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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 36503 | . . . 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 36507 | . . . 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 37138 | . . . 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 17665 | . . 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 2920 | 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 1536 ∈ wcel 2113 ‘cfv 6358 (class class class)co 7159 Basecbs 16486 joincjn 17557 meetcmee 17558 Latclat 17658 Atomscatm 36403 HLchlt 36490 LHypclh 37124 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-reu 3148 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-lub 17587 df-glb 17588 df-join 17589 df-meet 17590 df-lat 17659 df-ats 36407 df-atl 36438 df-cvlat 36462 df-hlat 36491 df-lhyp 37128 |
This theorem is referenced by: cdleme15b 37415 cdleme17b 37427 |
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