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Theorem smfval 28376

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 28368	. . . . 5 ⊢ ·_𝑠OLD = (2^nd ∘ 1^st )
3	2	fveq1i 6665	. . . 4 ⊢ ( ·_𝑠OLD ‘𝑈) = ((2^nd ∘ 1^st )‘𝑈)
4		fo1st 7703	. . . . . 6 ⊢ 1^st :V–onto→V
5		fof 6584	. . . . . 6 ⊢ (1^st :V–onto→V → 1^st :V⟶V)
6	4, 5	ax-mp 5	. . . . 5 ⊢ 1^st :V⟶V
7		fvco3 6754	. . . . 5 ⊢ ((1^st :V⟶V ∧ 𝑈 ∈ V) → ((2^nd ∘ 1^st )‘𝑈) = (2^nd ‘(1^st ‘𝑈)))
8	6, 7	mpan 688	. . . 4 ⊢ (𝑈 ∈ V → ((2^nd ∘ 1^st )‘𝑈) = (2^nd ‘(1^st ‘𝑈)))
9	3, 8	syl5eq 2868	. . 3 ⊢ (𝑈 ∈ V → ( ·_𝑠OLD ‘𝑈) = (2^nd ‘(1^st ‘𝑈)))
10		fvprc 6657	. . . 4 ⊢ (¬ 𝑈 ∈ V → ( ·_𝑠OLD ‘𝑈) = ∅)
11		fvprc 6657	. . . . . 6 ⊢ (¬ 𝑈 ∈ V → (1^st ‘𝑈) = ∅)
12	11	fveq2d 6668	. . . . 5 ⊢ (¬ 𝑈 ∈ V → (2^nd ‘(1^st ‘𝑈)) = (2^nd ‘∅))
13		2nd0 7690	. . . . 5 ⊢ (2^nd ‘∅) = ∅
14	12, 13	syl6req 2873	. . . 4 ⊢ (¬ 𝑈 ∈ V → ∅ = (2^nd ‘(1^st ‘𝑈)))
15	10, 14	eqtrd 2856	. . 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 2844	1 ⊢ 𝑆 = (2^nd ‘(1^st ‘𝑈))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 = wceq 1533 ∈ wcel 2110 Vcvv 3494 ∅c0 4290 ∘ ccom 5553 ⟶wf 6345 –onto→wfo 6347 ‘cfv 6349 1^st c1st 7681 2^nd c2nd 7682 ·_𝑠OLD cns 28358

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-fo 6355 df-fv 6357 df-1st 7683 df-2nd 7684 df-sm 28368

This theorem is referenced by: nvvop 28380 nvsf 28390 nvscl 28397 nvsid 28398 nvsass 28399 nvdi 28401 nvdir 28402 nv2 28403 nv0 28408 nvsz 28409 nvinv 28410 nvtri 28441 cnnvs 28451 phop 28589 ipdirilem 28600 h2hsm 28746 hhsssm 29029

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