axhvdistr2-zf - Hilbert Space Explorer

	Hilbert Space Explorer		< Previous Next > Nearby theorems
Mirrors > Home > HSE Home > Th. List > axhvdistr2-zf		Structured version Visualization version GIF version

Theorem axhvdistr2-zf 29381

Description: Derive Axiom ax-hvdistr2 29399 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 29359	. . . 4 ⊢ ℋ = (BaseSet‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
3		axhil.1	. . . . 5 ⊢ 𝑈 = ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩
4	3	fveq2i 6795	. . . 4 ⊢ (BaseSet‘𝑈) = (BaseSet‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
5	2, 4	eqtr4i 2764	. . 3 ⊢ ℋ = (BaseSet‘𝑈)
6	1	hlnvi 29282	. . . 4 ⊢ 𝑈 ∈ NrmCVec
7	3, 6	h2hva 29364	. . 3 ⊢ +_ℎ = ( +_𝑣 ‘𝑈)
8	3, 6	h2hsm 29365	. . 3 ⊢ ·_ℎ = ( ·_𝑠OLD ‘𝑈)
9	5, 7, 8	hldir 29298	. 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 686	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 1085 = wceq 1537 ∈ wcel 2101 ⟨cop 4570 ‘cfv 6447 (class class class)co 7295 ℂcc 10897 + caddc 10902 BaseSetcba 28976 CHil_OLDchlo 29275 ℋchba 29309 +_ℎ cva 29310 ·_ℎ csm 29311 norm_ℎcno 29313

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2103 ax-9 2111 ax-10 2132 ax-11 2149 ax-12 2166 ax-ext 2704 ax-rep 5212 ax-sep 5226 ax-nul 5233 ax-pr 5355 ax-un 7608

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3an 1087 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2063 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2884 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3223 df-rab 3224 df-v 3436 df-sbc 3719 df-csb 3835 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-nul 4260 df-if 4463 df-sn 4565 df-pr 4567 df-op 4571 df-uni 4842 df-iun 4929 df-br 5078 df-opab 5140 df-mpt 5161 df-id 5491 df-xp 5597 df-rel 5598 df-cnv 5599 df-co 5600 df-dm 5601 df-rn 5602 df-res 5603 df-ima 5604 df-iota 6399 df-fun 6449 df-fn 6450 df-f 6451 df-f1 6452 df-fo 6453 df-f1o 6454 df-fv 6455 df-ov 7298 df-oprab 7299 df-1st 7851 df-2nd 7852 df-vc 28949 df-nv 28982 df-va 28985 df-ba 28986 df-sm 28987 df-0v 28988 df-nmcv 28990 df-cbn 29253 df-hlo 29276 df-hba 29359

This theorem is referenced by: (None)

W3C validator