Theorem 0vfval 30592

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 3485	. 2 ⊢ (𝑈 ∈ 𝑉 → 𝑈 ∈ V)
2		fo1st 8013	. . . . . . 7 ⊢ 1st :V–onto→V
3		fofn 6797	. . . . . . 7 ⊢ (1st :V–onto→V → 1st Fn V)
4	2, 3	ax-mp 5	. . . . . 6 ⊢ 1st Fn V
5		ssv 3988	. . . . . 6 ⊢ ran 1st ⊆ V
6		fnco 6661	. . . . . 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 30581	. . . . . 6 ⊢ +𝑣 = (1st ∘ 1st )
9	8	fneq1i 6640	. . . . 5 ⊢ ( +𝑣 Fn V ↔ (1st ∘ 1st ) Fn V)
10	7, 9	mpbir 231	. . . 4 ⊢ +𝑣 Fn V
11		fvco2 6981	. . . 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 30584	. . . . 5 ⊢ 0vec = (GId ∘ +𝑣 )
15	14	fveq1i 6882	. . . 4 ⊢ (0vec‘𝑈) = ((GId ∘ +𝑣 )‘𝑈)
16	13, 15	eqtri 2759	. . 3 ⊢ 𝑍 = ((GId ∘ +𝑣 )‘𝑈)
17		0vfval.2	. . . 4 ⊢ 𝐺 = ( +𝑣 ‘𝑈)
18	17	fveq2i 6884	. . 3 ⊢ (GId‘𝐺) = (GId‘( +𝑣 ‘𝑈))
19	12, 16, 18	3eqtr4g 2796	. 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 3464   ⊆ wss 3931  ran crn 5660   ∘ ccom 5663   Fn wfn 6531  –onto→wfo 6534  ‘cfv 6536  1^st c1st 7991  GIdcgi 30476   +_𝑣 cpv 30571  0_veccn0v 30574

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 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407  ax-un 7734

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 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-opab 5187  df-mpt 5207  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-fo 6542  df-fv 6544  df-1st 7993  df-va 30581  df-0v 30584

This theorem is referenced by:  nvi  30600  nvzcl  30620  nv0rid  30621  nv0lid  30622  nv0  30623  nvsz  30624  nvrinv  30637  nvlinv  30638  hh0v  31154