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| Mirrors > Home > MPE Home > Th. List > Mathboxes > dihjatcclem3 | Structured version Visualization version GIF version | ||
| Description: Lemma for dihjatcc 41424. (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 40021 | . . . . . 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 40581 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐶 ∈ 𝐴 ∧ ¬ 𝐶 ≤ 𝑊) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝐺 ∈ 𝑇) | 
| 12 | 1, 7, 8, 11 | syl3anc 1373 | . . . 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 40581 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐶 ∈ 𝐴 ∧ ¬ 𝐶 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → 𝐷 ∈ 𝑇) | 
| 16 | 1, 7, 13, 15 | syl3anc 1373 | . . . . 5 ⊢ (𝜑 → 𝐷 ∈ 𝑇) | 
| 17 | 4, 9 | ltrncnv 40148 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐷 ∈ 𝑇) → ◡𝐷 ∈ 𝑇) | 
| 18 | 1, 16, 17 | syl2anc 584 | . . . 4 ⊢ (𝜑 → ◡𝐷 ∈ 𝑇) | 
| 19 | 4, 9 | ltrnco 40721 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇 ∧ ◡𝐷 ∈ 𝑇) → (𝐺 ∘ ◡𝐷) ∈ 𝑇) | 
| 20 | 1, 12, 18, 19 | syl3anc 1373 | . . 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 40165 | . . 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 1373 | . 2 ⊢ (𝜑 → (𝑅‘(𝐺 ∘ ◡𝐷)) = ((𝑄 ∨ ((𝐺 ∘ ◡𝐷)‘𝑄)) ∧ 𝑊)) | 
| 26 | 13 | simpld 494 | . . . . . . . 8 ⊢ (𝜑 → 𝑄 ∈ 𝐴) | 
| 27 | 2, 3, 4, 9 | ltrncoval 40147 | . . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐺 ∈ 𝑇 ∧ ◡𝐷 ∈ 𝑇) ∧ 𝑄 ∈ 𝐴) → ((𝐺 ∘ ◡𝐷)‘𝑄) = (𝐺‘(◡𝐷‘𝑄))) | 
| 28 | 1, 12, 18, 26, 27 | syl121anc 1377 | . . . . . . 7 ⊢ (𝜑 → ((𝐺 ∘ ◡𝐷)‘𝑄) = (𝐺‘(◡𝐷‘𝑄))) | 
| 29 | 2, 3, 4, 9, 14 | ltrniotacnvval 40584 | . . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐶 ∈ 𝐴 ∧ ¬ 𝐶 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (◡𝐷‘𝑄) = 𝐶) | 
| 30 | 1, 7, 13, 29 | syl3anc 1373 | . . . . . . . . 9 ⊢ (𝜑 → (◡𝐷‘𝑄) = 𝐶) | 
| 31 | 30 | fveq2d 6910 | . . . . . . . 8 ⊢ (𝜑 → (𝐺‘(◡𝐷‘𝑄)) = (𝐺‘𝐶)) | 
| 32 | 2, 3, 4, 9, 10 | ltrniotaval 40583 | . . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐶 ∈ 𝐴 ∧ ¬ 𝐶 ≤ 𝑊) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝐺‘𝐶) = 𝑃) | 
| 33 | 1, 7, 8, 32 | syl3anc 1373 | . . . . . . . 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 7447 | . . . . 5 ⊢ (𝜑 → (𝑄 ∨ ((𝐺 ∘ ◡𝐷)‘𝑄)) = (𝑄 ∨ 𝑃)) | 
| 37 | 1 | simpld 494 | . . . . . 6 ⊢ (𝜑 → 𝐾 ∈ HL) | 
| 38 | 8 | simpld 494 | . . . . . 6 ⊢ (𝜑 → 𝑃 ∈ 𝐴) | 
| 39 | 21, 3 | hlatjcom 39369 | . . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑃)) | 
| 40 | 37, 38, 26, 39 | syl3anc 1373 | . . . . 5 ⊢ (𝜑 → (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑃)) | 
| 41 | 36, 40 | eqtr4d 2780 | . . . 4 ⊢ (𝜑 → (𝑄 ∨ ((𝐺 ∘ ◡𝐷)‘𝑄)) = (𝑃 ∨ 𝑄)) | 
| 42 | 41 | oveq1d 7446 | . . 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 395 = wceq 1540 ∈ wcel 2108 class class class wbr 5143 ◡ccnv 5684 ∘ ccom 5689 ‘cfv 6561 ℩crio 7387 (class class class)co 7431 Basecbs 17247 lecple 17304 occoc 17305 joincjn 18357 meetcmee 18358 LSSumclsm 19652 Atomscatm 39264 HLchlt 39351 LHypclh 39986 LTrncltrn 40103 trLctrl 40160 TEndoctendo 40754 DVecHcdvh 41080 DIsoHcdih 41230 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-riotaBAD 38954 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-undef 8298 df-map 8868 df-proset 18340 df-poset 18359 df-plt 18375 df-lub 18391 df-glb 18392 df-join 18393 df-meet 18394 df-p0 18470 df-p1 18471 df-lat 18477 df-clat 18544 df-oposet 39177 df-ol 39179 df-oml 39180 df-covers 39267 df-ats 39268 df-atl 39299 df-cvlat 39323 df-hlat 39352 df-llines 39500 df-lplanes 39501 df-lvols 39502 df-lines 39503 df-psubsp 39505 df-pmap 39506 df-padd 39798 df-lhyp 39990 df-laut 39991 df-ldil 40106 df-ltrn 40107 df-trl 40161 | 
| This theorem is referenced by: dihjatcclem4 41423 | 
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