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Theorem axhvmulid-zf 31583

Description: Derive Axiom ax-hvmulid 31601 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 31564	. . . 4 ⊢ ℋ = (BaseSet‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
3		axhil.1	. . . . 5 ⊢ 𝑈 = ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩
4	3	fveq2i 6886	. . . 4 ⊢ (BaseSet‘𝑈) = (BaseSet‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
5	2, 4	eqtr4i 2787	. . 3 ⊢ ℋ = (BaseSet‘𝑈)
6	1	hlnvi 31487	. . . 4 ⊢ 𝑈 ∈ NrmCVec
7	3, 6	h2hsm 31570	. . 3 ⊢ ·_ℎ = ( ·_𝑠OLD ‘𝑈)
8	5, 7	hlmulid 31500	. 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 2145 ⟨cop 4590 ‘cfv 6537 (class class class)co 7418 1c1 11194 BaseSetcba 31181 CHil_OLDchlo 31480 ℋchba 31514 +_ℎ cva 31515 ·_ℎ csm 31516 norm_ℎcno 31518

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 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pr 5391 ax-un 7749

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 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-ral 3078 df-rex 3088 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4280 df-if 4483 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-id 5546 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-ov 7421 df-oprab 7422 df-1st 7999 df-2nd 8000 df-vc 31154 df-nv 31187 df-va 31190 df-ba 31191 df-sm 31192 df-0v 31193 df-nmcv 31195 df-cbn 31458 df-hlo 31481 df-hba 31564

This theorem is used by: (None)

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