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Theorem axhvmulid-zf 31636

Description: Derive Axiom ax-hvmulid 31654 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 31617	. . . 4 ⊢ ℋ = (BaseSet‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
3		axhil.1	. . . . 5 ⊢ 𝑈 = ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩
4	3	fveq2i 6917	. . . 4 ⊢ (BaseSet‘𝑈) = (BaseSet‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
5	2, 4	eqtr4i 2789	. . 3 ⊢ ℋ = (BaseSet‘𝑈)
6	1	hlnvi 31540	. . . 4 ⊢ 𝑈 ∈ NrmCVec
7	3, 6	h2hsm 31623	. . 3 ⊢ ·_ℎ = ( ·_𝑠OLD ‘𝑈)
8	5, 7	hlmulid 31553	. 2 ⊢ ((𝑈 ∈ CHil_OLD ∧ 𝐴 ∈ ℋ) → (1 ·_ℎ 𝐴) = 𝐴)
9	1, 8	mpan 703	1 ⊢ (𝐴 ∈ ℋ → (1 ·_ℎ 𝐴) = 𝐴)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 = wceq 1570 ∈ wcel 2147 ⟨cop 4838 ‘cfv 6564 (class class class)co 7449 1c1 11238 BaseSetcba 31234 CHil_OLDchlo 31533 ℋchba 31567 +_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-rep 5236 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-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 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-iun 4956 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-f1 6569 df-fo 6570 df-f1o 6571 df-fv 6572 df-ov 7452 df-oprab 7453 df-1st 8033 df-2nd 8034 df-vc 31207 df-nv 31240 df-va 31243 df-ba 31244 df-sm 31245 df-0v 31246 df-nmcv 31248 df-cbn 31511 df-hlo 31534 df-hba 31617

This theorem is used by: (None)

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