lnopsubmuli - Hilbert Space Explorer

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Theorem lnopsubmuli 31922

Description: Subtraction/product 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
lnopsubmuli	⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝑇‘(𝐵 −_ℎ (𝐴 ·_ℎ 𝐶))) = ((𝑇‘𝐵) −_ℎ (𝐴 ·_ℎ (𝑇‘𝐶))))

**Proof of Theorem lnopsubmuli**
Step	Hyp	Ref	Expression
1		hvmulcl 30960	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐶 ∈ ℋ) → (𝐴 ·_ℎ 𝐶) ∈ ℋ)
2		lnopl.1	. . . . . 6 ⊢ 𝑇 ∈ LinOp
3	2	lnopsubi 31921	. . . . 5 ⊢ ((𝐵 ∈ ℋ ∧ (𝐴 ·_ℎ 𝐶) ∈ ℋ) → (𝑇‘(𝐵 −_ℎ (𝐴 ·_ℎ 𝐶))) = ((𝑇‘𝐵) −_ℎ (𝑇‘(𝐴 ·_ℎ 𝐶))))
4	1, 3	sylan2 593	. . . 4 ⊢ ((𝐵 ∈ ℋ ∧ (𝐴 ∈ ℂ ∧ 𝐶 ∈ ℋ)) → (𝑇‘(𝐵 −_ℎ (𝐴 ·_ℎ 𝐶))) = ((𝑇‘𝐵) −_ℎ (𝑇‘(𝐴 ·_ℎ 𝐶))))
5	4	3impb 1114	. . 3 ⊢ ((𝐵 ∈ ℋ ∧ 𝐴 ∈ ℂ ∧ 𝐶 ∈ ℋ) → (𝑇‘(𝐵 −_ℎ (𝐴 ·_ℎ 𝐶))) = ((𝑇‘𝐵) −_ℎ (𝑇‘(𝐴 ·_ℎ 𝐶))))
6	5	3com12 1123	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝑇‘(𝐵 −_ℎ (𝐴 ·_ℎ 𝐶))) = ((𝑇‘𝐵) −_ℎ (𝑇‘(𝐴 ·_ℎ 𝐶))))
7	2	lnopmuli 31919	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐶 ∈ ℋ) → (𝑇‘(𝐴 ·_ℎ 𝐶)) = (𝐴 ·_ℎ (𝑇‘𝐶)))
8	7	oveq2d 7429	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((𝑇‘𝐵) −_ℎ (𝑇‘(𝐴 ·_ℎ 𝐶))) = ((𝑇‘𝐵) −_ℎ (𝐴 ·_ℎ (𝑇‘𝐶))))
9	8	3adant2 1131	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝑇‘𝐵) −_ℎ (𝑇‘(𝐴 ·_ℎ 𝐶))) = ((𝑇‘𝐵) −_ℎ (𝐴 ·_ℎ (𝑇‘𝐶))))
10	6, 9	eqtrd 2769	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝑇‘(𝐵 −_ℎ (𝐴 ·_ℎ 𝐶))) = ((𝑇‘𝐵) −_ℎ (𝐴 ·_ℎ (𝑇‘𝐶))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1539 ∈ wcel 2107 ‘cfv 6541 (class class class)co 7413 ℂcc 11135 ℋchba 30866 ·_ℎ csm 30868 −_ℎ cmv 30872 LinOpclo 30894

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2706 ax-sep 5276 ax-nul 5286 ax-pow 5345 ax-pr 5412 ax-un 7737 ax-resscn 11194 ax-1cn 11195 ax-icn 11196 ax-addcl 11197 ax-addrcl 11198 ax-mulcl 11199 ax-mulrcl 11200 ax-mulcom 11201 ax-addass 11202 ax-mulass 11203 ax-distr 11204 ax-i2m1 11205 ax-1ne0 11206 ax-1rid 11207 ax-rnegex 11208 ax-rrecex 11209 ax-cnre 11210 ax-pre-lttri 11211 ax-pre-lttrn 11212 ax-pre-ltadd 11213 ax-hilex 30946 ax-hfvadd 30947 ax-hvass 30949 ax-hv0cl 30950 ax-hvaddid 30951 ax-hfvmul 30952 ax-hvmulid 30953 ax-hvdistr2 30956 ax-hvmul0 30957

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2808 df-nfc 2884 df-ne 2932 df-nel 3036 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4888 df-iun 4973 df-br 5124 df-opab 5186 df-mpt 5206 df-id 5558 df-po 5572 df-so 5573 df-xp 5671 df-rel 5672 df-cnv 5673 df-co 5674 df-dm 5675 df-rn 5676 df-res 5677 df-ima 5678 df-iota 6494 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7370 df-ov 7416 df-oprab 7417 df-mpo 7418 df-er 8727 df-map 8850 df-en 8968 df-dom 8969 df-sdom 8970 df-pnf 11279 df-mnf 11280 df-ltxr 11282 df-sub 11476 df-neg 11477 df-hvsub 30918 df-lnop 31788

This theorem is referenced by: lnopeq0lem1 31952 lnophmlem2 31964

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