Theorem 0vfval 30693

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 3463	. 2 ⊢ (𝑈 ∈ 𝑉 → 𝑈 ∈ V)
2		fo1st 7963	. . . . . . 7 ⊢ 1st :V–onto→V
3		fofn 6756	. . . . . . 7 ⊢ (1st :V–onto→V → 1st Fn V)
4	2, 3	ax-mp 5	. . . . . 6 ⊢ 1st Fn V
5		ssv 3960	. . . . . 6 ⊢ ran 1st ⊆ V
6		fnco 6618	. . . . . 6 ⊢ ((1st Fn V ∧ 1st Fn V ∧ ran 1st ⊆ V) → (1st ∘ 1st ) Fn V)
7	4, 4, 5, 6	mp3an 1464	. . . . 5 ⊢ (1st ∘ 1st ) Fn V
8		df-va 30682	. . . . . 6 ⊢ +𝑣 = (1st ∘ 1st )
9	8	fneq1i 6597	. . . . 5 ⊢ ( +𝑣 Fn V ↔ (1st ∘ 1st ) Fn V)
10	7, 9	mpbir 231	. . . 4 ⊢ +𝑣 Fn V
11		fvco2 6939	. . . 4 ⊢ (( +𝑣 Fn V ∧ 𝑈 ∈ V) → ((GId ∘ +𝑣 )‘𝑈) = (GId‘( +𝑣 ‘𝑈)))
12	10, 11	mpan 691	. . 3 ⊢ (𝑈 ∈ V → ((GId ∘ +𝑣 )‘𝑈) = (GId‘( +𝑣 ‘𝑈)))
13		0vfval.5	. . . 4 ⊢ 𝑍 = (0vec‘𝑈)
14		df-0v 30685	. . . . 5 ⊢ 0vec = (GId ∘ +𝑣 )
15	14	fveq1i 6843	. . . 4 ⊢ (0vec‘𝑈) = ((GId ∘ +𝑣 )‘𝑈)
16	13, 15	eqtri 2760	. . 3 ⊢ 𝑍 = ((GId ∘ +𝑣 )‘𝑈)
17		0vfval.2	. . . 4 ⊢ 𝐺 = ( +𝑣 ‘𝑈)
18	17	fveq2i 6845	. . 3 ⊢ (GId‘𝐺) = (GId‘( +𝑣 ‘𝑈))
19	12, 16, 18	3eqtr4g 2797	. 2 ⊢ (𝑈 ∈ V → 𝑍 = (GId‘𝐺))
20	1, 19	syl 17	1 ⊢ (𝑈 ∈ 𝑉 → 𝑍 = (GId‘𝐺))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  Vcvv 3442   ⊆ wss 3903  ran crn 5633   ∘ ccom 5636   Fn wfn 6495  –onto→wfo 6498  ‘cfv 6500  1^st c1st 7941  GIdcgi 30577   +_𝑣 cpv 30672  0_veccn0v 30675

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-fo 6506  df-fv 6508  df-1st 7943  df-va 30682  df-0v 30685

This theorem is referenced by:  nvi  30701  nvzcl  30721  nv0rid  30722  nv0lid  30723  nv0  30724  nvsz  30725  nvrinv  30738  nvlinv  30739  hh0v  31255