0vfval - Metamath Proof Explorer

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Theorem 0vfval 31214

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 8022	. . . . . . 7 ⊢ 1^st :V–onto→V
3		fofn 6799	. . . . . . 7 ⊢ (1^st :V–onto→V → 1^st Fn V)
4	2, 3	ax-mp 5	. . . . . 6 ⊢ 1^st Fn V
5		ssv 3932	. . . . . 6 ⊢ ran 1^st ⊆ V
6		fnco 6658	. . . . . 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 31203	. . . . . 6 ⊢ +_𝑣 = (1^st ∘ 1^st )
9	8	fneq1i 6637	. . . . 5 ⊢ ( +_𝑣 Fn V ↔ (1^st ∘ 1^st ) Fn V)
10	7, 9	mpbir 234	. . . 4 ⊢ +_𝑣 Fn V
11		fvco2 6983	. . . 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 31206	. . . . 5 ⊢ 0_vec = (GId ∘ +_𝑣 )
15	14	fveq1i 6887	. . . 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 6889	. . 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 1569 ∈ wcel 2144 Vcvv 3450 ⊆ wss 3895 ran crn 5651 ∘ ccom 5654 Fn wfn 6534 –onto→wfo 6537 ‘cfv 6539 1^st c1st 8000 GIdcgi 31098 +_𝑣 cpv 31193 0_veccn0v 31196

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-sep 5248 ax-nul 5259 ax-pr 5390 ax-un 7752

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 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-ss 3896 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-sn 4585 df-pr 4587 df-op 4835 df-uni 4867 df-br 5103 df-opab 5167 df-mpt 5186 df-id 5545 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-iota 6494 df-fun 6541 df-fn 6542 df-f 6543 df-fo 6545 df-fv 6547 df-1st 8002 df-va 31203 df-0v 31206

This theorem is used by: nvi 31222 nvzcl 31242 nv0rid 31243 nv0lid 31244 nv0 31245 nvsz 31246 nvrinv 31259 nvlinv 31260 hh0v 31776

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