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Theorem nvmval 29758

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 29733	. . 3 ⊢ (𝑈 ∈ NrmCVec → 𝐺 ∈ GrpOp)
3		nvmval.1	. . . . 5 ⊢ 𝑋 = (BaseSet‘𝑈)
4	3, 1	bafval 29720	. . . 4 ⊢ 𝑋 = ran 𝐺
5		eqid 2731	. . . 4 ⊢ (inv‘𝐺) = (inv‘𝐺)
6		eqid 2731	. . . 4 ⊢ ( /_𝑔 ‘𝐺) = ( /_𝑔 ‘𝐺)
7	4, 5, 6	grpodivval 29651	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴( /_𝑔 ‘𝐺)𝐵) = (𝐴𝐺((inv‘𝐺)‘𝐵)))
8	2, 7	syl3an1 1163	. 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 29757	. 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 29755	. . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐵 ∈ 𝑋) → (-1𝑆𝐵) = ((inv‘𝐺)‘𝐵))
13	12	3adant2 1131	. . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (-1𝑆𝐵) = ((inv‘𝐺)‘𝐵))
14	13	oveq2d 7409	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺(-1𝑆𝐵)) = (𝐴𝐺((inv‘𝐺)‘𝐵)))
15	8, 10, 14	3eqtr4d 2781	1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝑀𝐵) = (𝐴𝐺(-1𝑆𝐵)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ‘cfv 6532 (class class class)co 7393 1c1 11093 -cneg 11427 GrpOpcgr 29605 invcgn 29607 /_𝑔 cgs 29608 NrmCVeccnv 29700 +_𝑣 cpv 29701 BaseSetcba 29702 ·_𝑠OLD cns 29703 −_𝑣 cnsb 29705

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 2702 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 ax-resscn 11149 ax-1cn 11150 ax-icn 11151 ax-addcl 11152 ax-addrcl 11153 ax-mulcl 11154 ax-mulrcl 11155 ax-mulcom 11156 ax-addass 11157 ax-mulass 11158 ax-distr 11159 ax-i2m1 11160 ax-1ne0 11161 ax-1rid 11162 ax-rnegex 11163 ax-rrecex 11164 ax-cnre 11165 ax-pre-lttri 11166 ax-pre-lttrn 11167 ax-pre-ltadd 11168

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 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-po 5581 df-so 5582 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-riota 7349 df-ov 7396 df-oprab 7397 df-mpo 7398 df-1st 7957 df-2nd 7958 df-er 8686 df-en 8923 df-dom 8924 df-sdom 8925 df-pnf 11232 df-mnf 11233 df-ltxr 11235 df-sub 11428 df-neg 11429 df-grpo 29609 df-gid 29610 df-ginv 29611 df-gdiv 29612 df-ablo 29661 df-vc 29675 df-nv 29708 df-va 29711 df-ba 29712 df-sm 29713 df-0v 29714 df-vs 29715 df-nmcv 29716

This theorem is referenced by: nvmval2 29759 nvmdi 29764 nvpncan2 29769 nvaddsub4 29773 nvmtri 29787 imsdval2 29803 nvnd 29804 ipval3 29825 sspmval 29849 isph 29938 dipsubdir 29964

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