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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 26838 | . . . . 5 ⊢ −𝑣 = ( /𝑔 ∘ +𝑣 ) | |
2 | 1 | fveq1i 6104 | . . . 4 ⊢ ( −𝑣 ‘𝑈) = (( /𝑔 ∘ +𝑣 )‘𝑈) |
3 | fo1st 7079 | . . . . . . . 8 ⊢ 1st :V–onto→V | |
4 | fof 6028 | . . . . . . . 8 ⊢ (1st :V–onto→V → 1st :V⟶V) | |
5 | 3, 4 | ax-mp 5 | . . . . . . 7 ⊢ 1st :V⟶V |
6 | fco 5971 | . . . . . . 7 ⊢ ((1st :V⟶V ∧ 1st :V⟶V) → (1st ∘ 1st ):V⟶V) | |
7 | 5, 5, 6 | mp2an 704 | . . . . . 6 ⊢ (1st ∘ 1st ):V⟶V |
8 | df-va 26834 | . . . . . . 7 ⊢ +𝑣 = (1st ∘ 1st ) | |
9 | 8 | feq1i 5949 | . . . . . 6 ⊢ ( +𝑣 :V⟶V ↔ (1st ∘ 1st ):V⟶V) |
10 | 7, 9 | mpbir 220 | . . . . 5 ⊢ +𝑣 :V⟶V |
11 | fvco3 6185 | . . . . 5 ⊢ (( +𝑣 :V⟶V ∧ 𝑈 ∈ V) → (( /𝑔 ∘ +𝑣 )‘𝑈) = ( /𝑔 ‘( +𝑣 ‘𝑈))) | |
12 | 10, 11 | mpan 702 | . . . 4 ⊢ (𝑈 ∈ V → (( /𝑔 ∘ +𝑣 )‘𝑈) = ( /𝑔 ‘( +𝑣 ‘𝑈))) |
13 | 2, 12 | syl5eq 2656 | . . 3 ⊢ (𝑈 ∈ V → ( −𝑣 ‘𝑈) = ( /𝑔 ‘( +𝑣 ‘𝑈))) |
14 | 0ngrp 26749 | . . . . . 6 ⊢ ¬ ∅ ∈ GrpOp | |
15 | vex 3176 | . . . . . . . . . 10 ⊢ 𝑔 ∈ V | |
16 | 15 | rnex 6992 | . . . . . . . . 9 ⊢ ran 𝑔 ∈ V |
17 | 16, 16 | mpt2ex 7136 | . . . . . . . 8 ⊢ (𝑥 ∈ ran 𝑔, 𝑦 ∈ ran 𝑔 ↦ (𝑥𝑔((inv‘𝑔)‘𝑦))) ∈ V |
18 | df-gdiv 26734 | . . . . . . . 8 ⊢ /𝑔 = (𝑔 ∈ GrpOp ↦ (𝑥 ∈ ran 𝑔, 𝑦 ∈ ran 𝑔 ↦ (𝑥𝑔((inv‘𝑔)‘𝑦)))) | |
19 | 17, 18 | dmmpti 5936 | . . . . . . 7 ⊢ dom /𝑔 = GrpOp |
20 | 19 | eleq2i 2680 | . . . . . 6 ⊢ (∅ ∈ dom /𝑔 ↔ ∅ ∈ GrpOp) |
21 | 14, 20 | mtbir 312 | . . . . 5 ⊢ ¬ ∅ ∈ dom /𝑔 |
22 | ndmfv 6128 | . . . . 5 ⊢ (¬ ∅ ∈ dom /𝑔 → ( /𝑔 ‘∅) = ∅) | |
23 | 21, 22 | mp1i 13 | . . . 4 ⊢ (¬ 𝑈 ∈ V → ( /𝑔 ‘∅) = ∅) |
24 | fvprc 6097 | . . . . 5 ⊢ (¬ 𝑈 ∈ V → ( +𝑣 ‘𝑈) = ∅) | |
25 | 24 | fveq2d 6107 | . . . 4 ⊢ (¬ 𝑈 ∈ V → ( /𝑔 ‘( +𝑣 ‘𝑈)) = ( /𝑔 ‘∅)) |
26 | fvprc 6097 | . . . 4 ⊢ (¬ 𝑈 ∈ V → ( −𝑣 ‘𝑈) = ∅) | |
27 | 23, 25, 26 | 3eqtr4rd 2655 | . . 3 ⊢ (¬ 𝑈 ∈ V → ( −𝑣 ‘𝑈) = ( /𝑔 ‘( +𝑣 ‘𝑈))) |
28 | 13, 27 | pm2.61i 175 | . 2 ⊢ ( −𝑣 ‘𝑈) = ( /𝑔 ‘( +𝑣 ‘𝑈)) |
29 | vsfval.3 | . 2 ⊢ 𝑀 = ( −𝑣 ‘𝑈) | |
30 | vsfval.2 | . . 3 ⊢ 𝐺 = ( +𝑣 ‘𝑈) | |
31 | 30 | fveq2i 6106 | . 2 ⊢ ( /𝑔 ‘𝐺) = ( /𝑔 ‘( +𝑣 ‘𝑈)) |
32 | 28, 29, 31 | 3eqtr4i 2642 | 1 ⊢ 𝑀 = ( /𝑔 ‘𝐺) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∅c0 3874 dom cdm 5038 ran crn 5039 ∘ ccom 5042 ⟶wf 5800 –onto→wfo 5802 ‘cfv 5804 (class class class)co 6549 ↦ cmpt2 6551 1st c1st 7057 GrpOpcgr 26727 invcgn 26729 /𝑔 cgs 26730 +𝑣 cpv 26824 −𝑣 cnsb 26828 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-grpo 26731 df-gdiv 26734 df-va 26834 df-vs 26838 |
This theorem is referenced by: nvm 26880 nvmfval 26883 nvnnncan1 26886 nvaddsub 26894 |
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