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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > cdlemn11a | 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 |
---|---|
cdlemn11a | ⊢ (((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)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1129 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ (𝐽‘𝑁) ⊆ ((𝐽‘𝑄) ⊕ (𝐼‘𝑋))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
2 | cdlemn11a.l | . . . . . . 7 ⊢ ≤ = (le‘𝐾) | |
3 | cdlemn11a.a | . . . . . . 7 ⊢ 𝐴 = (Atoms‘𝐾) | |
4 | cdlemn11a.h | . . . . . . 7 ⊢ 𝐻 = (LHyp‘𝐾) | |
5 | cdlemn11a.p | . . . . . . 7 ⊢ 𝑃 = ((oc‘𝐾)‘𝑊) | |
6 | 2, 3, 4, 5 | lhpocnel2 36707 | . . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) |
7 | 6 | 3ad2ant1 1126 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ (𝐽‘𝑁) ⊆ ((𝐽‘𝑄) ⊕ (𝐼‘𝑋))) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) |
8 | simp22 1200 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ (𝐽‘𝑁) ⊆ ((𝐽‘𝑄) ⊕ (𝐼‘𝑋))) → (𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊)) | |
9 | cdlemn11a.t | . . . . . 6 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
10 | cdlemn11a.g | . . . . . 6 ⊢ 𝐺 = (℩ℎ ∈ 𝑇 (ℎ‘𝑃) = 𝑁) | |
11 | 2, 3, 4, 9, 10 | ltrniotacl 37267 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊)) → 𝐺 ∈ 𝑇) |
12 | 1, 7, 8, 11 | syl3anc 1364 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ (𝐽‘𝑁) ⊆ ((𝐽‘𝑄) ⊕ (𝐼‘𝑋))) → 𝐺 ∈ 𝑇) |
13 | fvresi 6805 | . . . 4 ⊢ (𝐺 ∈ 𝑇 → (( I ↾ 𝑇)‘𝐺) = 𝐺) | |
14 | 12, 13 | syl 17 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ (𝐽‘𝑁) ⊆ ((𝐽‘𝑄) ⊕ (𝐼‘𝑋))) → (( I ↾ 𝑇)‘𝐺) = 𝐺) |
15 | 14 | eqcomd 2803 | . 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)‘𝐺)) |
16 | cdlemn11a.e | . . . 4 ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊) | |
17 | 4, 9, 16 | tendoidcl 37457 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( I ↾ 𝑇) ∈ 𝐸) |
18 | 17 | 3ad2ant1 1126 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) ∧ (𝐽‘𝑁) ⊆ ((𝐽‘𝑄) ⊕ (𝐼‘𝑋))) → ( I ↾ 𝑇) ∈ 𝐸) |
19 | cdlemn11a.J | . . . 4 ⊢ 𝐽 = ((DIsoC‘𝐾)‘𝑊) | |
20 | riotaex 6988 | . . . . 5 ⊢ (℩ℎ ∈ 𝑇 (ℎ‘𝑃) = 𝑁) ∈ V | |
21 | 10, 20 | eqeltri 2881 | . . . 4 ⊢ 𝐺 ∈ V |
22 | 9 | fvexi 6559 | . . . . 5 ⊢ 𝑇 ∈ V |
23 | resiexg 7482 | . . . . 5 ⊢ (𝑇 ∈ V → ( I ↾ 𝑇) ∈ V) | |
24 | 22, 23 | ax-mp 5 | . . . 4 ⊢ ( I ↾ 𝑇) ∈ V |
25 | 2, 3, 4, 5, 9, 16, 19, 10, 21, 24 | dicopelval2 37869 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊)) → (〈𝐺, ( I ↾ 𝑇)〉 ∈ (𝐽‘𝑁) ↔ (𝐺 = (( I ↾ 𝑇)‘𝐺) ∧ ( I ↾ 𝑇) ∈ 𝐸))) |
26 | 1, 8, 25 | syl2anc 584 | . 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) ↔ (𝐺 = (( I ↾ 𝑇)‘𝐺) ∧ ( I ↾ 𝑇) ∈ 𝐸))) |
27 | 15, 18, 26 | mpbir2and 709 | 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‘𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 ∧ wa 396 ∧ w3a 1080 = wceq 1525 ∈ wcel 2083 Vcvv 3440 ⊆ wss 3865 〈cop 4484 class class class wbr 4968 ↦ cmpt 5047 I cid 5354 ↾ cres 5452 ‘cfv 6232 ℩crio 6983 (class class class)co 7023 Basecbs 16316 +gcplusg 16398 lecple 16405 occoc 16406 joincjn 17387 LSSumclsm 18493 Atomscatm 35951 HLchlt 36038 LHypclh 36672 LTrncltrn 36789 trLctrl 36846 TEndoctendo 37440 DVecHcdvh 37766 DIsoBcdib 37826 DIsoCcdic 37860 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-13 2346 ax-ext 2771 ax-rep 5088 ax-sep 5101 ax-nul 5108 ax-pow 5164 ax-pr 5228 ax-un 7326 ax-riotaBAD 35641 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1528 df-ex 1766 df-nf 1770 df-sb 2045 df-mo 2578 df-eu 2614 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ne 2987 df-ral 3112 df-rex 3113 df-reu 3114 df-rmo 3115 df-rab 3116 df-v 3442 df-sbc 3712 df-csb 3818 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-nul 4218 df-if 4388 df-pw 4461 df-sn 4479 df-pr 4481 df-op 4485 df-uni 4752 df-iun 4833 df-iin 4834 df-br 4969 df-opab 5031 df-mpt 5048 df-id 5355 df-xp 5456 df-rel 5457 df-cnv 5458 df-co 5459 df-dm 5460 df-rn 5461 df-res 5462 df-ima 5463 df-iota 6196 df-fun 6234 df-fn 6235 df-f 6236 df-f1 6237 df-fo 6238 df-f1o 6239 df-fv 6240 df-riota 6984 df-ov 7026 df-oprab 7027 df-mpo 7028 df-1st 7552 df-2nd 7553 df-undef 7797 df-map 8265 df-proset 17371 df-poset 17389 df-plt 17401 df-lub 17417 df-glb 17418 df-join 17419 df-meet 17420 df-p0 17482 df-p1 17483 df-lat 17489 df-clat 17551 df-oposet 35864 df-ol 35866 df-oml 35867 df-covers 35954 df-ats 35955 df-atl 35986 df-cvlat 36010 df-hlat 36039 df-llines 36186 df-lplanes 36187 df-lvols 36188 df-lines 36189 df-psubsp 36191 df-pmap 36192 df-padd 36484 df-lhyp 36676 df-laut 36677 df-ldil 36792 df-ltrn 36793 df-trl 36847 df-tendo 37443 df-dic 37861 |
This theorem is referenced by: cdlemn11b 37896 |
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