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 2823 | . . . . . 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 2823 | . . . . . 6 ⊢ (LFnl‘𝑈) = (LFnl‘𝑈) | |
7 | eqid 2823 | . . . . . 6 ⊢ (LKer‘𝑈) = (LKer‘𝑈) | |
8 | eqid 2823 | . . . . . 6 ⊢ (LDual‘𝑈) = (LDual‘𝑈) | |
9 | lcdsca.k | . . . . . 6 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
10 | 2, 3, 4, 5, 6, 7, 8, 9 | lcdval 38727 | . . . . 5 ⊢ (𝜑 → 𝐶 = ((LDual‘𝑈) ↾s {𝑓 ∈ (LFnl‘𝑈) ∣ (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑓))) = ((LKer‘𝑈)‘𝑓)})) |
11 | 10 | fveq2d 6676 | . . . 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 6685 | . . . . . 6 ⊢ (LFnl‘𝑈) ∈ V | |
13 | 12 | rabex 5237 | . . . . 5 ⊢ {𝑓 ∈ (LFnl‘𝑈) ∣ (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑓))) = ((LKer‘𝑈)‘𝑓)} ∈ V |
14 | eqid 2823 | . . . . . 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 2823 | . . . . . 6 ⊢ (Scalar‘(LDual‘𝑈)) = (Scalar‘(LDual‘𝑈)) | |
16 | 14, 15 | resssca 16652 | . . . . 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 | syl6eqr 2876 | . . 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 38248 | . . . 4 ⊢ (𝜑 → 𝑈 ∈ LMod) |
22 | 19, 20, 8, 15, 21 | ldualsca 36270 | . . 3 ⊢ (𝜑 → (Scalar‘(LDual‘𝑈)) = 𝑂) |
23 | 18, 22 | eqtrd 2858 | . 2 ⊢ (𝜑 → (Scalar‘𝐶) = 𝑂) |
24 | 1, 23 | syl5eq 2870 | 1 ⊢ (𝜑 → 𝑅 = 𝑂) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 {crab 3144 Vcvv 3496 ‘cfv 6357 (class class class)co 7158 ↾s cress 16486 Scalarcsca 16570 opprcoppr 19374 LModclmod 19636 LFnlclfn 36195 LKerclk 36223 LDualcld 36261 HLchlt 36488 LHypclh 37122 DVecHcdvh 38216 ocHcoch 38485 LCDualclcd 38724 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-riotaBAD 36091 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-iin 4924 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-of 7411 df-om 7583 df-1st 7691 df-2nd 7692 df-tpos 7894 df-undef 7941 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 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 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-sca 16583 df-vsca 16584 df-0g 16717 df-proset 17540 df-poset 17558 df-plt 17570 df-lub 17586 df-glb 17587 df-join 17588 df-meet 17589 df-p0 17651 df-p1 17652 df-lat 17658 df-clat 17720 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-grp 18108 df-minusg 18109 df-mgp 19242 df-ur 19254 df-ring 19301 df-oppr 19375 df-dvdsr 19393 df-unit 19394 df-invr 19424 df-dvr 19435 df-drng 19506 df-lmod 19638 df-lvec 19877 df-ldual 36262 df-oposet 36314 df-ol 36316 df-oml 36317 df-covers 36404 df-ats 36405 df-atl 36436 df-cvlat 36460 df-hlat 36489 df-llines 36636 df-lplanes 36637 df-lvols 36638 df-lines 36639 df-psubsp 36641 df-pmap 36642 df-padd 36934 df-lhyp 37126 df-laut 37127 df-ldil 37242 df-ltrn 37243 df-trl 37297 df-tendo 37893 df-edring 37895 df-dvech 38217 df-lcdual 38725 |
This theorem is referenced by: lcdsbase 38738 lcdsadd 38739 lcdsmul 38740 lcd0 38746 lcd1 38747 lcdneg 38748 lcdvsub 38755 |
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