lnopmulsubi - Hilbert Space Explorer

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Theorem lnopmulsubi 31229

Description: Product/subtraction property of a linear Hilbert space operator. (Contributed by NM, 2-Jul-2005.) (New usage is discouraged.)

**Hypothesis**
Ref	Expression
lnopl.1	⊢ 𝑇 ∈ LinOp

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

**Proof of Theorem lnopmulsubi**
Step	Hyp	Ref	Expression
1		hvmulcl 30266	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) → (𝐴 ·_ℎ 𝐵) ∈ ℋ)
2		lnopl.1	. . . 4 ⊢ 𝑇 ∈ LinOp
3	2	lnopsubi 31227	. . 3 ⊢ (((𝐴 ·_ℎ 𝐵) ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝑇‘((𝐴 ·_ℎ 𝐵) −_ℎ 𝐶)) = ((𝑇‘(𝐴 ·_ℎ 𝐵)) −_ℎ (𝑇‘𝐶)))
4	1, 3	stoic3 1779	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝑇‘((𝐴 ·_ℎ 𝐵) −_ℎ 𝐶)) = ((𝑇‘(𝐴 ·_ℎ 𝐵)) −_ℎ (𝑇‘𝐶)))
5	2	lnopmuli 31225	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) → (𝑇‘(𝐴 ·_ℎ 𝐵)) = (𝐴 ·_ℎ (𝑇‘𝐵)))
6	5	3adant3 1133	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝑇‘(𝐴 ·_ℎ 𝐵)) = (𝐴 ·_ℎ (𝑇‘𝐵)))
7	6	oveq1d 7424	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝑇‘(𝐴 ·_ℎ 𝐵)) −_ℎ (𝑇‘𝐶)) = ((𝐴 ·_ℎ (𝑇‘𝐵)) −_ℎ (𝑇‘𝐶)))
8	4, 7	eqtrd 2773	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝑇‘((𝐴 ·_ℎ 𝐵) −_ℎ 𝐶)) = ((𝐴 ·_ℎ (𝑇‘𝐵)) −_ℎ (𝑇‘𝐶)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ‘cfv 6544 (class class class)co 7409 ℂcc 11108 ℋchba 30172 ·_ℎ csm 30174 −_ℎ cmv 30178 LinOpclo 30200

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-hilex 30252 ax-hfvadd 30253 ax-hvass 30255 ax-hv0cl 30256 ax-hvaddid 30257 ax-hfvmul 30258 ax-hvmulid 30259 ax-hvdistr2 30262 ax-hvmul0 30263

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-po 5589 df-so 5590 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-er 8703 df-map 8822 df-en 8940 df-dom 8941 df-sdom 8942 df-pnf 11250 df-mnf 11251 df-ltxr 11253 df-sub 11446 df-neg 11447 df-hvsub 30224 df-lnop 31094

This theorem is referenced by: lnophmlem2 31270

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