lnopsubi - Hilbert Space Explorer

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

Theorem lnopsubi 30916

Description: Subtraction property for a linear Hilbert space operator. (Contributed by NM, 1-Jul-2005.) (New usage is discouraged.)

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

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

**Proof of Theorem lnopsubi**
Step	Hyp	Ref	Expression
1		neg1cn 12267	. . 3 ⊢ -1 ∈ ℂ
2		lnopl.1	. . . 4 ⊢ 𝑇 ∈ LinOp
3	2	lnopaddmuli 30915	. . 3 ⊢ ((-1 ∈ ℂ ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝑇‘(𝐴 +_ℎ (-1 ·_ℎ 𝐵))) = ((𝑇‘𝐴) +_ℎ (-1 ·_ℎ (𝑇‘𝐵))))
4	1, 3	mp3an1 1448	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝑇‘(𝐴 +_ℎ (-1 ·_ℎ 𝐵))) = ((𝑇‘𝐴) +_ℎ (-1 ·_ℎ (𝑇‘𝐵))))
5		hvsubval 29958	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 −_ℎ 𝐵) = (𝐴 +_ℎ (-1 ·_ℎ 𝐵)))
6	5	fveq2d 6846	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝑇‘(𝐴 −_ℎ 𝐵)) = (𝑇‘(𝐴 +_ℎ (-1 ·_ℎ 𝐵))))
7	2	lnopfi 30911	. . . 4 ⊢ 𝑇: ℋ⟶ ℋ
8	7	ffvelcdmi 7034	. . 3 ⊢ (𝐴 ∈ ℋ → (𝑇‘𝐴) ∈ ℋ)
9	7	ffvelcdmi 7034	. . 3 ⊢ (𝐵 ∈ ℋ → (𝑇‘𝐵) ∈ ℋ)
10		hvsubval 29958	. . 3 ⊢ (((𝑇‘𝐴) ∈ ℋ ∧ (𝑇‘𝐵) ∈ ℋ) → ((𝑇‘𝐴) −_ℎ (𝑇‘𝐵)) = ((𝑇‘𝐴) +_ℎ (-1 ·_ℎ (𝑇‘𝐵))))
11	8, 9, 10	syl2an 596	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝑇‘𝐴) −_ℎ (𝑇‘𝐵)) = ((𝑇‘𝐴) +_ℎ (-1 ·_ℎ (𝑇‘𝐵))))
12	4, 6, 11	3eqtr4d 2786	1 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝑇‘(𝐴 −_ℎ 𝐵)) = ((𝑇‘𝐴) −_ℎ (𝑇‘𝐵)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ‘cfv 6496 (class class class)co 7357 ℂcc 11049 1c1 11052 -cneg 11386 ℋchba 29861 +_ℎ cva 29862 ·_ℎ csm 29863 −_ℎ cmv 29867 LinOpclo 29889

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-hilex 29941 ax-hfvadd 29942 ax-hvass 29944 ax-hv0cl 29945 ax-hvaddid 29946 ax-hfvmul 29947 ax-hvmulid 29948 ax-hvdistr2 29951 ax-hvmul0 29952

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-id 5531 df-po 5545 df-so 5546 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-er 8648 df-map 8767 df-en 8884 df-dom 8885 df-sdom 8886 df-pnf 11191 df-mnf 11192 df-ltxr 11194 df-sub 11387 df-neg 11388 df-hvsub 29913 df-lnop 30783

This theorem is referenced by: lnopsubmuli 30917 lnopmulsubi 30918 hoddii 30931 lnopeq0lem1 30947 lnophmlem2 30959 lnopconi 30976

W3C validator