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Theorem lnfnsubi 29481

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 11500	. . 3 ⊢ -1 ∈ ℂ
2		lnfnl.1	. . . 4 ⊢ 𝑇 ∈ LinFn
3	2	lnfnaddmuli 29480	. . 3 ⊢ ((-1 ∈ ℂ ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝑇‘(𝐴 +_ℎ (-1 ·_ℎ 𝐵))) = ((𝑇‘𝐴) + (-1 · (𝑇‘𝐵))))
4	1, 3	mp3an1 1521	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝑇‘(𝐴 +_ℎ (-1 ·_ℎ 𝐵))) = ((𝑇‘𝐴) + (-1 · (𝑇‘𝐵))))
5		hvsubval 28449	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 −_ℎ 𝐵) = (𝐴 +_ℎ (-1 ·_ℎ 𝐵)))
6	5	fveq2d 6452	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝑇‘(𝐴 −_ℎ 𝐵)) = (𝑇‘(𝐴 +_ℎ (-1 ·_ℎ 𝐵))))
7	2	lnfnfi 29476	. . . 4 ⊢ 𝑇: ℋ⟶ℂ
8	7	ffvelrni 6624	. . 3 ⊢ (𝐴 ∈ ℋ → (𝑇‘𝐴) ∈ ℂ)
9	7	ffvelrni 6624	. . 3 ⊢ (𝐵 ∈ ℋ → (𝑇‘𝐵) ∈ ℂ)
10		mulm1 10818	. . . . . 6 ⊢ ((𝑇‘𝐵) ∈ ℂ → (-1 · (𝑇‘𝐵)) = -(𝑇‘𝐵))
11	10	oveq2d 6940	. . . . 5 ⊢ ((𝑇‘𝐵) ∈ ℂ → ((𝑇‘𝐴) + (-1 · (𝑇‘𝐵))) = ((𝑇‘𝐴) + -(𝑇‘𝐵)))
12	11	adantl 475	. . . 4 ⊢ (((𝑇‘𝐴) ∈ ℂ ∧ (𝑇‘𝐵) ∈ ℂ) → ((𝑇‘𝐴) + (-1 · (𝑇‘𝐵))) = ((𝑇‘𝐴) + -(𝑇‘𝐵)))
13		negsub 10673	. . . 4 ⊢ (((𝑇‘𝐴) ∈ ℂ ∧ (𝑇‘𝐵) ∈ ℂ) → ((𝑇‘𝐴) + -(𝑇‘𝐵)) = ((𝑇‘𝐴) − (𝑇‘𝐵)))
14	12, 13	eqtr2d 2815	. . 3 ⊢ (((𝑇‘𝐴) ∈ ℂ ∧ (𝑇‘𝐵) ∈ ℂ) → ((𝑇‘𝐴) − (𝑇‘𝐵)) = ((𝑇‘𝐴) + (-1 · (𝑇‘𝐵))))
15	8, 9, 14	syl2an 589	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝑇‘𝐴) − (𝑇‘𝐵)) = ((𝑇‘𝐴) + (-1 · (𝑇‘𝐵))))
16	4, 6, 15	3eqtr4d 2824	1 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝑇‘(𝐴 −_ℎ 𝐵)) = ((𝑇‘𝐴) − (𝑇‘𝐵)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2107 ‘cfv 6137 (class class class)co 6924 ℂcc 10272 1c1 10275 + caddc 10277 · cmul 10279 − cmin 10608 -cneg 10609 ℋchba 28352 +_ℎ cva 28353 ·_ℎ csm 28354 −_ℎ cmv 28358 LinFnclf 28387

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-cnex 10330 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-hilex 28432 ax-hv0cl 28436 ax-hvaddid 28437 ax-hfvmul 28438 ax-hvmulid 28439

This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4674 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-id 5263 df-po 5276 df-so 5277 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-er 8028 df-map 8144 df-en 8244 df-dom 8245 df-sdom 8246 df-pnf 10415 df-mnf 10416 df-ltxr 10418 df-sub 10610 df-neg 10611 df-hvsub 28404 df-lnfn 29283

This theorem is referenced by: lnfnconi 29490 riesz3i 29497

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