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Mirrors > Home > MPE Home > Th. List > 0vfval | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
0vfval.2 | ⊢ 𝐺 = ( +𝑣 ‘𝑈) |
0vfval.5 | ⊢ 𝑍 = (0vec‘𝑈) |
Ref | Expression |
---|---|
0vfval | ⊢ (𝑈 ∈ 𝑉 → 𝑍 = (GId‘𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3185 | . 2 ⊢ (𝑈 ∈ 𝑉 → 𝑈 ∈ V) | |
2 | fo1st 7079 | . . . . . . 7 ⊢ 1st :V–onto→V | |
3 | fofn 6030 | . . . . . . 7 ⊢ (1st :V–onto→V → 1st Fn V) | |
4 | 2, 3 | ax-mp 5 | . . . . . 6 ⊢ 1st Fn V |
5 | ssv 3588 | . . . . . 6 ⊢ ran 1st ⊆ V | |
6 | fnco 5913 | . . . . . 6 ⊢ ((1st Fn V ∧ 1st Fn V ∧ ran 1st ⊆ V) → (1st ∘ 1st ) Fn V) | |
7 | 4, 4, 5, 6 | mp3an 1416 | . . . . 5 ⊢ (1st ∘ 1st ) Fn V |
8 | df-va 26834 | . . . . . 6 ⊢ +𝑣 = (1st ∘ 1st ) | |
9 | 8 | fneq1i 5899 | . . . . 5 ⊢ ( +𝑣 Fn V ↔ (1st ∘ 1st ) Fn V) |
10 | 7, 9 | mpbir 220 | . . . 4 ⊢ +𝑣 Fn V |
11 | fvco2 6183 | . . . 4 ⊢ (( +𝑣 Fn V ∧ 𝑈 ∈ V) → ((GId ∘ +𝑣 )‘𝑈) = (GId‘( +𝑣 ‘𝑈))) | |
12 | 10, 11 | mpan 702 | . . 3 ⊢ (𝑈 ∈ V → ((GId ∘ +𝑣 )‘𝑈) = (GId‘( +𝑣 ‘𝑈))) |
13 | 0vfval.5 | . . . 4 ⊢ 𝑍 = (0vec‘𝑈) | |
14 | df-0v 26837 | . . . . 5 ⊢ 0vec = (GId ∘ +𝑣 ) | |
15 | 14 | fveq1i 6104 | . . . 4 ⊢ (0vec‘𝑈) = ((GId ∘ +𝑣 )‘𝑈) |
16 | 13, 15 | eqtri 2632 | . . 3 ⊢ 𝑍 = ((GId ∘ +𝑣 )‘𝑈) |
17 | 0vfval.2 | . . . 4 ⊢ 𝐺 = ( +𝑣 ‘𝑈) | |
18 | 17 | fveq2i 6106 | . . 3 ⊢ (GId‘𝐺) = (GId‘( +𝑣 ‘𝑈)) |
19 | 12, 16, 18 | 3eqtr4g 2669 | . 2 ⊢ (𝑈 ∈ V → 𝑍 = (GId‘𝐺)) |
20 | 1, 19 | syl 17 | 1 ⊢ (𝑈 ∈ 𝑉 → 𝑍 = (GId‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ⊆ wss 3540 ran crn 5039 ∘ ccom 5042 Fn wfn 5799 –onto→wfo 5802 ‘cfv 5804 1st c1st 7057 GIdcgi 26728 +𝑣 cpv 26824 0veccn0v 26827 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fo 5810 df-fv 5812 df-1st 7059 df-va 26834 df-0v 26837 |
This theorem is referenced by: nvi 26853 nvzcl 26873 nv0rid 26874 nv0lid 26875 nv0 26876 nvsz 26877 nvrinv 26890 nvlinv 26891 hh0v 27409 |
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