phop - Metamath Proof Explorer

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

Theorem phop 30889

Description: A complex inner product space in terms of ordered pair components. (Contributed by NM, 2-Apr-2007.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
phop.2	⊢ 𝐺 = ( +_𝑣 ‘𝑈)
phop.4	⊢ 𝑆 = ( ·_𝑠OLD ‘𝑈)
phop.6	⊢ 𝑁 = (norm_CV‘𝑈)

**Assertion**
Ref	Expression
phop	⊢ (𝑈 ∈ CPreHil_OLD → 𝑈 = ⟨⟨𝐺, 𝑆⟩, 𝑁⟩)

**Proof of Theorem phop**
Step	Hyp	Ref	Expression
1		phrel 30886	. . 3 ⊢ Rel CPreHil_OLD
2		1st2nd 7992	. . 3 ⊢ ((Rel CPreHil_OLD ∧ 𝑈 ∈ CPreHil_OLD) → 𝑈 = ⟨(1^st ‘𝑈), (2^nd ‘𝑈)⟩)
3	1, 2	mpan 691	. 2 ⊢ (𝑈 ∈ CPreHil_OLD → 𝑈 = ⟨(1^st ‘𝑈), (2^nd ‘𝑈)⟩)
4		phop.6	. . . . 5 ⊢ 𝑁 = (norm_CV‘𝑈)
5	4	nmcvfval 30678	. . . 4 ⊢ 𝑁 = (2^nd ‘𝑈)
6	5	opeq2i 4820	. . 3 ⊢ ⟨(1^st ‘𝑈), 𝑁⟩ = ⟨(1^st ‘𝑈), (2^nd ‘𝑈)⟩
7		phnv 30885	. . . . 5 ⊢ (𝑈 ∈ CPreHil_OLD → 𝑈 ∈ NrmCVec)
8		eqid 2736	. . . . . 6 ⊢ (1^st ‘𝑈) = (1^st ‘𝑈)
9	8	nvvc 30686	. . . . 5 ⊢ (𝑈 ∈ NrmCVec → (1^st ‘𝑈) ∈ CVec_OLD)
10		vcrel 30631	. . . . . . 7 ⊢ Rel CVec_OLD
11		1st2nd 7992	. . . . . . 7 ⊢ ((Rel CVec_OLD ∧ (1^st ‘𝑈) ∈ CVec_OLD) → (1^st ‘𝑈) = ⟨(1^st ‘(1^st ‘𝑈)), (2^nd ‘(1^st ‘𝑈))⟩)
12	10, 11	mpan 691	. . . . . 6 ⊢ ((1^st ‘𝑈) ∈ CVec_OLD → (1^st ‘𝑈) = ⟨(1^st ‘(1^st ‘𝑈)), (2^nd ‘(1^st ‘𝑈))⟩)
13		phop.2	. . . . . . . 8 ⊢ 𝐺 = ( +_𝑣 ‘𝑈)
14	13	vafval 30674	. . . . . . 7 ⊢ 𝐺 = (1^st ‘(1^st ‘𝑈))
15		phop.4	. . . . . . . 8 ⊢ 𝑆 = ( ·_𝑠OLD ‘𝑈)
16	15	smfval 30676	. . . . . . 7 ⊢ 𝑆 = (2^nd ‘(1^st ‘𝑈))
17	14, 16	opeq12i 4821	. . . . . 6 ⊢ ⟨𝐺, 𝑆⟩ = ⟨(1^st ‘(1^st ‘𝑈)), (2^nd ‘(1^st ‘𝑈))⟩
18	12, 17	eqtr4di 2789	. . . . 5 ⊢ ((1^st ‘𝑈) ∈ CVec_OLD → (1^st ‘𝑈) = ⟨𝐺, 𝑆⟩)
19	7, 9, 18	3syl 18	. . . 4 ⊢ (𝑈 ∈ CPreHil_OLD → (1^st ‘𝑈) = ⟨𝐺, 𝑆⟩)
20	19	opeq1d 4822	. . 3 ⊢ (𝑈 ∈ CPreHil_OLD → ⟨(1^st ‘𝑈), 𝑁⟩ = ⟨⟨𝐺, 𝑆⟩, 𝑁⟩)
21	6, 20	eqtr3id 2785	. 2 ⊢ (𝑈 ∈ CPreHil_OLD → ⟨(1^st ‘𝑈), (2^nd ‘𝑈)⟩ = ⟨⟨𝐺, 𝑆⟩, 𝑁⟩)
22	3, 21	eqtrd 2771	1 ⊢ (𝑈 ∈ CPreHil_OLD → 𝑈 = ⟨⟨𝐺, 𝑆⟩, 𝑁⟩)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2114 ⟨cop 4573 Rel wrel 5636 ‘cfv 6498 1^st c1st 7940 2^nd c2nd 7941 CVec_OLDcvc 30629 NrmCVeccnv 30655 +_𝑣 cpv 30656 ·_𝑠OLD cns 30658 norm_CVcnmcv 30661 CPreHil_OLDccphlo 30883

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pr 5375 ax-un 7689

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-ral 3052 df-rex 3062 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-nul 4274 df-if 4467 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4851 df-iun 4935 df-br 5086 df-opab 5148 df-mpt 5167 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 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-ov 7370 df-oprab 7371 df-1st 7942 df-2nd 7943 df-vc 30630 df-nv 30663 df-va 30666 df-ba 30667 df-sm 30668 df-0v 30669 df-nmcv 30671 df-ph 30884

This theorem is referenced by: phpar 30895

W3C validator