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Theorem nvmval 30624

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 30599	. . 3 ⊢ (𝑈 ∈ NrmCVec → 𝐺 ∈ GrpOp)
3		nvmval.1	. . . . 5 ⊢ 𝑋 = (BaseSet‘𝑈)
4	3, 1	bafval 30586	. . . 4 ⊢ 𝑋 = ran 𝐺
5		eqid 2733	. . . 4 ⊢ (inv‘𝐺) = (inv‘𝐺)
6		eqid 2733	. . . 4 ⊢ ( /_𝑔 ‘𝐺) = ( /_𝑔 ‘𝐺)
7	4, 5, 6	grpodivval 30517	. . 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 30623	. 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 30621	. . . 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 7368	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺(-1𝑆𝐵)) = (𝐴𝐺((inv‘𝐺)‘𝐵)))
15	8, 10, 14	3eqtr4d 2778	1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝑀𝐵) = (𝐴𝐺(-1𝑆𝐵)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ w3a 1086 = wceq 1541 ∈ wcel 2113 ‘cfv 6486 (class class class)co 7352 1c1 11014 -cneg 11352 GrpOpcgr 30471 invcgn 30473 /_𝑔 cgs 30474 NrmCVeccnv 30566 +_𝑣 cpv 30567 BaseSetcba 30568 ·_𝑠OLD cns 30569 −_𝑣 cnsb 30571

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-rep 5219 ax-sep 5236 ax-nul 5246 ax-pow 5305 ax-pr 5372 ax-un 7674 ax-resscn 11070 ax-1cn 11071 ax-icn 11072 ax-addcl 11073 ax-addrcl 11074 ax-mulcl 11075 ax-mulrcl 11076 ax-mulcom 11077 ax-addass 11078 ax-mulass 11079 ax-distr 11080 ax-i2m1 11081 ax-1ne0 11082 ax-1rid 11083 ax-rnegex 11084 ax-rrecex 11085 ax-cnre 11086 ax-pre-lttri 11087 ax-pre-lttrn 11088 ax-pre-ltadd 11089

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-nel 3034 df-ral 3049 df-rex 3058 df-reu 3348 df-rab 3397 df-v 3439 df-sbc 3738 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-nul 4283 df-if 4475 df-pw 4551 df-sn 4576 df-pr 4578 df-op 4582 df-uni 4859 df-iun 4943 df-br 5094 df-opab 5156 df-mpt 5175 df-id 5514 df-po 5527 df-so 5528 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-riota 7309 df-ov 7355 df-oprab 7356 df-mpo 7357 df-1st 7927 df-2nd 7928 df-er 8628 df-en 8876 df-dom 8877 df-sdom 8878 df-pnf 11155 df-mnf 11156 df-ltxr 11158 df-sub 11353 df-neg 11354 df-grpo 30475 df-gid 30476 df-ginv 30477 df-gdiv 30478 df-ablo 30527 df-vc 30541 df-nv 30574 df-va 30577 df-ba 30578 df-sm 30579 df-0v 30580 df-vs 30581 df-nmcv 30582

This theorem is referenced by: nvmval2 30625 nvmdi 30630 nvpncan2 30635 nvaddsub4 30639 nvmtri 30653 imsdval2 30669 nvnd 30670 ipval3 30691 sspmval 30715 isph 30804 dipsubdir 30830

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