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Mirrors > Home > MPE Home > Th. List > Mathboxes > cdlemc | Structured version Visualization version GIF version |
Description: Lemma C in [Crawley] p. 113. (Contributed by NM, 26-May-2012.) |
Ref | Expression |
---|---|
cdlemc3.l | ⊢ ≤ = (le‘𝐾) |
cdlemc3.j | ⊢ ∨ = (join‘𝐾) |
cdlemc3.m | ⊢ ∧ = (meet‘𝐾) |
cdlemc3.a | ⊢ 𝐴 = (Atoms‘𝐾) |
cdlemc3.h | ⊢ 𝐻 = (LHyp‘𝐾) |
cdlemc3.t | ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
cdlemc3.r | ⊢ 𝑅 = ((trL‘𝐾)‘𝑊) |
Ref | Expression |
---|---|
cdlemc | ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃))) → (𝐹‘𝑄) = ((𝑄 ∨ (𝑅‘𝐹)) ∧ ((𝐹‘𝑃) ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl1 1188 | . . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃))) ∧ (𝐹‘𝑃) = 𝑃) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
2 | simpl2 1189 | . . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃))) ∧ (𝐹‘𝑃) = 𝑃) → (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))) | |
3 | simpr 488 | . . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃))) ∧ (𝐹‘𝑃) = 𝑃) → (𝐹‘𝑃) = 𝑃) | |
4 | cdlemc3.l | . . . 4 ⊢ ≤ = (le‘𝐾) | |
5 | cdlemc3.j | . . . 4 ⊢ ∨ = (join‘𝐾) | |
6 | cdlemc3.m | . . . 4 ⊢ ∧ = (meet‘𝐾) | |
7 | cdlemc3.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
8 | cdlemc3.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
9 | cdlemc3.t | . . . 4 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
10 | cdlemc3.r | . . . 4 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊) | |
11 | 4, 5, 6, 7, 8, 9, 10 | cdlemc6 37492 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹‘𝑃) = 𝑃) → (𝐹‘𝑄) = ((𝑄 ∨ (𝑅‘𝐹)) ∧ ((𝐹‘𝑃) ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊)))) |
12 | 1, 2, 3, 11 | syl3anc 1368 | . 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃))) ∧ (𝐹‘𝑃) = 𝑃) → (𝐹‘𝑄) = ((𝑄 ∨ (𝑅‘𝐹)) ∧ ((𝐹‘𝑃) ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊)))) |
13 | simpl1 1188 | . . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃))) ∧ (𝐹‘𝑃) ≠ 𝑃) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
14 | simpl2 1189 | . . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃))) ∧ (𝐹‘𝑃) ≠ 𝑃) → (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))) | |
15 | simpl3 1190 | . . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃))) ∧ (𝐹‘𝑃) ≠ 𝑃) → ¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃))) | |
16 | simpr 488 | . . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃))) ∧ (𝐹‘𝑃) ≠ 𝑃) → (𝐹‘𝑃) ≠ 𝑃) | |
17 | 4, 5, 6, 7, 8, 9, 10 | cdlemc5 37491 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃)) ∧ (𝐹‘𝑃) ≠ 𝑃)) → (𝐹‘𝑄) = ((𝑄 ∨ (𝑅‘𝐹)) ∧ ((𝐹‘𝑃) ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊)))) |
18 | 13, 14, 15, 16, 17 | syl112anc 1371 | . 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃))) ∧ (𝐹‘𝑃) ≠ 𝑃) → (𝐹‘𝑄) = ((𝑄 ∨ (𝑅‘𝐹)) ∧ ((𝐹‘𝑃) ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊)))) |
19 | 12, 18 | pm2.61dane 3074 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃))) → (𝐹‘𝑄) = ((𝑄 ∨ (𝑅‘𝐹)) ∧ ((𝐹‘𝑃) ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊)))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 class class class wbr 5030 ‘cfv 6324 (class class class)co 7135 lecple 16564 joincjn 17546 meetcmee 17547 Atomscatm 36559 HLchlt 36646 LHypclh 37280 LTrncltrn 37397 trLctrl 37454 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-1st 7671 df-2nd 7672 df-map 8391 df-proset 17530 df-poset 17548 df-plt 17560 df-lub 17576 df-glb 17577 df-join 17578 df-meet 17579 df-p0 17641 df-p1 17642 df-lat 17648 df-clat 17710 df-oposet 36472 df-ol 36474 df-oml 36475 df-covers 36562 df-ats 36563 df-atl 36594 df-cvlat 36618 df-hlat 36647 df-llines 36794 df-psubsp 36799 df-pmap 36800 df-padd 37092 df-lhyp 37284 df-laut 37285 df-ldil 37400 df-ltrn 37401 df-trl 37455 |
This theorem is referenced by: cdlemd6 37499 cdlemg4e 37910 cdlemg43 38026 |
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