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 2740 | . . . . . 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 2740 | . . . . . 6 ⊢ (LFnl‘𝑈) = (LFnl‘𝑈) | |
7 | eqid 2740 | . . . . . 6 ⊢ (LKer‘𝑈) = (LKer‘𝑈) | |
8 | eqid 2740 | . . . . . 6 ⊢ (LDual‘𝑈) = (LDual‘𝑈) | |
9 | lcdsca.k | . . . . . 6 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
10 | 2, 3, 4, 5, 6, 7, 8, 9 | lcdval 39612 | . . . . 5 ⊢ (𝜑 → 𝐶 = ((LDual‘𝑈) ↾s {𝑓 ∈ (LFnl‘𝑈) ∣ (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑓))) = ((LKer‘𝑈)‘𝑓)})) |
11 | 10 | fveq2d 6775 | . . . 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 6784 | . . . . . 6 ⊢ (LFnl‘𝑈) ∈ V | |
13 | 12 | rabex 5260 | . . . . 5 ⊢ {𝑓 ∈ (LFnl‘𝑈) ∣ (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑓))) = ((LKer‘𝑈)‘𝑓)} ∈ V |
14 | eqid 2740 | . . . . . 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 2740 | . . . . . 6 ⊢ (Scalar‘(LDual‘𝑈)) = (Scalar‘(LDual‘𝑈)) | |
16 | 14, 15 | resssca 17064 | . . . . 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 2798 | . . 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 39133 | . . . 4 ⊢ (𝜑 → 𝑈 ∈ LMod) |
22 | 19, 20, 8, 15, 21 | ldualsca 37155 | . . 3 ⊢ (𝜑 → (Scalar‘(LDual‘𝑈)) = 𝑂) |
23 | 18, 22 | eqtrd 2780 | . 2 ⊢ (𝜑 → (Scalar‘𝐶) = 𝑂) |
24 | 1, 23 | eqtrid 2792 | 1 ⊢ (𝜑 → 𝑅 = 𝑂) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1542 ∈ wcel 2110 {crab 3070 Vcvv 3431 ‘cfv 6432 (class class class)co 7272 ↾s cress 16952 Scalarcsca 16976 opprcoppr 19872 LModclmod 20134 LFnlclfn 37080 LKerclk 37108 LDualcld 37146 HLchlt 37373 LHypclh 38007 DVecHcdvh 39101 ocHcoch 39370 LCDualclcd 39609 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7583 ax-cnex 10938 ax-resscn 10939 ax-1cn 10940 ax-icn 10941 ax-addcl 10942 ax-addrcl 10943 ax-mulcl 10944 ax-mulrcl 10945 ax-mulcom 10946 ax-addass 10947 ax-mulass 10948 ax-distr 10949 ax-i2m1 10950 ax-1ne0 10951 ax-1rid 10952 ax-rnegex 10953 ax-rrecex 10954 ax-cnre 10955 ax-pre-lttri 10956 ax-pre-lttrn 10957 ax-pre-ltadd 10958 ax-pre-mulgt0 10959 ax-riotaBAD 36976 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4846 df-iun 4932 df-iin 4933 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-riota 7229 df-ov 7275 df-oprab 7276 df-mpo 7277 df-of 7528 df-om 7708 df-1st 7825 df-2nd 7826 df-tpos 8034 df-undef 8081 df-frecs 8089 df-wrecs 8120 df-recs 8194 df-rdg 8233 df-1o 8289 df-er 8490 df-map 8609 df-en 8726 df-dom 8727 df-sdom 8728 df-fin 8729 df-pnf 11022 df-mnf 11023 df-xr 11024 df-ltxr 11025 df-le 11026 df-sub 11218 df-neg 11219 df-nn 11985 df-2 12047 df-3 12048 df-4 12049 df-5 12050 df-6 12051 df-n0 12245 df-z 12331 df-uz 12594 df-fz 13251 df-struct 16859 df-sets 16876 df-slot 16894 df-ndx 16906 df-base 16924 df-ress 16953 df-plusg 16986 df-mulr 16987 df-sca 16989 df-vsca 16990 df-0g 17163 df-proset 18024 df-poset 18042 df-plt 18059 df-lub 18075 df-glb 18076 df-join 18077 df-meet 18078 df-p0 18154 df-p1 18155 df-lat 18161 df-clat 18228 df-mgm 18337 df-sgrp 18386 df-mnd 18397 df-grp 18591 df-minusg 18592 df-mgp 19732 df-ur 19749 df-ring 19796 df-oppr 19873 df-dvdsr 19894 df-unit 19895 df-invr 19925 df-dvr 19936 df-drng 20004 df-lmod 20136 df-lvec 20376 df-ldual 37147 df-oposet 37199 df-ol 37201 df-oml 37202 df-covers 37289 df-ats 37290 df-atl 37321 df-cvlat 37345 df-hlat 37374 df-llines 37521 df-lplanes 37522 df-lvols 37523 df-lines 37524 df-psubsp 37526 df-pmap 37527 df-padd 37819 df-lhyp 38011 df-laut 38012 df-ldil 38127 df-ltrn 38128 df-trl 38182 df-tendo 38778 df-edring 38780 df-dvech 39102 df-lcdual 39610 |
This theorem is referenced by: lcdsbase 39623 lcdsadd 39624 lcdsmul 39625 lcd0 39631 lcd1 39632 lcdneg 39633 lcdvsub 39640 |
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