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Theorem nvmval 28525

Description: Value of vector subtraction on a normed complex vector space. (Contributed by NM, 11-Sep-2007.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
nvmval.1	⊢ 𝑋 = (BaseSet‘𝑈)
nvmval.2	⊢ 𝐺 = ( +_𝑣 ‘𝑈)
nvmval.4	⊢ 𝑆 = ( ·_𝑠OLD ‘𝑈)
nvmval.3	⊢ 𝑀 = ( −_𝑣 ‘𝑈)

**Assertion**
Ref	Expression
nvmval	⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝑀𝐵) = (𝐴𝐺(-1𝑆𝐵)))

**Proof of Theorem nvmval**
Step	Hyp	Ref	Expression
1		nvmval.2	. . . 4 ⊢ 𝐺 = ( +_𝑣 ‘𝑈)
2	1	nvgrp 28500	. . 3 ⊢ (𝑈 ∈ NrmCVec → 𝐺 ∈ GrpOp)
3		nvmval.1	. . . . 5 ⊢ 𝑋 = (BaseSet‘𝑈)
4	3, 1	bafval 28487	. . . 4 ⊢ 𝑋 = ran 𝐺
5		eqid 2759	. . . 4 ⊢ (inv‘𝐺) = (inv‘𝐺)
6		eqid 2759	. . . 4 ⊢ ( /_𝑔 ‘𝐺) = ( /_𝑔 ‘𝐺)
7	4, 5, 6	grpodivval 28418	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴( /_𝑔 ‘𝐺)𝐵) = (𝐴𝐺((inv‘𝐺)‘𝐵)))
8	2, 7	syl3an1 1161	. 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 28524	. 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 28522	. . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐵 ∈ 𝑋) → (-1𝑆𝐵) = ((inv‘𝐺)‘𝐵))
13	12	3adant2 1129	. . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (-1𝑆𝐵) = ((inv‘𝐺)‘𝐵))
14	13	oveq2d 7167	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺(-1𝑆𝐵)) = (𝐴𝐺((inv‘𝐺)‘𝐵)))
15	8, 10, 14	3eqtr4d 2804	1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝑀𝐵) = (𝐴𝐺(-1𝑆𝐵)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ w3a 1085 = wceq 1539 ∈ wcel 2112 ‘cfv 6336 (class class class)co 7151 1c1 10577 -cneg 10910 GrpOpcgr 28372 invcgn 28374 /_𝑔 cgs 28375 NrmCVeccnv 28467 +_𝑣 cpv 28468 BaseSetcba 28469 ·_𝑠OLD cns 28470 −_𝑣 cnsb 28472

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2159 ax-12 2176 ax-ext 2730 ax-rep 5157 ax-sep 5170 ax-nul 5177 ax-pow 5235 ax-pr 5299 ax-un 7460 ax-resscn 10633 ax-1cn 10634 ax-icn 10635 ax-addcl 10636 ax-addrcl 10637 ax-mulcl 10638 ax-mulrcl 10639 ax-mulcom 10640 ax-addass 10641 ax-mulass 10642 ax-distr 10643 ax-i2m1 10644 ax-1ne0 10645 ax-1rid 10646 ax-rnegex 10647 ax-rrecex 10648 ax-cnre 10649 ax-pre-lttri 10650 ax-pre-lttrn 10651 ax-pre-ltadd 10652

This theorem depends on definitions: df-bi 210 df-an 401 df-or 846 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2071 df-mo 2558 df-eu 2589 df-clab 2737 df-cleq 2751 df-clel 2831 df-nfc 2902 df-ne 2953 df-nel 3057 df-ral 3076 df-rex 3077 df-reu 3078 df-rab 3080 df-v 3412 df-sbc 3698 df-csb 3807 df-dif 3862 df-un 3864 df-in 3866 df-ss 3876 df-nul 4227 df-if 4422 df-pw 4497 df-sn 4524 df-pr 4526 df-op 4530 df-uni 4800 df-iun 4886 df-br 5034 df-opab 5096 df-mpt 5114 df-id 5431 df-po 5444 df-so 5445 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-iota 6295 df-fun 6338 df-fn 6339 df-f 6340 df-f1 6341 df-fo 6342 df-f1o 6343 df-fv 6344 df-riota 7109 df-ov 7154 df-oprab 7155 df-mpo 7156 df-1st 7694 df-2nd 7695 df-er 8300 df-en 8529 df-dom 8530 df-sdom 8531 df-pnf 10716 df-mnf 10717 df-ltxr 10719 df-sub 10911 df-neg 10912 df-grpo 28376 df-gid 28377 df-ginv 28378 df-gdiv 28379 df-ablo 28428 df-vc 28442 df-nv 28475 df-va 28478 df-ba 28479 df-sm 28480 df-0v 28481 df-vs 28482 df-nmcv 28483

This theorem is referenced by: nvmval2 28526 nvmdi 28531 nvpncan2 28536 nvaddsub4 28540 nvmtri 28554 imsdval2 28570 nvnd 28571 ipval3 28592 sspmval 28616 isph 28705 dipsubdir 28731

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