0vfval - Metamath Proof Explorer

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Theorem 0vfval 27461

Description: Value of the function for the zero vector on a normed complex vector space. (Contributed by NM, 24-Apr-2007.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
0vfval.2	⊢ 𝐺 = ( +_𝑣 ‘𝑈)
0vfval.5	⊢ 𝑍 = (0_vec‘𝑈)

**Assertion**
Ref	Expression
0vfval	⊢ (𝑈 ∈ 𝑉 → 𝑍 = (GId‘𝐺))

**Proof of Theorem 0vfval**
Step	Hyp	Ref	Expression
1		elex 3212	. 2 ⊢ (𝑈 ∈ 𝑉 → 𝑈 ∈ V)
2		fo1st 7188	. . . . . . 7 ⊢ 1^st :V–onto→V
3		fofn 6117	. . . . . . 7 ⊢ (1^st :V–onto→V → 1^st Fn V)
4	2, 3	ax-mp 5	. . . . . 6 ⊢ 1^st Fn V
5		ssv 3625	. . . . . 6 ⊢ ran 1^st ⊆ V
6		fnco 5999	. . . . . 6 ⊢ ((1^st Fn V ∧ 1^st Fn V ∧ ran 1^st ⊆ V) → (1^st ∘ 1^st ) Fn V)
7	4, 4, 5, 6	mp3an 1424	. . . . 5 ⊢ (1^st ∘ 1^st ) Fn V
8		df-va 27450	. . . . . 6 ⊢ +_𝑣 = (1^st ∘ 1^st )
9	8	fneq1i 5985	. . . . 5 ⊢ ( +_𝑣 Fn V ↔ (1^st ∘ 1^st ) Fn V)
10	7, 9	mpbir 221	. . . 4 ⊢ +_𝑣 Fn V
11		fvco2 6273	. . . 4 ⊢ (( +_𝑣 Fn V ∧ 𝑈 ∈ V) → ((GId ∘ +_𝑣 )‘𝑈) = (GId‘( +_𝑣 ‘𝑈)))
12	10, 11	mpan 706	. . 3 ⊢ (𝑈 ∈ V → ((GId ∘ +_𝑣 )‘𝑈) = (GId‘( +_𝑣 ‘𝑈)))
13		0vfval.5	. . . 4 ⊢ 𝑍 = (0_vec‘𝑈)
14		df-0v 27453	. . . . 5 ⊢ 0_vec = (GId ∘ +_𝑣 )
15	14	fveq1i 6192	. . . 4 ⊢ (0_vec‘𝑈) = ((GId ∘ +_𝑣 )‘𝑈)
16	13, 15	eqtri 2644	. . 3 ⊢ 𝑍 = ((GId ∘ +_𝑣 )‘𝑈)
17		0vfval.2	. . . 4 ⊢ 𝐺 = ( +_𝑣 ‘𝑈)
18	17	fveq2i 6194	. . 3 ⊢ (GId‘𝐺) = (GId‘( +_𝑣 ‘𝑈))
19	12, 16, 18	3eqtr4g 2681	. 2 ⊢ (𝑈 ∈ V → 𝑍 = (GId‘𝐺))
20	1, 19	syl 17	1 ⊢ (𝑈 ∈ 𝑉 → 𝑍 = (GId‘𝐺))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 Vcvv 3200 ⊆ wss 3574 ran crn 5115 ∘ ccom 5118 Fn wfn 5883 –onto→wfo 5886 ‘cfv 5888 1^st c1st 7166 GIdcgi 27344 +_𝑣 cpv 27440 0_veccn0v 27443

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-fo 5894 df-fv 5896 df-1st 7168 df-va 27450 df-0v 27453

This theorem is referenced by: nvi 27469 nvzcl 27489 nv0rid 27490 nv0lid 27491 nv0 27492 nvsz 27493 nvrinv 27506 nvlinv 27507 hh0v 28025

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