nvsass - Metamath Proof Explorer

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

Theorem nvsass 30714

Description: Associative law for the scalar product of a normed complex vector space. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
nvscl.1	⊢ 𝑋 = (BaseSet‘𝑈)
nvscl.4	⊢ 𝑆 = ( ·_𝑠OLD ‘𝑈)

**Assertion**
Ref	Expression
nvsass	⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ 𝑋)) → ((𝐴 · 𝐵)𝑆𝐶) = (𝐴𝑆(𝐵𝑆𝐶)))

**Proof of Theorem nvsass**
Step	Hyp	Ref	Expression
1		eqid 2737	. . 3 ⊢ (1^st ‘𝑈) = (1^st ‘𝑈)
2	1	nvvc 30701	. 2 ⊢ (𝑈 ∈ NrmCVec → (1^st ‘𝑈) ∈ CVec_OLD)
3		eqid 2737	. . . 4 ⊢ ( +_𝑣 ‘𝑈) = ( +_𝑣 ‘𝑈)
4	3	vafval 30689	. . 3 ⊢ ( +_𝑣 ‘𝑈) = (1^st ‘(1^st ‘𝑈))
5		nvscl.4	. . . 4 ⊢ 𝑆 = ( ·_𝑠OLD ‘𝑈)
6	5	smfval 30691	. . 3 ⊢ 𝑆 = (2^nd ‘(1^st ‘𝑈))
7		nvscl.1	. . . 4 ⊢ 𝑋 = (BaseSet‘𝑈)
8	7, 3	bafval 30690	. . 3 ⊢ 𝑋 = ran ( +_𝑣 ‘𝑈)
9	4, 6, 8	vcass 30653	. 2 ⊢ (((1^st ‘𝑈) ∈ CVec_OLD ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ 𝑋)) → ((𝐴 · 𝐵)𝑆𝐶) = (𝐴𝑆(𝐵𝑆𝐶)))
10	2, 9	sylan 581	1 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ 𝑋)) → ((𝐴 · 𝐵)𝑆𝐶) = (𝐴𝑆(𝐵𝑆𝐶)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ‘cfv 6492 (class class class)co 7360 1^st c1st 7933 ℂcc 11027 · cmul 11034 CVec_OLDcvc 30644 NrmCVeccnv 30670 +_𝑣 cpv 30671 BaseSetcba 30672 ·_𝑠OLD cns 30673

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pr 5370 ax-un 7682

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-ral 3053 df-rex 3063 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-nul 4275 df-if 4468 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-id 5519 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-ov 7363 df-oprab 7364 df-1st 7935 df-2nd 7936 df-vc 30645 df-nv 30678 df-va 30681 df-ba 30682 df-sm 30683 df-0v 30684 df-nmcv 30686

This theorem is referenced by: nvscom 30715 nvmul0or 30736 nvpi 30753 smcnlem 30783 ipval3 30795 ipdirilem 30915 ipasslem2 30918 ipasslem4 30920 ipasslem5 30921 ipasslem10 30925 ipasslem11 30926 minvecolem2 30961 hlmulass 30992

W3C validator