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Theorem lnfnsubi 30127

Description: Subtraction property for a linear Hilbert space functional. (Contributed by NM, 13-Feb-2006.) (New usage is discouraged.)

**Hypothesis**
Ref	Expression
lnfnl.1	⊢ 𝑇 ∈ LinFn

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

**Proof of Theorem lnfnsubi**
Step	Hyp	Ref	Expression
1		neg1cn 11944	. . 3 ⊢ -1 ∈ ℂ
2		lnfnl.1	. . . 4 ⊢ 𝑇 ∈ LinFn
3	2	lnfnaddmuli 30126	. . 3 ⊢ ((-1 ∈ ℂ ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝑇‘(𝐴 +_ℎ (-1 ·_ℎ 𝐵))) = ((𝑇‘𝐴) + (-1 · (𝑇‘𝐵))))
4	1, 3	mp3an1 1450	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝑇‘(𝐴 +_ℎ (-1 ·_ℎ 𝐵))) = ((𝑇‘𝐴) + (-1 · (𝑇‘𝐵))))
5		hvsubval 29097	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 −_ℎ 𝐵) = (𝐴 +_ℎ (-1 ·_ℎ 𝐵)))
6	5	fveq2d 6721	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝑇‘(𝐴 −_ℎ 𝐵)) = (𝑇‘(𝐴 +_ℎ (-1 ·_ℎ 𝐵))))
7	2	lnfnfi 30122	. . . 4 ⊢ 𝑇: ℋ⟶ℂ
8	7	ffvelrni 6903	. . 3 ⊢ (𝐴 ∈ ℋ → (𝑇‘𝐴) ∈ ℂ)
9	7	ffvelrni 6903	. . 3 ⊢ (𝐵 ∈ ℋ → (𝑇‘𝐵) ∈ ℂ)
10		mulm1 11273	. . . . . 6 ⊢ ((𝑇‘𝐵) ∈ ℂ → (-1 · (𝑇‘𝐵)) = -(𝑇‘𝐵))
11	10	oveq2d 7229	. . . . 5 ⊢ ((𝑇‘𝐵) ∈ ℂ → ((𝑇‘𝐴) + (-1 · (𝑇‘𝐵))) = ((𝑇‘𝐴) + -(𝑇‘𝐵)))
12	11	adantl 485	. . . 4 ⊢ (((𝑇‘𝐴) ∈ ℂ ∧ (𝑇‘𝐵) ∈ ℂ) → ((𝑇‘𝐴) + (-1 · (𝑇‘𝐵))) = ((𝑇‘𝐴) + -(𝑇‘𝐵)))
13		negsub 11126	. . . 4 ⊢ (((𝑇‘𝐴) ∈ ℂ ∧ (𝑇‘𝐵) ∈ ℂ) → ((𝑇‘𝐴) + -(𝑇‘𝐵)) = ((𝑇‘𝐴) − (𝑇‘𝐵)))
14	12, 13	eqtr2d 2778	. . 3 ⊢ (((𝑇‘𝐴) ∈ ℂ ∧ (𝑇‘𝐵) ∈ ℂ) → ((𝑇‘𝐴) − (𝑇‘𝐵)) = ((𝑇‘𝐴) + (-1 · (𝑇‘𝐵))))
15	8, 9, 14	syl2an 599	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝑇‘𝐴) − (𝑇‘𝐵)) = ((𝑇‘𝐴) + (-1 · (𝑇‘𝐵))))
16	4, 6, 15	3eqtr4d 2787	1 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝑇‘(𝐴 −_ℎ 𝐵)) = ((𝑇‘𝐴) − (𝑇‘𝐵)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2110 ‘cfv 6380 (class class class)co 7213 ℂcc 10727 1c1 10730 + caddc 10732 · cmul 10734 − cmin 11062 -cneg 11063 ℋchba 29000 +_ℎ cva 29001 ·_ℎ csm 29002 −_ℎ cmv 29006 LinFnclf 29035

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-hilex 29080 ax-hv0cl 29084 ax-hvaddid 29085 ax-hfvmul 29086 ax-hvmulid 29087

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-id 5455 df-po 5468 df-so 5469 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-er 8391 df-map 8510 df-en 8627 df-dom 8628 df-sdom 8629 df-pnf 10869 df-mnf 10870 df-ltxr 10872 df-sub 11064 df-neg 11065 df-hvsub 29052 df-lnfn 29929

This theorem is referenced by: lnfnconi 30136 riesz3i 30143

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