0vfval - Metamath Proof Explorer

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Theorem 0vfval 31201

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 3472	. 2 ⊢ (𝑈 ∈ 𝑉 → 𝑈 ∈ V)
2		fo1st 8019	. . . . . . 7 ⊢ 1^st :V–onto→V
3		fofn 6796	. . . . . . 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 6655	. . . . . 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 31190	. . . . . 6 ⊢ +_𝑣 = (1^st ∘ 1^st )
9	8	fneq1i 6634	. . . . 5 ⊢ ( +_𝑣 Fn V ↔ (1^st ∘ 1^st ) Fn V)
10	7, 9	mpbir 234	. . . 4 ⊢ +_𝑣 Fn V
11		fvco2 6980	. . . 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 31193	. . . . 5 ⊢ 0_vec = (GId ∘ +_𝑣 )
15	14	fveq1i 6884	. . . 4 ⊢ (0_vec‘𝑈) = ((GId ∘ +_𝑣 )‘𝑈)
16	13, 15	eqtri 2784	. . 3 ⊢ 𝑍 = ((GId ∘ +_𝑣 )‘𝑈)
17		0vfval.2	. . . 4 ⊢ 𝐺 = ( +_𝑣 ‘𝑈)
18	17	fveq2i 6886	. . 3 ⊢ (GId‘𝐺) = (GId‘( +_𝑣 ‘𝑈))
19	12, 16, 18	3eqtr4g 2821	. 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 3451 ⊆ wss 3899 ran crn 5652 ∘ ccom 5655 Fn wfn 6532 –onto→wfo 6535 ‘cfv 6537 1^st c1st 7997 GIdcgi 31085 +_𝑣 cpv 31180 0_veccn0v 31183

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 2733 ax-sep 5249 ax-nul 5260 ax-pr 5391 ax-un 7749

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 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-ral 3078 df-rex 3088 df-rab 3414 df-v 3453 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 5546 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-fo 6543 df-fv 6545 df-1st 7999 df-va 31190 df-0v 31193

This theorem is used by: nvi 31209 nvzcl 31229 nv0rid 31230 nv0lid 31231 nv0 31232 nvsz 31233 nvrinv 31246 nvlinv 31247 hh0v 31763

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