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Theorem h2hsm 31134

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 2761	. . . 4 ⊢ ( ·_𝑠OLD ‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩) = ( ·_𝑠OLD ‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
2	1	smfval 30764	. . 3 ⊢ ( ·_𝑠OLD ‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩) = (2^nd ‘(1^st ‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩))
3		opex 5428	. . . . 5 ⊢ ⟨ +_ℎ , ·_ℎ ⟩ ∈ V
4		h2h.1	. . . . . . . 8 ⊢ 𝑈 = ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩
5		h2h.2	. . . . . . . 8 ⊢ 𝑈 ∈ NrmCVec
6	4, 5	eqeltrri 2858	. . . . . . 7 ⊢ ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩ ∈ NrmCVec
7		nvex 30770	. . . . . . 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 1153	. . . . 5 ⊢ norm_ℎ ∈ V
10	3, 9	op1st 7972	. . . 4 ⊢ (1^st ‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩) = ⟨ +_ℎ , ·_ℎ ⟩
11	10	fveq2i 6864	. . 3 ⊢ (2^nd ‘(1^st ‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)) = (2^nd ‘⟨ +_ℎ , ·_ℎ ⟩)
12	8	simp1i 1151	. . . 4 ⊢ +_ℎ ∈ V
13	8	simp2i 1152	. . . 4 ⊢ ·_ℎ ∈ V
14	12, 13	op2nd 7973	. . 3 ⊢ (2^nd ‘⟨ +_ℎ , ·_ℎ ⟩) = ·_ℎ
15	2, 11, 14	3eqtrri 2789	. 2 ⊢ ·_ℎ = ( ·_𝑠OLD ‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
16	4	fveq2i 6864	. 2 ⊢ ( ·_𝑠OLD ‘𝑈) = ( ·_𝑠OLD ‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
17	15, 16	eqtr4i 2787	1 ⊢ ·_ℎ = ( ·_𝑠OLD ‘𝑈)

Colors of variables: wff setvar class

Syntax hints: ∧ w3a 1097 = wceq 1559 ∈ wcel 2141 Vcvv 3453 ⟨cop 4585 ‘cfv 6515 1^st c1st 7962 2^nd c2nd 7963 NrmCVeccnv 30743 ·_𝑠OLD cns 30746 +_ℎ cva 31079 ·_ℎ csm 31080 norm_ℎcno 31082

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-sep 5243 ax-nul 5253 ax-pr 5387 ax-un 7712

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-ral 3076 df-rex 3086 df-rab 3414 df-v 3455 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-nul 4284 df-if 4478 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4863 df-br 5098 df-opab 5160 df-mpt 5179 df-id 5538 df-xp 5649 df-rel 5650 df-cnv 5651 df-co 5652 df-dm 5653 df-rn 5654 df-res 5655 df-ima 5656 df-iota 6471 df-fun 6517 df-fn 6518 df-f 6519 df-fo 6521 df-fv 6523 df-oprab 7394 df-1st 7964 df-2nd 7965 df-vc 30718 df-nv 30751 df-sm 30756

This theorem is referenced by: h2hvs 31136 axhfvmul-zf 31146 axhvmulid-zf 31147 axhvmulass-zf 31148 axhvdistr1-zf 31149 axhvdistr2-zf 31150 axhvmul0-zf 31151 axhis3-zf 31155 hhsm 31328

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