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Theorem h2hsm 31623

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 2763	. . . 4 ⊢ ( ·_𝑠OLD ‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩) = ( ·_𝑠OLD ‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
2	1	smfval 31253	. . 3 ⊢ ( ·_𝑠OLD ‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩) = (2^nd ‘(1^st ‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩))
3		opex 5437	. . . . 5 ⊢ ⟨ +_ℎ , ·_ℎ ⟩ ∈ V
4		h2h.1	. . . . . . . 8 ⊢ 𝑈 = ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩
5		h2h.2	. . . . . . . 8 ⊢ 𝑈 ∈ NrmCVec
6	4, 5	eqeltrri 2861	. . . . . . 7 ⊢ ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩ ∈ NrmCVec
7		nvex 31259	. . . . . . 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 1159	. . . . 5 ⊢ norm_ℎ ∈ V
10	3, 9	op1st 8041	. . . 4 ⊢ (1^st ‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩) = ⟨ +_ℎ , ·_ℎ ⟩
11	10	fveq2i 6917	. . 3 ⊢ (2^nd ‘(1^st ‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)) = (2^nd ‘⟨ +_ℎ , ·_ℎ ⟩)
12	8	simp1i 1157	. . . 4 ⊢ +_ℎ ∈ V
13	8	simp2i 1158	. . . 4 ⊢ ·_ℎ ∈ V
14	12, 13	op2nd 8042	. . 3 ⊢ (2^nd ‘⟨ +_ℎ , ·_ℎ ⟩) = ·_ℎ
15	2, 11, 14	3eqtrri 2791	. 2 ⊢ ·_ℎ = ( ·_𝑠OLD ‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
16	4	fveq2i 6917	. 2 ⊢ ( ·_𝑠OLD ‘𝑈) = ( ·_𝑠OLD ‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
17	15, 16	eqtr4i 2789	1 ⊢ ·_ℎ = ( ·_𝑠OLD ‘𝑈)

Colors of variables: wff setvar class

This proof depends on syntax axioms: ∧ w3a 1103 = wceq 1570 ∈ wcel 2147 Vcvv 3454 ⟨cop 4838 ‘cfv 6564 1^st c1st 8031 2^nd c2nd 8032 NrmCVeccnv 31232 ·_𝑠OLD cns 31235 +_ℎ cva 31568 ·_ℎ csm 31569 norm_ℎcno 31571

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-sep 5253 ax-nul 5264 ax-pr 5396 ax-un 7781

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-ral 3081 df-rex 3091 df-rab 3417 df-v 3456 df-ss 3901 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-sn 4589 df-pr 4591 df-op 4839 df-uni 4871 df-br 5107 df-opab 5172 df-mpt 5191 df-id 5551 df-xp 5662 df-rel 5663 df-cnv 5664 df-co 5665 df-dm 5666 df-rn 5667 df-res 5668 df-ima 5669 df-iota 6518 df-fun 6566 df-fn 6567 df-f 6568 df-fo 6570 df-fv 6572 df-oprab 7453 df-1st 8033 df-2nd 8034 df-vc 31207 df-nv 31240 df-sm 31245

This theorem is used by: h2hvs 31625 axhfvmul-zf 31635 axhvmulid-zf 31636 axhvmulass-zf 31637 axhvdistr1-zf 31638 axhvdistr2-zf 31639 axhvmul0-zf 31640 axhis3-zf 31644 hhsm 31817

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