lnopmul - Hilbert Space Explorer

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Theorem lnopmul 31790

Description: Multiplicative property of a linear Hilbert space operator. (Contributed by NM, 13-Aug-2006.) (New usage is discouraged.)

**Assertion**
Ref	Expression
lnopmul	⊢ ((𝑇 ∈ LinOp ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) → (𝑇‘(𝐴 ·_ℎ 𝐵)) = (𝐴 ·_ℎ (𝑇‘𝐵)))

**Proof of Theorem lnopmul**
Step	Hyp	Ref	Expression
1		ax-hv0cl 30826	. . . 4 ⊢ 0_ℎ ∈ ℋ
2		lnopl 31737	. . . 4 ⊢ (((𝑇 ∈ LinOp ∧ 𝐴 ∈ ℂ) ∧ (𝐵 ∈ ℋ ∧ 0_ℎ ∈ ℋ)) → (𝑇‘((𝐴 ·_ℎ 𝐵) +_ℎ 0_ℎ)) = ((𝐴 ·_ℎ (𝑇‘𝐵)) +_ℎ (𝑇‘0_ℎ)))
3	1, 2	mpanr2 703	. . 3 ⊢ (((𝑇 ∈ LinOp ∧ 𝐴 ∈ ℂ) ∧ 𝐵 ∈ ℋ) → (𝑇‘((𝐴 ·_ℎ 𝐵) +_ℎ 0_ℎ)) = ((𝐴 ·_ℎ (𝑇‘𝐵)) +_ℎ (𝑇‘0_ℎ)))
4	3	3impa 1108	. 2 ⊢ ((𝑇 ∈ LinOp ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) → (𝑇‘((𝐴 ·_ℎ 𝐵) +_ℎ 0_ℎ)) = ((𝐴 ·_ℎ (𝑇‘𝐵)) +_ℎ (𝑇‘0_ℎ)))
5		hvmulcl 30836	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) → (𝐴 ·_ℎ 𝐵) ∈ ℋ)
6		ax-hvaddid 30827	. . . . 5 ⊢ ((𝐴 ·_ℎ 𝐵) ∈ ℋ → ((𝐴 ·_ℎ 𝐵) +_ℎ 0_ℎ) = (𝐴 ·_ℎ 𝐵))
7	5, 6	syl 17	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) → ((𝐴 ·_ℎ 𝐵) +_ℎ 0_ℎ) = (𝐴 ·_ℎ 𝐵))
8	7	3adant1 1128	. . 3 ⊢ ((𝑇 ∈ LinOp ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) → ((𝐴 ·_ℎ 𝐵) +_ℎ 0_ℎ) = (𝐴 ·_ℎ 𝐵))
9	8	fveq2d 6901	. 2 ⊢ ((𝑇 ∈ LinOp ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) → (𝑇‘((𝐴 ·_ℎ 𝐵) +_ℎ 0_ℎ)) = (𝑇‘(𝐴 ·_ℎ 𝐵)))
10		lnop0 31789	. . . . 5 ⊢ (𝑇 ∈ LinOp → (𝑇‘0_ℎ) = 0_ℎ)
11	10	oveq2d 7436	. . . 4 ⊢ (𝑇 ∈ LinOp → ((𝐴 ·_ℎ (𝑇‘𝐵)) +_ℎ (𝑇‘0_ℎ)) = ((𝐴 ·_ℎ (𝑇‘𝐵)) +_ℎ 0_ℎ))
12	11	3ad2ant1 1131	. . 3 ⊢ ((𝑇 ∈ LinOp ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) → ((𝐴 ·_ℎ (𝑇‘𝐵)) +_ℎ (𝑇‘0_ℎ)) = ((𝐴 ·_ℎ (𝑇‘𝐵)) +_ℎ 0_ℎ))
13		lnopf 31682	. . . . . . . 8 ⊢ (𝑇 ∈ LinOp → 𝑇: ℋ⟶ ℋ)
14	13	ffvelcdmda 7094	. . . . . . 7 ⊢ ((𝑇 ∈ LinOp ∧ 𝐵 ∈ ℋ) → (𝑇‘𝐵) ∈ ℋ)
15		hvmulcl 30836	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (𝑇‘𝐵) ∈ ℋ) → (𝐴 ·_ℎ (𝑇‘𝐵)) ∈ ℋ)
16	14, 15	sylan2 592	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (𝑇 ∈ LinOp ∧ 𝐵 ∈ ℋ)) → (𝐴 ·_ℎ (𝑇‘𝐵)) ∈ ℋ)
17	16	3impb 1113	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑇 ∈ LinOp ∧ 𝐵 ∈ ℋ) → (𝐴 ·_ℎ (𝑇‘𝐵)) ∈ ℋ)
18	17	3com12 1121	. . . 4 ⊢ ((𝑇 ∈ LinOp ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) → (𝐴 ·_ℎ (𝑇‘𝐵)) ∈ ℋ)
19		ax-hvaddid 30827	. . . 4 ⊢ ((𝐴 ·_ℎ (𝑇‘𝐵)) ∈ ℋ → ((𝐴 ·_ℎ (𝑇‘𝐵)) +_ℎ 0_ℎ) = (𝐴 ·_ℎ (𝑇‘𝐵)))
20	18, 19	syl 17	. . 3 ⊢ ((𝑇 ∈ LinOp ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) → ((𝐴 ·_ℎ (𝑇‘𝐵)) +_ℎ 0_ℎ) = (𝐴 ·_ℎ (𝑇‘𝐵)))
21	12, 20	eqtrd 2768	. 2 ⊢ ((𝑇 ∈ LinOp ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) → ((𝐴 ·_ℎ (𝑇‘𝐵)) +_ℎ (𝑇‘0_ℎ)) = (𝐴 ·_ℎ (𝑇‘𝐵)))
22	4, 9, 21	3eqtr3d 2776	1 ⊢ ((𝑇 ∈ LinOp ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) → (𝑇‘(𝐴 ·_ℎ 𝐵)) = (𝐴 ·_ℎ (𝑇‘𝐵)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 ‘cfv 6548 (class class class)co 7420 ℂcc 11137 ℋchba 30742 +_ℎ cva 30743 ·_ℎ csm 30744 0_ℎc0v 30747 LinOpclo 30770

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-hilex 30822 ax-hfvadd 30823 ax-hvass 30825 ax-hv0cl 30826 ax-hvaddid 30827 ax-hfvmul 30828 ax-hvmulid 30829 ax-hvdistr2 30832 ax-hvmul0 30833

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-po 5590 df-so 5591 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-er 8725 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11281 df-mnf 11282 df-ltxr 11284 df-sub 11477 df-neg 11478 df-hvsub 30794 df-lnop 31664

This theorem is referenced by: lnopmuli 31795 homco2 31800

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