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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 28382 | . . . . 5 ⊢ −𝑣 = ( /𝑔 ∘ +𝑣 ) | |
2 | 1 | fveq1i 6646 | . . . 4 ⊢ ( −𝑣 ‘𝑈) = (( /𝑔 ∘ +𝑣 )‘𝑈) |
3 | fo1st 7691 | . . . . . . . 8 ⊢ 1st :V–onto→V | |
4 | fof 6565 | . . . . . . . 8 ⊢ (1st :V–onto→V → 1st :V⟶V) | |
5 | 3, 4 | ax-mp 5 | . . . . . . 7 ⊢ 1st :V⟶V |
6 | fco 6505 | . . . . . . 7 ⊢ ((1st :V⟶V ∧ 1st :V⟶V) → (1st ∘ 1st ):V⟶V) | |
7 | 5, 5, 6 | mp2an 691 | . . . . . 6 ⊢ (1st ∘ 1st ):V⟶V |
8 | df-va 28378 | . . . . . . 7 ⊢ +𝑣 = (1st ∘ 1st ) | |
9 | 8 | feq1i 6478 | . . . . . 6 ⊢ ( +𝑣 :V⟶V ↔ (1st ∘ 1st ):V⟶V) |
10 | 7, 9 | mpbir 234 | . . . . 5 ⊢ +𝑣 :V⟶V |
11 | fvco3 6737 | . . . . 5 ⊢ (( +𝑣 :V⟶V ∧ 𝑈 ∈ V) → (( /𝑔 ∘ +𝑣 )‘𝑈) = ( /𝑔 ‘( +𝑣 ‘𝑈))) | |
12 | 10, 11 | mpan 689 | . . . 4 ⊢ (𝑈 ∈ V → (( /𝑔 ∘ +𝑣 )‘𝑈) = ( /𝑔 ‘( +𝑣 ‘𝑈))) |
13 | 2, 12 | syl5eq 2845 | . . 3 ⊢ (𝑈 ∈ V → ( −𝑣 ‘𝑈) = ( /𝑔 ‘( +𝑣 ‘𝑈))) |
14 | 0ngrp 28294 | . . . . . 6 ⊢ ¬ ∅ ∈ GrpOp | |
15 | vex 3444 | . . . . . . . . . 10 ⊢ 𝑔 ∈ V | |
16 | 15 | rnex 7599 | . . . . . . . . 9 ⊢ ran 𝑔 ∈ V |
17 | 16, 16 | mpoex 7760 | . . . . . . . 8 ⊢ (𝑥 ∈ ran 𝑔, 𝑦 ∈ ran 𝑔 ↦ (𝑥𝑔((inv‘𝑔)‘𝑦))) ∈ V |
18 | df-gdiv 28279 | . . . . . . . 8 ⊢ /𝑔 = (𝑔 ∈ GrpOp ↦ (𝑥 ∈ ran 𝑔, 𝑦 ∈ ran 𝑔 ↦ (𝑥𝑔((inv‘𝑔)‘𝑦)))) | |
19 | 17, 18 | dmmpti 6464 | . . . . . . 7 ⊢ dom /𝑔 = GrpOp |
20 | 19 | eleq2i 2881 | . . . . . 6 ⊢ (∅ ∈ dom /𝑔 ↔ ∅ ∈ GrpOp) |
21 | 14, 20 | mtbir 326 | . . . . 5 ⊢ ¬ ∅ ∈ dom /𝑔 |
22 | ndmfv 6675 | . . . . 5 ⊢ (¬ ∅ ∈ dom /𝑔 → ( /𝑔 ‘∅) = ∅) | |
23 | 21, 22 | mp1i 13 | . . . 4 ⊢ (¬ 𝑈 ∈ V → ( /𝑔 ‘∅) = ∅) |
24 | fvprc 6638 | . . . . 5 ⊢ (¬ 𝑈 ∈ V → ( +𝑣 ‘𝑈) = ∅) | |
25 | 24 | fveq2d 6649 | . . . 4 ⊢ (¬ 𝑈 ∈ V → ( /𝑔 ‘( +𝑣 ‘𝑈)) = ( /𝑔 ‘∅)) |
26 | fvprc 6638 | . . . 4 ⊢ (¬ 𝑈 ∈ V → ( −𝑣 ‘𝑈) = ∅) | |
27 | 23, 25, 26 | 3eqtr4rd 2844 | . . 3 ⊢ (¬ 𝑈 ∈ V → ( −𝑣 ‘𝑈) = ( /𝑔 ‘( +𝑣 ‘𝑈))) |
28 | 13, 27 | pm2.61i 185 | . 2 ⊢ ( −𝑣 ‘𝑈) = ( /𝑔 ‘( +𝑣 ‘𝑈)) |
29 | vsfval.3 | . 2 ⊢ 𝑀 = ( −𝑣 ‘𝑈) | |
30 | vsfval.2 | . . 3 ⊢ 𝐺 = ( +𝑣 ‘𝑈) | |
31 | 30 | fveq2i 6648 | . 2 ⊢ ( /𝑔 ‘𝐺) = ( /𝑔 ‘( +𝑣 ‘𝑈)) |
32 | 28, 29, 31 | 3eqtr4i 2831 | 1 ⊢ 𝑀 = ( /𝑔 ‘𝐺) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ∅c0 4243 dom cdm 5519 ran crn 5520 ∘ ccom 5523 ⟶wf 6320 –onto→wfo 6322 ‘cfv 6324 (class class class)co 7135 ∈ cmpo 7137 1st c1st 7669 GrpOpcgr 28272 invcgn 28274 /𝑔 cgs 28275 +𝑣 cpv 28368 −𝑣 cnsb 28372 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-1st 7671 df-2nd 7672 df-grpo 28276 df-gdiv 28279 df-va 28378 df-vs 28382 |
This theorem is referenced by: nvm 28424 nvmfval 28427 nvnnncan1 28430 nvaddsub 28438 |
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