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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 3492 | . 2 ⊢ (𝑈 ∈ 𝑉 → 𝑈 ∈ V) | |
2 | fo1st 7991 | . . . . . . 7 ⊢ 1st :V–onto→V | |
3 | fofn 6804 | . . . . . . 7 ⊢ (1st :V–onto→V → 1st Fn V) | |
4 | 2, 3 | ax-mp 5 | . . . . . 6 ⊢ 1st Fn V |
5 | ssv 4005 | . . . . . 6 ⊢ ran 1st ⊆ V | |
6 | fnco 6664 | . . . . . 6 ⊢ ((1st Fn V ∧ 1st Fn V ∧ ran 1st ⊆ V) → (1st ∘ 1st ) Fn V) | |
7 | 4, 4, 5, 6 | mp3an 1461 | . . . . 5 ⊢ (1st ∘ 1st ) Fn V |
8 | df-va 29835 | . . . . . 6 ⊢ +𝑣 = (1st ∘ 1st ) | |
9 | 8 | fneq1i 6643 | . . . . 5 ⊢ ( +𝑣 Fn V ↔ (1st ∘ 1st ) Fn V) |
10 | 7, 9 | mpbir 230 | . . . 4 ⊢ +𝑣 Fn V |
11 | fvco2 6985 | . . . 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 29838 | . . . . 5 ⊢ 0vec = (GId ∘ +𝑣 ) | |
15 | 14 | fveq1i 6889 | . . . 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 6891 | . . 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)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 Vcvv 3474 ⊆ wss 3947 ran crn 5676 ∘ ccom 5679 Fn wfn 6535 –onto→wfo 6538 ‘cfv 6540 1st c1st 7969 GIdcgi 29730 +𝑣 cpv 29825 0veccn0v 29828 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pr 5426 ax-un 7721 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-fo 6546 df-fv 6548 df-1st 7971 df-va 29835 df-0v 29838 |
This theorem is referenced by: nvi 29854 nvzcl 29874 nv0rid 29875 nv0lid 29876 nv0 29877 nvsz 29878 nvrinv 29891 nvlinv 29892 hh0v 30408 |
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