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Theorem axhvmulid-zf 30958

Description: Derive Axiom ax-hvmulid 30976 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)

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

**Assertion**
Ref	Expression
axhvmulid-zf	⊢ (𝐴 ∈ ℋ → (1 ·_ℎ 𝐴) = 𝐴)

**Proof of Theorem axhvmulid-zf**
Step	Hyp	Ref	Expression
1		axhil.2	. 2 ⊢ 𝑈 ∈ CHil_OLD
2		df-hba 30939	. . . 4 ⊢ ℋ = (BaseSet‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
3		axhil.1	. . . . 5 ⊢ 𝑈 = ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩
4	3	fveq2i 6820	. . . 4 ⊢ (BaseSet‘𝑈) = (BaseSet‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
5	2, 4	eqtr4i 2756	. . 3 ⊢ ℋ = (BaseSet‘𝑈)
6	1	hlnvi 30862	. . . 4 ⊢ 𝑈 ∈ NrmCVec
7	3, 6	h2hsm 30945	. . 3 ⊢ ·_ℎ = ( ·_𝑠OLD ‘𝑈)
8	5, 7	hlmulid 30875	. 2 ⊢ ((𝑈 ∈ CHil_OLD ∧ 𝐴 ∈ ℋ) → (1 ·_ℎ 𝐴) = 𝐴)
9	1, 8	mpan 690	1 ⊢ (𝐴 ∈ ℋ → (1 ·_ℎ 𝐴) = 𝐴)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2110 ⟨cop 4580 ‘cfv 6477 (class class class)co 7341 1c1 10999 BaseSetcba 30556 CHil_OLDchlo 30855 ℋchba 30889 +_ℎ cva 30890 ·_ℎ csm 30891 norm_ℎcno 30893

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2112 ax-9 2120 ax-10 2143 ax-11 2159 ax-12 2179 ax-ext 2702 ax-rep 5215 ax-sep 5232 ax-nul 5242 ax-pr 5368 ax-un 7663

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-ral 3046 df-rex 3055 df-reu 3345 df-rab 3394 df-v 3436 df-sbc 3740 df-csb 3849 df-dif 3903 df-un 3905 df-in 3907 df-ss 3917 df-nul 4282 df-if 4474 df-sn 4575 df-pr 4577 df-op 4581 df-uni 4858 df-iun 4941 df-br 5090 df-opab 5152 df-mpt 5171 df-id 5509 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-iota 6433 df-fun 6479 df-fn 6480 df-f 6481 df-f1 6482 df-fo 6483 df-f1o 6484 df-fv 6485 df-ov 7344 df-oprab 7345 df-1st 7916 df-2nd 7917 df-vc 30529 df-nv 30562 df-va 30565 df-ba 30566 df-sm 30567 df-0v 30568 df-nmcv 30570 df-cbn 30833 df-hlo 30856 df-hba 30939

This theorem is referenced by: (None)

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