0vfval - Metamath Proof Explorer

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Theorem 0vfval 31087

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 3471	. 2 ⊢ (𝑈 ∈ 𝑉 → 𝑈 ∈ V)
2		fo1st 8006	. . . . . . 7 ⊢ 1^st :V–onto→V
3		fofn 6791	. . . . . . 7 ⊢ (1^st :V–onto→V → 1^st Fn V)
4	2, 3	ax-mp 5	. . . . . 6 ⊢ 1^st Fn V
5		ssv 3955	. . . . . 6 ⊢ ran 1^st ⊆ V
6		fnco 6650	. . . . . 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 1490	. . . . 5 ⊢ (1^st ∘ 1^st ) Fn V
8		df-va 31076	. . . . . 6 ⊢ +_𝑣 = (1^st ∘ 1^st )
9	8	fneq1i 6629	. . . . 5 ⊢ ( +_𝑣 Fn V ↔ (1^st ∘ 1^st ) Fn V)
10	7, 9	mpbir 234	. . . 4 ⊢ +_𝑣 Fn V
11		fvco2 6975	. . . 4 ⊢ (( +_𝑣 Fn V ∧ 𝑈 ∈ V) → ((GId ∘ +_𝑣 )‘𝑈) = (GId‘( +_𝑣 ‘𝑈)))
12	10, 11	mpan 703	. . 3 ⊢ (𝑈 ∈ V → ((GId ∘ +_𝑣 )‘𝑈) = (GId‘( +_𝑣 ‘𝑈)))
13		0vfval.5	. . . 4 ⊢ 𝑍 = (0_vec‘𝑈)
14		df-0v 31079	. . . . 5 ⊢ 0_vec = (GId ∘ +_𝑣 )
15	14	fveq1i 6879	. . . 4 ⊢ (0_vec‘𝑈) = ((GId ∘ +_𝑣 )‘𝑈)
16	13, 15	eqtri 2783	. . 3 ⊢ 𝑍 = ((GId ∘ +_𝑣 )‘𝑈)
17		0vfval.2	. . . 4 ⊢ 𝐺 = ( +_𝑣 ‘𝑈)
18	17	fveq2i 6881	. . 3 ⊢ (GId‘𝐺) = (GId‘( +_𝑣 ‘𝑈))
19	12, 16, 18	3eqtr4g 2820	. 2 ⊢ (𝑈 ∈ V → 𝑍 = (GId‘𝐺))
20	1, 19	syl 18	1 ⊢ (𝑈 ∈ 𝑉 → 𝑍 = (GId‘𝐺))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 = wceq 1570 ∈ wcel 2145 Vcvv 3450 ⊆ wss 3899 ran crn 5656 ∘ ccom 5659 Fn wfn 6528 –onto→wfo 6531 ‘cfv 6533 1^st c1st 7984 GIdcgi 30971 +_𝑣 cpv 31066 0_veccn0v 31069

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 2732 ax-sep 5251 ax-nul 5263 ax-pr 5398 ax-un 7736

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 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-ral 3077 df-rex 3087 df-rab 3413 df-v 3452 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-br 5104 df-opab 5168 df-mpt 5187 df-id 5550 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-fo 6539 df-fv 6541 df-1st 7986 df-va 31076 df-0v 31079

This theorem is used by: nvi 31095 nvzcl 31115 nv0rid 31116 nv0lid 31117 nv0 31118 nvsz 31119 nvrinv 31132 nvlinv 31133 hh0v 31649

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