Theorem 0vfval 30434

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 3490	. 2 ⊢ (𝑈 ∈ 𝑉 → 𝑈 ∈ V)
2		fo1st 8017	. . . . . . 7 ⊢ 1st :V–onto→V
3		fofn 6816	. . . . . . 7 ⊢ (1st :V–onto→V → 1st Fn V)
4	2, 3	ax-mp 5	. . . . . 6 ⊢ 1st Fn V
5		ssv 4004	. . . . . 6 ⊢ ran 1st ⊆ V
6		fnco 6675	. . . . . 6 ⊢ ((1st Fn V ∧ 1st Fn V ∧ ran 1st ⊆ V) → (1st ∘ 1st ) Fn V)
7	4, 4, 5, 6	mp3an 1457	. . . . 5 ⊢ (1st ∘ 1st ) Fn V
8		df-va 30423	. . . . . 6 ⊢ +𝑣 = (1st ∘ 1st )
9	8	fneq1i 6654	. . . . 5 ⊢ ( +𝑣 Fn V ↔ (1st ∘ 1st ) Fn V)
10	7, 9	mpbir 230	. . . 4 ⊢ +𝑣 Fn V
11		fvco2 6998	. . . 4 ⊢ (( +𝑣 Fn V ∧ 𝑈 ∈ V) → ((GId ∘ +𝑣 )‘𝑈) = (GId‘( +𝑣 ‘𝑈)))
12	10, 11	mpan 688	. . 3 ⊢ (𝑈 ∈ V → ((GId ∘ +𝑣 )‘𝑈) = (GId‘( +𝑣 ‘𝑈)))
13		0vfval.5	. . . 4 ⊢ 𝑍 = (0vec‘𝑈)
14		df-0v 30426	. . . . 5 ⊢ 0vec = (GId ∘ +𝑣 )
15	14	fveq1i 6901	. . . 4 ⊢ (0vec‘𝑈) = ((GId ∘ +𝑣 )‘𝑈)
16	13, 15	eqtri 2755	. . 3 ⊢ 𝑍 = ((GId ∘ +𝑣 )‘𝑈)
17		0vfval.2	. . . 4 ⊢ 𝐺 = ( +𝑣 ‘𝑈)
18	17	fveq2i 6903	. . 3 ⊢ (GId‘𝐺) = (GId‘( +𝑣 ‘𝑈))
19	12, 16, 18	3eqtr4g 2792	. 2 ⊢ (𝑈 ∈ V → 𝑍 = (GId‘𝐺))
20	1, 19	syl 17	1 ⊢ (𝑈 ∈ 𝑉 → 𝑍 = (GId‘𝐺))

Syntax hints:   → wi 4   = wceq 1533   ∈ wcel 2098  Vcvv 3471   ⊆ wss 3947  ran crn 5681   ∘ ccom 5684   Fn wfn 6546  –onto→wfo 6549  ‘cfv 6551  1^st c1st 7995  GIdcgi 30318   +_𝑣 cpv 30413  0_veccn0v 30416

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2698  ax-sep 5301  ax-nul 5308  ax-pr 5431  ax-un 7744

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-br 5151  df-opab 5213  df-mpt 5234  df-id 5578  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-res 5692  df-ima 5693  df-iota 6503  df-fun 6553  df-fn 6554  df-f 6555  df-fo 6557  df-fv 6559  df-1st 7997  df-va 30423  df-0v 30426

This theorem is referenced by:  nvi  30442  nvzcl  30462  nv0rid  30463  nv0lid  30464  nv0  30465  nvsz  30466  nvrinv  30479  nvlinv  30480  hh0v  30996