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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 3515 | . 2 ⊢ (𝑈 ∈ 𝑉 → 𝑈 ∈ V) | |
2 | fo1st 7712 | . . . . . . 7 ⊢ 1st :V–onto→V | |
3 | fofn 6595 | . . . . . . 7 ⊢ (1st :V–onto→V → 1st Fn V) | |
4 | 2, 3 | ax-mp 5 | . . . . . 6 ⊢ 1st Fn V |
5 | ssv 3994 | . . . . . 6 ⊢ ran 1st ⊆ V | |
6 | fnco 6468 | . . . . . 6 ⊢ ((1st Fn V ∧ 1st Fn V ∧ ran 1st ⊆ V) → (1st ∘ 1st ) Fn V) | |
7 | 4, 4, 5, 6 | mp3an 1457 | . . . . 5 ⊢ (1st ∘ 1st ) Fn V |
8 | df-va 28375 | . . . . . 6 ⊢ +𝑣 = (1st ∘ 1st ) | |
9 | 8 | fneq1i 6453 | . . . . 5 ⊢ ( +𝑣 Fn V ↔ (1st ∘ 1st ) Fn V) |
10 | 7, 9 | mpbir 233 | . . . 4 ⊢ +𝑣 Fn V |
11 | fvco2 6761 | . . . 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 28378 | . . . . 5 ⊢ 0vec = (GId ∘ +𝑣 ) | |
15 | 14 | fveq1i 6674 | . . . 4 ⊢ (0vec‘𝑈) = ((GId ∘ +𝑣 )‘𝑈) |
16 | 13, 15 | eqtri 2847 | . . 3 ⊢ 𝑍 = ((GId ∘ +𝑣 )‘𝑈) |
17 | 0vfval.2 | . . . 4 ⊢ 𝐺 = ( +𝑣 ‘𝑈) | |
18 | 17 | fveq2i 6676 | . . 3 ⊢ (GId‘𝐺) = (GId‘( +𝑣 ‘𝑈)) |
19 | 12, 16, 18 | 3eqtr4g 2884 | . 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 1536 ∈ wcel 2113 Vcvv 3497 ⊆ wss 3939 ran crn 5559 ∘ ccom 5562 Fn wfn 6353 –onto→wfo 6356 ‘cfv 6358 1st c1st 7690 GIdcgi 28270 +𝑣 cpv 28365 0veccn0v 28368 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-fo 6364 df-fv 6366 df-1st 7692 df-va 28375 df-0v 28378 |
This theorem is referenced by: nvi 28394 nvzcl 28414 nv0rid 28415 nv0lid 28416 nv0 28417 nvsz 28418 nvrinv 28431 nvlinv 28432 hh0v 28948 |
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