nvvop - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > nvvop		Structured version Visualization version GIF version

Theorem nvvop 30588

Description: The vector space component of a normed complex vector space is an ordered pair of the underlying group and a scalar product. (Contributed by NM, 28-Nov-2006.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
nvvop.1	⊢ 𝑊 = (1^st ‘𝑈)
nvvop.2	⊢ 𝐺 = ( +_𝑣 ‘𝑈)
nvvop.4	⊢ 𝑆 = ( ·_𝑠OLD ‘𝑈)

**Assertion**
Ref	Expression
nvvop	⊢ (𝑈 ∈ NrmCVec → 𝑊 = ⟨𝐺, 𝑆⟩)

**Proof of Theorem nvvop**
Step	Hyp	Ref	Expression
1		vcrel 30539	. . 3 ⊢ Rel CVec_OLD
2		nvss 30572	. . . . 5 ⊢ NrmCVec ⊆ (CVec_OLD × V)
3		nvvop.1	. . . . . . . 8 ⊢ 𝑊 = (1^st ‘𝑈)
4		eqid 2729	. . . . . . . 8 ⊢ (norm_CV‘𝑈) = (norm_CV‘𝑈)
5	3, 4	nvop2 30587	. . . . . . 7 ⊢ (𝑈 ∈ NrmCVec → 𝑈 = ⟨𝑊, (norm_CV‘𝑈)⟩)
6	5	eleq1d 2813	. . . . . 6 ⊢ (𝑈 ∈ NrmCVec → (𝑈 ∈ NrmCVec ↔ ⟨𝑊, (norm_CV‘𝑈)⟩ ∈ NrmCVec))
7	6	ibi 267	. . . . 5 ⊢ (𝑈 ∈ NrmCVec → ⟨𝑊, (norm_CV‘𝑈)⟩ ∈ NrmCVec)
8	2, 7	sselid 3941	. . . 4 ⊢ (𝑈 ∈ NrmCVec → ⟨𝑊, (norm_CV‘𝑈)⟩ ∈ (CVec_OLD × V))
9		opelxp1 5673	. . . 4 ⊢ (⟨𝑊, (norm_CV‘𝑈)⟩ ∈ (CVec_OLD × V) → 𝑊 ∈ CVec_OLD)
10	8, 9	syl 17	. . 3 ⊢ (𝑈 ∈ NrmCVec → 𝑊 ∈ CVec_OLD)
11		1st2nd 7997	. . 3 ⊢ ((Rel CVec_OLD ∧ 𝑊 ∈ CVec_OLD) → 𝑊 = ⟨(1^st ‘𝑊), (2^nd ‘𝑊)⟩)
12	1, 10, 11	sylancr 587	. 2 ⊢ (𝑈 ∈ NrmCVec → 𝑊 = ⟨(1^st ‘𝑊), (2^nd ‘𝑊)⟩)
13		nvvop.2	. . . . 5 ⊢ 𝐺 = ( +_𝑣 ‘𝑈)
14	13	vafval 30582	. . . 4 ⊢ 𝐺 = (1^st ‘(1^st ‘𝑈))
15	3	fveq2i 6843	. . . 4 ⊢ (1^st ‘𝑊) = (1^st ‘(1^st ‘𝑈))
16	14, 15	eqtr4i 2755	. . 3 ⊢ 𝐺 = (1^st ‘𝑊)
17		nvvop.4	. . . . 5 ⊢ 𝑆 = ( ·_𝑠OLD ‘𝑈)
18	17	smfval 30584	. . . 4 ⊢ 𝑆 = (2^nd ‘(1^st ‘𝑈))
19	3	fveq2i 6843	. . . 4 ⊢ (2^nd ‘𝑊) = (2^nd ‘(1^st ‘𝑈))
20	18, 19	eqtr4i 2755	. . 3 ⊢ 𝑆 = (2^nd ‘𝑊)
21	16, 20	opeq12i 4838	. 2 ⊢ ⟨𝐺, 𝑆⟩ = ⟨(1^st ‘𝑊), (2^nd ‘𝑊)⟩
22	12, 21	eqtr4di 2782	1 ⊢ (𝑈 ∈ NrmCVec → 𝑊 = ⟨𝐺, 𝑆⟩)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2109 Vcvv 3444 ⟨cop 4591 × cxp 5629 Rel wrel 5636 ‘cfv 6499 1^st c1st 7945 2^nd c2nd 7946 CVec_OLDcvc 30537 NrmCVeccnv 30563 +_𝑣 cpv 30564 ·_𝑠OLD cns 30566 norm_CVcnmcv 30569

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5246 ax-nul 5256 ax-pr 5382 ax-un 7691

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-ral 3045 df-rex 3054 df-rab 3403 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4293 df-if 4485 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4868 df-br 5103 df-opab 5165 df-mpt 5184 df-id 5526 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-fo 6505 df-fv 6507 df-oprab 7373 df-1st 7947 df-2nd 7948 df-vc 30538 df-nv 30571 df-va 30574 df-sm 30576 df-nmcv 30579

This theorem is referenced by: nvi 30593 nvvc 30594 nvop 30655

W3C validator