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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 2826 | . . . . 5 ⊢ (+g‘𝑈) = (+g‘𝑈) | |
13 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 | cdlemn4 37274 | . . . 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 4410 | . . 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 6438 | . 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 1172 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
17 | 5, 8, 16 | dvhlmod 37186 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) → 𝑈 ∈ LMod) |
18 | 2, 3, 5, 4 | lhpocnel2 36095 | . . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) |
19 | 18 | 3ad2ant1 1169 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) |
20 | simp2 1173 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) | |
21 | 2, 3, 5, 6, 9 | ltrniotacl 36655 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → 𝐹 ∈ 𝑇) |
22 | 16, 19, 20, 21 | syl3anc 1496 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) → 𝐹 ∈ 𝑇) |
23 | eqid 2826 | . . . . . 6 ⊢ ((TEndo‘𝐾)‘𝑊) = ((TEndo‘𝐾)‘𝑊) | |
24 | 5, 6, 23 | tendoidcl 36845 | . . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( I ↾ 𝑇) ∈ ((TEndo‘𝐾)‘𝑊)) |
25 | 24 | 3ad2ant1 1169 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) → ( I ↾ 𝑇) ∈ ((TEndo‘𝐾)‘𝑊)) |
26 | eqid 2826 | . . . . 5 ⊢ (Base‘𝑈) = (Base‘𝑈) | |
27 | 5, 6, 23, 8, 26 | dvhelvbasei 37164 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ ( I ↾ 𝑇) ∈ ((TEndo‘𝐾)‘𝑊))) → 〈𝐹, ( I ↾ 𝑇)〉 ∈ (Base‘𝑈)) |
28 | 16, 22, 25, 27 | syl12anc 872 | . . 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 36655 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) → 𝐽 ∈ 𝑇) |
30 | 1, 5, 6, 23, 7 | tendo0cl 36866 | . . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑂 ∈ ((TEndo‘𝐾)‘𝑊)) |
31 | 30 | 3ad2ant1 1169 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) → 𝑂 ∈ ((TEndo‘𝐾)‘𝑊)) |
32 | 5, 6, 23, 8, 26 | dvhelvbasei 37164 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐽 ∈ 𝑇 ∧ 𝑂 ∈ ((TEndo‘𝐾)‘𝑊))) → 〈𝐽, 𝑂〉 ∈ (Base‘𝑈)) |
33 | 16, 29, 31, 32 | syl12anc 872 | . . 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 19457 | . . 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 1496 | . 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 3865 | 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 386 ∧ w3a 1113 = wceq 1658 ∈ wcel 2166 ⊆ wss 3799 {csn 4398 〈cop 4404 class class class wbr 4874 ↦ cmpt 4953 I cid 5250 ↾ cres 5345 ‘cfv 6124 ℩crio 6866 (class class class)co 6906 Basecbs 16223 +gcplusg 16306 lecple 16313 occoc 16314 LSSumclsm 18401 LModclmod 19220 LSpanclspn 19331 Atomscatm 35339 HLchlt 35426 LHypclh 36060 LTrncltrn 36177 TEndoctendo 36828 DVecHcdvh 37154 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-rep 4995 ax-sep 5006 ax-nul 5014 ax-pow 5066 ax-pr 5128 ax-un 7210 ax-cnex 10309 ax-resscn 10310 ax-1cn 10311 ax-icn 10312 ax-addcl 10313 ax-addrcl 10314 ax-mulcl 10315 ax-mulrcl 10316 ax-mulcom 10317 ax-addass 10318 ax-mulass 10319 ax-distr 10320 ax-i2m1 10321 ax-1ne0 10322 ax-1rid 10323 ax-rnegex 10324 ax-rrecex 10325 ax-cnre 10326 ax-pre-lttri 10327 ax-pre-lttrn 10328 ax-pre-ltadd 10329 ax-pre-mulgt0 10330 ax-riotaBAD 35029 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-fal 1672 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ne 3001 df-nel 3104 df-ral 3123 df-rex 3124 df-reu 3125 df-rmo 3126 df-rab 3127 df-v 3417 df-sbc 3664 df-csb 3759 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-pss 3815 df-nul 4146 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4660 df-int 4699 df-iun 4743 df-iin 4744 df-br 4875 df-opab 4937 df-mpt 4954 df-tr 4977 df-id 5251 df-eprel 5256 df-po 5264 df-so 5265 df-fr 5302 df-we 5304 df-xp 5349 df-rel 5350 df-cnv 5351 df-co 5352 df-dm 5353 df-rn 5354 df-res 5355 df-ima 5356 df-pred 5921 df-ord 5967 df-on 5968 df-lim 5969 df-suc 5970 df-iota 6087 df-fun 6126 df-fn 6127 df-f 6128 df-f1 6129 df-fo 6130 df-f1o 6131 df-fv 6132 df-riota 6867 df-ov 6909 df-oprab 6910 df-mpt2 6911 df-om 7328 df-1st 7429 df-2nd 7430 df-tpos 7618 df-undef 7665 df-wrecs 7673 df-recs 7735 df-rdg 7773 df-1o 7827 df-oadd 7831 df-er 8010 df-map 8125 df-en 8224 df-dom 8225 df-sdom 8226 df-fin 8227 df-pnf 10394 df-mnf 10395 df-xr 10396 df-ltxr 10397 df-le 10398 df-sub 10588 df-neg 10589 df-nn 11352 df-2 11415 df-3 11416 df-4 11417 df-5 11418 df-6 11419 df-n0 11620 df-z 11706 df-uz 11970 df-fz 12621 df-struct 16225 df-ndx 16226 df-slot 16227 df-base 16229 df-sets 16230 df-ress 16231 df-plusg 16319 df-mulr 16320 df-sca 16322 df-vsca 16323 df-0g 16456 df-proset 17282 df-poset 17300 df-plt 17312 df-lub 17328 df-glb 17329 df-join 17330 df-meet 17331 df-p0 17393 df-p1 17394 df-lat 17400 df-clat 17462 df-mgm 17596 df-sgrp 17638 df-mnd 17649 df-submnd 17690 df-grp 17780 df-minusg 17781 df-sbg 17782 df-subg 17943 df-cntz 18101 df-lsm 18403 df-cmn 18549 df-abl 18550 df-mgp 18845 df-ur 18857 df-ring 18904 df-oppr 18978 df-dvdsr 18996 df-unit 18997 df-invr 19027 df-dvr 19038 df-drng 19106 df-lmod 19222 df-lss 19290 df-lsp 19332 df-lvec 19463 df-oposet 35252 df-ol 35254 df-oml 35255 df-covers 35342 df-ats 35343 df-atl 35374 df-cvlat 35398 df-hlat 35427 df-llines 35574 df-lplanes 35575 df-lvols 35576 df-lines 35577 df-psubsp 35579 df-pmap 35580 df-padd 35872 df-lhyp 36064 df-laut 36065 df-ldil 36180 df-ltrn 36181 df-trl 36235 df-tendo 36831 df-edring 36833 df-dvech 37155 |
This theorem is referenced by: cdlemn5pre 37276 |
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