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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 28388 | . . 3 ⊢ (𝑈 ∈ NrmCVec → 𝐺 ∈ GrpOp) |
3 | nvmval.1 | . . . . 5 ⊢ 𝑋 = (BaseSet‘𝑈) | |
4 | 3, 1 | bafval 28375 | . . . 4 ⊢ 𝑋 = ran 𝐺 |
5 | eqid 2821 | . . . 4 ⊢ (inv‘𝐺) = (inv‘𝐺) | |
6 | nvmval.3 | . . . . 5 ⊢ 𝑀 = ( −𝑣 ‘𝑈) | |
7 | 1, 6 | vsfval 28404 | . . . 4 ⊢ 𝑀 = ( /𝑔 ‘𝐺) |
8 | 4, 5, 7 | grpodivfval 28305 | . . 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 28410 | . . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑦 ∈ 𝑋) → (-1𝑆𝑦) = ((inv‘𝐺)‘𝑦)) |
12 | 11 | 3adant2 1127 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (-1𝑆𝑦) = ((inv‘𝐺)‘𝑦)) |
13 | 12 | oveq2d 7166 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥𝐺(-1𝑆𝑦)) = (𝑥𝐺((inv‘𝐺)‘𝑦))) |
14 | 13 | mpoeq3dva 7225 | . 2 ⊢ (𝑈 ∈ NrmCVec → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥𝐺(-1𝑆𝑦))) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥𝐺((inv‘𝐺)‘𝑦)))) |
15 | 9, 14 | eqtr4d 2859 | 1 ⊢ (𝑈 ∈ NrmCVec → 𝑀 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥𝐺(-1𝑆𝑦)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ‘cfv 6349 (class class class)co 7150 ∈ cmpo 7152 1c1 10532 -cneg 10865 GrpOpcgr 28260 invcgn 28262 NrmCVeccnv 28355 +𝑣 cpv 28356 BaseSetcba 28357 ·𝑠OLD cns 28358 −𝑣 cnsb 28360 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-po 5468 df-so 5469 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-1st 7683 df-2nd 7684 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-ltxr 10674 df-sub 10866 df-neg 10867 df-grpo 28264 df-gid 28265 df-ginv 28266 df-gdiv 28267 df-ablo 28316 df-vc 28330 df-nv 28363 df-va 28366 df-ba 28367 df-sm 28368 df-0v 28369 df-vs 28370 df-nmcv 28371 |
This theorem is referenced by: nvmf 28416 cnnvm 28453 vmcn 28470 h2hvs 28748 |
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