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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 2738 | . . . 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 2738 | . . . 4 ⊢ (LFnl‘𝑈) = (LFnl‘𝑈) | |
6 | eqid 2738 | . . . 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 39611 | . . 3 ⊢ (𝜑 → 𝐶 = (𝐷 ↾s {𝑓 ∈ (LFnl‘𝑈) ∣ (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑓))) = ((LKer‘𝑈)‘𝑓)})) |
10 | 9 | fveq2d 6770 | . 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 6779 | . . . 4 ⊢ (LFnl‘𝑈) ∈ V | |
13 | 12 | rabex 5254 | . . 3 ⊢ {𝑓 ∈ (LFnl‘𝑈) ∣ (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑓))) = ((LKer‘𝑈)‘𝑓)} ∈ V |
14 | eqid 2738 | . . . 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 17064 | . . 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 2803 | 1 ⊢ (𝜑 → ∙ = · ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 {crab 3068 Vcvv 3429 ‘cfv 6426 (class class class)co 7267 ↾s cress 16951 ·𝑠 cvsca 16976 LFnlclfn 37079 LKerclk 37107 LDualcld 37145 HLchlt 37372 LHypclh 38006 DVecHcdvh 39100 ocHcoch 39369 LCDualclcd 39608 |
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 5208 ax-sep 5221 ax-nul 5228 ax-pow 5286 ax-pr 5350 ax-un 7578 ax-cnex 10937 ax-resscn 10938 ax-1cn 10939 ax-icn 10940 ax-addcl 10941 ax-addrcl 10942 ax-mulcl 10943 ax-mulrcl 10944 ax-mulcom 10945 ax-addass 10946 ax-mulass 10947 ax-distr 10948 ax-i2m1 10949 ax-1ne0 10950 ax-1rid 10951 ax-rnegex 10952 ax-rrecex 10953 ax-cnre 10954 ax-pre-lttri 10955 ax-pre-lttrn 10956 ax-pre-ltadd 10957 ax-pre-mulgt0 10958 |
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-reu 3071 df-rab 3073 df-v 3431 df-sbc 3716 df-csb 3832 df-dif 3889 df-un 3891 df-in 3893 df-ss 3903 df-pss 3905 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5074 df-opab 5136 df-mpt 5157 df-tr 5191 df-id 5484 df-eprel 5490 df-po 5498 df-so 5499 df-fr 5539 df-we 5541 df-xp 5590 df-rel 5591 df-cnv 5592 df-co 5593 df-dm 5594 df-rn 5595 df-res 5596 df-ima 5597 df-pred 6195 df-ord 6262 df-on 6263 df-lim 6264 df-suc 6265 df-iota 6384 df-fun 6428 df-fn 6429 df-f 6430 df-f1 6431 df-fo 6432 df-f1o 6433 df-fv 6434 df-riota 7224 df-ov 7270 df-oprab 7271 df-mpo 7272 df-om 7703 df-2nd 7821 df-frecs 8084 df-wrecs 8115 df-recs 8189 df-rdg 8228 df-er 8485 df-en 8721 df-dom 8722 df-sdom 8723 df-pnf 11021 df-mnf 11022 df-xr 11023 df-ltxr 11024 df-le 11025 df-sub 11217 df-neg 11218 df-nn 11984 df-2 12046 df-3 12047 df-4 12048 df-5 12049 df-6 12050 df-sets 16875 df-slot 16893 df-ndx 16905 df-base 16923 df-ress 16952 df-vsca 16989 df-lcdual 39609 |
This theorem is referenced by: lcdvsval 39626 lcdlkreq2N 39645 |
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