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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dihjatcclem3 | Structured version Visualization version GIF version |
Description: Lemma for dihjatcc 39816. (Contributed by NM, 28-Sep-2014.) |
Ref | Expression |
---|---|
dihjatcclem.b | ⊢ 𝐵 = (Base‘𝐾) |
dihjatcclem.l | ⊢ ≤ = (le‘𝐾) |
dihjatcclem.h | ⊢ 𝐻 = (LHyp‘𝐾) |
dihjatcclem.j | ⊢ ∨ = (join‘𝐾) |
dihjatcclem.m | ⊢ ∧ = (meet‘𝐾) |
dihjatcclem.a | ⊢ 𝐴 = (Atoms‘𝐾) |
dihjatcclem.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
dihjatcclem.s | ⊢ ⊕ = (LSSum‘𝑈) |
dihjatcclem.i | ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) |
dihjatcclem.v | ⊢ 𝑉 = ((𝑃 ∨ 𝑄) ∧ 𝑊) |
dihjatcclem.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
dihjatcclem.p | ⊢ (𝜑 → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) |
dihjatcclem.q | ⊢ (𝜑 → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) |
dihjatcc.w | ⊢ 𝐶 = ((oc‘𝐾)‘𝑊) |
dihjatcc.t | ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
dihjatcc.r | ⊢ 𝑅 = ((trL‘𝐾)‘𝑊) |
dihjatcc.e | ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊) |
dihjatcc.g | ⊢ 𝐺 = (℩𝑑 ∈ 𝑇 (𝑑‘𝐶) = 𝑃) |
dihjatcc.dd | ⊢ 𝐷 = (℩𝑑 ∈ 𝑇 (𝑑‘𝐶) = 𝑄) |
Ref | Expression |
---|---|
dihjatcclem3 | ⊢ (𝜑 → (𝑅‘(𝐺 ∘ ◡𝐷)) = 𝑉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dihjatcclem.k | . . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
2 | dihjatcclem.l | . . . . . . 7 ⊢ ≤ = (le‘𝐾) | |
3 | dihjatcclem.a | . . . . . . 7 ⊢ 𝐴 = (Atoms‘𝐾) | |
4 | dihjatcclem.h | . . . . . . 7 ⊢ 𝐻 = (LHyp‘𝐾) | |
5 | dihjatcc.w | . . . . . . 7 ⊢ 𝐶 = ((oc‘𝐾)‘𝑊) | |
6 | 2, 3, 4, 5 | lhpocnel2 38413 | . . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝐶 ∈ 𝐴 ∧ ¬ 𝐶 ≤ 𝑊)) |
7 | 1, 6 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝐶 ∈ 𝐴 ∧ ¬ 𝐶 ≤ 𝑊)) |
8 | dihjatcclem.p | . . . . 5 ⊢ (𝜑 → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) | |
9 | dihjatcc.t | . . . . . 6 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
10 | dihjatcc.g | . . . . . 6 ⊢ 𝐺 = (℩𝑑 ∈ 𝑇 (𝑑‘𝐶) = 𝑃) | |
11 | 2, 3, 4, 9, 10 | ltrniotacl 38973 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐶 ∈ 𝐴 ∧ ¬ 𝐶 ≤ 𝑊) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝐺 ∈ 𝑇) |
12 | 1, 7, 8, 11 | syl3anc 1371 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ 𝑇) |
13 | dihjatcclem.q | . . . . . 6 ⊢ (𝜑 → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) | |
14 | dihjatcc.dd | . . . . . . 7 ⊢ 𝐷 = (℩𝑑 ∈ 𝑇 (𝑑‘𝐶) = 𝑄) | |
15 | 2, 3, 4, 9, 14 | ltrniotacl 38973 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐶 ∈ 𝐴 ∧ ¬ 𝐶 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → 𝐷 ∈ 𝑇) |
16 | 1, 7, 13, 15 | syl3anc 1371 | . . . . 5 ⊢ (𝜑 → 𝐷 ∈ 𝑇) |
17 | 4, 9 | ltrncnv 38540 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐷 ∈ 𝑇) → ◡𝐷 ∈ 𝑇) |
18 | 1, 16, 17 | syl2anc 584 | . . . 4 ⊢ (𝜑 → ◡𝐷 ∈ 𝑇) |
19 | 4, 9 | ltrnco 39113 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇 ∧ ◡𝐷 ∈ 𝑇) → (𝐺 ∘ ◡𝐷) ∈ 𝑇) |
20 | 1, 12, 18, 19 | syl3anc 1371 | . . 3 ⊢ (𝜑 → (𝐺 ∘ ◡𝐷) ∈ 𝑇) |
21 | dihjatcclem.j | . . . 4 ⊢ ∨ = (join‘𝐾) | |
22 | dihjatcclem.m | . . . 4 ⊢ ∧ = (meet‘𝐾) | |
23 | dihjatcc.r | . . . 4 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊) | |
24 | 2, 21, 22, 3, 4, 9, 23 | trlval2 38557 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐺 ∘ ◡𝐷) ∈ 𝑇 ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝑅‘(𝐺 ∘ ◡𝐷)) = ((𝑄 ∨ ((𝐺 ∘ ◡𝐷)‘𝑄)) ∧ 𝑊)) |
25 | 1, 20, 13, 24 | syl3anc 1371 | . 2 ⊢ (𝜑 → (𝑅‘(𝐺 ∘ ◡𝐷)) = ((𝑄 ∨ ((𝐺 ∘ ◡𝐷)‘𝑄)) ∧ 𝑊)) |
26 | 13 | simpld 495 | . . . . . . . 8 ⊢ (𝜑 → 𝑄 ∈ 𝐴) |
27 | 2, 3, 4, 9 | ltrncoval 38539 | . . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐺 ∈ 𝑇 ∧ ◡𝐷 ∈ 𝑇) ∧ 𝑄 ∈ 𝐴) → ((𝐺 ∘ ◡𝐷)‘𝑄) = (𝐺‘(◡𝐷‘𝑄))) |
28 | 1, 12, 18, 26, 27 | syl121anc 1375 | . . . . . . 7 ⊢ (𝜑 → ((𝐺 ∘ ◡𝐷)‘𝑄) = (𝐺‘(◡𝐷‘𝑄))) |
29 | 2, 3, 4, 9, 14 | ltrniotacnvval 38976 | . . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐶 ∈ 𝐴 ∧ ¬ 𝐶 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (◡𝐷‘𝑄) = 𝐶) |
30 | 1, 7, 13, 29 | syl3anc 1371 | . . . . . . . . 9 ⊢ (𝜑 → (◡𝐷‘𝑄) = 𝐶) |
31 | 30 | fveq2d 6843 | . . . . . . . 8 ⊢ (𝜑 → (𝐺‘(◡𝐷‘𝑄)) = (𝐺‘𝐶)) |
32 | 2, 3, 4, 9, 10 | ltrniotaval 38975 | . . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐶 ∈ 𝐴 ∧ ¬ 𝐶 ≤ 𝑊) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝐺‘𝐶) = 𝑃) |
33 | 1, 7, 8, 32 | syl3anc 1371 | . . . . . . . 8 ⊢ (𝜑 → (𝐺‘𝐶) = 𝑃) |
34 | 31, 33 | eqtrd 2777 | . . . . . . 7 ⊢ (𝜑 → (𝐺‘(◡𝐷‘𝑄)) = 𝑃) |
35 | 28, 34 | eqtrd 2777 | . . . . . 6 ⊢ (𝜑 → ((𝐺 ∘ ◡𝐷)‘𝑄) = 𝑃) |
36 | 35 | oveq2d 7367 | . . . . 5 ⊢ (𝜑 → (𝑄 ∨ ((𝐺 ∘ ◡𝐷)‘𝑄)) = (𝑄 ∨ 𝑃)) |
37 | 1 | simpld 495 | . . . . . 6 ⊢ (𝜑 → 𝐾 ∈ HL) |
38 | 8 | simpld 495 | . . . . . 6 ⊢ (𝜑 → 𝑃 ∈ 𝐴) |
39 | 21, 3 | hlatjcom 37761 | . . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑃)) |
40 | 37, 38, 26, 39 | syl3anc 1371 | . . . . 5 ⊢ (𝜑 → (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑃)) |
41 | 36, 40 | eqtr4d 2780 | . . . 4 ⊢ (𝜑 → (𝑄 ∨ ((𝐺 ∘ ◡𝐷)‘𝑄)) = (𝑃 ∨ 𝑄)) |
42 | 41 | oveq1d 7366 | . . 3 ⊢ (𝜑 → ((𝑄 ∨ ((𝐺 ∘ ◡𝐷)‘𝑄)) ∧ 𝑊) = ((𝑃 ∨ 𝑄) ∧ 𝑊)) |
43 | dihjatcclem.v | . . 3 ⊢ 𝑉 = ((𝑃 ∨ 𝑄) ∧ 𝑊) | |
44 | 42, 43 | eqtr4di 2795 | . 2 ⊢ (𝜑 → ((𝑄 ∨ ((𝐺 ∘ ◡𝐷)‘𝑄)) ∧ 𝑊) = 𝑉) |
45 | 25, 44 | eqtrd 2777 | 1 ⊢ (𝜑 → (𝑅‘(𝐺 ∘ ◡𝐷)) = 𝑉) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 class class class wbr 5103 ◡ccnv 5630 ∘ ccom 5635 ‘cfv 6493 ℩crio 7306 (class class class)co 7351 Basecbs 17042 lecple 17099 occoc 17100 joincjn 18159 meetcmee 18160 LSSumclsm 19374 Atomscatm 37656 HLchlt 37743 LHypclh 38378 LTrncltrn 38495 trLctrl 38552 TEndoctendo 39146 DVecHcdvh 39472 DIsoHcdih 39622 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 ax-riotaBAD 37346 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-iin 4955 df-br 5104 df-opab 5166 df-mpt 5187 df-id 5529 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7307 df-ov 7354 df-oprab 7355 df-mpo 7356 df-1st 7913 df-2nd 7914 df-undef 8196 df-map 8725 df-proset 18143 df-poset 18161 df-plt 18178 df-lub 18194 df-glb 18195 df-join 18196 df-meet 18197 df-p0 18273 df-p1 18274 df-lat 18280 df-clat 18347 df-oposet 37569 df-ol 37571 df-oml 37572 df-covers 37659 df-ats 37660 df-atl 37691 df-cvlat 37715 df-hlat 37744 df-llines 37892 df-lplanes 37893 df-lvols 37894 df-lines 37895 df-psubsp 37897 df-pmap 37898 df-padd 38190 df-lhyp 38382 df-laut 38383 df-ldil 38498 df-ltrn 38499 df-trl 38553 |
This theorem is referenced by: dihjatcclem4 39815 |
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