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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > cdlemn11b | Structured version Visualization version GIF version |
Description: Part of proof of Lemma N of [Crawley] p. 121 line 37. (Contributed by NM, 27-Feb-2014.) |
Ref | Expression |
---|---|
cdlemn11a.b | ⊢ 𝐵 = (Base‘𝐾) |
cdlemn11a.l | ⊢ ≤ = (le‘𝐾) |
cdlemn11a.j | ⊢ ∨ = (join‘𝐾) |
cdlemn11a.a | ⊢ 𝐴 = (Atoms‘𝐾) |
cdlemn11a.h | ⊢ 𝐻 = (LHyp‘𝐾) |
cdlemn11a.p | ⊢ 𝑃 = ((oc‘𝐾)‘𝑊) |
cdlemn11a.o | ⊢ 𝑂 = (ℎ ∈ 𝑇 ↦ ( I ↾ 𝐵)) |
cdlemn11a.t | ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
cdlemn11a.r | ⊢ 𝑅 = ((trL‘𝐾)‘𝑊) |
cdlemn11a.e | ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊) |
cdlemn11a.i | ⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊) |
cdlemn11a.J | ⊢ 𝐽 = ((DIsoC‘𝐾)‘𝑊) |
cdlemn11a.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
cdlemn11a.d | ⊢ + = (+g‘𝑈) |
cdlemn11a.s | ⊢ ⊕ = (LSSum‘𝑈) |
cdlemn11a.f | ⊢ 𝐹 = (℩ℎ ∈ 𝑇 (ℎ‘𝑃) = 𝑄) |
cdlemn11a.g | ⊢ 𝐺 = (℩ℎ ∈ 𝑇 (ℎ‘𝑃) = 𝑁) |
Ref | Expression |
---|---|
cdlemn11b | ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ (𝐽‘𝑁) ⊆ ((𝐽‘𝑄) ⊕ (𝐼‘𝑋))) → 〈𝐺, ( I ↾ 𝑇)〉 ∈ ((𝐽‘𝑄) ⊕ (𝐼‘𝑋))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp3 1135 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ (𝐽‘𝑁) ⊆ ((𝐽‘𝑄) ⊕ (𝐼‘𝑋))) → (𝐽‘𝑁) ⊆ ((𝐽‘𝑄) ⊕ (𝐼‘𝑋))) | |
2 | cdlemn11a.b | . . 3 ⊢ 𝐵 = (Base‘𝐾) | |
3 | cdlemn11a.l | . . 3 ⊢ ≤ = (le‘𝐾) | |
4 | cdlemn11a.j | . . 3 ⊢ ∨ = (join‘𝐾) | |
5 | cdlemn11a.a | . . 3 ⊢ 𝐴 = (Atoms‘𝐾) | |
6 | cdlemn11a.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
7 | cdlemn11a.p | . . 3 ⊢ 𝑃 = ((oc‘𝐾)‘𝑊) | |
8 | cdlemn11a.o | . . 3 ⊢ 𝑂 = (ℎ ∈ 𝑇 ↦ ( I ↾ 𝐵)) | |
9 | cdlemn11a.t | . . 3 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
10 | cdlemn11a.r | . . 3 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊) | |
11 | cdlemn11a.e | . . 3 ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊) | |
12 | cdlemn11a.i | . . 3 ⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊) | |
13 | cdlemn11a.J | . . 3 ⊢ 𝐽 = ((DIsoC‘𝐾)‘𝑊) | |
14 | cdlemn11a.u | . . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
15 | cdlemn11a.d | . . 3 ⊢ + = (+g‘𝑈) | |
16 | cdlemn11a.s | . . 3 ⊢ ⊕ = (LSSum‘𝑈) | |
17 | cdlemn11a.f | . . 3 ⊢ 𝐹 = (℩ℎ ∈ 𝑇 (ℎ‘𝑃) = 𝑄) | |
18 | cdlemn11a.g | . . 3 ⊢ 𝐺 = (℩ℎ ∈ 𝑇 (ℎ‘𝑃) = 𝑁) | |
19 | 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18 | cdlemn11a 40810 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ (𝐽‘𝑁) ⊆ ((𝐽‘𝑄) ⊕ (𝐼‘𝑋))) → 〈𝐺, ( I ↾ 𝑇)〉 ∈ (𝐽‘𝑁)) |
20 | 1, 19 | sseldd 3977 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ (𝐽‘𝑁) ⊆ ((𝐽‘𝑄) ⊕ (𝐼‘𝑋))) → 〈𝐺, ( I ↾ 𝑇)〉 ∈ ((𝐽‘𝑄) ⊕ (𝐼‘𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ⊆ wss 3944 〈cop 4636 class class class wbr 5149 ↦ cmpt 5232 I cid 5575 ↾ cres 5680 ‘cfv 6549 ℩crio 7374 (class class class)co 7419 Basecbs 17183 +gcplusg 17236 lecple 17243 occoc 17244 joincjn 18306 LSSumclsm 19601 Atomscatm 38865 HLchlt 38952 LHypclh 39587 LTrncltrn 39704 trLctrl 39761 TEndoctendo 40355 DVecHcdvh 40681 DIsoBcdib 40741 DIsoCcdic 40775 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-riotaBAD 38555 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-iin 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-1st 7994 df-2nd 7995 df-undef 8279 df-map 8847 df-proset 18290 df-poset 18308 df-plt 18325 df-lub 18341 df-glb 18342 df-join 18343 df-meet 18344 df-p0 18420 df-p1 18421 df-lat 18427 df-clat 18494 df-oposet 38778 df-ol 38780 df-oml 38781 df-covers 38868 df-ats 38869 df-atl 38900 df-cvlat 38924 df-hlat 38953 df-llines 39101 df-lplanes 39102 df-lvols 39103 df-lines 39104 df-psubsp 39106 df-pmap 39107 df-padd 39399 df-lhyp 39591 df-laut 39592 df-ldil 39707 df-ltrn 39708 df-trl 39762 df-tendo 40358 df-dic 40776 |
This theorem is referenced by: cdlemn11c 40812 |
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