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Mirrors > Home > MPE Home > Th. List > lssvscl | Structured version Visualization version GIF version |
Description: Closure of scalar product in a subspace. (Contributed by NM, 11-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) |
Ref | Expression |
---|---|
lssvscl.f | ⊢ 𝐹 = (Scalar‘𝑊) |
lssvscl.t | ⊢ · = ( ·𝑠 ‘𝑊) |
lssvscl.b | ⊢ 𝐵 = (Base‘𝐹) |
lssvscl.s | ⊢ 𝑆 = (LSubSp‘𝑊) |
Ref | Expression |
---|---|
lssvscl | ⊢ (((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈)) → (𝑋 · 𝑌) ∈ 𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpll 765 | . . 3 ⊢ (((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈)) → 𝑊 ∈ LMod) | |
2 | simprl 769 | . . . 4 ⊢ (((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈)) → 𝑋 ∈ 𝐵) | |
3 | eqid 2731 | . . . . . 6 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
4 | lssvscl.s | . . . . . 6 ⊢ 𝑆 = (LSubSp‘𝑊) | |
5 | 3, 4 | lssel 20497 | . . . . 5 ⊢ ((𝑈 ∈ 𝑆 ∧ 𝑌 ∈ 𝑈) → 𝑌 ∈ (Base‘𝑊)) |
6 | 5 | ad2ant2l 744 | . . . 4 ⊢ (((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈)) → 𝑌 ∈ (Base‘𝑊)) |
7 | lssvscl.f | . . . . 5 ⊢ 𝐹 = (Scalar‘𝑊) | |
8 | lssvscl.t | . . . . 5 ⊢ · = ( ·𝑠 ‘𝑊) | |
9 | lssvscl.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐹) | |
10 | 3, 7, 8, 9 | lmodvscl 20438 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ (Base‘𝑊)) → (𝑋 · 𝑌) ∈ (Base‘𝑊)) |
11 | 1, 2, 6, 10 | syl3anc 1371 | . . 3 ⊢ (((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈)) → (𝑋 · 𝑌) ∈ (Base‘𝑊)) |
12 | eqid 2731 | . . . 4 ⊢ (+g‘𝑊) = (+g‘𝑊) | |
13 | eqid 2731 | . . . 4 ⊢ (0g‘𝑊) = (0g‘𝑊) | |
14 | 3, 12, 13 | lmod0vrid 20452 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ (𝑋 · 𝑌) ∈ (Base‘𝑊)) → ((𝑋 · 𝑌)(+g‘𝑊)(0g‘𝑊)) = (𝑋 · 𝑌)) |
15 | 1, 11, 14 | syl2anc 584 | . 2 ⊢ (((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈)) → ((𝑋 · 𝑌)(+g‘𝑊)(0g‘𝑊)) = (𝑋 · 𝑌)) |
16 | simplr 767 | . . 3 ⊢ (((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈)) → 𝑈 ∈ 𝑆) | |
17 | simprr 771 | . . 3 ⊢ (((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈)) → 𝑌 ∈ 𝑈) | |
18 | 13, 4 | lss0cl 20506 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) → (0g‘𝑊) ∈ 𝑈) |
19 | 18 | adantr 481 | . . 3 ⊢ (((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈)) → (0g‘𝑊) ∈ 𝑈) |
20 | 7, 9, 12, 8, 4 | lsscl 20502 | . . 3 ⊢ ((𝑈 ∈ 𝑆 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈 ∧ (0g‘𝑊) ∈ 𝑈)) → ((𝑋 · 𝑌)(+g‘𝑊)(0g‘𝑊)) ∈ 𝑈) |
21 | 16, 2, 17, 19, 20 | syl13anc 1372 | . 2 ⊢ (((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈)) → ((𝑋 · 𝑌)(+g‘𝑊)(0g‘𝑊)) ∈ 𝑈) |
22 | 15, 21 | eqeltrrd 2833 | 1 ⊢ (((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈)) → (𝑋 · 𝑌) ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ‘cfv 6532 (class class class)co 7393 Basecbs 17126 +gcplusg 17179 Scalarcsca 17182 ·𝑠 cvsca 17183 0gc0g 17367 LModclmod 20420 LSubSpclss 20491 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 ax-cnex 11148 ax-resscn 11149 ax-1cn 11150 ax-icn 11151 ax-addcl 11152 ax-addrcl 11153 ax-mulcl 11154 ax-mulrcl 11155 ax-mulcom 11156 ax-addass 11157 ax-mulass 11158 ax-distr 11159 ax-i2m1 11160 ax-1ne0 11161 ax-1rid 11162 ax-rnegex 11163 ax-rrecex 11164 ax-cnre 11165 ax-pre-lttri 11166 ax-pre-lttrn 11167 ax-pre-ltadd 11168 ax-pre-mulgt0 11169 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6289 df-ord 6356 df-on 6357 df-lim 6358 df-suc 6359 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-riota 7349 df-ov 7396 df-oprab 7397 df-mpo 7398 df-om 7839 df-1st 7957 df-2nd 7958 df-frecs 8248 df-wrecs 8279 df-recs 8353 df-rdg 8392 df-er 8686 df-en 8923 df-dom 8924 df-sdom 8925 df-pnf 11232 df-mnf 11233 df-xr 11234 df-ltxr 11235 df-le 11236 df-sub 11428 df-neg 11429 df-nn 12195 df-2 12257 df-sets 17079 df-slot 17097 df-ndx 17109 df-base 17127 df-plusg 17192 df-0g 17369 df-mgm 18543 df-sgrp 18592 df-mnd 18603 df-grp 18797 df-minusg 18798 df-sbg 18799 df-mgp 19947 df-ur 19964 df-ring 20016 df-lmod 20422 df-lss 20492 |
This theorem is referenced by: lssvnegcl 20516 islss3 20519 islss4 20522 lspsneli 20561 lspsn 20562 lmhmima 20607 lmhmpreima 20608 reslmhm 20612 lsmcl 20643 pj1lmhm 20660 lssvs0or 20672 lspfixed 20690 lspexch 20691 lspsolv 20705 lidlmcl 20788 frlmssuvc1 21282 frlmsslsp 21284 mplbas2 21525 lssnlm 24147 minveclem2 24872 pjthlem1 24883 eqgvscpbl 32327 lindsunlem 32545 lshpkrlem5 37787 ldualssvscl 37831 dochkr1 40152 dochkr1OLDN 40153 lclkrlem2o 40195 lcfrlem5 40220 lcdlssvscl 40280 hgmapvvlem3 40599 gsumlsscl 46705 |
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