lnopsubi - Hilbert Space Explorer

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Theorem lnopsubi 32112

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 12166	. . 3 ⊢ -1 ∈ ℂ
2		lnopl.1	. . . 4 ⊢ 𝑇 ∈ LinOp
3	2	lnopaddmuli 32111	. . 3 ⊢ ((-1 ∈ ℂ ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝑇‘(𝐴 +_ℎ (-1 ·_ℎ 𝐵))) = ((𝑇‘𝐴) +_ℎ (-1 ·_ℎ (𝑇‘𝐵))))
4	1, 3	mp3an1 1459	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝑇‘(𝐴 +_ℎ (-1 ·_ℎ 𝐵))) = ((𝑇‘𝐴) +_ℎ (-1 ·_ℎ (𝑇‘𝐵))))
5		hvsubval 31154	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 −_ℎ 𝐵) = (𝐴 +_ℎ (-1 ·_ℎ 𝐵)))
6	5	fveq2d 6856	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝑇‘(𝐴 −_ℎ 𝐵)) = (𝑇‘(𝐴 +_ℎ (-1 ·_ℎ 𝐵))))
7	2	lnopfi 32107	. . . 4 ⊢ 𝑇: ℋ⟶ ℋ
8	7	ffvelcdmi 7049	. . 3 ⊢ (𝐴 ∈ ℋ → (𝑇‘𝐴) ∈ ℋ)
9	7	ffvelcdmi 7049	. . 3 ⊢ (𝐵 ∈ ℋ → (𝑇‘𝐵) ∈ ℋ)
10		hvsubval 31154	. . 3 ⊢ (((𝑇‘𝐴) ∈ ℋ ∧ (𝑇‘𝐵) ∈ ℋ) → ((𝑇‘𝐴) −_ℎ (𝑇‘𝐵)) = ((𝑇‘𝐴) +_ℎ (-1 ·_ℎ (𝑇‘𝐵))))
11	8, 9, 10	syl2an 604	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝑇‘𝐴) −_ℎ (𝑇‘𝐵)) = ((𝑇‘𝐴) +_ℎ (-1 ·_ℎ (𝑇‘𝐵))))
12	4, 6, 11	3eqtr4d 2797	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 398 = wceq 1550 ∈ wcel 2132 ‘cfv 6506 (class class class)co 7381 ℂcc 11057 1c1 11060 -cneg 11401 ℋchba 31057 +_ℎ cva 31058 ·_ℎ csm 31059 −_ℎ cmv 31063 LinOpclo 31085

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1805 ax-4 1819 ax-5 1920 ax-6 1977 ax-7 2018 ax-8 2134 ax-9 2142 ax-10 2165 ax-11 2181 ax-12 2202 ax-ext 2724 ax-sep 5236 ax-nul 5246 ax-pow 5312 ax-pr 5380 ax-un 7703 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-hilex 31137 ax-hfvadd 31138 ax-hvass 31140 ax-hv0cl 31141 ax-hvaddid 31142 ax-hfvmul 31143 ax-hvmulid 31144 ax-hvdistr2 31147 ax-hvmul0 31148

This theorem depends on definitions: df-bi 209 df-an 399 df-or 857 df-3or 1096 df-3an 1097 df-tru 1553 df-fal 1563 df-ex 1790 df-nf 1794 df-sb 2081 df-mo 2556 df-eu 2586 df-clab 2731 df-cleq 2744 df-clel 2827 df-nfc 2901 df-ne 2948 df-nel 3052 df-ral 3067 df-rex 3077 df-reu 3358 df-rab 3405 df-v 3446 df-sbc 3736 df-csb 3844 df-dif 3898 df-un 3900 df-in 3902 df-ss 3912 df-nul 4277 df-if 4471 df-pw 4547 df-sn 4573 df-pr 4575 df-op 4579 df-uni 4856 df-iun 4941 df-br 5091 df-opab 5153 df-mpt 5172 df-id 5531 df-po 5544 df-so 5545 df-xp 5642 df-rel 5643 df-cnv 5644 df-co 5645 df-dm 5646 df-rn 5647 df-res 5648 df-ima 5649 df-iota 6462 df-fun 6508 df-fn 6509 df-f 6510 df-f1 6511 df-fo 6512 df-f1o 6513 df-fv 6514 df-riota 7338 df-ov 7384 df-oprab 7385 df-mpo 7386 df-er 8662 df-map 8794 df-en 8913 df-dom 8914 df-sdom 8915 df-pnf 11204 df-mnf 11205 df-ltxr 11207 df-sub 11402 df-neg 11403 df-hvsub 31109 df-lnop 31979

This theorem is referenced by: lnopsubmuli 32113 lnopmulsubi 32114 hoddii 32127 lnopeq0lem1 32143 lnophmlem2 32155 lnopconi 32172

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