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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 30618 | . . . . 5 ⊢ −𝑣 = ( /𝑔 ∘ +𝑣 ) | |
| 2 | 1 | fveq1i 6907 | . . . 4 ⊢ ( −𝑣 ‘𝑈) = (( /𝑔 ∘ +𝑣 )‘𝑈) |
| 3 | fo1st 8034 | . . . . . . . 8 ⊢ 1st :V–onto→V | |
| 4 | fof 6820 | . . . . . . . 8 ⊢ (1st :V–onto→V → 1st :V⟶V) | |
| 5 | 3, 4 | ax-mp 5 | . . . . . . 7 ⊢ 1st :V⟶V |
| 6 | fco 6760 | . . . . . . 7 ⊢ ((1st :V⟶V ∧ 1st :V⟶V) → (1st ∘ 1st ):V⟶V) | |
| 7 | 5, 5, 6 | mp2an 692 | . . . . . 6 ⊢ (1st ∘ 1st ):V⟶V |
| 8 | df-va 30614 | . . . . . . 7 ⊢ +𝑣 = (1st ∘ 1st ) | |
| 9 | 8 | feq1i 6727 | . . . . . 6 ⊢ ( +𝑣 :V⟶V ↔ (1st ∘ 1st ):V⟶V) |
| 10 | 7, 9 | mpbir 231 | . . . . 5 ⊢ +𝑣 :V⟶V |
| 11 | fvco3 7008 | . . . . 5 ⊢ (( +𝑣 :V⟶V ∧ 𝑈 ∈ V) → (( /𝑔 ∘ +𝑣 )‘𝑈) = ( /𝑔 ‘( +𝑣 ‘𝑈))) | |
| 12 | 10, 11 | mpan 690 | . . . 4 ⊢ (𝑈 ∈ V → (( /𝑔 ∘ +𝑣 )‘𝑈) = ( /𝑔 ‘( +𝑣 ‘𝑈))) |
| 13 | 2, 12 | eqtrid 2789 | . . 3 ⊢ (𝑈 ∈ V → ( −𝑣 ‘𝑈) = ( /𝑔 ‘( +𝑣 ‘𝑈))) |
| 14 | 0ngrp 30530 | . . . . . 6 ⊢ ¬ ∅ ∈ GrpOp | |
| 15 | vex 3484 | . . . . . . . . . 10 ⊢ 𝑔 ∈ V | |
| 16 | 15 | rnex 7932 | . . . . . . . . 9 ⊢ ran 𝑔 ∈ V |
| 17 | 16, 16 | mpoex 8104 | . . . . . . . 8 ⊢ (𝑥 ∈ ran 𝑔, 𝑦 ∈ ran 𝑔 ↦ (𝑥𝑔((inv‘𝑔)‘𝑦))) ∈ V |
| 18 | df-gdiv 30515 | . . . . . . . 8 ⊢ /𝑔 = (𝑔 ∈ GrpOp ↦ (𝑥 ∈ ran 𝑔, 𝑦 ∈ ran 𝑔 ↦ (𝑥𝑔((inv‘𝑔)‘𝑦)))) | |
| 19 | 17, 18 | dmmpti 6712 | . . . . . . 7 ⊢ dom /𝑔 = GrpOp |
| 20 | 19 | eleq2i 2833 | . . . . . 6 ⊢ (∅ ∈ dom /𝑔 ↔ ∅ ∈ GrpOp) |
| 21 | 14, 20 | mtbir 323 | . . . . 5 ⊢ ¬ ∅ ∈ dom /𝑔 |
| 22 | ndmfv 6941 | . . . . 5 ⊢ (¬ ∅ ∈ dom /𝑔 → ( /𝑔 ‘∅) = ∅) | |
| 23 | 21, 22 | mp1i 13 | . . . 4 ⊢ (¬ 𝑈 ∈ V → ( /𝑔 ‘∅) = ∅) |
| 24 | fvprc 6898 | . . . . 5 ⊢ (¬ 𝑈 ∈ V → ( +𝑣 ‘𝑈) = ∅) | |
| 25 | 24 | fveq2d 6910 | . . . 4 ⊢ (¬ 𝑈 ∈ V → ( /𝑔 ‘( +𝑣 ‘𝑈)) = ( /𝑔 ‘∅)) |
| 26 | fvprc 6898 | . . . 4 ⊢ (¬ 𝑈 ∈ V → ( −𝑣 ‘𝑈) = ∅) | |
| 27 | 23, 25, 26 | 3eqtr4rd 2788 | . . 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 6909 | . 2 ⊢ ( /𝑔 ‘𝐺) = ( /𝑔 ‘( +𝑣 ‘𝑈)) |
| 32 | 28, 29, 31 | 3eqtr4i 2775 | 1 ⊢ 𝑀 = ( /𝑔 ‘𝐺) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∅c0 4333 dom cdm 5685 ran crn 5686 ∘ ccom 5689 ⟶wf 6557 –onto→wfo 6559 ‘cfv 6561 (class class class)co 7431 ∈ cmpo 7433 1st c1st 8012 GrpOpcgr 30508 invcgn 30510 /𝑔 cgs 30511 +𝑣 cpv 30604 −𝑣 cnsb 30608 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-grpo 30512 df-gdiv 30515 df-va 30614 df-vs 30618 |
| This theorem is referenced by: nvm 30660 nvmfval 30663 nvnnncan1 30666 nvaddsub 30674 |
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