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Mirrors > Home > MPE Home > Th. List > Mathboxes > dihjatcclem3 | Structured version Visualization version GIF version |
Description: Lemma for dihjatcc 38560. (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 37157 | . . . . . 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 37717 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐶 ∈ 𝐴 ∧ ¬ 𝐶 ≤ 𝑊) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝐺 ∈ 𝑇) |
12 | 1, 7, 8, 11 | syl3anc 1367 | . . . 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 37717 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐶 ∈ 𝐴 ∧ ¬ 𝐶 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → 𝐷 ∈ 𝑇) |
16 | 1, 7, 13, 15 | syl3anc 1367 | . . . . 5 ⊢ (𝜑 → 𝐷 ∈ 𝑇) |
17 | 4, 9 | ltrncnv 37284 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐷 ∈ 𝑇) → ◡𝐷 ∈ 𝑇) |
18 | 1, 16, 17 | syl2anc 586 | . . . 4 ⊢ (𝜑 → ◡𝐷 ∈ 𝑇) |
19 | 4, 9 | ltrnco 37857 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇 ∧ ◡𝐷 ∈ 𝑇) → (𝐺 ∘ ◡𝐷) ∈ 𝑇) |
20 | 1, 12, 18, 19 | syl3anc 1367 | . . 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 37301 | . . 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 1367 | . 2 ⊢ (𝜑 → (𝑅‘(𝐺 ∘ ◡𝐷)) = ((𝑄 ∨ ((𝐺 ∘ ◡𝐷)‘𝑄)) ∧ 𝑊)) |
26 | 13 | simpld 497 | . . . . . . . 8 ⊢ (𝜑 → 𝑄 ∈ 𝐴) |
27 | 2, 3, 4, 9 | ltrncoval 37283 | . . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐺 ∈ 𝑇 ∧ ◡𝐷 ∈ 𝑇) ∧ 𝑄 ∈ 𝐴) → ((𝐺 ∘ ◡𝐷)‘𝑄) = (𝐺‘(◡𝐷‘𝑄))) |
28 | 1, 12, 18, 26, 27 | syl121anc 1371 | . . . . . . 7 ⊢ (𝜑 → ((𝐺 ∘ ◡𝐷)‘𝑄) = (𝐺‘(◡𝐷‘𝑄))) |
29 | 2, 3, 4, 9, 14 | ltrniotacnvval 37720 | . . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐶 ∈ 𝐴 ∧ ¬ 𝐶 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (◡𝐷‘𝑄) = 𝐶) |
30 | 1, 7, 13, 29 | syl3anc 1367 | . . . . . . . . 9 ⊢ (𝜑 → (◡𝐷‘𝑄) = 𝐶) |
31 | 30 | fveq2d 6676 | . . . . . . . 8 ⊢ (𝜑 → (𝐺‘(◡𝐷‘𝑄)) = (𝐺‘𝐶)) |
32 | 2, 3, 4, 9, 10 | ltrniotaval 37719 | . . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐶 ∈ 𝐴 ∧ ¬ 𝐶 ≤ 𝑊) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝐺‘𝐶) = 𝑃) |
33 | 1, 7, 8, 32 | syl3anc 1367 | . . . . . . . 8 ⊢ (𝜑 → (𝐺‘𝐶) = 𝑃) |
34 | 31, 33 | eqtrd 2858 | . . . . . . 7 ⊢ (𝜑 → (𝐺‘(◡𝐷‘𝑄)) = 𝑃) |
35 | 28, 34 | eqtrd 2858 | . . . . . 6 ⊢ (𝜑 → ((𝐺 ∘ ◡𝐷)‘𝑄) = 𝑃) |
36 | 35 | oveq2d 7174 | . . . . 5 ⊢ (𝜑 → (𝑄 ∨ ((𝐺 ∘ ◡𝐷)‘𝑄)) = (𝑄 ∨ 𝑃)) |
37 | 1 | simpld 497 | . . . . . 6 ⊢ (𝜑 → 𝐾 ∈ HL) |
38 | 8 | simpld 497 | . . . . . 6 ⊢ (𝜑 → 𝑃 ∈ 𝐴) |
39 | 21, 3 | hlatjcom 36506 | . . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑃)) |
40 | 37, 38, 26, 39 | syl3anc 1367 | . . . . 5 ⊢ (𝜑 → (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑃)) |
41 | 36, 40 | eqtr4d 2861 | . . . 4 ⊢ (𝜑 → (𝑄 ∨ ((𝐺 ∘ ◡𝐷)‘𝑄)) = (𝑃 ∨ 𝑄)) |
42 | 41 | oveq1d 7173 | . . 3 ⊢ (𝜑 → ((𝑄 ∨ ((𝐺 ∘ ◡𝐷)‘𝑄)) ∧ 𝑊) = ((𝑃 ∨ 𝑄) ∧ 𝑊)) |
43 | dihjatcclem.v | . . 3 ⊢ 𝑉 = ((𝑃 ∨ 𝑄) ∧ 𝑊) | |
44 | 42, 43 | syl6eqr 2876 | . 2 ⊢ (𝜑 → ((𝑄 ∨ ((𝐺 ∘ ◡𝐷)‘𝑄)) ∧ 𝑊) = 𝑉) |
45 | 25, 44 | eqtrd 2858 | 1 ⊢ (𝜑 → (𝑅‘(𝐺 ∘ ◡𝐷)) = 𝑉) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 class class class wbr 5068 ◡ccnv 5556 ∘ ccom 5561 ‘cfv 6357 ℩crio 7115 (class class class)co 7158 Basecbs 16485 lecple 16574 occoc 16575 joincjn 17556 meetcmee 17557 LSSumclsm 18761 Atomscatm 36401 HLchlt 36488 LHypclh 37122 LTrncltrn 37239 trLctrl 37296 TEndoctendo 37890 DVecHcdvh 38216 DIsoHcdih 38366 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-riotaBAD 36091 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-iin 4924 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-1st 7691 df-2nd 7692 df-undef 7941 df-map 8410 df-proset 17540 df-poset 17558 df-plt 17570 df-lub 17586 df-glb 17587 df-join 17588 df-meet 17589 df-p0 17651 df-p1 17652 df-lat 17658 df-clat 17720 df-oposet 36314 df-ol 36316 df-oml 36317 df-covers 36404 df-ats 36405 df-atl 36436 df-cvlat 36460 df-hlat 36489 df-llines 36636 df-lplanes 36637 df-lvols 36638 df-lines 36639 df-psubsp 36641 df-pmap 36642 df-padd 36934 df-lhyp 37126 df-laut 37127 df-ldil 37242 df-ltrn 37243 df-trl 37297 |
This theorem is referenced by: dihjatcclem4 38559 |
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