Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lcdsca | Structured version Visualization version GIF version |
Description: The ring of scalars of the closed kernel dual space. (Contributed by NM, 16-Mar-2015.) (Revised by Mario Carneiro, 6-May-2015.) |
Ref | Expression |
---|---|
lcdsca.h | ⊢ 𝐻 = (LHyp‘𝐾) |
lcdsca.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
lcdsca.f | ⊢ 𝐹 = (Scalar‘𝑈) |
lcdsca.o | ⊢ 𝑂 = (oppr‘𝐹) |
lcdsca.c | ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) |
lcdsca.r | ⊢ 𝑅 = (Scalar‘𝐶) |
lcdsca.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
Ref | Expression |
---|---|
lcdsca | ⊢ (𝜑 → 𝑅 = 𝑂) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lcdsca.r | . 2 ⊢ 𝑅 = (Scalar‘𝐶) | |
2 | lcdsca.h | . . . . . 6 ⊢ 𝐻 = (LHyp‘𝐾) | |
3 | eqid 2738 | . . . . . 6 ⊢ ((ocH‘𝐾)‘𝑊) = ((ocH‘𝐾)‘𝑊) | |
4 | lcdsca.c | . . . . . 6 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) | |
5 | lcdsca.u | . . . . . 6 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
6 | eqid 2738 | . . . . . 6 ⊢ (LFnl‘𝑈) = (LFnl‘𝑈) | |
7 | eqid 2738 | . . . . . 6 ⊢ (LKer‘𝑈) = (LKer‘𝑈) | |
8 | eqid 2738 | . . . . . 6 ⊢ (LDual‘𝑈) = (LDual‘𝑈) | |
9 | lcdsca.k | . . . . . 6 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
10 | 2, 3, 4, 5, 6, 7, 8, 9 | lcdval 39603 | . . . . 5 ⊢ (𝜑 → 𝐶 = ((LDual‘𝑈) ↾s {𝑓 ∈ (LFnl‘𝑈) ∣ (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑓))) = ((LKer‘𝑈)‘𝑓)})) |
11 | 10 | fveq2d 6778 | . . . 4 ⊢ (𝜑 → (Scalar‘𝐶) = (Scalar‘((LDual‘𝑈) ↾s {𝑓 ∈ (LFnl‘𝑈) ∣ (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑓))) = ((LKer‘𝑈)‘𝑓)}))) |
12 | fvex 6787 | . . . . . 6 ⊢ (LFnl‘𝑈) ∈ V | |
13 | 12 | rabex 5256 | . . . . 5 ⊢ {𝑓 ∈ (LFnl‘𝑈) ∣ (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑓))) = ((LKer‘𝑈)‘𝑓)} ∈ V |
14 | eqid 2738 | . . . . . 6 ⊢ ((LDual‘𝑈) ↾s {𝑓 ∈ (LFnl‘𝑈) ∣ (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑓))) = ((LKer‘𝑈)‘𝑓)}) = ((LDual‘𝑈) ↾s {𝑓 ∈ (LFnl‘𝑈) ∣ (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑓))) = ((LKer‘𝑈)‘𝑓)}) | |
15 | eqid 2738 | . . . . . 6 ⊢ (Scalar‘(LDual‘𝑈)) = (Scalar‘(LDual‘𝑈)) | |
16 | 14, 15 | resssca 17053 | . . . . 5 ⊢ ({𝑓 ∈ (LFnl‘𝑈) ∣ (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑓))) = ((LKer‘𝑈)‘𝑓)} ∈ V → (Scalar‘(LDual‘𝑈)) = (Scalar‘((LDual‘𝑈) ↾s {𝑓 ∈ (LFnl‘𝑈) ∣ (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑓))) = ((LKer‘𝑈)‘𝑓)}))) |
17 | 13, 16 | ax-mp 5 | . . . 4 ⊢ (Scalar‘(LDual‘𝑈)) = (Scalar‘((LDual‘𝑈) ↾s {𝑓 ∈ (LFnl‘𝑈) ∣ (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑓))) = ((LKer‘𝑈)‘𝑓)})) |
18 | 11, 17 | eqtr4di 2796 | . . 3 ⊢ (𝜑 → (Scalar‘𝐶) = (Scalar‘(LDual‘𝑈))) |
19 | lcdsca.f | . . . 4 ⊢ 𝐹 = (Scalar‘𝑈) | |
20 | lcdsca.o | . . . 4 ⊢ 𝑂 = (oppr‘𝐹) | |
21 | 2, 5, 9 | dvhlmod 39124 | . . . 4 ⊢ (𝜑 → 𝑈 ∈ LMod) |
22 | 19, 20, 8, 15, 21 | ldualsca 37146 | . . 3 ⊢ (𝜑 → (Scalar‘(LDual‘𝑈)) = 𝑂) |
23 | 18, 22 | eqtrd 2778 | . 2 ⊢ (𝜑 → (Scalar‘𝐶) = 𝑂) |
24 | 1, 23 | eqtrid 2790 | 1 ⊢ (𝜑 → 𝑅 = 𝑂) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 {crab 3068 Vcvv 3432 ‘cfv 6433 (class class class)co 7275 ↾s cress 16941 Scalarcsca 16965 opprcoppr 19861 LModclmod 20123 LFnlclfn 37071 LKerclk 37099 LDualcld 37137 HLchlt 37364 LHypclh 37998 DVecHcdvh 39092 ocHcoch 39361 LCDualclcd 39600 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 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 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-riotaBAD 36967 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-iun 4926 df-iin 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-of 7533 df-om 7713 df-1st 7831 df-2nd 7832 df-tpos 8042 df-undef 8089 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-er 8498 df-map 8617 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-n0 12234 df-z 12320 df-uz 12583 df-fz 13240 df-struct 16848 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-mulr 16976 df-sca 16978 df-vsca 16979 df-0g 17152 df-proset 18013 df-poset 18031 df-plt 18048 df-lub 18064 df-glb 18065 df-join 18066 df-meet 18067 df-p0 18143 df-p1 18144 df-lat 18150 df-clat 18217 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-grp 18580 df-minusg 18581 df-mgp 19721 df-ur 19738 df-ring 19785 df-oppr 19862 df-dvdsr 19883 df-unit 19884 df-invr 19914 df-dvr 19925 df-drng 19993 df-lmod 20125 df-lvec 20365 df-ldual 37138 df-oposet 37190 df-ol 37192 df-oml 37193 df-covers 37280 df-ats 37281 df-atl 37312 df-cvlat 37336 df-hlat 37365 df-llines 37512 df-lplanes 37513 df-lvols 37514 df-lines 37515 df-psubsp 37517 df-pmap 37518 df-padd 37810 df-lhyp 38002 df-laut 38003 df-ldil 38118 df-ltrn 38119 df-trl 38173 df-tendo 38769 df-edring 38771 df-dvech 39093 df-lcdual 39601 |
This theorem is referenced by: lcdsbase 39614 lcdsadd 39615 lcdsmul 39616 lcd0 39622 lcd1 39623 lcdneg 39624 lcdvsub 39631 |
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