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Mirrors > Home > MPE Home > Th. List > Mathboxes > ldualvsubcl | Structured version Visualization version GIF version |
Description: Closure of vector subtraction in the dual of a vector space. (Contributed by NM, 27-Feb-2015.) |
Ref | Expression |
---|---|
ldualvsubcl.f | ⊢ 𝐹 = (LFnl‘𝑊) |
ldualvsubcl.d | ⊢ 𝐷 = (LDual‘𝑊) |
ldualvsubcl.m | ⊢ − = (-g‘𝐷) |
ldualvsubcl.w | ⊢ (𝜑 → 𝑊 ∈ LMod) |
ldualvsubcl.g | ⊢ (𝜑 → 𝐺 ∈ 𝐹) |
ldualvsubcl.h | ⊢ (𝜑 → 𝐻 ∈ 𝐹) |
Ref | Expression |
---|---|
ldualvsubcl | ⊢ (𝜑 → (𝐺 − 𝐻) ∈ 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2824 | . . 3 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
2 | eqid 2824 | . . 3 ⊢ (invg‘(Scalar‘𝑊)) = (invg‘(Scalar‘𝑊)) | |
3 | eqid 2824 | . . 3 ⊢ (1r‘(Scalar‘𝑊)) = (1r‘(Scalar‘𝑊)) | |
4 | ldualvsubcl.f | . . 3 ⊢ 𝐹 = (LFnl‘𝑊) | |
5 | ldualvsubcl.d | . . 3 ⊢ 𝐷 = (LDual‘𝑊) | |
6 | eqid 2824 | . . 3 ⊢ (+g‘𝐷) = (+g‘𝐷) | |
7 | eqid 2824 | . . 3 ⊢ ( ·𝑠 ‘𝐷) = ( ·𝑠 ‘𝐷) | |
8 | ldualvsubcl.m | . . 3 ⊢ − = (-g‘𝐷) | |
9 | ldualvsubcl.w | . . 3 ⊢ (𝜑 → 𝑊 ∈ LMod) | |
10 | ldualvsubcl.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ 𝐹) | |
11 | ldualvsubcl.h | . . 3 ⊢ (𝜑 → 𝐻 ∈ 𝐹) | |
12 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 | ldualvsub 36295 | . 2 ⊢ (𝜑 → (𝐺 − 𝐻) = (𝐺(+g‘𝐷)(((invg‘(Scalar‘𝑊))‘(1r‘(Scalar‘𝑊)))( ·𝑠 ‘𝐷)𝐻))) |
13 | eqid 2824 | . . . 4 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊)) | |
14 | 1 | lmodring 19645 | . . . . . . 7 ⊢ (𝑊 ∈ LMod → (Scalar‘𝑊) ∈ Ring) |
15 | 9, 14 | syl 17 | . . . . . 6 ⊢ (𝜑 → (Scalar‘𝑊) ∈ Ring) |
16 | ringgrp 19305 | . . . . . 6 ⊢ ((Scalar‘𝑊) ∈ Ring → (Scalar‘𝑊) ∈ Grp) | |
17 | 15, 16 | syl 17 | . . . . 5 ⊢ (𝜑 → (Scalar‘𝑊) ∈ Grp) |
18 | 13, 3 | ringidcl 19321 | . . . . . 6 ⊢ ((Scalar‘𝑊) ∈ Ring → (1r‘(Scalar‘𝑊)) ∈ (Base‘(Scalar‘𝑊))) |
19 | 15, 18 | syl 17 | . . . . 5 ⊢ (𝜑 → (1r‘(Scalar‘𝑊)) ∈ (Base‘(Scalar‘𝑊))) |
20 | 13, 2 | grpinvcl 18154 | . . . . 5 ⊢ (((Scalar‘𝑊) ∈ Grp ∧ (1r‘(Scalar‘𝑊)) ∈ (Base‘(Scalar‘𝑊))) → ((invg‘(Scalar‘𝑊))‘(1r‘(Scalar‘𝑊))) ∈ (Base‘(Scalar‘𝑊))) |
21 | 17, 19, 20 | syl2anc 586 | . . . 4 ⊢ (𝜑 → ((invg‘(Scalar‘𝑊))‘(1r‘(Scalar‘𝑊))) ∈ (Base‘(Scalar‘𝑊))) |
22 | 4, 1, 13, 5, 7, 9, 21, 11 | ldualvscl 36279 | . . 3 ⊢ (𝜑 → (((invg‘(Scalar‘𝑊))‘(1r‘(Scalar‘𝑊)))( ·𝑠 ‘𝐷)𝐻) ∈ 𝐹) |
23 | 4, 5, 6, 9, 10, 22 | ldualvaddcl 36270 | . 2 ⊢ (𝜑 → (𝐺(+g‘𝐷)(((invg‘(Scalar‘𝑊))‘(1r‘(Scalar‘𝑊)))( ·𝑠 ‘𝐷)𝐻)) ∈ 𝐹) |
24 | 12, 23 | eqeltrd 2916 | 1 ⊢ (𝜑 → (𝐺 − 𝐻) ∈ 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2113 ‘cfv 6358 (class class class)co 7159 Basecbs 16486 +gcplusg 16568 Scalarcsca 16571 ·𝑠 cvsca 16572 Grpcgrp 18106 invgcminusg 18107 -gcsg 18108 1rcur 19254 Ringcrg 19300 LModclmod 19637 LFnlclfn 36197 LDualcld 36263 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-of 7412 df-om 7584 df-1st 7692 df-2nd 7693 df-tpos 7895 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-oadd 8109 df-er 8292 df-map 8411 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-nn 11642 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-n0 11901 df-z 11985 df-uz 12247 df-fz 12896 df-struct 16488 df-ndx 16489 df-slot 16490 df-base 16492 df-sets 16493 df-plusg 16581 df-mulr 16582 df-sca 16584 df-vsca 16585 df-0g 16718 df-mgm 17855 df-sgrp 17904 df-mnd 17915 df-grp 18109 df-minusg 18110 df-sbg 18111 df-cmn 18911 df-abl 18912 df-mgp 19243 df-ur 19255 df-ring 19302 df-oppr 19376 df-lmod 19639 df-lfl 36198 df-ldual 36264 |
This theorem is referenced by: lcfrlem3 38684 lcfrlem30 38712 |
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