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Mirrors > Home > MPE Home > Th. List > nvmval | Structured version Visualization version GIF version |
Description: Value of vector subtraction on a normed complex vector space. (Contributed by NM, 11-Sep-2007.) (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 |
---|---|
nvmval | ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝑀𝐵) = (𝐴𝐺(-1𝑆𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nvmval.2 | . . . 4 ⊢ 𝐺 = ( +𝑣 ‘𝑈) | |
2 | 1 | nvgrp 27600 | . . 3 ⊢ (𝑈 ∈ NrmCVec → 𝐺 ∈ GrpOp) |
3 | nvmval.1 | . . . . 5 ⊢ 𝑋 = (BaseSet‘𝑈) | |
4 | 3, 1 | bafval 27587 | . . . 4 ⊢ 𝑋 = ran 𝐺 |
5 | eqid 2651 | . . . 4 ⊢ (inv‘𝐺) = (inv‘𝐺) | |
6 | eqid 2651 | . . . 4 ⊢ ( /𝑔 ‘𝐺) = ( /𝑔 ‘𝐺) | |
7 | 4, 5, 6 | grpodivval 27517 | . . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴( /𝑔 ‘𝐺)𝐵) = (𝐴𝐺((inv‘𝐺)‘𝐵))) |
8 | 2, 7 | syl3an1 1399 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴( /𝑔 ‘𝐺)𝐵) = (𝐴𝐺((inv‘𝐺)‘𝐵))) |
9 | nvmval.3 | . . 3 ⊢ 𝑀 = ( −𝑣 ‘𝑈) | |
10 | 3, 1, 9, 6 | nvm 27624 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝑀𝐵) = (𝐴( /𝑔 ‘𝐺)𝐵)) |
11 | nvmval.4 | . . . . 5 ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) | |
12 | 3, 1, 11, 5 | nvinv 27622 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐵 ∈ 𝑋) → (-1𝑆𝐵) = ((inv‘𝐺)‘𝐵)) |
13 | 12 | 3adant2 1100 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (-1𝑆𝐵) = ((inv‘𝐺)‘𝐵)) |
14 | 13 | oveq2d 6706 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺(-1𝑆𝐵)) = (𝐴𝐺((inv‘𝐺)‘𝐵))) |
15 | 8, 10, 14 | 3eqtr4d 2695 | 1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝑀𝐵) = (𝐴𝐺(-1𝑆𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1054 = wceq 1523 ∈ wcel 2030 ‘cfv 5926 (class class class)co 6690 1c1 9975 -cneg 10305 GrpOpcgr 27471 invcgn 27473 /𝑔 cgs 27474 NrmCVeccnv 27567 +𝑣 cpv 27568 BaseSetcba 27569 ·𝑠OLD cns 27570 −𝑣 cnsb 27572 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-op 4217 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-id 5053 df-po 5064 df-so 5065 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-1st 7210 df-2nd 7211 df-er 7787 df-en 7998 df-dom 7999 df-sdom 8000 df-pnf 10114 df-mnf 10115 df-ltxr 10117 df-sub 10306 df-neg 10307 df-grpo 27475 df-gid 27476 df-ginv 27477 df-gdiv 27478 df-ablo 27527 df-vc 27542 df-nv 27575 df-va 27578 df-ba 27579 df-sm 27580 df-0v 27581 df-vs 27582 df-nmcv 27583 |
This theorem is referenced by: nvmval2 27626 nvmdi 27631 nvpncan2 27636 nvaddsub4 27640 nvmtri 27654 imsdval2 27670 nvnd 27671 ipval3 27692 sspmval 27716 isph 27805 dipsubdir 27831 |
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