bramul - Hilbert Space Explorer

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

Theorem bramul 31466

Description: Linearity property of bra for multiplication. (Contributed by NM, 23-May-2006.) (New usage is discouraged.)

**Assertion**
Ref	Expression
bramul	⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((bra‘𝐴)‘(𝐵 ·_ℎ 𝐶)) = (𝐵 · ((bra‘𝐴)‘𝐶)))

**Proof of Theorem bramul**
Step	Hyp	Ref	Expression
1		ax-his3 30604	. . 3 ⊢ ((𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ ∧ 𝐴 ∈ ℋ) → ((𝐵 ·_ℎ 𝐶) ·_ih 𝐴) = (𝐵 · (𝐶 ·_ih 𝐴)))
2	1	3comr 1123	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((𝐵 ·_ℎ 𝐶) ·_ih 𝐴) = (𝐵 · (𝐶 ·_ih 𝐴)))
3		hvmulcl 30533	. . . 4 ⊢ ((𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → (𝐵 ·_ℎ 𝐶) ∈ ℋ)
4		braval 31464	. . . 4 ⊢ ((𝐴 ∈ ℋ ∧ (𝐵 ·_ℎ 𝐶) ∈ ℋ) → ((bra‘𝐴)‘(𝐵 ·_ℎ 𝐶)) = ((𝐵 ·_ℎ 𝐶) ·_ih 𝐴))
5	3, 4	sylan2 591	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ (𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ)) → ((bra‘𝐴)‘(𝐵 ·_ℎ 𝐶)) = ((𝐵 ·_ℎ 𝐶) ·_ih 𝐴))
6	5	3impb 1113	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((bra‘𝐴)‘(𝐵 ·_ℎ 𝐶)) = ((𝐵 ·_ℎ 𝐶) ·_ih 𝐴))
7		braval 31464	. . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((bra‘𝐴)‘𝐶) = (𝐶 ·_ih 𝐴))
8	7	3adant2 1129	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((bra‘𝐴)‘𝐶) = (𝐶 ·_ih 𝐴))
9	8	oveq2d 7427	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → (𝐵 · ((bra‘𝐴)‘𝐶)) = (𝐵 · (𝐶 ·_ih 𝐴)))
10	2, 6, 9	3eqtr4d 2780	1 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((bra‘𝐴)‘(𝐵 ·_ℎ 𝐶)) = (𝐵 · ((bra‘𝐴)‘𝐶)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 ∧ w3a 1085 = wceq 1539 ∈ wcel 2104 ‘cfv 6542 (class class class)co 7411 ℂcc 11110 · cmul 11117 ℋchba 30439 ·_ℎ csm 30441 ·_ih csp 30442 bracbr 30476

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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pr 5426 ax-hilex 30519 ax-hfvmul 30525 ax-his3 30604

This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-ov 7414 df-bra 31370

This theorem is referenced by: bralnfn 31468

W3C validator