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Mirrors > Home > MPE Home > Th. List > Mathboxes > lcfrlem6 | Structured version Visualization version GIF version |
Description: Lemma for lcfr 41567. Closure of vector sum with colinear vectors. TODO: Move down 𝑁 definition so top hypotheses can be shared. (Contributed by NM, 10-Mar-2015.) |
Ref | Expression |
---|---|
lcfrlem6.h | ⊢ 𝐻 = (LHyp‘𝐾) |
lcfrlem6.o | ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) |
lcfrlem6.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
lcfrlem6.p | ⊢ + = (+g‘𝑈) |
lcfrlem6.n | ⊢ 𝑁 = (LSpan‘𝑈) |
lcfrlem6.l | ⊢ 𝐿 = (LKer‘𝑈) |
lcfrlem6.d | ⊢ 𝐷 = (LDual‘𝑈) |
lcfrlem6.q | ⊢ 𝑄 = (LSubSp‘𝐷) |
lcfrlem6.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
lcfrlem6.g | ⊢ (𝜑 → 𝐺 ∈ 𝑄) |
lcfrlem6.e | ⊢ 𝐸 = ∪ 𝑔 ∈ 𝐺 ( ⊥ ‘(𝐿‘𝑔)) |
lcfrlem6.x | ⊢ (𝜑 → 𝑋 ∈ 𝐸) |
lcfrlem6.y | ⊢ (𝜑 → 𝑌 ∈ 𝐸) |
lcfrlem6.en | ⊢ (𝜑 → (𝑁‘{𝑋}) = (𝑁‘{𝑌})) |
Ref | Expression |
---|---|
lcfrlem6 | ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝐸) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lcfrlem6.x | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐸) | |
2 | lcfrlem6.e | . . . . . 6 ⊢ 𝐸 = ∪ 𝑔 ∈ 𝐺 ( ⊥ ‘(𝐿‘𝑔)) | |
3 | 1, 2 | eleqtrdi 2848 | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ ∪ 𝑔 ∈ 𝐺 ( ⊥ ‘(𝐿‘𝑔))) |
4 | eliun 4999 | . . . . 5 ⊢ (𝑋 ∈ ∪ 𝑔 ∈ 𝐺 ( ⊥ ‘(𝐿‘𝑔)) ↔ ∃𝑔 ∈ 𝐺 𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔))) | |
5 | 3, 4 | sylib 218 | . . . 4 ⊢ (𝜑 → ∃𝑔 ∈ 𝐺 𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔))) |
6 | lcfrlem6.h | . . . . . . . . . 10 ⊢ 𝐻 = (LHyp‘𝐾) | |
7 | lcfrlem6.u | . . . . . . . . . 10 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
8 | lcfrlem6.k | . . . . . . . . . 10 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
9 | 6, 7, 8 | dvhlmod 41092 | . . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ LMod) |
10 | 9 | adantr 480 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺) → 𝑈 ∈ LMod) |
11 | 10 | adantr 480 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐺) ∧ 𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔))) → 𝑈 ∈ LMod) |
12 | 8 | adantr 480 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
13 | eqid 2734 | . . . . . . . . . 10 ⊢ (Base‘𝑈) = (Base‘𝑈) | |
14 | eqid 2734 | . . . . . . . . . 10 ⊢ (LFnl‘𝑈) = (LFnl‘𝑈) | |
15 | lcfrlem6.l | . . . . . . . . . 10 ⊢ 𝐿 = (LKer‘𝑈) | |
16 | lcfrlem6.g | . . . . . . . . . . . 12 ⊢ (𝜑 → 𝐺 ∈ 𝑄) | |
17 | eqid 2734 | . . . . . . . . . . . . 13 ⊢ (Base‘𝐷) = (Base‘𝐷) | |
18 | lcfrlem6.q | . . . . . . . . . . . . 13 ⊢ 𝑄 = (LSubSp‘𝐷) | |
19 | 17, 18 | lssel 20952 | . . . . . . . . . . . 12 ⊢ ((𝐺 ∈ 𝑄 ∧ 𝑔 ∈ 𝐺) → 𝑔 ∈ (Base‘𝐷)) |
20 | 16, 19 | sylan 580 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺) → 𝑔 ∈ (Base‘𝐷)) |
21 | lcfrlem6.d | . . . . . . . . . . . . 13 ⊢ 𝐷 = (LDual‘𝑈) | |
22 | 14, 21, 17, 9 | ldualvbase 39107 | . . . . . . . . . . . 12 ⊢ (𝜑 → (Base‘𝐷) = (LFnl‘𝑈)) |
23 | 22 | adantr 480 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺) → (Base‘𝐷) = (LFnl‘𝑈)) |
24 | 20, 23 | eleqtrd 2840 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺) → 𝑔 ∈ (LFnl‘𝑈)) |
25 | 13, 14, 15, 10, 24 | lkrssv 39077 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺) → (𝐿‘𝑔) ⊆ (Base‘𝑈)) |
26 | eqid 2734 | . . . . . . . . . 10 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈) | |
27 | lcfrlem6.o | . . . . . . . . . 10 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) | |
28 | 6, 7, 13, 26, 27 | dochlss 41336 | . . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐿‘𝑔) ⊆ (Base‘𝑈)) → ( ⊥ ‘(𝐿‘𝑔)) ∈ (LSubSp‘𝑈)) |
29 | 12, 25, 28 | syl2anc 584 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺) → ( ⊥ ‘(𝐿‘𝑔)) ∈ (LSubSp‘𝑈)) |
30 | 29 | adantr 480 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐺) ∧ 𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔))) → ( ⊥ ‘(𝐿‘𝑔)) ∈ (LSubSp‘𝑈)) |
31 | simpr 484 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐺) ∧ 𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔))) → 𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔))) | |
32 | lcfrlem6.en | . . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑁‘{𝑋}) = (𝑁‘{𝑌})) | |
33 | 32 | adantr 480 | . . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺) → (𝑁‘{𝑋}) = (𝑁‘{𝑌})) |
34 | 33 | adantr 480 | . . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐺) ∧ (𝑁‘{𝑋}) ⊆ ( ⊥ ‘(𝐿‘𝑔))) → (𝑁‘{𝑋}) = (𝑁‘{𝑌})) |
35 | simpr 484 | . . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐺) ∧ (𝑁‘{𝑋}) ⊆ ( ⊥ ‘(𝐿‘𝑔))) → (𝑁‘{𝑋}) ⊆ ( ⊥ ‘(𝐿‘𝑔))) | |
36 | 34, 35 | eqsstrrd 4034 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐺) ∧ (𝑁‘{𝑋}) ⊆ ( ⊥ ‘(𝐿‘𝑔))) → (𝑁‘{𝑌}) ⊆ ( ⊥ ‘(𝐿‘𝑔))) |
37 | 36 | ex 412 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺) → ((𝑁‘{𝑋}) ⊆ ( ⊥ ‘(𝐿‘𝑔)) → (𝑁‘{𝑌}) ⊆ ( ⊥ ‘(𝐿‘𝑔)))) |
38 | lcfrlem6.n | . . . . . . . . . 10 ⊢ 𝑁 = (LSpan‘𝑈) | |
39 | 6, 27, 7, 13, 15, 21, 18, 2, 8, 16, 1 | lcfrlem4 41527 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝑈)) |
40 | 39 | adantr 480 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺) → 𝑋 ∈ (Base‘𝑈)) |
41 | 13, 26, 38, 10, 29, 40 | ellspsn5b 21010 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺) → (𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔)) ↔ (𝑁‘{𝑋}) ⊆ ( ⊥ ‘(𝐿‘𝑔)))) |
42 | lcfrlem6.y | . . . . . . . . . . . 12 ⊢ (𝜑 → 𝑌 ∈ 𝐸) | |
43 | 6, 27, 7, 13, 15, 21, 18, 2, 8, 16, 42 | lcfrlem4 41527 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝑈)) |
44 | 43 | adantr 480 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺) → 𝑌 ∈ (Base‘𝑈)) |
45 | 13, 26, 38, 10, 29, 44 | ellspsn5b 21010 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺) → (𝑌 ∈ ( ⊥ ‘(𝐿‘𝑔)) ↔ (𝑁‘{𝑌}) ⊆ ( ⊥ ‘(𝐿‘𝑔)))) |
46 | 37, 41, 45 | 3imtr4d 294 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺) → (𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔)) → 𝑌 ∈ ( ⊥ ‘(𝐿‘𝑔)))) |
47 | 46 | imp 406 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐺) ∧ 𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔))) → 𝑌 ∈ ( ⊥ ‘(𝐿‘𝑔))) |
48 | lcfrlem6.p | . . . . . . . 8 ⊢ + = (+g‘𝑈) | |
49 | 48, 26 | lssvacl 20958 | . . . . . . 7 ⊢ (((𝑈 ∈ LMod ∧ ( ⊥ ‘(𝐿‘𝑔)) ∈ (LSubSp‘𝑈)) ∧ (𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔)) ∧ 𝑌 ∈ ( ⊥ ‘(𝐿‘𝑔)))) → (𝑋 + 𝑌) ∈ ( ⊥ ‘(𝐿‘𝑔))) |
50 | 11, 30, 31, 47, 49 | syl22anc 839 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐺) ∧ 𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔))) → (𝑋 + 𝑌) ∈ ( ⊥ ‘(𝐿‘𝑔))) |
51 | 50 | ex 412 | . . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺) → (𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔)) → (𝑋 + 𝑌) ∈ ( ⊥ ‘(𝐿‘𝑔)))) |
52 | 51 | reximdva 3165 | . . . 4 ⊢ (𝜑 → (∃𝑔 ∈ 𝐺 𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔)) → ∃𝑔 ∈ 𝐺 (𝑋 + 𝑌) ∈ ( ⊥ ‘(𝐿‘𝑔)))) |
53 | 5, 52 | mpd 15 | . . 3 ⊢ (𝜑 → ∃𝑔 ∈ 𝐺 (𝑋 + 𝑌) ∈ ( ⊥ ‘(𝐿‘𝑔))) |
54 | eliun 4999 | . . 3 ⊢ ((𝑋 + 𝑌) ∈ ∪ 𝑔 ∈ 𝐺 ( ⊥ ‘(𝐿‘𝑔)) ↔ ∃𝑔 ∈ 𝐺 (𝑋 + 𝑌) ∈ ( ⊥ ‘(𝐿‘𝑔))) | |
55 | 53, 54 | sylibr 234 | . 2 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ ∪ 𝑔 ∈ 𝐺 ( ⊥ ‘(𝐿‘𝑔))) |
56 | 55, 2 | eleqtrrdi 2849 | 1 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝐸) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1536 ∈ wcel 2105 ∃wrex 3067 ⊆ wss 3962 {csn 4630 ∪ ciun 4995 ‘cfv 6562 (class class class)co 7430 Basecbs 17244 +gcplusg 17297 LModclmod 20874 LSubSpclss 20946 LSpanclspn 20986 LFnlclfn 39038 LKerclk 39066 LDualcld 39104 HLchlt 39331 LHypclh 39966 DVecHcdvh 41060 ocHcoch 41329 |
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-of 7696 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-lfl 39039 df-lkr 39067 df-ldual 39105 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-disoa 41011 df-dvech 41061 df-dib 41121 df-dic 41155 df-dih 41211 df-doch 41330 |
This theorem is referenced by: lcfrlem41 41565 |
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