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Theorem lnopsubi 29754

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 11754	. . 3 ⊢ -1 ∈ ℂ
2		lnopl.1	. . . 4 ⊢ 𝑇 ∈ LinOp
3	2	lnopaddmuli 29753	. . 3 ⊢ ((-1 ∈ ℂ ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝑇‘(𝐴 +_ℎ (-1 ·_ℎ 𝐵))) = ((𝑇‘𝐴) +_ℎ (-1 ·_ℎ (𝑇‘𝐵))))
4	1, 3	mp3an1 1444	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝑇‘(𝐴 +_ℎ (-1 ·_ℎ 𝐵))) = ((𝑇‘𝐴) +_ℎ (-1 ·_ℎ (𝑇‘𝐵))))
5		hvsubval 28796	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 −_ℎ 𝐵) = (𝐴 +_ℎ (-1 ·_ℎ 𝐵)))
6	5	fveq2d 6677	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝑇‘(𝐴 −_ℎ 𝐵)) = (𝑇‘(𝐴 +_ℎ (-1 ·_ℎ 𝐵))))
7	2	lnopfi 29749	. . . 4 ⊢ 𝑇: ℋ⟶ ℋ
8	7	ffvelrni 6853	. . 3 ⊢ (𝐴 ∈ ℋ → (𝑇‘𝐴) ∈ ℋ)
9	7	ffvelrni 6853	. . 3 ⊢ (𝐵 ∈ ℋ → (𝑇‘𝐵) ∈ ℋ)
10		hvsubval 28796	. . 3 ⊢ (((𝑇‘𝐴) ∈ ℋ ∧ (𝑇‘𝐵) ∈ ℋ) → ((𝑇‘𝐴) −_ℎ (𝑇‘𝐵)) = ((𝑇‘𝐴) +_ℎ (-1 ·_ℎ (𝑇‘𝐵))))
11	8, 9, 10	syl2an 597	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝑇‘𝐴) −_ℎ (𝑇‘𝐵)) = ((𝑇‘𝐴) +_ℎ (-1 ·_ℎ (𝑇‘𝐵))))
12	4, 6, 11	3eqtr4d 2869	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 1536 ∈ wcel 2113 ‘cfv 6358 (class class class)co 7159 ℂcc 10538 1c1 10541 -cneg 10874 ℋchba 28699 +_ℎ cva 28700 ·_ℎ csm 28701 −_ℎ cmv 28705 LinOpclo 28727

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-hilex 28779 ax-hfvadd 28780 ax-hvass 28782 ax-hv0cl 28783 ax-hvaddid 28784 ax-hfvmul 28785 ax-hvmulid 28786 ax-hvdistr2 28789 ax-hvmul0 28790

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-po 5477 df-so 5478 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-er 8292 df-map 8411 df-en 8513 df-dom 8514 df-sdom 8515 df-pnf 10680 df-mnf 10681 df-ltxr 10683 df-sub 10875 df-neg 10876 df-hvsub 28751 df-lnop 29621

This theorem is referenced by: lnopsubmuli 29755 lnopmulsubi 29756 hoddii 29769 lnopeq0lem1 29785 lnophmlem2 29797 lnopconi 29814

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