ho2times - Hilbert Space Explorer

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Theorem ho2times 31673

Description: Two times a Hilbert space operator. (Contributed by NM, 26-Aug-2006.) (New usage is discouraged.)

**Assertion**
Ref	Expression
ho2times	⊢ (𝑇: ℋ⟶ ℋ → (2 ·_op 𝑇) = (𝑇 +_op 𝑇))

**Proof of Theorem ho2times**
Step	Hyp	Ref	Expression
1		df-2 12305	. . . 4 ⊢ 2 = (1 + 1)
2	1	oveq1i 7426	. . 3 ⊢ (2 ·_op 𝑇) = ((1 + 1) ·_op 𝑇)
3		ax-1cn 11196	. . . 4 ⊢ 1 ∈ ℂ
4		hoadddir 31658	. . . 4 ⊢ ((1 ∈ ℂ ∧ 1 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → ((1 + 1) ·_op 𝑇) = ((1 ·_op 𝑇) +_op (1 ·_op 𝑇)))
5	3, 3, 4	mp3an12 1447	. . 3 ⊢ (𝑇: ℋ⟶ ℋ → ((1 + 1) ·_op 𝑇) = ((1 ·_op 𝑇) +_op (1 ·_op 𝑇)))
6	2, 5	eqtrid 2777	. 2 ⊢ (𝑇: ℋ⟶ ℋ → (2 ·_op 𝑇) = ((1 ·_op 𝑇) +_op (1 ·_op 𝑇)))
7		hoadddi 31657	. . . 4 ⊢ ((1 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → (1 ·_op (𝑇 +_op 𝑇)) = ((1 ·_op 𝑇) +_op (1 ·_op 𝑇)))
8	3, 7	mp3an1 1444	. . 3 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → (1 ·_op (𝑇 +_op 𝑇)) = ((1 ·_op 𝑇) +_op (1 ·_op 𝑇)))
9	8	anidms 565	. 2 ⊢ (𝑇: ℋ⟶ ℋ → (1 ·_op (𝑇 +_op 𝑇)) = ((1 ·_op 𝑇) +_op (1 ·_op 𝑇)))
10		hoaddcl 31612	. . . 4 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → (𝑇 +_op 𝑇): ℋ⟶ ℋ)
11	10	anidms 565	. . 3 ⊢ (𝑇: ℋ⟶ ℋ → (𝑇 +_op 𝑇): ℋ⟶ ℋ)
12		homullid 31654	. . 3 ⊢ ((𝑇 +_op 𝑇): ℋ⟶ ℋ → (1 ·_op (𝑇 +_op 𝑇)) = (𝑇 +_op 𝑇))
13	11, 12	syl 17	. 2 ⊢ (𝑇: ℋ⟶ ℋ → (1 ·_op (𝑇 +_op 𝑇)) = (𝑇 +_op 𝑇))
14	6, 9, 13	3eqtr2d 2771	1 ⊢ (𝑇: ℋ⟶ ℋ → (2 ·_op 𝑇) = (𝑇 +_op 𝑇))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ⟶wf 6539 (class class class)co 7416 ℂcc 11136 1c1 11139 + caddc 11141 2c2 12297 ℋchba 30773 +_op chos 30792 ·_op chot 30793

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-1cn 11196 ax-addcl 11198 ax-hilex 30853 ax-hfvadd 30854 ax-hfvmul 30859 ax-hvmulid 30860 ax-hvdistr1 30862 ax-hvdistr2 30863

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7419 df-oprab 7420 df-mpo 7421 df-map 8845 df-2 12305 df-hosum 31584 df-homul 31585

This theorem is referenced by: opsqrlem6 31999

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