Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > nvdi | Structured version Visualization version GIF version |
Description: Distributive law for the scalar product of a complex vector space. (Contributed by NM, 4-Dec-2007.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nvdi.1 | ⊢ 𝑋 = (BaseSet‘𝑈) |
nvdi.2 | ⊢ 𝐺 = ( +𝑣 ‘𝑈) |
nvdi.4 | ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) |
Ref | Expression |
---|---|
nvdi | ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝑆(𝐵𝐺𝐶)) = ((𝐴𝑆𝐵)𝐺(𝐴𝑆𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2738 | . . 3 ⊢ (1st ‘𝑈) = (1st ‘𝑈) | |
2 | 1 | nvvc 28963 | . 2 ⊢ (𝑈 ∈ NrmCVec → (1st ‘𝑈) ∈ CVecOLD) |
3 | nvdi.2 | . . . 4 ⊢ 𝐺 = ( +𝑣 ‘𝑈) | |
4 | 3 | vafval 28951 | . . 3 ⊢ 𝐺 = (1st ‘(1st ‘𝑈)) |
5 | nvdi.4 | . . . 4 ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) | |
6 | 5 | smfval 28953 | . . 3 ⊢ 𝑆 = (2nd ‘(1st ‘𝑈)) |
7 | nvdi.1 | . . . 4 ⊢ 𝑋 = (BaseSet‘𝑈) | |
8 | 7, 3 | bafval 28952 | . . 3 ⊢ 𝑋 = ran 𝐺 |
9 | 4, 6, 8 | vcdi 28913 | . 2 ⊢ (((1st ‘𝑈) ∈ CVecOLD ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝑆(𝐵𝐺𝐶)) = ((𝐴𝑆𝐵)𝐺(𝐴𝑆𝐶))) |
10 | 2, 9 | sylan 580 | 1 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝑆(𝐵𝐺𝐶)) = ((𝐴𝑆𝐵)𝐺(𝐴𝑆𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 ‘cfv 6427 (class class class)co 7268 1st c1st 7819 ℂcc 10857 CVecOLDcvc 28906 NrmCVeccnv 28932 +𝑣 cpv 28933 BaseSetcba 28934 ·𝑠OLD cns 28935 |
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 5209 ax-sep 5222 ax-nul 5229 ax-pr 5351 ax-un 7579 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 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-ral 3069 df-rex 3070 df-reu 3071 df-rab 3073 df-v 3432 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4258 df-if 4461 df-sn 4563 df-pr 4565 df-op 4569 df-uni 4841 df-iun 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5485 df-xp 5591 df-rel 5592 df-cnv 5593 df-co 5594 df-dm 5595 df-rn 5596 df-res 5597 df-ima 5598 df-iota 6385 df-fun 6429 df-fn 6430 df-f 6431 df-f1 6432 df-fo 6433 df-f1o 6434 df-fv 6435 df-ov 7271 df-oprab 7272 df-1st 7821 df-2nd 7822 df-vc 28907 df-nv 28940 df-va 28943 df-ba 28944 df-sm 28945 df-0v 28946 df-nmcv 28948 |
This theorem is referenced by: nvmdi 28996 nvaddsub4 29005 nvdif 29014 nvpi 29015 ipdirilem 29177 hldi 29255 |
Copyright terms: Public domain | W3C validator |