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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 2824 | . . . 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 2824 | . . . 4 ⊢ (LFnl‘𝑈) = (LFnl‘𝑈) | |
6 | eqid 2824 | . . . 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 38729 | . . 3 ⊢ (𝜑 → 𝐶 = (𝐷 ↾s {𝑓 ∈ (LFnl‘𝑈) ∣ (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑓))) = ((LKer‘𝑈)‘𝑓)})) |
10 | 9 | fveq2d 6677 | . 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 6686 | . . . 4 ⊢ (LFnl‘𝑈) ∈ V | |
13 | 12 | rabex 5238 | . . 3 ⊢ {𝑓 ∈ (LFnl‘𝑈) ∣ (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑓))) = ((LKer‘𝑈)‘𝑓)} ∈ V |
14 | eqid 2824 | . . . 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 16654 | . . 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 2884 | 1 ⊢ (𝜑 → ∙ = · ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 {crab 3145 Vcvv 3497 ‘cfv 6358 (class class class)co 7159 ↾s cress 16487 ·𝑠 cvsca 16572 LFnlclfn 36197 LKerclk 36225 LDualcld 36263 HLchlt 36490 LHypclh 37124 DVecHcdvh 38218 ocHcoch 38487 LCDualclcd 38726 |
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-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-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-om 7584 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-er 8292 df-en 8513 df-dom 8514 df-sdom 8515 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-ndx 16489 df-slot 16490 df-base 16492 df-sets 16493 df-ress 16494 df-vsca 16585 df-lcdual 38727 |
This theorem is referenced by: lcdvsval 38744 lcdlkreq2N 38763 |
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