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Theorem smfval 30866

Description: Value of the function for the scalar multiplication operation on a normed complex vector space. (Contributed by NM, 24-Apr-2007.) (New usage is discouraged.)

**Hypothesis**
Ref	Expression
smfval.4	⊢ 𝑆 = ( ·_𝑠OLD ‘𝑈)

**Assertion**
Ref	Expression
smfval	⊢ 𝑆 = (2^nd ‘(1^st ‘𝑈))

**Proof of Theorem smfval**
Step	Hyp	Ref	Expression
1		smfval.4	. 2 ⊢ 𝑆 = ( ·_𝑠OLD ‘𝑈)
2		df-sm 30858	. . . . 5 ⊢ ·_𝑠OLD = (2^nd ∘ 1^st )
3	2	fveq1i 6872	. . . 4 ⊢ ( ·_𝑠OLD ‘𝑈) = ((2^nd ∘ 1^st )‘𝑈)
4		fo1st 7994	. . . . . 6 ⊢ 1^st :V–onto→V
5		fof 6782	. . . . . 6 ⊢ (1^st :V–onto→V → 1^st :V⟶V)
6	4, 5	ax-mp 5	. . . . 5 ⊢ 1^st :V⟶V
7		fvco3 6971	. . . . 5 ⊢ ((1^st :V⟶V ∧ 𝑈 ∈ V) → ((2^nd ∘ 1^st )‘𝑈) = (2^nd ‘(1^st ‘𝑈)))
8	6, 7	mpan 702	. . . 4 ⊢ (𝑈 ∈ V → ((2^nd ∘ 1^st )‘𝑈) = (2^nd ‘(1^st ‘𝑈)))
9	3, 8	eqtrid 2812	. . 3 ⊢ (𝑈 ∈ V → ( ·_𝑠OLD ‘𝑈) = (2^nd ‘(1^st ‘𝑈)))
10		fvprc 6863	. . . 4 ⊢ (¬ 𝑈 ∈ V → ( ·_𝑠OLD ‘𝑈) = ∅)
11		fvprc 6863	. . . . . 6 ⊢ (¬ 𝑈 ∈ V → (1^st ‘𝑈) = ∅)
12	11	fveq2d 6875	. . . . 5 ⊢ (¬ 𝑈 ∈ V → (2^nd ‘(1^st ‘𝑈)) = (2^nd ‘∅))
13		2nd0 7981	. . . . 5 ⊢ (2^nd ‘∅) = ∅
14	12, 13	eqtr2di 2817	. . . 4 ⊢ (¬ 𝑈 ∈ V → ∅ = (2^nd ‘(1^st ‘𝑈)))
15	10, 14	eqtrd 2800	. . 3 ⊢ (¬ 𝑈 ∈ V → ( ·_𝑠OLD ‘𝑈) = (2^nd ‘(1^st ‘𝑈)))
16	9, 15	pm2.61i 184	. 2 ⊢ ( ·_𝑠OLD ‘𝑈) = (2^nd ‘(1^st ‘𝑈))
17	1, 16	eqtri 2788	1 ⊢ 𝑆 = (2^nd ‘(1^st ‘𝑈))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 = wceq 1563 ∈ wcel 2145 Vcvv 3457 ∅c0 4288 ∘ ccom 5656 ⟶wf 6521 –onto→wfo 6523 ‘cfv 6525 1^st c1st 7972 2^nd c2nd 7973 ·_𝑠OLD cns 30848

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1818 ax-4 1832 ax-5 1933 ax-6 1990 ax-7 2031 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2737 ax-sep 5251 ax-nul 5261 ax-pr 5395 ax-un 7722

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3an 1103 df-tru 1566 df-fal 1576 df-ex 1803 df-nf 1807 df-sb 2094 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-ral 3080 df-rex 3090 df-rab 3418 df-v 3459 df-dif 3910 df-un 3912 df-in 3914 df-ss 3924 df-nul 4289 df-if 4484 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4869 df-br 5106 df-opab 5168 df-mpt 5187 df-id 5547 df-xp 5658 df-rel 5659 df-cnv 5660 df-co 5661 df-dm 5662 df-rn 5663 df-res 5664 df-ima 5665 df-iota 6481 df-fun 6527 df-fn 6528 df-f 6529 df-fo 6531 df-fv 6533 df-1st 7974 df-2nd 7975 df-sm 30858

This theorem is referenced by: nvvop 30870 nvsf 30880 nvscl 30887 nvsid 30888 nvsass 30889 nvdi 30891 nvdir 30892 nv2 30893 nv0 30898 nvsz 30899 nvinv 30900 nvtri 30931 cnnvs 30941 phop 31079 ipdirilem 31090 h2hsm 31236 hhsssm 31519

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