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Theorem axhvmulid-zf 31596

Description: Derive Axiom ax-hvmulid 31614 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 31577	. . . 4 ⊢ ℋ = (BaseSet‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
3		axhil.1	. . . . 5 ⊢ 𝑈 = ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩
4	3	fveq2i 6889	. . . 4 ⊢ (BaseSet‘𝑈) = (BaseSet‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
5	2, 4	eqtr4i 2786	. . 3 ⊢ ℋ = (BaseSet‘𝑈)
6	1	hlnvi 31500	. . . 4 ⊢ 𝑈 ∈ NrmCVec
7	3, 6	h2hsm 31583	. . 3 ⊢ ·_ℎ = ( ·_𝑠OLD ‘𝑈)
8	5, 7	hlmulid 31513	. 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 1569 ∈ wcel 2144 ⟨cop 4834 ‘cfv 6539 (class class class)co 7421 1c1 11205 BaseSetcba 31194 CHil_OLDchlo 31493 ℋchba 31527 +_ℎ cva 31528 ·_ℎ csm 31529 norm_ℎcno 31531

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pr 5390 ax-un 7752

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-ral 3077 df-rex 3087 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-sn 4585 df-pr 4587 df-op 4835 df-uni 4867 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-id 5545 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-iota 6494 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-ov 7424 df-oprab 7425 df-1st 8002 df-2nd 8003 df-vc 31167 df-nv 31200 df-va 31203 df-ba 31204 df-sm 31205 df-0v 31206 df-nmcv 31208 df-cbn 31471 df-hlo 31494 df-hba 31577

This theorem is used by: (None)

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