Theorem 0vfval 30535

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 = (0vec‘𝑈)

Assertion

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

Proof of Theorem 0vfval

Step	Hyp	Ref	Expression
1		elex 3468	. 2 ⊢ (𝑈 ∈ 𝑉 → 𝑈 ∈ V)
2		fo1st 7988	. . . . . . 7 ⊢ 1st :V–onto→V
3		fofn 6774	. . . . . . 7 ⊢ (1st :V–onto→V → 1st Fn V)
4	2, 3	ax-mp 5	. . . . . 6 ⊢ 1st Fn V
5		ssv 3971	. . . . . 6 ⊢ ran 1st ⊆ V
6		fnco 6636	. . . . . 6 ⊢ ((1st Fn V ∧ 1st Fn V ∧ ran 1st ⊆ V) → (1st ∘ 1st ) Fn V)
7	4, 4, 5, 6	mp3an 1463	. . . . 5 ⊢ (1st ∘ 1st ) Fn V
8		df-va 30524	. . . . . 6 ⊢ +𝑣 = (1st ∘ 1st )
9	8	fneq1i 6615	. . . . 5 ⊢ ( +𝑣 Fn V ↔ (1st ∘ 1st ) Fn V)
10	7, 9	mpbir 231	. . . 4 ⊢ +𝑣 Fn V
11		fvco2 6958	. . . 4 ⊢ (( +𝑣 Fn V ∧ 𝑈 ∈ V) → ((GId ∘ +𝑣 )‘𝑈) = (GId‘( +𝑣 ‘𝑈)))
12	10, 11	mpan 690	. . 3 ⊢ (𝑈 ∈ V → ((GId ∘ +𝑣 )‘𝑈) = (GId‘( +𝑣 ‘𝑈)))
13		0vfval.5	. . . 4 ⊢ 𝑍 = (0vec‘𝑈)
14		df-0v 30527	. . . . 5 ⊢ 0vec = (GId ∘ +𝑣 )
15	14	fveq1i 6859	. . . 4 ⊢ (0vec‘𝑈) = ((GId ∘ +𝑣 )‘𝑈)
16	13, 15	eqtri 2752	. . 3 ⊢ 𝑍 = ((GId ∘ +𝑣 )‘𝑈)
17		0vfval.2	. . . 4 ⊢ 𝐺 = ( +𝑣 ‘𝑈)
18	17	fveq2i 6861	. . 3 ⊢ (GId‘𝐺) = (GId‘( +𝑣 ‘𝑈))
19	12, 16, 18	3eqtr4g 2789	. 2 ⊢ (𝑈 ∈ V → 𝑍 = (GId‘𝐺))
20	1, 19	syl 17	1 ⊢ (𝑈 ∈ 𝑉 → 𝑍 = (GId‘𝐺))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  Vcvv 3447   ⊆ wss 3914  ran crn 5639   ∘ ccom 5642   Fn wfn 6506  –onto→wfo 6509  ‘cfv 6511  1^st c1st 7966  GIdcgi 30419   +_𝑣 cpv 30514  0_veccn0v 30517

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-fo 6517  df-fv 6519  df-1st 7968  df-va 30524  df-0v 30527

This theorem is referenced by:  nvi  30543  nvzcl  30563  nv0rid  30564  nv0lid  30565  nv0  30566  nvsz  30567  nvrinv  30580  nvlinv  30581  hh0v  31097