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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > cdleme0aa | Structured version Visualization version GIF version |
Description: Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 14-Jun-2012.) |
Ref | Expression |
---|---|
cdleme0.l | ⊢ ≤ = (le‘𝐾) |
cdleme0.j | ⊢ ∨ = (join‘𝐾) |
cdleme0.m | ⊢ ∧ = (meet‘𝐾) |
cdleme0.a | ⊢ 𝐴 = (Atoms‘𝐾) |
cdleme0.h | ⊢ 𝐻 = (LHyp‘𝐾) |
cdleme0.u | ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊) |
cdleme0.b | ⊢ 𝐵 = (Base‘𝐾) |
Ref | Expression |
---|---|
cdleme0aa | ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → 𝑈 ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cdleme0.u | . 2 ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊) | |
2 | simp1l 1190 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → 𝐾 ∈ HL) | |
3 | 2 | hllatd 36050 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → 𝐾 ∈ Lat) |
4 | cdleme0.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐾) | |
5 | cdleme0.a | . . . . . 6 ⊢ 𝐴 = (Atoms‘𝐾) | |
6 | 4, 5 | atbase 35975 | . . . . 5 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ 𝐵) |
7 | 6 | 3ad2ant2 1127 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → 𝑃 ∈ 𝐵) |
8 | 4, 5 | atbase 35975 | . . . . 5 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ 𝐵) |
9 | 8 | 3ad2ant3 1128 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → 𝑄 ∈ 𝐵) |
10 | cdleme0.j | . . . . 5 ⊢ ∨ = (join‘𝐾) | |
11 | 4, 10 | latjcl 17490 | . . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵) → (𝑃 ∨ 𝑄) ∈ 𝐵) |
12 | 3, 7, 9, 11 | syl3anc 1364 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ∨ 𝑄) ∈ 𝐵) |
13 | simp1r 1191 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → 𝑊 ∈ 𝐻) | |
14 | cdleme0.h | . . . . 5 ⊢ 𝐻 = (LHyp‘𝐾) | |
15 | 4, 14 | lhpbase 36684 | . . . 4 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ 𝐵) |
16 | 13, 15 | syl 17 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → 𝑊 ∈ 𝐵) |
17 | cdleme0.m | . . . 4 ⊢ ∧ = (meet‘𝐾) | |
18 | 4, 17 | latmcl 17491 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑄) ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → ((𝑃 ∨ 𝑄) ∧ 𝑊) ∈ 𝐵) |
19 | 3, 12, 16, 18 | syl3anc 1364 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → ((𝑃 ∨ 𝑄) ∧ 𝑊) ∈ 𝐵) |
20 | 1, 19 | syl5eqel 2887 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → 𝑈 ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1080 = wceq 1522 ∈ wcel 2081 ‘cfv 6225 (class class class)co 7016 Basecbs 16312 lecple 16401 joincjn 17383 meetcmee 17384 Latclat 17484 Atomscatm 35949 HLchlt 36036 LHypclh 36670 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-rep 5081 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-ral 3110 df-rex 3111 df-reu 3112 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-op 4479 df-uni 4746 df-iun 4827 df-br 4963 df-opab 5025 df-mpt 5042 df-id 5348 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-f1 6230 df-fo 6231 df-f1o 6232 df-fv 6233 df-riota 6977 df-ov 7019 df-oprab 7020 df-lub 17413 df-glb 17414 df-join 17415 df-meet 17416 df-lat 17485 df-ats 35953 df-atl 35984 df-cvlat 36008 df-hlat 36037 df-lhyp 36674 |
This theorem is referenced by: cdleme1b 36912 cdleme5 36926 cdleme9 36939 cdleme11g 36951 cdleme11 36956 cdleme35fnpq 37135 cdleme42e 37165 cdlemeg46frv 37211 cdlemg2fv2 37286 cdlemg2m 37290 |
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