes > cdleme9b | Structured version Visualization version GIF version |
Description: Utility lemma for Lemma E in [Crawley] p. 113. (Contributed by NM, 9-Oct-2012.) |
Ref | Expression |
---|---|
cdleme9b.b | ⊢ 𝐵 = (Base‘𝐾) |
cdleme9b.j | ⊢ ∨ = (join‘𝐾) |
cdleme9b.m | ⊢ ∧ = (meet‘𝐾) |
cdleme9b.a | ⊢ 𝐴 = (Atoms‘𝐾) |
cdleme9b.h | ⊢ 𝐻 = (LHyp‘𝐾) |
cdleme9b.c | ⊢ 𝐶 = ((𝑃 ∨ 𝑆) ∧ 𝑊) |
Ref | Expression |
---|---|
cdleme9b | ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑊 ∈ 𝐻)) → 𝐶 ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cdleme9b.c | . 2 ⊢ 𝐶 = ((𝑃 ∨ 𝑆) ∧ 𝑊) | |
2 | hllat 36501 | . . . 4 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat) | |
3 | 2 | adantr 483 | . . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑊 ∈ 𝐻)) → 𝐾 ∈ Lat) |
4 | cdleme9b.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐾) | |
5 | cdleme9b.j | . . . . 5 ⊢ ∨ = (join‘𝐾) | |
6 | cdleme9b.a | . . . . 5 ⊢ 𝐴 = (Atoms‘𝐾) | |
7 | 4, 5, 6 | hlatjcl 36505 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) → (𝑃 ∨ 𝑆) ∈ 𝐵) |
8 | 7 | 3adant3r3 1180 | . . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑊 ∈ 𝐻)) → (𝑃 ∨ 𝑆) ∈ 𝐵) |
9 | simpr3 1192 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑊 ∈ 𝐻)) → 𝑊 ∈ 𝐻) | |
10 | cdleme9b.h | . . . . 5 ⊢ 𝐻 = (LHyp‘𝐾) | |
11 | 4, 10 | lhpbase 37136 | . . . 4 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ 𝐵) |
12 | 9, 11 | syl 17 | . . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑊 ∈ 𝐻)) → 𝑊 ∈ 𝐵) |
13 | cdleme9b.m | . . . 4 ⊢ ∧ = (meet‘𝐾) | |
14 | 4, 13 | latmcl 17664 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑆) ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → ((𝑃 ∨ 𝑆) ∧ 𝑊) ∈ 𝐵) |
15 | 3, 8, 12, 14 | syl3anc 1367 | . 2 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑊 ∈ 𝐻)) → ((𝑃 ∨ 𝑆) ∧ 𝑊) ∈ 𝐵) |
16 | 1, 15 | eqeltrid 2919 | 1 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑊 ∈ 𝐻)) → 𝐶 ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ‘cfv 6357 (class class class)co 7158 Basecbs 16485 joincjn 17556 meetcmee 17557 Latclat 17657 Atomscatm 36401 HLchlt 36488 LHypclh 37122 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-lub 17586 df-glb 17587 df-join 17588 df-meet 17589 df-lat 17658 df-ats 36405 df-atl 36436 df-cvlat 36460 df-hlat 36489 df-lhyp 37126 |
This theorem is referenced by: cdleme15b 37413 cdleme17b 37425 |
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