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Theorem nvmval 30671

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 30646	. . 3 ⊢ (𝑈 ∈ NrmCVec → 𝐺 ∈ GrpOp)
3		nvmval.1	. . . . 5 ⊢ 𝑋 = (BaseSet‘𝑈)
4	3, 1	bafval 30633	. . . 4 ⊢ 𝑋 = ran 𝐺
5		eqid 2735	. . . 4 ⊢ (inv‘𝐺) = (inv‘𝐺)
6		eqid 2735	. . . 4 ⊢ ( /_𝑔 ‘𝐺) = ( /_𝑔 ‘𝐺)
7	4, 5, 6	grpodivval 30564	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴( /_𝑔 ‘𝐺)𝐵) = (𝐴𝐺((inv‘𝐺)‘𝐵)))
8	2, 7	syl3an1 1162	. 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 30670	. 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 30668	. . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐵 ∈ 𝑋) → (-1𝑆𝐵) = ((inv‘𝐺)‘𝐵))
13	12	3adant2 1130	. . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (-1𝑆𝐵) = ((inv‘𝐺)‘𝐵))
14	13	oveq2d 7447	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺(-1𝑆𝐵)) = (𝐴𝐺((inv‘𝐺)‘𝐵)))
15	8, 10, 14	3eqtr4d 2785	1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝑀𝐵) = (𝐴𝐺(-1𝑆𝐵)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ‘cfv 6563 (class class class)co 7431 1c1 11154 -cneg 11491 GrpOpcgr 30518 invcgn 30520 /_𝑔 cgs 30521 NrmCVeccnv 30613 +_𝑣 cpv 30614 BaseSetcba 30615 ·_𝑠OLD cns 30616 −_𝑣 cnsb 30618

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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-po 5597 df-so 5598 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-ltxr 11298 df-sub 11492 df-neg 11493 df-grpo 30522 df-gid 30523 df-ginv 30524 df-gdiv 30525 df-ablo 30574 df-vc 30588 df-nv 30621 df-va 30624 df-ba 30625 df-sm 30626 df-0v 30627 df-vs 30628 df-nmcv 30629

This theorem is referenced by: nvmval2 30672 nvmdi 30677 nvpncan2 30682 nvaddsub4 30686 nvmtri 30700 imsdval2 30716 nvnd 30717 ipval3 30738 sspmval 30762 isph 30851 dipsubdir 30877

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