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Theorem smfval 31240

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 31232	. . . . 5 ⊢ ·_𝑠OLD = (2^nd ∘ 1^st )
3	2	fveq1i 6912	. . . 4 ⊢ ( ·_𝑠OLD ‘𝑈) = ((2^nd ∘ 1^st )‘𝑈)
4		fo1st 8048	. . . . . 6 ⊢ 1^st :V⟶ontoV
5		fof 6821	. . . . . 6 ⊢ (1^st :V⟶ontoV → 1^st :V⟶V)
6	4, 5	ax-mp 5	. . . . 5 ⊢ 1^st :V⟶V
7		fvco3 7011	. . . . 5 ⊢ ((1^st :V⟶V ∧ 𝑈 ∈ V) → ((2^nd ∘ 1^st )‘𝑈) = (2^nd ‘(1^st ‘𝑈)))
8	6, 7	mpan 703	. . . 4 ⊢ (𝑈 ∈ V → ((2^nd ∘ 1^st )‘𝑈) = (2^nd ‘(1^st ‘𝑈)))
9	3, 8	eqtrid 2810	. . 3 ⊢ (𝑈 ∈ V → ( ·_𝑠OLD ‘𝑈) = (2^nd ‘(1^st ‘𝑈)))
10		fvprc 6903	. . . 4 ⊢ (¬ 𝑈 ∈ V → ( ·_𝑠OLD ‘𝑈) = ∅)
11		fvprc 6903	. . . . . 6 ⊢ (¬ 𝑈 ∈ V → (1^st ‘𝑈) = ∅)
12	11	fveq2d 6915	. . . . 5 ⊢ (¬ 𝑈 ∈ V → (2^nd ‘(1^st ‘𝑈)) = (2^nd ‘∅))
13		2nd0 8035	. . . . 5 ⊢ (2^nd ‘∅) = ∅
14	12, 13	eqtr2di 2815	. . . 4 ⊢ (¬ 𝑈 ∈ V → ∅ = (2^nd ‘(1^st ‘𝑈)))
15	10, 14	eqtrd 2798	. . 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 2786	1 ⊢ 𝑆 = (2^nd ‘(1^st ‘𝑈))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 = wceq 1570 ∈ wcel 2147 Vcvv 3454 ∅c0 4281 ∘ ccom 5660 ⟶wf 6557 ⟶ontowfo 6559 ‘cfv 6561 1^st c1st 8026 2^nd c2nd 8027 ·_𝑠OLD cns 31222

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 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-sep 5253 ax-nul 5264 ax-pr 5396 ax-un 7778

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 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-ral 3081 df-rex 3091 df-rab 3417 df-v 3456 df-ss 3901 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-sn 4589 df-pr 4591 df-op 4839 df-uni 4871 df-br 5107 df-opab 5172 df-mpt 5191 df-id 5551 df-xp 5662 df-rel 5663 df-cnv 5664 df-co 5665 df-dm 5666 df-rn 5667 df-res 5668 df-ima 5669 df-iota 6515 df-fun 6563 df-fn 6564 df-f 6565 df-fo 6567 df-fv 6569 df-1st 8028 df-2nd 8029 df-sm 31232

This theorem is used by: nvvop 31244 nvsf 31254 nvscl 31261 nvsid 31262 nvsass 31263 nvdi 31265 nvdir 31266 nv2 31267 nv0 31272 nvsz 31273 nvinv 31274 nvtri 31305 cnnvs 31315 phop 31453 ipdirilem 31464 h2hsm 31610 hhsssm 31893

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