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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 2724 | . . . . 5 ⊢ (+g‘𝑈) = (+g‘𝑈) | |
13 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 | cdlemn4 40559 | . . . 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 4632 | . . 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 6885 | . 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 1133 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
17 | 5, 8, 16 | dvhlmod 40471 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) → 𝑈 ∈ LMod) |
18 | 2, 3, 5, 4 | lhpocnel2 39380 | . . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) |
19 | 18 | 3ad2ant1 1130 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) |
20 | simp2 1134 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) | |
21 | 2, 3, 5, 6, 9 | ltrniotacl 39940 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → 𝐹 ∈ 𝑇) |
22 | 16, 19, 20, 21 | syl3anc 1368 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) → 𝐹 ∈ 𝑇) |
23 | eqid 2724 | . . . . . 6 ⊢ ((TEndo‘𝐾)‘𝑊) = ((TEndo‘𝐾)‘𝑊) | |
24 | 5, 6, 23 | tendoidcl 40130 | . . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( I ↾ 𝑇) ∈ ((TEndo‘𝐾)‘𝑊)) |
25 | 24 | 3ad2ant1 1130 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) → ( I ↾ 𝑇) ∈ ((TEndo‘𝐾)‘𝑊)) |
26 | eqid 2724 | . . . . 5 ⊢ (Base‘𝑈) = (Base‘𝑈) | |
27 | 5, 6, 23, 8, 26 | dvhelvbasei 40449 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ ( I ↾ 𝑇) ∈ ((TEndo‘𝐾)‘𝑊))) → 〈𝐹, ( I ↾ 𝑇)〉 ∈ (Base‘𝑈)) |
28 | 16, 22, 25, 27 | syl12anc 834 | . . 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 39940 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) → 𝐽 ∈ 𝑇) |
30 | 1, 5, 6, 23, 7 | tendo0cl 40151 | . . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑂 ∈ ((TEndo‘𝐾)‘𝑊)) |
31 | 30 | 3ad2ant1 1130 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) → 𝑂 ∈ ((TEndo‘𝐾)‘𝑊)) |
32 | 5, 6, 23, 8, 26 | dvhelvbasei 40449 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐽 ∈ 𝑇 ∧ 𝑂 ∈ ((TEndo‘𝐾)‘𝑊))) → 〈𝐽, 𝑂〉 ∈ (Base‘𝑈)) |
33 | 16, 29, 31, 32 | syl12anc 834 | . . 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 20935 | . . 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 1368 | . 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 4012 | 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 1084 = wceq 1533 ∈ wcel 2098 ⊆ wss 3940 {csn 4620 〈cop 4626 class class class wbr 5138 ↦ cmpt 5221 I cid 5563 ↾ cres 5668 ‘cfv 6533 ℩crio 7356 (class class class)co 7401 Basecbs 17143 +gcplusg 17196 lecple 17203 occoc 17204 LSSumclsm 19544 LModclmod 20696 LSpanclspn 20808 Atomscatm 38623 HLchlt 38710 LHypclh 39345 LTrncltrn 39462 TEndoctendo 40113 DVecHcdvh 40439 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-riotaBAD 38313 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-tp 4625 df-op 4627 df-uni 4900 df-int 4941 df-iun 4989 df-iin 4990 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-om 7849 df-1st 7968 df-2nd 7969 df-tpos 8206 df-undef 8253 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-er 8699 df-map 8818 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-nn 12210 df-2 12272 df-3 12273 df-4 12274 df-5 12275 df-6 12276 df-n0 12470 df-z 12556 df-uz 12820 df-fz 13482 df-struct 17079 df-sets 17096 df-slot 17114 df-ndx 17126 df-base 17144 df-ress 17173 df-plusg 17209 df-mulr 17210 df-sca 17212 df-vsca 17213 df-0g 17386 df-proset 18250 df-poset 18268 df-plt 18285 df-lub 18301 df-glb 18302 df-join 18303 df-meet 18304 df-p0 18380 df-p1 18381 df-lat 18387 df-clat 18454 df-mgm 18563 df-sgrp 18642 df-mnd 18658 df-submnd 18704 df-grp 18856 df-minusg 18857 df-sbg 18858 df-subg 19040 df-cntz 19223 df-lsm 19546 df-cmn 19692 df-abl 19693 df-mgp 20030 df-rng 20048 df-ur 20077 df-ring 20130 df-oppr 20226 df-dvdsr 20249 df-unit 20250 df-invr 20280 df-dvr 20293 df-drng 20579 df-lmod 20698 df-lss 20769 df-lsp 20809 df-lvec 20941 df-oposet 38536 df-ol 38538 df-oml 38539 df-covers 38626 df-ats 38627 df-atl 38658 df-cvlat 38682 df-hlat 38711 df-llines 38859 df-lplanes 38860 df-lvols 38861 df-lines 38862 df-psubsp 38864 df-pmap 38865 df-padd 39157 df-lhyp 39349 df-laut 39350 df-ldil 39465 df-ltrn 39466 df-trl 39520 df-tendo 40116 df-edring 40118 df-dvech 40440 |
This theorem is referenced by: cdlemn5pre 40561 |
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