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Mirrors > Home > MPE Home > Th. List > vsfval | Structured version Visualization version GIF version |
Description: Value of the function for the vector subtraction operation on a normed complex vector space. (Contributed by NM, 15-Feb-2008.) (Revised by Mario Carneiro, 27-Dec-2014.) (New usage is discouraged.) |
Ref | Expression |
---|---|
vsfval.2 | ⊢ 𝐺 = ( +𝑣 ‘𝑈) |
vsfval.3 | ⊢ 𝑀 = ( −𝑣 ‘𝑈) |
Ref | Expression |
---|---|
vsfval | ⊢ 𝑀 = ( /𝑔 ‘𝐺) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-vs 30501 | . . . . 5 ⊢ −𝑣 = ( /𝑔 ∘ +𝑣 ) | |
2 | 1 | fveq1i 6897 | . . . 4 ⊢ ( −𝑣 ‘𝑈) = (( /𝑔 ∘ +𝑣 )‘𝑈) |
3 | fo1st 8014 | . . . . . . . 8 ⊢ 1st :V–onto→V | |
4 | fof 6810 | . . . . . . . 8 ⊢ (1st :V–onto→V → 1st :V⟶V) | |
5 | 3, 4 | ax-mp 5 | . . . . . . 7 ⊢ 1st :V⟶V |
6 | fco 6747 | . . . . . . 7 ⊢ ((1st :V⟶V ∧ 1st :V⟶V) → (1st ∘ 1st ):V⟶V) | |
7 | 5, 5, 6 | mp2an 690 | . . . . . 6 ⊢ (1st ∘ 1st ):V⟶V |
8 | df-va 30497 | . . . . . . 7 ⊢ +𝑣 = (1st ∘ 1st ) | |
9 | 8 | feq1i 6714 | . . . . . 6 ⊢ ( +𝑣 :V⟶V ↔ (1st ∘ 1st ):V⟶V) |
10 | 7, 9 | mpbir 230 | . . . . 5 ⊢ +𝑣 :V⟶V |
11 | fvco3 6996 | . . . . 5 ⊢ (( +𝑣 :V⟶V ∧ 𝑈 ∈ V) → (( /𝑔 ∘ +𝑣 )‘𝑈) = ( /𝑔 ‘( +𝑣 ‘𝑈))) | |
12 | 10, 11 | mpan 688 | . . . 4 ⊢ (𝑈 ∈ V → (( /𝑔 ∘ +𝑣 )‘𝑈) = ( /𝑔 ‘( +𝑣 ‘𝑈))) |
13 | 2, 12 | eqtrid 2777 | . . 3 ⊢ (𝑈 ∈ V → ( −𝑣 ‘𝑈) = ( /𝑔 ‘( +𝑣 ‘𝑈))) |
14 | 0ngrp 30413 | . . . . . 6 ⊢ ¬ ∅ ∈ GrpOp | |
15 | vex 3465 | . . . . . . . . . 10 ⊢ 𝑔 ∈ V | |
16 | 15 | rnex 7918 | . . . . . . . . 9 ⊢ ran 𝑔 ∈ V |
17 | 16, 16 | mpoex 8084 | . . . . . . . 8 ⊢ (𝑥 ∈ ran 𝑔, 𝑦 ∈ ran 𝑔 ↦ (𝑥𝑔((inv‘𝑔)‘𝑦))) ∈ V |
18 | df-gdiv 30398 | . . . . . . . 8 ⊢ /𝑔 = (𝑔 ∈ GrpOp ↦ (𝑥 ∈ ran 𝑔, 𝑦 ∈ ran 𝑔 ↦ (𝑥𝑔((inv‘𝑔)‘𝑦)))) | |
19 | 17, 18 | dmmpti 6700 | . . . . . . 7 ⊢ dom /𝑔 = GrpOp |
20 | 19 | eleq2i 2817 | . . . . . 6 ⊢ (∅ ∈ dom /𝑔 ↔ ∅ ∈ GrpOp) |
21 | 14, 20 | mtbir 322 | . . . . 5 ⊢ ¬ ∅ ∈ dom /𝑔 |
22 | ndmfv 6931 | . . . . 5 ⊢ (¬ ∅ ∈ dom /𝑔 → ( /𝑔 ‘∅) = ∅) | |
23 | 21, 22 | mp1i 13 | . . . 4 ⊢ (¬ 𝑈 ∈ V → ( /𝑔 ‘∅) = ∅) |
24 | fvprc 6888 | . . . . 5 ⊢ (¬ 𝑈 ∈ V → ( +𝑣 ‘𝑈) = ∅) | |
25 | 24 | fveq2d 6900 | . . . 4 ⊢ (¬ 𝑈 ∈ V → ( /𝑔 ‘( +𝑣 ‘𝑈)) = ( /𝑔 ‘∅)) |
26 | fvprc 6888 | . . . 4 ⊢ (¬ 𝑈 ∈ V → ( −𝑣 ‘𝑈) = ∅) | |
27 | 23, 25, 26 | 3eqtr4rd 2776 | . . 3 ⊢ (¬ 𝑈 ∈ V → ( −𝑣 ‘𝑈) = ( /𝑔 ‘( +𝑣 ‘𝑈))) |
28 | 13, 27 | pm2.61i 182 | . 2 ⊢ ( −𝑣 ‘𝑈) = ( /𝑔 ‘( +𝑣 ‘𝑈)) |
29 | vsfval.3 | . 2 ⊢ 𝑀 = ( −𝑣 ‘𝑈) | |
30 | vsfval.2 | . . 3 ⊢ 𝐺 = ( +𝑣 ‘𝑈) | |
31 | 30 | fveq2i 6899 | . 2 ⊢ ( /𝑔 ‘𝐺) = ( /𝑔 ‘( +𝑣 ‘𝑈)) |
32 | 28, 29, 31 | 3eqtr4i 2763 | 1 ⊢ 𝑀 = ( /𝑔 ‘𝐺) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1533 ∈ wcel 2098 Vcvv 3461 ∅c0 4322 dom cdm 5678 ran crn 5679 ∘ ccom 5682 ⟶wf 6545 –onto→wfo 6547 ‘cfv 6549 (class class class)co 7419 ∈ cmpo 7421 1st c1st 7992 GrpOpcgr 30391 invcgn 30393 /𝑔 cgs 30394 +𝑣 cpv 30487 −𝑣 cnsb 30491 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-ov 7422 df-oprab 7423 df-mpo 7424 df-1st 7994 df-2nd 7995 df-grpo 30395 df-gdiv 30398 df-va 30497 df-vs 30501 |
This theorem is referenced by: nvm 30543 nvmfval 30546 nvnnncan1 30549 nvaddsub 30557 |
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