0vfval - Metamath Proof Explorer

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Theorem 0vfval 30987

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 3478	. 2 ⊢ (𝑈 ∈ 𝑉 → 𝑈 ∈ V)
2		fo1st 8008	. . . . . . 7 ⊢ 1^st :V–onto→V
3		fofn 6798	. . . . . . 7 ⊢ (1^st :V–onto→V → 1^st Fn V)
4	2, 3	ax-mp 5	. . . . . 6 ⊢ 1^st Fn V
5		ssv 3962	. . . . . 6 ⊢ ran 1^st ⊆ V
6		fnco 6657	. . . . . 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 30976	. . . . . 6 ⊢ +_𝑣 = (1^st ∘ 1^st )
9	8	fneq1i 6636	. . . . 5 ⊢ ( +_𝑣 Fn V ↔ (1^st ∘ 1^st ) Fn V)
10	7, 9	mpbir 234	. . . 4 ⊢ +_𝑣 Fn V
11		fvco2 6982	. . . 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 30979	. . . . 5 ⊢ 0_vec = (GId ∘ +_𝑣 )
15	14	fveq1i 6886	. . . 4 ⊢ (0_vec‘𝑈) = ((GId ∘ +_𝑣 )‘𝑈)
16	13, 15	eqtri 2788	. . 3 ⊢ 𝑍 = ((GId ∘ +_𝑣 )‘𝑈)
17		0vfval.2	. . . 4 ⊢ 𝐺 = ( +_𝑣 ‘𝑈)
18	17	fveq2i 6888	. . 3 ⊢ (GId‘𝐺) = (GId‘( +_𝑣 ‘𝑈))
19	12, 16, 18	3eqtr4g 2825	. 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 2146 Vcvv 3457 ⊆ wss 3906 ran crn 5664 ∘ ccom 5667 Fn wfn 6535 –onto→wfo 6538 ‘cfv 6540 1^st c1st 7986 GIdcgi 30871 +_𝑣 cpv 30966 0_veccn0v 30969

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 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-sep 5259 ax-nul 5271 ax-pr 5406 ax-un 7738

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 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-ral 3082 df-rex 3092 df-rab 3419 df-v 3459 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-nul 4287 df-if 4490 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4875 df-br 5112 df-opab 5176 df-mpt 5195 df-id 5558 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-fo 6546 df-fv 6548 df-1st 7988 df-va 30976 df-0v 30979

This theorem is used by: nvi 30995 nvzcl 31015 nv0rid 31016 nv0lid 31017 nv0 31018 nvsz 31019 nvrinv 31032 nvlinv 31033 hh0v 31549

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