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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 3454 | . 2 ⊢ (𝑈 ∈ 𝑉 → 𝑈 ∈ V) | |
2 | fo1st 7568 | . . . . . . 7 ⊢ 1st :V–onto→V | |
3 | fofn 6463 | . . . . . . 7 ⊢ (1st :V–onto→V → 1st Fn V) | |
4 | 2, 3 | ax-mp 5 | . . . . . 6 ⊢ 1st Fn V |
5 | ssv 3914 | . . . . . 6 ⊢ ran 1st ⊆ V | |
6 | fnco 6338 | . . . . . 6 ⊢ ((1st Fn V ∧ 1st Fn V ∧ ran 1st ⊆ V) → (1st ∘ 1st ) Fn V) | |
7 | 4, 4, 5, 6 | mp3an 1453 | . . . . 5 ⊢ (1st ∘ 1st ) Fn V |
8 | df-va 28055 | . . . . . 6 ⊢ +𝑣 = (1st ∘ 1st ) | |
9 | 8 | fneq1i 6323 | . . . . 5 ⊢ ( +𝑣 Fn V ↔ (1st ∘ 1st ) Fn V) |
10 | 7, 9 | mpbir 232 | . . . 4 ⊢ +𝑣 Fn V |
11 | fvco2 6628 | . . . 4 ⊢ (( +𝑣 Fn V ∧ 𝑈 ∈ V) → ((GId ∘ +𝑣 )‘𝑈) = (GId‘( +𝑣 ‘𝑈))) | |
12 | 10, 11 | mpan 686 | . . 3 ⊢ (𝑈 ∈ V → ((GId ∘ +𝑣 )‘𝑈) = (GId‘( +𝑣 ‘𝑈))) |
13 | 0vfval.5 | . . . 4 ⊢ 𝑍 = (0vec‘𝑈) | |
14 | df-0v 28058 | . . . . 5 ⊢ 0vec = (GId ∘ +𝑣 ) | |
15 | 14 | fveq1i 6542 | . . . 4 ⊢ (0vec‘𝑈) = ((GId ∘ +𝑣 )‘𝑈) |
16 | 13, 15 | eqtri 2818 | . . 3 ⊢ 𝑍 = ((GId ∘ +𝑣 )‘𝑈) |
17 | 0vfval.2 | . . . 4 ⊢ 𝐺 = ( +𝑣 ‘𝑈) | |
18 | 17 | fveq2i 6544 | . . 3 ⊢ (GId‘𝐺) = (GId‘( +𝑣 ‘𝑈)) |
19 | 12, 16, 18 | 3eqtr4g 2855 | . 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 1522 ∈ wcel 2080 Vcvv 3436 ⊆ wss 3861 ran crn 5447 ∘ ccom 5450 Fn wfn 6223 –onto→wfo 6226 ‘cfv 6228 1st c1st 7546 GIdcgi 27950 +𝑣 cpv 28045 0veccn0v 28048 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1778 ax-4 1792 ax-5 1889 ax-6 1948 ax-7 1993 ax-8 2082 ax-9 2090 ax-10 2111 ax-11 2125 ax-12 2140 ax-13 2343 ax-ext 2768 ax-sep 5097 ax-nul 5104 ax-pow 5160 ax-pr 5224 ax-un 7322 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1525 df-ex 1763 df-nf 1767 df-sb 2042 df-mo 2575 df-eu 2611 df-clab 2775 df-cleq 2787 df-clel 2862 df-nfc 2934 df-ne 2984 df-ral 3109 df-rex 3110 df-rab 3113 df-v 3438 df-sbc 3708 df-dif 3864 df-un 3866 df-in 3868 df-ss 3876 df-nul 4214 df-if 4384 df-sn 4475 df-pr 4477 df-op 4481 df-uni 4748 df-br 4965 df-opab 5027 df-mpt 5044 df-id 5351 df-xp 5452 df-rel 5453 df-cnv 5454 df-co 5455 df-dm 5456 df-rn 5457 df-res 5458 df-ima 5459 df-iota 6192 df-fun 6230 df-fn 6231 df-f 6232 df-fo 6234 df-fv 6236 df-1st 7548 df-va 28055 df-0v 28058 |
This theorem is referenced by: nvi 28074 nvzcl 28094 nv0rid 28095 nv0lid 28096 nv0 28097 nvsz 28098 nvrinv 28111 nvlinv 28112 hh0v 28628 |
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