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Theorem h2hsm 28755

Description: The scalar product operation of Hilbert space. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
h2h.1	⊢ 𝑈 = ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩
h2h.2	⊢ 𝑈 ∈ NrmCVec

**Assertion**
Ref	Expression
h2hsm	⊢ ·_ℎ = ( ·_𝑠OLD ‘𝑈)

**Proof of Theorem h2hsm**
Step	Hyp	Ref	Expression
1		eqid 2824	. . . 4 ⊢ ( ·_𝑠OLD ‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩) = ( ·_𝑠OLD ‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
2	1	smfval 28385	. . 3 ⊢ ( ·_𝑠OLD ‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩) = (2^nd ‘(1^st ‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩))
3		opex 5359	. . . . 5 ⊢ ⟨ +_ℎ , ·_ℎ ⟩ ∈ V
4		h2h.1	. . . . . . . 8 ⊢ 𝑈 = ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩
5		h2h.2	. . . . . . . 8 ⊢ 𝑈 ∈ NrmCVec
6	4, 5	eqeltrri 2913	. . . . . . 7 ⊢ ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩ ∈ NrmCVec
7		nvex 28391	. . . . . . 7 ⊢ (⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩ ∈ NrmCVec → ( +_ℎ ∈ V ∧ ·_ℎ ∈ V ∧ norm_ℎ ∈ V))
8	6, 7	ax-mp 5	. . . . . 6 ⊢ ( +_ℎ ∈ V ∧ ·_ℎ ∈ V ∧ norm_ℎ ∈ V)
9	8	simp3i 1137	. . . . 5 ⊢ norm_ℎ ∈ V
10	3, 9	op1st 7700	. . . 4 ⊢ (1^st ‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩) = ⟨ +_ℎ , ·_ℎ ⟩
11	10	fveq2i 6676	. . 3 ⊢ (2^nd ‘(1^st ‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)) = (2^nd ‘⟨ +_ℎ , ·_ℎ ⟩)
12	8	simp1i 1135	. . . 4 ⊢ +_ℎ ∈ V
13	8	simp2i 1136	. . . 4 ⊢ ·_ℎ ∈ V
14	12, 13	op2nd 7701	. . 3 ⊢ (2^nd ‘⟨ +_ℎ , ·_ℎ ⟩) = ·_ℎ
15	2, 11, 14	3eqtrri 2852	. 2 ⊢ ·_ℎ = ( ·_𝑠OLD ‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
16	4	fveq2i 6676	. 2 ⊢ ( ·_𝑠OLD ‘𝑈) = ( ·_𝑠OLD ‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
17	15, 16	eqtr4i 2850	1 ⊢ ·_ℎ = ( ·_𝑠OLD ‘𝑈)

Colors of variables: wff setvar class

Syntax hints: ∧ w3a 1083 = wceq 1536 ∈ wcel 2113 Vcvv 3497 ⟨cop 4576 ‘cfv 6358 1^st c1st 7690 2^nd c2nd 7691 NrmCVeccnv 28364 ·_𝑠OLD cns 28367 +_ℎ cva 28700 ·_ℎ csm 28701 norm_ℎcno 28703

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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-fo 6364 df-fv 6366 df-oprab 7163 df-1st 7692 df-2nd 7693 df-vc 28339 df-nv 28372 df-sm 28377

This theorem is referenced by: h2hvs 28757 axhfvmul-zf 28767 axhvmulid-zf 28768 axhvmulass-zf 28769 axhvdistr1-zf 28770 axhvdistr2-zf 28771 axhvmul0-zf 28772 axhis3-zf 28776 hhsm 28949

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