lnopsubmuli - Hilbert Space Explorer

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

Theorem lnopsubmuli 32034

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 31072	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐶 ∈ ℋ) → (𝐴 ·_ℎ 𝐶) ∈ ℋ)
2		lnopl.1	. . . . . 6 ⊢ 𝑇 ∈ LinOp
3	2	lnopsubi 32033	. . . . 5 ⊢ ((𝐵 ∈ ℋ ∧ (𝐴 ·_ℎ 𝐶) ∈ ℋ) → (𝑇‘(𝐵 −_ℎ (𝐴 ·_ℎ 𝐶))) = ((𝑇‘𝐵) −_ℎ (𝑇‘(𝐴 ·_ℎ 𝐶))))
4	1, 3	sylan2 594	. . . 4 ⊢ ((𝐵 ∈ ℋ ∧ (𝐴 ∈ ℂ ∧ 𝐶 ∈ ℋ)) → (𝑇‘(𝐵 −_ℎ (𝐴 ·_ℎ 𝐶))) = ((𝑇‘𝐵) −_ℎ (𝑇‘(𝐴 ·_ℎ 𝐶))))
5	4	3impb 1115	. . 3 ⊢ ((𝐵 ∈ ℋ ∧ 𝐴 ∈ ℂ ∧ 𝐶 ∈ ℋ) → (𝑇‘(𝐵 −_ℎ (𝐴 ·_ℎ 𝐶))) = ((𝑇‘𝐵) −_ℎ (𝑇‘(𝐴 ·_ℎ 𝐶))))
6	5	3com12 1124	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝑇‘(𝐵 −_ℎ (𝐴 ·_ℎ 𝐶))) = ((𝑇‘𝐵) −_ℎ (𝑇‘(𝐴 ·_ℎ 𝐶))))
7	2	lnopmuli 32031	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐶 ∈ ℋ) → (𝑇‘(𝐴 ·_ℎ 𝐶)) = (𝐴 ·_ℎ (𝑇‘𝐶)))
8	7	oveq2d 7372	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((𝑇‘𝐵) −_ℎ (𝑇‘(𝐴 ·_ℎ 𝐶))) = ((𝑇‘𝐵) −_ℎ (𝐴 ·_ℎ (𝑇‘𝐶))))
9	8	3adant2 1132	. 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 2770	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 1087 = wceq 1542 ∈ wcel 2114 ‘cfv 6487 (class class class)co 7356 ℂcc 11025 ℋchba 30978 ·_ℎ csm 30980 −_ℎ cmv 30984 LinOpclo 31006

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2184 ax-ext 2707 ax-sep 5220 ax-nul 5230 ax-pow 5296 ax-pr 5364 ax-un 7678 ax-resscn 11084 ax-1cn 11085 ax-icn 11086 ax-addcl 11087 ax-addrcl 11088 ax-mulcl 11089 ax-mulrcl 11090 ax-mulcom 11091 ax-addass 11092 ax-mulass 11093 ax-distr 11094 ax-i2m1 11095 ax-1ne0 11096 ax-1rid 11097 ax-rnegex 11098 ax-rrecex 11099 ax-cnre 11100 ax-pre-lttri 11101 ax-pre-lttrn 11102 ax-pre-ltadd 11103 ax-hilex 31058 ax-hfvadd 31059 ax-hvass 31061 ax-hv0cl 31062 ax-hvaddid 31063 ax-hfvmul 31064 ax-hvmulid 31065 ax-hvdistr2 31068 ax-hvmul0 31069

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2538 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2810 df-nfc 2884 df-ne 2931 df-nel 3035 df-ral 3050 df-rex 3060 df-reu 3341 df-rab 3388 df-v 3429 df-sbc 3726 df-csb 3834 df-dif 3888 df-un 3890 df-in 3892 df-ss 3902 df-nul 4264 df-if 4457 df-pw 4533 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4841 df-iun 4925 df-br 5075 df-opab 5137 df-mpt 5156 df-id 5515 df-po 5528 df-so 5529 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-iota 6443 df-fun 6489 df-fn 6490 df-f 6491 df-f1 6492 df-fo 6493 df-f1o 6494 df-fv 6495 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-er 8632 df-map 8764 df-en 8883 df-dom 8884 df-sdom 8885 df-pnf 11170 df-mnf 11171 df-ltxr 11173 df-sub 11368 df-neg 11369 df-hvsub 31030 df-lnop 31900

This theorem is referenced by: lnopeq0lem1 32064 lnophmlem2 32076

W3C validator