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Theorem lnfnsubi 32135

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 12135	. . 3 ⊢ -1 ∈ ℂ
2		lnfnl.1	. . . 4 ⊢ 𝑇 ∈ LinFn
3	2	lnfnaddmuli 32134	. . 3 ⊢ ((-1 ∈ ℂ ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝑇‘(𝐴 +_ℎ (-1 ·_ℎ 𝐵))) = ((𝑇‘𝐴) + (-1 · (𝑇‘𝐵))))
4	1, 3	mp3an1 1456	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝑇‘(𝐴 +_ℎ (-1 ·_ℎ 𝐵))) = ((𝑇‘𝐴) + (-1 · (𝑇‘𝐵))))
5		hvsubval 31105	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 −_ℎ 𝐵) = (𝐴 +_ℎ (-1 ·_ℎ 𝐵)))
6	5	fveq2d 6831	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝑇‘(𝐴 −_ℎ 𝐵)) = (𝑇‘(𝐴 +_ℎ (-1 ·_ℎ 𝐵))))
7	2	lnfnfi 32130	. . . 4 ⊢ 𝑇: ℋ⟶ℂ
8	7	ffvelcdmi 7024	. . 3 ⊢ (𝐴 ∈ ℋ → (𝑇‘𝐴) ∈ ℂ)
9	7	ffvelcdmi 7024	. . 3 ⊢ (𝐵 ∈ ℋ → (𝑇‘𝐵) ∈ ℂ)
10		mulm1 11582	. . . . . 6 ⊢ ((𝑇‘𝐵) ∈ ℂ → (-1 · (𝑇‘𝐵)) = -(𝑇‘𝐵))
11	10	oveq2d 7372	. . . . 5 ⊢ ((𝑇‘𝐵) ∈ ℂ → ((𝑇‘𝐴) + (-1 · (𝑇‘𝐵))) = ((𝑇‘𝐴) + -(𝑇‘𝐵)))
12	11	adantl 482	. . . 4 ⊢ (((𝑇‘𝐴) ∈ ℂ ∧ (𝑇‘𝐵) ∈ ℂ) → ((𝑇‘𝐴) + (-1 · (𝑇‘𝐵))) = ((𝑇‘𝐴) + -(𝑇‘𝐵)))
13		negsub 11433	. . . 4 ⊢ (((𝑇‘𝐴) ∈ ℂ ∧ (𝑇‘𝐵) ∈ ℂ) → ((𝑇‘𝐴) + -(𝑇‘𝐵)) = ((𝑇‘𝐴) − (𝑇‘𝐵)))
14	12, 13	eqtr2d 2775	. . 3 ⊢ (((𝑇‘𝐴) ∈ ℂ ∧ (𝑇‘𝐵) ∈ ℂ) → ((𝑇‘𝐴) − (𝑇‘𝐵)) = ((𝑇‘𝐴) + (-1 · (𝑇‘𝐵))))
15	8, 9, 14	syl2an 602	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝑇‘𝐴) − (𝑇‘𝐵)) = ((𝑇‘𝐴) + (-1 · (𝑇‘𝐵))))
16	4, 6, 15	3eqtr4d 2784	1 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝑇‘(𝐴 −_ℎ 𝐵)) = ((𝑇‘𝐴) − (𝑇‘𝐵)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1547 ∈ wcel 2119 ‘cfv 6485 (class class class)co 7356 ℂcc 11027 1c1 11030 + caddc 11032 · cmul 11034 − cmin 11368 -cneg 11369 ℋchba 31008 +_ℎ cva 31009 ·_ℎ csm 31010 −_ℎ cmv 31014 LinFnclf 31043

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-sep 5218 ax-nul 5228 ax-pow 5294 ax-pr 5362 ax-un 7678 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-hilex 31088 ax-hv0cl 31092 ax-hvaddid 31093 ax-hfvmul 31094 ax-hvmulid 31095

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-nel 3039 df-ral 3054 df-rex 3064 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4262 df-if 4455 df-pw 4531 df-sn 4556 df-pr 4558 df-op 4562 df-uni 4839 df-iun 4923 df-br 5073 df-opab 5135 df-mpt 5154 df-id 5513 df-po 5526 df-so 5527 df-xp 5624 df-rel 5625 df-cnv 5626 df-co 5627 df-dm 5628 df-rn 5629 df-res 5630 df-ima 5631 df-iota 6441 df-fun 6487 df-fn 6488 df-f 6489 df-f1 6490 df-fo 6491 df-f1o 6492 df-fv 6493 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-er 8633 df-map 8765 df-en 8884 df-dom 8885 df-sdom 8886 df-pnf 11172 df-mnf 11173 df-ltxr 11175 df-sub 11370 df-neg 11371 df-hvsub 31060 df-lnfn 31937

This theorem is referenced by: lnfnconi 32144 riesz3i 32151

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