Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > cdlemefr32fvaN | Structured version Visualization version GIF version |
Description: Part of proof of Lemma E in [Crawley] p. 113. Value of 𝐹 at an atom not under 𝑊. TODO: FIX COMMENT. (Contributed by NM, 29-Mar-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
cdlemefr27.b | ⊢ 𝐵 = (Base‘𝐾) |
cdlemefr27.l | ⊢ ≤ = (le‘𝐾) |
cdlemefr27.j | ⊢ ∨ = (join‘𝐾) |
cdlemefr27.m | ⊢ ∧ = (meet‘𝐾) |
cdlemefr27.a | ⊢ 𝐴 = (Atoms‘𝐾) |
cdlemefr27.h | ⊢ 𝐻 = (LHyp‘𝐾) |
cdlemefr27.u | ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊) |
cdlemefr27.c | ⊢ 𝐶 = ((𝑠 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑠) ∧ 𝑊))) |
cdlemefr27.n | ⊢ 𝑁 = if(𝑠 ≤ (𝑃 ∨ 𝑄), 𝐼, 𝐶) |
cdleme29fr.o | ⊢ 𝑂 = (℩𝑧 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑥 ∧ 𝑊)) = 𝑥) → 𝑧 = (𝑁 ∨ (𝑥 ∧ 𝑊)))) |
Ref | Expression |
---|---|
cdlemefr32fvaN | ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄)) → ⦋𝑅 / 𝑥⦌𝑂 = ⦋𝑅 / 𝑠⦌𝑁) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cdlemefr27.b | . 2 ⊢ 𝐵 = (Base‘𝐾) | |
2 | cdlemefr27.l | . 2 ⊢ ≤ = (le‘𝐾) | |
3 | cdlemefr27.j | . 2 ⊢ ∨ = (join‘𝐾) | |
4 | cdlemefr27.m | . 2 ⊢ ∧ = (meet‘𝐾) | |
5 | cdlemefr27.a | . 2 ⊢ 𝐴 = (Atoms‘𝐾) | |
6 | cdlemefr27.h | . 2 ⊢ 𝐻 = (LHyp‘𝐾) | |
7 | breq1 5077 | . . 3 ⊢ (𝑠 = 𝑅 → (𝑠 ≤ (𝑃 ∨ 𝑄) ↔ 𝑅 ≤ (𝑃 ∨ 𝑄))) | |
8 | 7 | notbid 318 | . 2 ⊢ (𝑠 = 𝑅 → (¬ 𝑠 ≤ (𝑃 ∨ 𝑄) ↔ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄))) |
9 | simp11 1202 | . . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 ≠ 𝑄 ∧ (𝑠 ∈ 𝐴 ∧ (¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ (𝑃 ∨ 𝑄)))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
10 | simp12l 1285 | . . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 ≠ 𝑄 ∧ (𝑠 ∈ 𝐴 ∧ (¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ (𝑃 ∨ 𝑄)))) → 𝑃 ∈ 𝐴) | |
11 | simp13l 1287 | . . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 ≠ 𝑄 ∧ (𝑠 ∈ 𝐴 ∧ (¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ (𝑃 ∨ 𝑄)))) → 𝑄 ∈ 𝐴) | |
12 | simp3l 1200 | . . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 ≠ 𝑄 ∧ (𝑠 ∈ 𝐴 ∧ (¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ (𝑃 ∨ 𝑄)))) → 𝑠 ∈ 𝐴) | |
13 | simp3rr 1246 | . . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 ≠ 𝑄 ∧ (𝑠 ∈ 𝐴 ∧ (¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ (𝑃 ∨ 𝑄)))) → ¬ 𝑠 ≤ (𝑃 ∨ 𝑄)) | |
14 | simp2 1136 | . . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 ≠ 𝑄 ∧ (𝑠 ∈ 𝐴 ∧ (¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ (𝑃 ∨ 𝑄)))) → 𝑃 ≠ 𝑄) | |
15 | cdlemefr27.u | . . . 4 ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊) | |
16 | cdlemefr27.c | . . . 4 ⊢ 𝐶 = ((𝑠 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑠) ∧ 𝑊))) | |
17 | cdlemefr27.n | . . . 4 ⊢ 𝑁 = if(𝑠 ≤ (𝑃 ∨ 𝑄), 𝐼, 𝐶) | |
18 | 1, 2, 3, 4, 5, 6, 15, 16, 17 | cdlemefr27cl 38403 | . . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ (𝑃 ∨ 𝑄) ∧ 𝑃 ≠ 𝑄)) → 𝑁 ∈ 𝐵) |
19 | 9, 10, 11, 12, 13, 14, 18 | syl33anc 1384 | . 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 ≠ 𝑄 ∧ (𝑠 ∈ 𝐴 ∧ (¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ (𝑃 ∨ 𝑄)))) → 𝑁 ∈ 𝐵) |
20 | 1, 2, 3, 4, 5, 6, 15, 16, 17 | cdlemefr32snb 38405 | . 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄)) → ⦋𝑅 / 𝑠⦌𝑁 ∈ 𝐵) |
21 | cdleme29fr.o | . 2 ⊢ 𝑂 = (℩𝑧 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑥 ∧ 𝑊)) = 𝑥) → 𝑧 = (𝑁 ∨ (𝑥 ∧ 𝑊)))) | |
22 | 1, 2, 3, 4, 5, 6, 8, 19, 20, 21 | cdlemefrs32fva 38400 | 1 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄)) → ⦋𝑅 / 𝑥⦌𝑂 = ⦋𝑅 / 𝑠⦌𝑁) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 ∀wral 3064 ⦋csb 3832 ifcif 4460 class class class wbr 5074 ‘cfv 6427 ℩crio 7224 (class class class)co 7268 Basecbs 16900 lecple 16957 joincjn 18017 meetcmee 18018 Atomscatm 37263 HLchlt 37350 LHypclh 37984 |
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 5209 ax-sep 5222 ax-nul 5229 ax-pow 5287 ax-pr 5351 ax-un 7579 |
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-rmo 3072 df-rab 3073 df-v 3432 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4258 df-if 4461 df-pw 4536 df-sn 4563 df-pr 4565 df-op 4569 df-uni 4841 df-iun 4927 df-iin 4928 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5485 df-xp 5591 df-rel 5592 df-cnv 5593 df-co 5594 df-dm 5595 df-rn 5596 df-res 5597 df-ima 5598 df-iota 6385 df-fun 6429 df-fn 6430 df-f 6431 df-f1 6432 df-fo 6433 df-f1o 6434 df-fv 6435 df-riota 7225 df-ov 7271 df-oprab 7272 df-mpo 7273 df-1st 7821 df-2nd 7822 df-proset 18001 df-poset 18019 df-plt 18036 df-lub 18052 df-glb 18053 df-join 18054 df-meet 18055 df-p0 18131 df-p1 18132 df-lat 18138 df-clat 18205 df-oposet 37176 df-ol 37178 df-oml 37179 df-covers 37266 df-ats 37267 df-atl 37298 df-cvlat 37322 df-hlat 37351 df-lines 37501 df-psubsp 37503 df-pmap 37504 df-padd 37796 df-lhyp 37988 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |