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Theorem nvsf 28018

Description: Mapping for the scalar multiplication operation. (Contributed by NM, 28-Jan-2008.) (New usage is discouraged.)

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

**Assertion**
Ref	Expression
nvsf	⊢ (𝑈 ∈ NrmCVec → 𝑆:(ℂ × 𝑋)⟶𝑋)

**Proof of Theorem nvsf**
Step	Hyp	Ref	Expression
1		eqid 2825	. . 3 ⊢ (1^st ‘𝑈) = (1^st ‘𝑈)
2	1	nvvc 28014	. 2 ⊢ (𝑈 ∈ NrmCVec → (1^st ‘𝑈) ∈ CVec_OLD)
3		eqid 2825	. . . 4 ⊢ ( +_𝑣 ‘𝑈) = ( +_𝑣 ‘𝑈)
4	3	vafval 28002	. . 3 ⊢ ( +_𝑣 ‘𝑈) = (1^st ‘(1^st ‘𝑈))
5		nvsf.4	. . . 4 ⊢ 𝑆 = ( ·_𝑠OLD ‘𝑈)
6	5	smfval 28004	. . 3 ⊢ 𝑆 = (2^nd ‘(1^st ‘𝑈))
7		nvsf.1	. . . 4 ⊢ 𝑋 = (BaseSet‘𝑈)
8	7, 3	bafval 28003	. . 3 ⊢ 𝑋 = ran ( +_𝑣 ‘𝑈)
9	4, 6, 8	vcsm 27961	. 2 ⊢ ((1^st ‘𝑈) ∈ CVec_OLD → 𝑆:(ℂ × 𝑋)⟶𝑋)
10	2, 9	syl 17	1 ⊢ (𝑈 ∈ NrmCVec → 𝑆:(ℂ × 𝑋)⟶𝑋)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1656 ∈ wcel 2164 × cxp 5340 ⟶wf 6119 ‘cfv 6123 1^st c1st 7426 ℂcc 10250 CVec_OLDcvc 27957 NrmCVeccnv 27983 +_𝑣 cpv 27984 BaseSetcba 27985 ·_𝑠OLD cns 27986

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209

This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4659 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-id 5250 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-ov 6908 df-oprab 6909 df-1st 7428 df-2nd 7429 df-vc 27958 df-nv 27991 df-va 27994 df-ba 27995 df-sm 27996 df-0v 27997 df-nmcv 27999

This theorem is referenced by: nvinvfval 28039 smcnlem 28096 ssps 28129 hlmulf 28304

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