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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 1134 | . 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 38345 | . 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 3970 | 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 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ⊆ wss 3938 〈cop 4575 class class class wbr 5068 ↦ cmpt 5148 I cid 5461 ↾ cres 5559 ‘cfv 6357 ℩crio 7115 (class class class)co 7158 Basecbs 16485 +gcplusg 16567 lecple 16574 occoc 16575 joincjn 17556 LSSumclsm 18761 Atomscatm 36401 HLchlt 36488 LHypclh 37122 LTrncltrn 37239 trLctrl 37296 TEndoctendo 37890 DVecHcdvh 38216 DIsoBcdib 38276 DIsoCcdic 38310 |
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 df-tendo 37893 df-dic 38311 |
This theorem is referenced by: cdlemn11c 38347 |
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