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Mirrors > Home > MPE Home > Th. List > Mathboxes > cdlemn4a | Structured version Visualization version GIF version |
Description: Part of proof of Lemma N of [Crawley] p. 121 line 32. (Contributed by NM, 24-Feb-2014.) |
Ref | Expression |
---|---|
cdlemn4.b | ⊢ 𝐵 = (Base‘𝐾) |
cdlemn4.l | ⊢ ≤ = (le‘𝐾) |
cdlemn4.a | ⊢ 𝐴 = (Atoms‘𝐾) |
cdlemn4.p | ⊢ 𝑃 = ((oc‘𝐾)‘𝑊) |
cdlemn4.h | ⊢ 𝐻 = (LHyp‘𝐾) |
cdlemn4.t | ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
cdlemn4.o | ⊢ 𝑂 = (ℎ ∈ 𝑇 ↦ ( I ↾ 𝐵)) |
cdlemn4.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
cdlemn4.f | ⊢ 𝐹 = (℩ℎ ∈ 𝑇 (ℎ‘𝑃) = 𝑄) |
cdlemn4.g | ⊢ 𝐺 = (℩ℎ ∈ 𝑇 (ℎ‘𝑃) = 𝑅) |
cdlemn4.j | ⊢ 𝐽 = (℩ℎ ∈ 𝑇 (ℎ‘𝑄) = 𝑅) |
cdlemn4a.n | ⊢ 𝑁 = (LSpan‘𝑈) |
cdlemn4a.s | ⊢ ⊕ = (LSSum‘𝑈) |
Ref | Expression |
---|---|
cdlemn4a | ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) → (𝑁‘{〈𝐺, ( I ↾ 𝑇)〉}) ⊆ ((𝑁‘{〈𝐹, ( I ↾ 𝑇)〉}) ⊕ (𝑁‘{〈𝐽, 𝑂〉}))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cdlemn4.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐾) | |
2 | cdlemn4.l | . . . . 5 ⊢ ≤ = (le‘𝐾) | |
3 | cdlemn4.a | . . . . 5 ⊢ 𝐴 = (Atoms‘𝐾) | |
4 | cdlemn4.p | . . . . 5 ⊢ 𝑃 = ((oc‘𝐾)‘𝑊) | |
5 | cdlemn4.h | . . . . 5 ⊢ 𝐻 = (LHyp‘𝐾) | |
6 | cdlemn4.t | . . . . 5 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
7 | cdlemn4.o | . . . . 5 ⊢ 𝑂 = (ℎ ∈ 𝑇 ↦ ( I ↾ 𝐵)) | |
8 | cdlemn4.u | . . . . 5 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
9 | cdlemn4.f | . . . . 5 ⊢ 𝐹 = (℩ℎ ∈ 𝑇 (ℎ‘𝑃) = 𝑄) | |
10 | cdlemn4.g | . . . . 5 ⊢ 𝐺 = (℩ℎ ∈ 𝑇 (ℎ‘𝑃) = 𝑅) | |
11 | cdlemn4.j | . . . . 5 ⊢ 𝐽 = (℩ℎ ∈ 𝑇 (ℎ‘𝑄) = 𝑅) | |
12 | eqid 2734 | . . . . 5 ⊢ (+g‘𝑈) = (+g‘𝑈) | |
13 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 | cdlemn4 41180 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) → 〈𝐺, ( I ↾ 𝑇)〉 = (〈𝐹, ( I ↾ 𝑇)〉(+g‘𝑈)〈𝐽, 𝑂〉)) |
14 | 13 | sneqd 4642 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) → {〈𝐺, ( I ↾ 𝑇)〉} = {(〈𝐹, ( I ↾ 𝑇)〉(+g‘𝑈)〈𝐽, 𝑂〉)}) |
15 | 14 | fveq2d 6910 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) → (𝑁‘{〈𝐺, ( I ↾ 𝑇)〉}) = (𝑁‘{(〈𝐹, ( I ↾ 𝑇)〉(+g‘𝑈)〈𝐽, 𝑂〉)})) |
16 | simp1 1135 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
17 | 5, 8, 16 | dvhlmod 41092 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) → 𝑈 ∈ LMod) |
18 | 2, 3, 5, 4 | lhpocnel2 40001 | . . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) |
19 | 18 | 3ad2ant1 1132 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) |
20 | simp2 1136 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) | |
21 | 2, 3, 5, 6, 9 | ltrniotacl 40561 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → 𝐹 ∈ 𝑇) |
22 | 16, 19, 20, 21 | syl3anc 1370 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) → 𝐹 ∈ 𝑇) |
23 | eqid 2734 | . . . . . 6 ⊢ ((TEndo‘𝐾)‘𝑊) = ((TEndo‘𝐾)‘𝑊) | |
24 | 5, 6, 23 | tendoidcl 40751 | . . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( I ↾ 𝑇) ∈ ((TEndo‘𝐾)‘𝑊)) |
25 | 24 | 3ad2ant1 1132 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) → ( I ↾ 𝑇) ∈ ((TEndo‘𝐾)‘𝑊)) |
26 | eqid 2734 | . . . . 5 ⊢ (Base‘𝑈) = (Base‘𝑈) | |
27 | 5, 6, 23, 8, 26 | dvhelvbasei 41070 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ ( I ↾ 𝑇) ∈ ((TEndo‘𝐾)‘𝑊))) → 〈𝐹, ( I ↾ 𝑇)〉 ∈ (Base‘𝑈)) |
28 | 16, 22, 25, 27 | syl12anc 837 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) → 〈𝐹, ( I ↾ 𝑇)〉 ∈ (Base‘𝑈)) |
29 | 2, 3, 5, 6, 11 | ltrniotacl 40561 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) → 𝐽 ∈ 𝑇) |
30 | 1, 5, 6, 23, 7 | tendo0cl 40772 | . . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑂 ∈ ((TEndo‘𝐾)‘𝑊)) |
31 | 30 | 3ad2ant1 1132 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) → 𝑂 ∈ ((TEndo‘𝐾)‘𝑊)) |
32 | 5, 6, 23, 8, 26 | dvhelvbasei 41070 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐽 ∈ 𝑇 ∧ 𝑂 ∈ ((TEndo‘𝐾)‘𝑊))) → 〈𝐽, 𝑂〉 ∈ (Base‘𝑈)) |
33 | 16, 29, 31, 32 | syl12anc 837 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) → 〈𝐽, 𝑂〉 ∈ (Base‘𝑈)) |
34 | cdlemn4a.n | . . . 4 ⊢ 𝑁 = (LSpan‘𝑈) | |
35 | cdlemn4a.s | . . . 4 ⊢ ⊕ = (LSSum‘𝑈) | |
36 | 26, 12, 34, 35 | lspsntri 21113 | . . 3 ⊢ ((𝑈 ∈ LMod ∧ 〈𝐹, ( I ↾ 𝑇)〉 ∈ (Base‘𝑈) ∧ 〈𝐽, 𝑂〉 ∈ (Base‘𝑈)) → (𝑁‘{(〈𝐹, ( I ↾ 𝑇)〉(+g‘𝑈)〈𝐽, 𝑂〉)}) ⊆ ((𝑁‘{〈𝐹, ( I ↾ 𝑇)〉}) ⊕ (𝑁‘{〈𝐽, 𝑂〉}))) |
37 | 17, 28, 33, 36 | syl3anc 1370 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) → (𝑁‘{(〈𝐹, ( I ↾ 𝑇)〉(+g‘𝑈)〈𝐽, 𝑂〉)}) ⊆ ((𝑁‘{〈𝐹, ( I ↾ 𝑇)〉}) ⊕ (𝑁‘{〈𝐽, 𝑂〉}))) |
38 | 15, 37 | eqsstrd 4033 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) → (𝑁‘{〈𝐺, ( I ↾ 𝑇)〉}) ⊆ ((𝑁‘{〈𝐹, ( I ↾ 𝑇)〉}) ⊕ (𝑁‘{〈𝐽, 𝑂〉}))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1536 ∈ wcel 2105 ⊆ wss 3962 {csn 4630 〈cop 4636 class class class wbr 5147 ↦ cmpt 5230 I cid 5581 ↾ cres 5690 ‘cfv 6562 ℩crio 7386 (class class class)co 7430 Basecbs 17244 +gcplusg 17297 lecple 17304 occoc 17305 LSSumclsm 19666 LModclmod 20874 LSpanclspn 20986 Atomscatm 39244 HLchlt 39331 LHypclh 39966 LTrncltrn 40083 TEndoctendo 40734 DVecHcdvh 41060 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-riotaBAD 38934 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-iin 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-tpos 8249 df-undef 8296 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-er 8743 df-map 8866 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-nn 12264 df-2 12326 df-3 12327 df-4 12328 df-5 12329 df-6 12330 df-n0 12524 df-z 12611 df-uz 12876 df-fz 13544 df-struct 17180 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17245 df-ress 17274 df-plusg 17310 df-mulr 17311 df-sca 17313 df-vsca 17314 df-0g 17487 df-proset 18351 df-poset 18370 df-plt 18387 df-lub 18403 df-glb 18404 df-join 18405 df-meet 18406 df-p0 18482 df-p1 18483 df-lat 18489 df-clat 18556 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-submnd 18809 df-grp 18966 df-minusg 18967 df-sbg 18968 df-subg 19153 df-cntz 19347 df-lsm 19668 df-cmn 19814 df-abl 19815 df-mgp 20152 df-rng 20170 df-ur 20199 df-ring 20252 df-oppr 20350 df-dvdsr 20373 df-unit 20374 df-invr 20404 df-dvr 20417 df-drng 20747 df-lmod 20876 df-lss 20947 df-lsp 20987 df-lvec 21119 df-oposet 39157 df-ol 39159 df-oml 39160 df-covers 39247 df-ats 39248 df-atl 39279 df-cvlat 39303 df-hlat 39332 df-llines 39480 df-lplanes 39481 df-lvols 39482 df-lines 39483 df-psubsp 39485 df-pmap 39486 df-padd 39778 df-lhyp 39970 df-laut 39971 df-ldil 40086 df-ltrn 40087 df-trl 40141 df-tendo 40737 df-edring 40739 df-dvech 41061 |
This theorem is referenced by: cdlemn5pre 41182 |
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