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Mirrors > Home > MPE Home > Th. List > Mathboxes > dihjatcclem3 | Structured version Visualization version GIF version |
Description: Lemma for dihjatcc 38550. (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 37147 | . . . . . 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 37707 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐶 ∈ 𝐴 ∧ ¬ 𝐶 ≤ 𝑊) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝐺 ∈ 𝑇) |
12 | 1, 7, 8, 11 | syl3anc 1366 | . . . 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 37707 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐶 ∈ 𝐴 ∧ ¬ 𝐶 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → 𝐷 ∈ 𝑇) |
16 | 1, 7, 13, 15 | syl3anc 1366 | . . . . 5 ⊢ (𝜑 → 𝐷 ∈ 𝑇) |
17 | 4, 9 | ltrncnv 37274 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐷 ∈ 𝑇) → ◡𝐷 ∈ 𝑇) |
18 | 1, 16, 17 | syl2anc 586 | . . . 4 ⊢ (𝜑 → ◡𝐷 ∈ 𝑇) |
19 | 4, 9 | ltrnco 37847 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇 ∧ ◡𝐷 ∈ 𝑇) → (𝐺 ∘ ◡𝐷) ∈ 𝑇) |
20 | 1, 12, 18, 19 | syl3anc 1366 | . . 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 37291 | . . 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 1366 | . 2 ⊢ (𝜑 → (𝑅‘(𝐺 ∘ ◡𝐷)) = ((𝑄 ∨ ((𝐺 ∘ ◡𝐷)‘𝑄)) ∧ 𝑊)) |
26 | 13 | simpld 497 | . . . . . . . 8 ⊢ (𝜑 → 𝑄 ∈ 𝐴) |
27 | 2, 3, 4, 9 | ltrncoval 37273 | . . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐺 ∈ 𝑇 ∧ ◡𝐷 ∈ 𝑇) ∧ 𝑄 ∈ 𝐴) → ((𝐺 ∘ ◡𝐷)‘𝑄) = (𝐺‘(◡𝐷‘𝑄))) |
28 | 1, 12, 18, 26, 27 | syl121anc 1370 | . . . . . . 7 ⊢ (𝜑 → ((𝐺 ∘ ◡𝐷)‘𝑄) = (𝐺‘(◡𝐷‘𝑄))) |
29 | 2, 3, 4, 9, 14 | ltrniotacnvval 37710 | . . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐶 ∈ 𝐴 ∧ ¬ 𝐶 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (◡𝐷‘𝑄) = 𝐶) |
30 | 1, 7, 13, 29 | syl3anc 1366 | . . . . . . . . 9 ⊢ (𝜑 → (◡𝐷‘𝑄) = 𝐶) |
31 | 30 | fveq2d 6667 | . . . . . . . 8 ⊢ (𝜑 → (𝐺‘(◡𝐷‘𝑄)) = (𝐺‘𝐶)) |
32 | 2, 3, 4, 9, 10 | ltrniotaval 37709 | . . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐶 ∈ 𝐴 ∧ ¬ 𝐶 ≤ 𝑊) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝐺‘𝐶) = 𝑃) |
33 | 1, 7, 8, 32 | syl3anc 1366 | . . . . . . . 8 ⊢ (𝜑 → (𝐺‘𝐶) = 𝑃) |
34 | 31, 33 | eqtrd 2854 | . . . . . . 7 ⊢ (𝜑 → (𝐺‘(◡𝐷‘𝑄)) = 𝑃) |
35 | 28, 34 | eqtrd 2854 | . . . . . 6 ⊢ (𝜑 → ((𝐺 ∘ ◡𝐷)‘𝑄) = 𝑃) |
36 | 35 | oveq2d 7164 | . . . . 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 36496 | . . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑃)) |
40 | 37, 38, 26, 39 | syl3anc 1366 | . . . . 5 ⊢ (𝜑 → (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑃)) |
41 | 36, 40 | eqtr4d 2857 | . . . 4 ⊢ (𝜑 → (𝑄 ∨ ((𝐺 ∘ ◡𝐷)‘𝑄)) = (𝑃 ∨ 𝑄)) |
42 | 41 | oveq1d 7163 | . . 3 ⊢ (𝜑 → ((𝑄 ∨ ((𝐺 ∘ ◡𝐷)‘𝑄)) ∧ 𝑊) = ((𝑃 ∨ 𝑄) ∧ 𝑊)) |
43 | dihjatcclem.v | . . 3 ⊢ 𝑉 = ((𝑃 ∨ 𝑄) ∧ 𝑊) | |
44 | 42, 43 | syl6eqr 2872 | . 2 ⊢ (𝜑 → ((𝑄 ∨ ((𝐺 ∘ ◡𝐷)‘𝑄)) ∧ 𝑊) = 𝑉) |
45 | 25, 44 | eqtrd 2854 | 1 ⊢ (𝜑 → (𝑅‘(𝐺 ∘ ◡𝐷)) = 𝑉) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1531 ∈ wcel 2108 class class class wbr 5057 ◡ccnv 5547 ∘ ccom 5552 ‘cfv 6348 ℩crio 7105 (class class class)co 7148 Basecbs 16475 lecple 16564 occoc 16565 joincjn 17546 meetcmee 17547 LSSumclsm 18751 Atomscatm 36391 HLchlt 36478 LHypclh 37112 LTrncltrn 37229 trLctrl 37286 TEndoctendo 37880 DVecHcdvh 38206 DIsoHcdih 38356 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-riotaBAD 36081 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-ral 3141 df-rex 3142 df-reu 3143 df-rmo 3144 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4831 df-iun 4912 df-iin 4913 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7106 df-ov 7151 df-oprab 7152 df-mpo 7153 df-1st 7681 df-2nd 7682 df-undef 7931 df-map 8400 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 36304 df-ol 36306 df-oml 36307 df-covers 36394 df-ats 36395 df-atl 36426 df-cvlat 36450 df-hlat 36479 df-llines 36626 df-lplanes 36627 df-lvols 36628 df-lines 36629 df-psubsp 36631 df-pmap 36632 df-padd 36924 df-lhyp 37116 df-laut 37117 df-ldil 37232 df-ltrn 37233 df-trl 37287 |
This theorem is referenced by: dihjatcclem4 38549 |
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