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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 29559 | . . 3 ⊢ (𝑈 ∈ NrmCVec → 𝐺 ∈ GrpOp) |
3 | nvmval.1 | . . . . 5 ⊢ 𝑋 = (BaseSet‘𝑈) | |
4 | 3, 1 | bafval 29546 | . . . 4 ⊢ 𝑋 = ran 𝐺 |
5 | eqid 2736 | . . . 4 ⊢ (inv‘𝐺) = (inv‘𝐺) | |
6 | nvmval.3 | . . . . 5 ⊢ 𝑀 = ( −𝑣 ‘𝑈) | |
7 | 1, 6 | vsfval 29575 | . . . 4 ⊢ 𝑀 = ( /𝑔 ‘𝐺) |
8 | 4, 5, 7 | grpodivfval 29476 | . . 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 29581 | . . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑦 ∈ 𝑋) → (-1𝑆𝑦) = ((inv‘𝐺)‘𝑦)) |
12 | 11 | 3adant2 1131 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (-1𝑆𝑦) = ((inv‘𝐺)‘𝑦)) |
13 | 12 | oveq2d 7373 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥𝐺(-1𝑆𝑦)) = (𝑥𝐺((inv‘𝐺)‘𝑦))) |
14 | 13 | mpoeq3dva 7434 | . 2 ⊢ (𝑈 ∈ NrmCVec → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥𝐺(-1𝑆𝑦))) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥𝐺((inv‘𝐺)‘𝑦)))) |
15 | 9, 14 | eqtr4d 2779 | 1 ⊢ (𝑈 ∈ NrmCVec → 𝑀 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥𝐺(-1𝑆𝑦)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ‘cfv 6496 (class class class)co 7357 ∈ cmpo 7359 1c1 11052 -cneg 11386 GrpOpcgr 29431 invcgn 29433 NrmCVeccnv 29526 +𝑣 cpv 29527 BaseSetcba 29528 ·𝑠OLD cns 29529 −𝑣 cnsb 29531 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-id 5531 df-po 5545 df-so 5546 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-1st 7921 df-2nd 7922 df-er 8648 df-en 8884 df-dom 8885 df-sdom 8886 df-pnf 11191 df-mnf 11192 df-ltxr 11194 df-sub 11387 df-neg 11388 df-grpo 29435 df-gid 29436 df-ginv 29437 df-gdiv 29438 df-ablo 29487 df-vc 29501 df-nv 29534 df-va 29537 df-ba 29538 df-sm 29539 df-0v 29540 df-vs 29541 df-nmcv 29542 |
This theorem is referenced by: nvmf 29587 cnnvm 29624 vmcn 29641 h2hvs 29919 |
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