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Theorem axhvdistr2-zf 30893

Description: Derive Axiom ax-hvdistr2 30911 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
axhvdistr2-zf	⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((𝐴 + 𝐵) ·_ℎ 𝐶) = ((𝐴 ·_ℎ 𝐶) +_ℎ (𝐵 ·_ℎ 𝐶)))

**Proof of Theorem axhvdistr2-zf**
Step	Hyp	Ref	Expression
1		axhil.2	. 2 ⊢ 𝑈 ∈ CHil_OLD
2		df-hba 30871	. . . 4 ⊢ ℋ = (BaseSet‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
3		axhil.1	. . . . 5 ⊢ 𝑈 = ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩
4	3	fveq2i 6843	. . . 4 ⊢ (BaseSet‘𝑈) = (BaseSet‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
5	2, 4	eqtr4i 2755	. . 3 ⊢ ℋ = (BaseSet‘𝑈)
6	1	hlnvi 30794	. . . 4 ⊢ 𝑈 ∈ NrmCVec
7	3, 6	h2hva 30876	. . 3 ⊢ +_ℎ = ( +_𝑣 ‘𝑈)
8	3, 6	h2hsm 30877	. . 3 ⊢ ·_ℎ = ( ·_𝑠OLD ‘𝑈)
9	5, 7, 8	hldir 30810	. 2 ⊢ ((𝑈 ∈ CHil_OLD ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ)) → ((𝐴 + 𝐵) ·_ℎ 𝐶) = ((𝐴 ·_ℎ 𝐶) +_ℎ (𝐵 ·_ℎ 𝐶)))
10	1, 9	mpan 690	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((𝐴 + 𝐵) ·_ℎ 𝐶) = ((𝐴 ·_ℎ 𝐶) +_ℎ (𝐵 ·_ℎ 𝐶)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ⟨cop 4591 ‘cfv 6499 (class class class)co 7369 ℂcc 11042 + caddc 11047 BaseSetcba 30488 CHil_OLDchlo 30787 ℋchba 30821 +_ℎ cva 30822 ·_ℎ csm 30823 norm_ℎcno 30825

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5229 ax-sep 5246 ax-nul 5256 ax-pr 5382 ax-un 7691

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-ral 3045 df-rex 3054 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4293 df-if 4485 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4868 df-iun 4953 df-br 5103 df-opab 5165 df-mpt 5184 df-id 5526 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-ov 7372 df-oprab 7373 df-1st 7947 df-2nd 7948 df-vc 30461 df-nv 30494 df-va 30497 df-ba 30498 df-sm 30499 df-0v 30500 df-nmcv 30502 df-cbn 30765 df-hlo 30788 df-hba 30871

This theorem is referenced by: (None)

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