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Mirrors > Home > MPE Home > Th. List > nvmfval | 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, 11-Sep-2007.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nvmval.1 | ⊢ 𝑋 = (BaseSet‘𝑈) |
nvmval.2 | ⊢ 𝐺 = ( +𝑣 ‘𝑈) |
nvmval.4 | ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) |
nvmval.3 | ⊢ 𝑀 = ( −𝑣 ‘𝑈) |
Ref | Expression |
---|---|
nvmfval | ⊢ (𝑈 ∈ NrmCVec → 𝑀 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥𝐺(-1𝑆𝑦)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nvmval.2 | . . . 4 ⊢ 𝐺 = ( +𝑣 ‘𝑈) | |
2 | 1 | nvgrp 28048 | . . 3 ⊢ (𝑈 ∈ NrmCVec → 𝐺 ∈ GrpOp) |
3 | nvmval.1 | . . . . 5 ⊢ 𝑋 = (BaseSet‘𝑈) | |
4 | 3, 1 | bafval 28035 | . . . 4 ⊢ 𝑋 = ran 𝐺 |
5 | eqid 2778 | . . . 4 ⊢ (inv‘𝐺) = (inv‘𝐺) | |
6 | nvmval.3 | . . . . 5 ⊢ 𝑀 = ( −𝑣 ‘𝑈) | |
7 | 1, 6 | vsfval 28064 | . . . 4 ⊢ 𝑀 = ( /𝑔 ‘𝐺) |
8 | 4, 5, 7 | grpodivfval 27965 | . . 3 ⊢ (𝐺 ∈ GrpOp → 𝑀 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥𝐺((inv‘𝐺)‘𝑦)))) |
9 | 2, 8 | syl 17 | . 2 ⊢ (𝑈 ∈ NrmCVec → 𝑀 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥𝐺((inv‘𝐺)‘𝑦)))) |
10 | nvmval.4 | . . . . . 6 ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) | |
11 | 3, 1, 10, 5 | nvinv 28070 | . . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑦 ∈ 𝑋) → (-1𝑆𝑦) = ((inv‘𝐺)‘𝑦)) |
12 | 11 | 3adant2 1122 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (-1𝑆𝑦) = ((inv‘𝐺)‘𝑦)) |
13 | 12 | oveq2d 6940 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥𝐺(-1𝑆𝑦)) = (𝑥𝐺((inv‘𝐺)‘𝑦))) |
14 | 13 | mpt2eq3dva 6998 | . 2 ⊢ (𝑈 ∈ NrmCVec → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥𝐺(-1𝑆𝑦))) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥𝐺((inv‘𝐺)‘𝑦)))) |
15 | 9, 14 | eqtr4d 2817 | 1 ⊢ (𝑈 ∈ NrmCVec → 𝑀 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥𝐺(-1𝑆𝑦)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1071 = wceq 1601 ∈ wcel 2107 ‘cfv 6137 (class class class)co 6924 ↦ cmpt2 6926 1c1 10275 -cneg 10609 GrpOpcgr 27920 invcgn 27922 NrmCVeccnv 28015 +𝑣 cpv 28016 BaseSetcba 28017 ·𝑠OLD cns 28018 −𝑣 cnsb 28020 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4674 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-id 5263 df-po 5276 df-so 5277 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-1st 7447 df-2nd 7448 df-er 8028 df-en 8244 df-dom 8245 df-sdom 8246 df-pnf 10415 df-mnf 10416 df-ltxr 10418 df-sub 10610 df-neg 10611 df-grpo 27924 df-gid 27925 df-ginv 27926 df-gdiv 27927 df-ablo 27976 df-vc 27990 df-nv 28023 df-va 28026 df-ba 28027 df-sm 28028 df-0v 28029 df-vs 28030 df-nmcv 28031 |
This theorem is referenced by: nvmf 28076 cnnvm 28113 vmcn 28130 h2hvs 28410 |
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