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Theorem lnfnsubi 29823

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 11752	. . 3 ⊢ -1 ∈ ℂ
2		lnfnl.1	. . . 4 ⊢ 𝑇 ∈ LinFn
3	2	lnfnaddmuli 29822	. . 3 ⊢ ((-1 ∈ ℂ ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝑇‘(𝐴 +_ℎ (-1 ·_ℎ 𝐵))) = ((𝑇‘𝐴) + (-1 · (𝑇‘𝐵))))
4	1, 3	mp3an1 1444	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝑇‘(𝐴 +_ℎ (-1 ·_ℎ 𝐵))) = ((𝑇‘𝐴) + (-1 · (𝑇‘𝐵))))
5		hvsubval 28793	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 −_ℎ 𝐵) = (𝐴 +_ℎ (-1 ·_ℎ 𝐵)))
6	5	fveq2d 6674	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝑇‘(𝐴 −_ℎ 𝐵)) = (𝑇‘(𝐴 +_ℎ (-1 ·_ℎ 𝐵))))
7	2	lnfnfi 29818	. . . 4 ⊢ 𝑇: ℋ⟶ℂ
8	7	ffvelrni 6850	. . 3 ⊢ (𝐴 ∈ ℋ → (𝑇‘𝐴) ∈ ℂ)
9	7	ffvelrni 6850	. . 3 ⊢ (𝐵 ∈ ℋ → (𝑇‘𝐵) ∈ ℂ)
10		mulm1 11081	. . . . . 6 ⊢ ((𝑇‘𝐵) ∈ ℂ → (-1 · (𝑇‘𝐵)) = -(𝑇‘𝐵))
11	10	oveq2d 7172	. . . . 5 ⊢ ((𝑇‘𝐵) ∈ ℂ → ((𝑇‘𝐴) + (-1 · (𝑇‘𝐵))) = ((𝑇‘𝐴) + -(𝑇‘𝐵)))
12	11	adantl 484	. . . 4 ⊢ (((𝑇‘𝐴) ∈ ℂ ∧ (𝑇‘𝐵) ∈ ℂ) → ((𝑇‘𝐴) + (-1 · (𝑇‘𝐵))) = ((𝑇‘𝐴) + -(𝑇‘𝐵)))
13		negsub 10934	. . . 4 ⊢ (((𝑇‘𝐴) ∈ ℂ ∧ (𝑇‘𝐵) ∈ ℂ) → ((𝑇‘𝐴) + -(𝑇‘𝐵)) = ((𝑇‘𝐴) − (𝑇‘𝐵)))
14	12, 13	eqtr2d 2857	. . 3 ⊢ (((𝑇‘𝐴) ∈ ℂ ∧ (𝑇‘𝐵) ∈ ℂ) → ((𝑇‘𝐴) − (𝑇‘𝐵)) = ((𝑇‘𝐴) + (-1 · (𝑇‘𝐵))))
15	8, 9, 14	syl2an 597	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝑇‘𝐴) − (𝑇‘𝐵)) = ((𝑇‘𝐴) + (-1 · (𝑇‘𝐵))))
16	4, 6, 15	3eqtr4d 2866	1 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝑇‘(𝐴 −_ℎ 𝐵)) = ((𝑇‘𝐴) − (𝑇‘𝐵)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ‘cfv 6355 (class class class)co 7156 ℂcc 10535 1c1 10538 + caddc 10540 · cmul 10542 − cmin 10870 -cneg 10871 ℋchba 28696 +_ℎ cva 28697 ·_ℎ csm 28698 −_ℎ cmv 28702 LinFnclf 28731

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-hilex 28776 ax-hv0cl 28780 ax-hvaddid 28781 ax-hfvmul 28782 ax-hvmulid 28783

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-po 5474 df-so 5475 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-er 8289 df-map 8408 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-ltxr 10680 df-sub 10872 df-neg 10873 df-hvsub 28748 df-lnfn 29625

This theorem is referenced by: lnfnconi 29832 riesz3i 29839

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