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Mirrors > Home > MPE Home > Th. List > Mathboxes > lcdvs | Structured version Visualization version GIF version |
Description: Scalar product for the closed kernel vector space dual. (Contributed by NM, 28-Mar-2015.) |
Ref | Expression |
---|---|
lcdvs.h | ⊢ 𝐻 = (LHyp‘𝐾) |
lcdvs.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
lcdvs.d | ⊢ 𝐷 = (LDual‘𝑈) |
lcdvs.t | ⊢ · = ( ·𝑠 ‘𝐷) |
lcdvs.c | ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) |
lcdvs.m | ⊢ ∙ = ( ·𝑠 ‘𝐶) |
lcdvs.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
Ref | Expression |
---|---|
lcdvs | ⊢ (𝜑 → ∙ = · ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lcdvs.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | eqid 2821 | . . . 4 ⊢ ((ocH‘𝐾)‘𝑊) = ((ocH‘𝐾)‘𝑊) | |
3 | lcdvs.c | . . . 4 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) | |
4 | lcdvs.u | . . . 4 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
5 | eqid 2821 | . . . 4 ⊢ (LFnl‘𝑈) = (LFnl‘𝑈) | |
6 | eqid 2821 | . . . 4 ⊢ (LKer‘𝑈) = (LKer‘𝑈) | |
7 | lcdvs.d | . . . 4 ⊢ 𝐷 = (LDual‘𝑈) | |
8 | lcdvs.k | . . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | lcdval 38719 | . . 3 ⊢ (𝜑 → 𝐶 = (𝐷 ↾s {𝑓 ∈ (LFnl‘𝑈) ∣ (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑓))) = ((LKer‘𝑈)‘𝑓)})) |
10 | 9 | fveq2d 6668 | . 2 ⊢ (𝜑 → ( ·𝑠 ‘𝐶) = ( ·𝑠 ‘(𝐷 ↾s {𝑓 ∈ (LFnl‘𝑈) ∣ (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑓))) = ((LKer‘𝑈)‘𝑓)}))) |
11 | lcdvs.m | . 2 ⊢ ∙ = ( ·𝑠 ‘𝐶) | |
12 | fvex 6677 | . . . 4 ⊢ (LFnl‘𝑈) ∈ V | |
13 | 12 | rabex 5227 | . . 3 ⊢ {𝑓 ∈ (LFnl‘𝑈) ∣ (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑓))) = ((LKer‘𝑈)‘𝑓)} ∈ V |
14 | eqid 2821 | . . . 4 ⊢ (𝐷 ↾s {𝑓 ∈ (LFnl‘𝑈) ∣ (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑓))) = ((LKer‘𝑈)‘𝑓)}) = (𝐷 ↾s {𝑓 ∈ (LFnl‘𝑈) ∣ (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑓))) = ((LKer‘𝑈)‘𝑓)}) | |
15 | lcdvs.t | . . . 4 ⊢ · = ( ·𝑠 ‘𝐷) | |
16 | 14, 15 | ressvsca 16645 | . . 3 ⊢ ({𝑓 ∈ (LFnl‘𝑈) ∣ (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑓))) = ((LKer‘𝑈)‘𝑓)} ∈ V → · = ( ·𝑠 ‘(𝐷 ↾s {𝑓 ∈ (LFnl‘𝑈) ∣ (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑓))) = ((LKer‘𝑈)‘𝑓)}))) |
17 | 13, 16 | ax-mp 5 | . 2 ⊢ · = ( ·𝑠 ‘(𝐷 ↾s {𝑓 ∈ (LFnl‘𝑈) ∣ (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑓))) = ((LKer‘𝑈)‘𝑓)})) |
18 | 10, 11, 17 | 3eqtr4g 2881 | 1 ⊢ (𝜑 → ∙ = · ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 {crab 3142 Vcvv 3494 ‘cfv 6349 (class class class)co 7150 ↾s cress 16478 ·𝑠 cvsca 16563 LFnlclfn 36187 LKerclk 36215 LDualcld 36253 HLchlt 36480 LHypclh 37114 DVecHcdvh 38208 ocHcoch 38477 LCDualclcd 38716 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-vsca 16576 df-lcdual 38717 |
This theorem is referenced by: lcdvsval 38734 lcdlkreq2N 38753 |
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