Theorem 0vfval 30765

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 3474	. 2 ⊢ (𝑈 ∈ 𝑉 → 𝑈 ∈ V)
2		fo1st 7984	. . . . . . 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 3958	. . . . . 6 ⊢ ran 1st ⊆ V
6		fnco 6633	. . . . . 6 ⊢ ((1st Fn V ∧ 1st Fn V ∧ ran 1st ⊆ V) → (1st ∘ 1st ) Fn V)
7	4, 4, 5, 6	mp3an 1481	. . . . 5 ⊢ (1st ∘ 1st ) Fn V
8		df-va 30754	. . . . . 6 ⊢ +𝑣 = (1st ∘ 1st )
9	8	fneq1i 6612	. . . . 5 ⊢ ( +𝑣 Fn V ↔ (1st ∘ 1st ) Fn V)
10	7, 9	mpbir 233	. . . 4 ⊢ +𝑣 Fn V
11		fvco2 6958	. . . 4 ⊢ (( +𝑣 Fn V ∧ 𝑈 ∈ V) → ((GId ∘ +𝑣 )‘𝑈) = (GId‘( +𝑣 ‘𝑈)))
12	10, 11	mpan 700	. . 3 ⊢ (𝑈 ∈ V → ((GId ∘ +𝑣 )‘𝑈) = (GId‘( +𝑣 ‘𝑈)))
13		0vfval.5	. . . 4 ⊢ 𝑍 = (0vec‘𝑈)
14		df-0v 30757	. . . . 5 ⊢ 0vec = (GId ∘ +𝑣 )
15	14	fveq1i 6862	. . . 4 ⊢ (0vec‘𝑈) = ((GId ∘ +𝑣 )‘𝑈)
16	13, 15	eqtri 2784	. . 3 ⊢ 𝑍 = ((GId ∘ +𝑣 )‘𝑈)
17		0vfval.2	. . . 4 ⊢ 𝐺 = ( +𝑣 ‘𝑈)
18	17	fveq2i 6864	. . 3 ⊢ (GId‘𝐺) = (GId‘( +𝑣 ‘𝑈))
19	12, 16, 18	3eqtr4g 2821	. 2 ⊢ (𝑈 ∈ V → 𝑍 = (GId‘𝐺))
20	1, 19	syl 17	1 ⊢ (𝑈 ∈ 𝑉 → 𝑍 = (GId‘𝐺))

Syntax hints:   → wi 4   = wceq 1559   ∈ wcel 2141  Vcvv 3453   ⊆ wss 3902  ran crn 5644   ∘ ccom 5647   Fn wfn 6510  –onto→wfo 6513  ‘cfv 6515  1^st c1st 7962  GIdcgi 30649   +_𝑣 cpv 30744  0_veccn0v 30747

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5243  ax-nul 5253  ax-pr 5387  ax-un 7712

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-opab 5160  df-mpt 5179  df-id 5538  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-iota 6471  df-fun 6517  df-fn 6518  df-f 6519  df-fo 6521  df-fv 6523  df-1st 7964  df-va 30754  df-0v 30757

This theorem is referenced by:  nvi  30773  nvzcl  30793  nv0rid  30794  nv0lid  30795  nv0  30796  nvsz  30797  nvrinv  30810  nvlinv  30811  hh0v  31327