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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 2823 | . . 3 ⊢ (1st ‘𝑈) = (1st ‘𝑈) | |
2 | 1 | nvvc 28394 | . 2 ⊢ (𝑈 ∈ NrmCVec → (1st ‘𝑈) ∈ CVecOLD) |
3 | nvdi.2 | . . . 4 ⊢ 𝐺 = ( +𝑣 ‘𝑈) | |
4 | 3 | vafval 28382 | . . 3 ⊢ 𝐺 = (1st ‘(1st ‘𝑈)) |
5 | nvdi.4 | . . . 4 ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) | |
6 | 5 | smfval 28384 | . . 3 ⊢ 𝑆 = (2nd ‘(1st ‘𝑈)) |
7 | nvdi.1 | . . . 4 ⊢ 𝑋 = (BaseSet‘𝑈) | |
8 | 7, 3 | bafval 28383 | . . 3 ⊢ 𝑋 = ran 𝐺 |
9 | 4, 6, 8 | vcdi 28344 | . 2 ⊢ (((1st ‘𝑈) ∈ CVecOLD ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝑆(𝐵𝐺𝐶)) = ((𝐴𝑆𝐵)𝐺(𝐴𝑆𝐶))) |
10 | 2, 9 | sylan 582 | 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 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ‘cfv 6357 (class class class)co 7158 1st c1st 7689 ℂcc 10537 CVecOLDcvc 28337 NrmCVeccnv 28363 +𝑣 cpv 28364 BaseSetcba 28365 ·𝑠OLD cns 28366 |
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 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 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-ral 3145 df-rex 3146 df-reu 3147 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-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 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-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-oprab 7162 df-1st 7691 df-2nd 7692 df-vc 28338 df-nv 28371 df-va 28374 df-ba 28375 df-sm 28376 df-0v 28377 df-nmcv 28379 |
This theorem is referenced by: nvmdi 28427 nvaddsub4 28436 nvdif 28445 nvpi 28446 ipdirilem 28608 hldi 28686 |
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