Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > lcfrlem6 | Structured version Visualization version GIF version |
Description: Lemma for lcfr 38736. 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 2923 | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ ∪ 𝑔 ∈ 𝐺 ( ⊥ ‘(𝐿‘𝑔))) |
4 | eliun 4923 | . . . . 5 ⊢ (𝑋 ∈ ∪ 𝑔 ∈ 𝐺 ( ⊥ ‘(𝐿‘𝑔)) ↔ ∃𝑔 ∈ 𝐺 𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔))) | |
5 | 3, 4 | sylib 220 | . . . 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 38261 | . . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ LMod) |
10 | 9 | adantr 483 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺) → 𝑈 ∈ LMod) |
11 | 10 | adantr 483 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐺) ∧ 𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔))) → 𝑈 ∈ LMod) |
12 | 8 | adantr 483 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
13 | eqid 2821 | . . . . . . . . . 10 ⊢ (Base‘𝑈) = (Base‘𝑈) | |
14 | eqid 2821 | . . . . . . . . . 10 ⊢ (LFnl‘𝑈) = (LFnl‘𝑈) | |
15 | lcfrlem6.l | . . . . . . . . . 10 ⊢ 𝐿 = (LKer‘𝑈) | |
16 | lcfrlem6.g | . . . . . . . . . . . 12 ⊢ (𝜑 → 𝐺 ∈ 𝑄) | |
17 | eqid 2821 | . . . . . . . . . . . . 13 ⊢ (Base‘𝐷) = (Base‘𝐷) | |
18 | lcfrlem6.q | . . . . . . . . . . . . 13 ⊢ 𝑄 = (LSubSp‘𝐷) | |
19 | 17, 18 | lssel 19709 | . . . . . . . . . . . 12 ⊢ ((𝐺 ∈ 𝑄 ∧ 𝑔 ∈ 𝐺) → 𝑔 ∈ (Base‘𝐷)) |
20 | 16, 19 | sylan 582 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺) → 𝑔 ∈ (Base‘𝐷)) |
21 | lcfrlem6.d | . . . . . . . . . . . . 13 ⊢ 𝐷 = (LDual‘𝑈) | |
22 | 14, 21, 17, 9 | ldualvbase 36277 | . . . . . . . . . . . 12 ⊢ (𝜑 → (Base‘𝐷) = (LFnl‘𝑈)) |
23 | 22 | adantr 483 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺) → (Base‘𝐷) = (LFnl‘𝑈)) |
24 | 20, 23 | eleqtrd 2915 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺) → 𝑔 ∈ (LFnl‘𝑈)) |
25 | 13, 14, 15, 10, 24 | lkrssv 36247 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺) → (𝐿‘𝑔) ⊆ (Base‘𝑈)) |
26 | eqid 2821 | . . . . . . . . . 10 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈) | |
27 | lcfrlem6.o | . . . . . . . . . 10 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) | |
28 | 6, 7, 13, 26, 27 | dochlss 38505 | . . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐿‘𝑔) ⊆ (Base‘𝑈)) → ( ⊥ ‘(𝐿‘𝑔)) ∈ (LSubSp‘𝑈)) |
29 | 12, 25, 28 | syl2anc 586 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺) → ( ⊥ ‘(𝐿‘𝑔)) ∈ (LSubSp‘𝑈)) |
30 | 29 | adantr 483 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐺) ∧ 𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔))) → ( ⊥ ‘(𝐿‘𝑔)) ∈ (LSubSp‘𝑈)) |
31 | simpr 487 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐺) ∧ 𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔))) → 𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔))) | |
32 | lcfrlem6.en | . . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑁‘{𝑋}) = (𝑁‘{𝑌})) | |
33 | 32 | adantr 483 | . . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺) → (𝑁‘{𝑋}) = (𝑁‘{𝑌})) |
34 | 33 | adantr 483 | . . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐺) ∧ (𝑁‘{𝑋}) ⊆ ( ⊥ ‘(𝐿‘𝑔))) → (𝑁‘{𝑋}) = (𝑁‘{𝑌})) |
35 | simpr 487 | . . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐺) ∧ (𝑁‘{𝑋}) ⊆ ( ⊥ ‘(𝐿‘𝑔))) → (𝑁‘{𝑋}) ⊆ ( ⊥ ‘(𝐿‘𝑔))) | |
36 | 34, 35 | eqsstrrd 4006 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐺) ∧ (𝑁‘{𝑋}) ⊆ ( ⊥ ‘(𝐿‘𝑔))) → (𝑁‘{𝑌}) ⊆ ( ⊥ ‘(𝐿‘𝑔))) |
37 | 36 | ex 415 | . . . . . . . . 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 38696 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝑈)) |
40 | 39 | adantr 483 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺) → 𝑋 ∈ (Base‘𝑈)) |
41 | 13, 26, 38, 10, 29, 40 | lspsnel5 19767 | . . . . . . . . 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 38696 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝑈)) |
44 | 43 | adantr 483 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺) → 𝑌 ∈ (Base‘𝑈)) |
45 | 13, 26, 38, 10, 29, 44 | lspsnel5 19767 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺) → (𝑌 ∈ ( ⊥ ‘(𝐿‘𝑔)) ↔ (𝑁‘{𝑌}) ⊆ ( ⊥ ‘(𝐿‘𝑔)))) |
46 | 37, 41, 45 | 3imtr4d 296 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺) → (𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔)) → 𝑌 ∈ ( ⊥ ‘(𝐿‘𝑔)))) |
47 | 46 | imp 409 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐺) ∧ 𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔))) → 𝑌 ∈ ( ⊥ ‘(𝐿‘𝑔))) |
48 | lcfrlem6.p | . . . . . . . 8 ⊢ + = (+g‘𝑈) | |
49 | 48, 26 | lssvacl 19726 | . . . . . . 7 ⊢ (((𝑈 ∈ LMod ∧ ( ⊥ ‘(𝐿‘𝑔)) ∈ (LSubSp‘𝑈)) ∧ (𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔)) ∧ 𝑌 ∈ ( ⊥ ‘(𝐿‘𝑔)))) → (𝑋 + 𝑌) ∈ ( ⊥ ‘(𝐿‘𝑔))) |
50 | 11, 30, 31, 47, 49 | syl22anc 836 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐺) ∧ 𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔))) → (𝑋 + 𝑌) ∈ ( ⊥ ‘(𝐿‘𝑔))) |
51 | 50 | ex 415 | . . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺) → (𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔)) → (𝑋 + 𝑌) ∈ ( ⊥ ‘(𝐿‘𝑔)))) |
52 | 51 | reximdva 3274 | . . . 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 4923 | . . 3 ⊢ ((𝑋 + 𝑌) ∈ ∪ 𝑔 ∈ 𝐺 ( ⊥ ‘(𝐿‘𝑔)) ↔ ∃𝑔 ∈ 𝐺 (𝑋 + 𝑌) ∈ ( ⊥ ‘(𝐿‘𝑔))) | |
55 | 53, 54 | sylibr 236 | . 2 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ ∪ 𝑔 ∈ 𝐺 ( ⊥ ‘(𝐿‘𝑔))) |
56 | 55, 2 | eleqtrrdi 2924 | 1 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝐸) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∃wrex 3139 ⊆ wss 3936 {csn 4567 ∪ ciun 4919 ‘cfv 6355 (class class class)co 7156 Basecbs 16483 +gcplusg 16565 LModclmod 19634 LSubSpclss 19703 LSpanclspn 19743 LFnlclfn 36208 LKerclk 36236 LDualcld 36274 HLchlt 36501 LHypclh 37135 DVecHcdvh 38229 ocHcoch 38498 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-riotaBAD 36104 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-iin 4922 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7409 df-om 7581 df-1st 7689 df-2nd 7690 df-tpos 7892 df-undef 7939 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-map 8408 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-n0 11899 df-z 11983 df-uz 12245 df-fz 12894 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-sca 16581 df-vsca 16582 df-0g 16715 df-proset 17538 df-poset 17556 df-plt 17568 df-lub 17584 df-glb 17585 df-join 17586 df-meet 17587 df-p0 17649 df-p1 17650 df-lat 17656 df-clat 17718 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-submnd 17957 df-grp 18106 df-minusg 18107 df-sbg 18108 df-subg 18276 df-cntz 18447 df-lsm 18761 df-cmn 18908 df-abl 18909 df-mgp 19240 df-ur 19252 df-ring 19299 df-oppr 19373 df-dvdsr 19391 df-unit 19392 df-invr 19422 df-dvr 19433 df-drng 19504 df-lmod 19636 df-lss 19704 df-lsp 19744 df-lvec 19875 df-lfl 36209 df-lkr 36237 df-ldual 36275 df-oposet 36327 df-ol 36329 df-oml 36330 df-covers 36417 df-ats 36418 df-atl 36449 df-cvlat 36473 df-hlat 36502 df-llines 36649 df-lplanes 36650 df-lvols 36651 df-lines 36652 df-psubsp 36654 df-pmap 36655 df-padd 36947 df-lhyp 37139 df-laut 37140 df-ldil 37255 df-ltrn 37256 df-trl 37310 df-tendo 37906 df-edring 37908 df-disoa 38180 df-dvech 38230 df-dib 38290 df-dic 38324 df-dih 38380 df-doch 38499 |
This theorem is referenced by: lcfrlem41 38734 |
Copyright terms: Public domain | W3C validator |