nvdir - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > nvdir		Structured version Visualization version GIF version

Theorem nvdir 28979

Description: Distributive law for the scalar product of a complex vector space. (Contributed by NM, 4-Dec-2007.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
nvdi.1	⊢ 𝑋 = (BaseSet‘𝑈)
nvdi.2	⊢ 𝐺 = ( +_𝑣 ‘𝑈)
nvdi.4	⊢ 𝑆 = ( ·_𝑠OLD ‘𝑈)

**Assertion**
Ref	Expression
nvdir	⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ 𝑋)) → ((𝐴 + 𝐵)𝑆𝐶) = ((𝐴𝑆𝐶)𝐺(𝐵𝑆𝐶)))

**Proof of Theorem nvdir**
Step	Hyp	Ref	Expression
1		eqid 2738	. . 3 ⊢ (1^st ‘𝑈) = (1^st ‘𝑈)
2	1	nvvc 28963	. 2 ⊢ (𝑈 ∈ NrmCVec → (1^st ‘𝑈) ∈ CVec_OLD)
3		nvdi.2	. . . 4 ⊢ 𝐺 = ( +_𝑣 ‘𝑈)
4	3	vafval 28951	. . 3 ⊢ 𝐺 = (1^st ‘(1^st ‘𝑈))
5		nvdi.4	. . . 4 ⊢ 𝑆 = ( ·_𝑠OLD ‘𝑈)
6	5	smfval 28953	. . 3 ⊢ 𝑆 = (2^nd ‘(1^st ‘𝑈))
7		nvdi.1	. . . 4 ⊢ 𝑋 = (BaseSet‘𝑈)
8	7, 3	bafval 28952	. . 3 ⊢ 𝑋 = ran 𝐺
9	4, 6, 8	vcdir 28914	. 2 ⊢ (((1^st ‘𝑈) ∈ CVec_OLD ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ 𝑋)) → ((𝐴 + 𝐵)𝑆𝐶) = ((𝐴𝑆𝐶)𝐺(𝐵𝑆𝐶)))
10	2, 9	sylan 580	1 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ 𝑋)) → ((𝐴 + 𝐵)𝑆𝐶) = ((𝐴𝑆𝐶)𝐺(𝐵𝑆𝐶)))

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 1^st c1st 7819 ℂcc 10857 + caddc 10862 CVec_OLDcvc 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: nvge0 29021 smcnlem 29045 ipidsq 29058 ip2i 29176 ipasslem1 29179 ipasslem11 29188 hldir 29256

W3C validator