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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 767 | . . 3 ⊢ (((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈)) → 𝑊 ∈ LMod) | |
| 2 | simprl 771 | . . . 4 ⊢ (((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈)) → 𝑋 ∈ 𝐵) | |
| 3 | eqid 2737 | . . . . . 6 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
| 4 | lssvscl.s | . . . . . 6 ⊢ 𝑆 = (LSubSp‘𝑊) | |
| 5 | 3, 4 | lssel 20935 | . . . . 5 ⊢ ((𝑈 ∈ 𝑆 ∧ 𝑌 ∈ 𝑈) → 𝑌 ∈ (Base‘𝑊)) |
| 6 | 5 | ad2ant2l 746 | . . . 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 20876 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ (Base‘𝑊)) → (𝑋 · 𝑌) ∈ (Base‘𝑊)) |
| 11 | 1, 2, 6, 10 | syl3anc 1373 | . . 3 ⊢ (((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈)) → (𝑋 · 𝑌) ∈ (Base‘𝑊)) |
| 12 | eqid 2737 | . . . 4 ⊢ (+g‘𝑊) = (+g‘𝑊) | |
| 13 | eqid 2737 | . . . 4 ⊢ (0g‘𝑊) = (0g‘𝑊) | |
| 14 | 3, 12, 13 | lmod0vrid 20891 | . . 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 769 | . . 3 ⊢ (((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈)) → 𝑈 ∈ 𝑆) | |
| 17 | simprr 773 | . . 3 ⊢ (((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈)) → 𝑌 ∈ 𝑈) | |
| 18 | 13, 4 | lss0cl 20945 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) → (0g‘𝑊) ∈ 𝑈) |
| 19 | 18 | adantr 480 | . . 3 ⊢ (((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈)) → (0g‘𝑊) ∈ 𝑈) |
| 20 | 7, 9, 12, 8, 4 | lsscl 20940 | . . 3 ⊢ ((𝑈 ∈ 𝑆 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈 ∧ (0g‘𝑊) ∈ 𝑈)) → ((𝑋 · 𝑌)(+g‘𝑊)(0g‘𝑊)) ∈ 𝑈) |
| 21 | 16, 2, 17, 19, 20 | syl13anc 1374 | . 2 ⊢ (((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈)) → ((𝑋 · 𝑌)(+g‘𝑊)(0g‘𝑊)) ∈ 𝑈) |
| 22 | 15, 21 | eqeltrrd 2842 | 1 ⊢ (((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈)) → (𝑋 · 𝑌) ∈ 𝑈) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 +gcplusg 17297 Scalarcsca 17300 ·𝑠 cvsca 17301 0gc0g 17484 LModclmod 20858 LSubSpclss 20929 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-plusg 17310 df-0g 17486 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-grp 18954 df-minusg 18955 df-sbg 18956 df-mgp 20138 df-ur 20179 df-ring 20232 df-lmod 20860 df-lss 20930 |
| This theorem is referenced by: lssvnegcl 20954 islss3 20957 islss4 20960 ellspsni 20999 lspsn 21000 lmhmima 21046 lmhmpreima 21047 reslmhm 21051 lsmcl 21082 pj1lmhm 21099 lssvs0or 21112 lspfixed 21130 lspexch 21131 lspsolv 21145 frlmssuvc1 21814 frlmsslsp 21816 mplbas2 22060 lssnlm 24722 minveclem2 25460 pjthlem1 25471 eqgvscpbl 33378 lindsunlem 33675 algextdeglem8 33765 lshpkrlem5 39115 ldualssvscl 39159 dochkr1 41480 dochkr1OLDN 41481 lclkrlem2o 41523 lcfrlem5 41548 lcdlssvscl 41608 hgmapvvlem3 41927 gsumlsscl 48296 |
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