nvsf - Metamath Proof Explorer

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Theorem nvsf 31214

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 2761	. . 3 ⊢ (1^st ‘𝑈) = (1^st ‘𝑈)
2	1	nvvc 31210	. 2 ⊢ (𝑈 ∈ NrmCVec → (1^st ‘𝑈) ∈ CVec_OLD)
3		eqid 2761	. . . 4 ⊢ ( +_𝑣 ‘𝑈) = ( +_𝑣 ‘𝑈)
4	3	vafval 31198	. . 3 ⊢ ( +_𝑣 ‘𝑈) = (1^st ‘(1^st ‘𝑈))
5		nvsf.4	. . . 4 ⊢ 𝑆 = ( ·_𝑠OLD ‘𝑈)
6	5	smfval 31200	. . 3 ⊢ 𝑆 = (2^nd ‘(1^st ‘𝑈))
7		nvsf.1	. . . 4 ⊢ 𝑋 = (BaseSet‘𝑈)
8	7, 3	bafval 31199	. . 3 ⊢ 𝑋 = ran ( +_𝑣 ‘𝑈)
9	4, 6, 8	vcsm 31157	. 2 ⊢ ((1^st ‘𝑈) ∈ CVec_OLD → 𝑆:(ℂ × 𝑋)⟶𝑋)
10	2, 9	syl 18	1 ⊢ (𝑈 ∈ NrmCVec → 𝑆:(ℂ × 𝑋)⟶𝑋)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 = wceq 1570 ∈ wcel 2145 × cxp 5649 ⟶wf 6533 ‘cfv 6537 1^st c1st 7997 ℂcc 11191 CVec_OLDcvc 31153 NrmCVeccnv 31179 +_𝑣 cpv 31180 BaseSetcba 31181 ·_𝑠OLD cns 31182

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pr 5391 ax-un 7749

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-ral 3078 df-rex 3088 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4280 df-if 4483 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-id 5546 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-ov 7421 df-oprab 7422 df-1st 7999 df-2nd 8000 df-vc 31154 df-nv 31187 df-va 31190 df-ba 31191 df-sm 31192 df-0v 31193 df-nmcv 31195

This theorem is used by: nvinvfval 31235 smcnlem 31292 ssps 31325 hlmulf 31499

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