lnfnaddi - Hilbert Space Explorer

	Hilbert Space Explorer		< Previous Next > Nearby theorems
Mirrors > Home > HSE Home > Th. List > lnfnaddi		Structured version Visualization version GIF version

Theorem lnfnaddi 32024

Description: Additive property of a linear Hilbert space functional. (Contributed by NM, 11-Feb-2006.) (New usage is discouraged.)

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

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

**Proof of Theorem lnfnaddi**
Step	Hyp	Ref	Expression
1		ax-1cn 11187	. . 3 ⊢ 1 ∈ ℂ
2		lnfnl.1	. . . 4 ⊢ 𝑇 ∈ LinFn
3	2	lnfnli 32021	. . 3 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝑇‘((1 ·_ℎ 𝐴) +_ℎ 𝐵)) = ((1 · (𝑇‘𝐴)) + (𝑇‘𝐵)))
4	1, 3	mp3an1 1450	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝑇‘((1 ·_ℎ 𝐴) +_ℎ 𝐵)) = ((1 · (𝑇‘𝐴)) + (𝑇‘𝐵)))
5		ax-hvmulid 30987	. . . 4 ⊢ (𝐴 ∈ ℋ → (1 ·_ℎ 𝐴) = 𝐴)
6	5	fvoveq1d 7427	. . 3 ⊢ (𝐴 ∈ ℋ → (𝑇‘((1 ·_ℎ 𝐴) +_ℎ 𝐵)) = (𝑇‘(𝐴 +_ℎ 𝐵)))
7	6	adantr 480	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝑇‘((1 ·_ℎ 𝐴) +_ℎ 𝐵)) = (𝑇‘(𝐴 +_ℎ 𝐵)))
8	2	lnfnfi 32022	. . . . . 6 ⊢ 𝑇: ℋ⟶ℂ
9	8	ffvelcdmi 7073	. . . . 5 ⊢ (𝐴 ∈ ℋ → (𝑇‘𝐴) ∈ ℂ)
10	9	mullidd 11253	. . . 4 ⊢ (𝐴 ∈ ℋ → (1 · (𝑇‘𝐴)) = (𝑇‘𝐴))
11	10	adantr 480	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (1 · (𝑇‘𝐴)) = (𝑇‘𝐴))
12	11	oveq1d 7420	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((1 · (𝑇‘𝐴)) + (𝑇‘𝐵)) = ((𝑇‘𝐴) + (𝑇‘𝐵)))
13	4, 7, 12	3eqtr3d 2778	1 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝑇‘(𝐴 +_ℎ 𝐵)) = ((𝑇‘𝐴) + (𝑇‘𝐵)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ‘cfv 6531 (class class class)co 7405 ℂcc 11127 1c1 11130 + caddc 11132 · cmul 11134 ℋchba 30900 +_ℎ cva 30901 ·_ℎ csm 30902 LinFnclf 30935

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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7729 ax-cnex 11185 ax-resscn 11186 ax-1cn 11187 ax-icn 11188 ax-addcl 11189 ax-mulcl 11191 ax-mulcom 11193 ax-mulass 11195 ax-distr 11196 ax-1rid 11199 ax-cnre 11202 ax-hilex 30980 ax-hvmulid 30987

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-ral 3052 df-rex 3061 df-rab 3416 df-v 3461 df-sbc 3766 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-op 4608 df-uni 4884 df-br 5120 df-opab 5182 df-id 5548 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-iota 6484 df-fun 6533 df-fn 6534 df-f 6535 df-fv 6539 df-ov 7408 df-oprab 7409 df-mpo 7410 df-map 8842 df-lnfn 31829

This theorem is referenced by: lnfnaddmuli 32026 nlelshi 32041

W3C validator