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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 3500 | . 2 ⊢ (𝑈 ∈ 𝑉 → 𝑈 ∈ V) | |
| 2 | fo1st 8035 | . . . . . . 7 ⊢ 1st :V–onto→V | |
| 3 | fofn 6821 | . . . . . . 7 ⊢ (1st :V–onto→V → 1st Fn V) | |
| 4 | 2, 3 | ax-mp 5 | . . . . . 6 ⊢ 1st Fn V | 
| 5 | ssv 4007 | . . . . . 6 ⊢ ran 1st ⊆ V | |
| 6 | fnco 6685 | . . . . . 6 ⊢ ((1st Fn V ∧ 1st Fn V ∧ ran 1st ⊆ V) → (1st ∘ 1st ) Fn V) | |
| 7 | 4, 4, 5, 6 | mp3an 1462 | . . . . 5 ⊢ (1st ∘ 1st ) Fn V | 
| 8 | df-va 30615 | . . . . . 6 ⊢ +𝑣 = (1st ∘ 1st ) | |
| 9 | 8 | fneq1i 6664 | . . . . 5 ⊢ ( +𝑣 Fn V ↔ (1st ∘ 1st ) Fn V) | 
| 10 | 7, 9 | mpbir 231 | . . . 4 ⊢ +𝑣 Fn V | 
| 11 | fvco2 7005 | . . . 4 ⊢ (( +𝑣 Fn V ∧ 𝑈 ∈ V) → ((GId ∘ +𝑣 )‘𝑈) = (GId‘( +𝑣 ‘𝑈))) | |
| 12 | 10, 11 | mpan 690 | . . 3 ⊢ (𝑈 ∈ V → ((GId ∘ +𝑣 )‘𝑈) = (GId‘( +𝑣 ‘𝑈))) | 
| 13 | 0vfval.5 | . . . 4 ⊢ 𝑍 = (0vec‘𝑈) | |
| 14 | df-0v 30618 | . . . . 5 ⊢ 0vec = (GId ∘ +𝑣 ) | |
| 15 | 14 | fveq1i 6906 | . . . 4 ⊢ (0vec‘𝑈) = ((GId ∘ +𝑣 )‘𝑈) | 
| 16 | 13, 15 | eqtri 2764 | . . 3 ⊢ 𝑍 = ((GId ∘ +𝑣 )‘𝑈) | 
| 17 | 0vfval.2 | . . . 4 ⊢ 𝐺 = ( +𝑣 ‘𝑈) | |
| 18 | 17 | fveq2i 6908 | . . 3 ⊢ (GId‘𝐺) = (GId‘( +𝑣 ‘𝑈)) | 
| 19 | 12, 16, 18 | 3eqtr4g 2801 | . 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 1539 ∈ wcel 2107 Vcvv 3479 ⊆ wss 3950 ran crn 5685 ∘ ccom 5688 Fn wfn 6555 –onto→wfo 6558 ‘cfv 6560 1st c1st 8013 GIdcgi 30510 +𝑣 cpv 30605 0veccn0v 30608 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-fo 6566 df-fv 6568 df-1st 8015 df-va 30615 df-0v 30618 | 
| This theorem is referenced by: nvi 30634 nvzcl 30654 nv0rid 30655 nv0lid 30656 nv0 30657 nvsz 30658 nvrinv 30671 nvlinv 30672 hh0v 31188 | 
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