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Theorem h2hsm 31356

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 2765	. . . 4 ⊢ ( ·_𝑠OLD ‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩) = ( ·_𝑠OLD ‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
2	1	smfval 30986	. . 3 ⊢ ( ·_𝑠OLD ‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩) = (2^nd ‘(1^st ‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩))
3		opex 5447	. . . . 5 ⊢ ⟨ +_ℎ , ·_ℎ ⟩ ∈ V
4		h2h.1	. . . . . . . 8 ⊢ 𝑈 = ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩
5		h2h.2	. . . . . . . 8 ⊢ 𝑈 ∈ NrmCVec
6	4, 5	eqeltrri 2862	. . . . . . 7 ⊢ ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩ ∈ NrmCVec
7		nvex 30992	. . . . . . 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 7996	. . . 4 ⊢ (1^st ‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩) = ⟨ +_ℎ , ·_ℎ ⟩
11	10	fveq2i 6888	. . 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 7997	. . 3 ⊢ (2^nd ‘⟨ +_ℎ , ·_ℎ ⟩) = ·_ℎ
15	2, 11, 14	3eqtrri 2793	. 2 ⊢ ·_ℎ = ( ·_𝑠OLD ‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
16	4	fveq2i 6888	. 2 ⊢ ( ·_𝑠OLD ‘𝑈) = ( ·_𝑠OLD ‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
17	15, 16	eqtr4i 2791	1 ⊢ ·_ℎ = ( ·_𝑠OLD ‘𝑈)

Colors of variables: wff setvar class

This proof depends on syntax axioms: ∧ w3a 1103 = wceq 1570 ∈ wcel 2146 Vcvv 3457 ⟨cop 4597 ‘cfv 6540 1^st c1st 7986 2^nd c2nd 7987 NrmCVeccnv 30965 ·_𝑠OLD cns 30968 +_ℎ cva 31301 ·_ℎ csm 31302 norm_ℎcno 31304

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 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-sep 5259 ax-nul 5271 ax-pr 5406 ax-un 7738

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 2100 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-ral 3082 df-rex 3092 df-rab 3419 df-v 3459 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-nul 4287 df-if 4490 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4875 df-br 5112 df-opab 5176 df-mpt 5195 df-id 5558 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-fo 6546 df-fv 6548 df-oprab 7420 df-1st 7988 df-2nd 7989 df-vc 30940 df-nv 30973 df-sm 30978

This theorem is used by: h2hvs 31358 axhfvmul-zf 31368 axhvmulid-zf 31369 axhvmulass-zf 31370 axhvdistr1-zf 31371 axhvdistr2-zf 31372 axhvmul0-zf 31373 axhis3-zf 31377 hhsm 31550

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