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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 38503 | . 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 3916 | 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 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ⊆ wss 3881 〈cop 4531 class class class wbr 5030 ↦ cmpt 5110 I cid 5424 ↾ cres 5521 ‘cfv 6324 ℩crio 7092 (class class class)co 7135 Basecbs 16475 +gcplusg 16557 lecple 16564 occoc 16565 joincjn 17546 LSSumclsm 18751 Atomscatm 36559 HLchlt 36646 LHypclh 37280 LTrncltrn 37397 trLctrl 37454 TEndoctendo 38048 DVecHcdvh 38374 DIsoBcdib 38434 DIsoCcdic 38468 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-riotaBAD 36249 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-1st 7671 df-2nd 7672 df-undef 7922 df-map 8391 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 36472 df-ol 36474 df-oml 36475 df-covers 36562 df-ats 36563 df-atl 36594 df-cvlat 36618 df-hlat 36647 df-llines 36794 df-lplanes 36795 df-lvols 36796 df-lines 36797 df-psubsp 36799 df-pmap 36800 df-padd 37092 df-lhyp 37284 df-laut 37285 df-ldil 37400 df-ltrn 37401 df-trl 37455 df-tendo 38051 df-dic 38469 |
This theorem is referenced by: cdlemn11c 38505 |
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