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| Description: Additive property of a linear Hilbert space functional. (Contributed by NM, 11-Feb-2006.) (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| lnfnl.1 | ⊢ 𝑇 ∈ LinFn | 
| Ref | Expression | 
|---|---|
| lnfnaddi | ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝑇‘(𝐴 +ℎ 𝐵)) = ((𝑇‘𝐴) + (𝑇‘𝐵))) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | ax-1cn 11213 | . . 3 ⊢ 1 ∈ ℂ | |
| 2 | lnfnl.1 | . . . 4 ⊢ 𝑇 ∈ LinFn | |
| 3 | 2 | lnfnli 32059 | . . 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 31025 | . . . 4 ⊢ (𝐴 ∈ ℋ → (1 ·ℎ 𝐴) = 𝐴) | |
| 6 | 5 | fvoveq1d 7453 | . . 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 32060 | . . . . . 6 ⊢ 𝑇: ℋ⟶ℂ | 
| 9 | 8 | ffvelcdmi 7103 | . . . . 5 ⊢ (𝐴 ∈ ℋ → (𝑇‘𝐴) ∈ ℂ) | 
| 10 | 9 | mullidd 11279 | . . . 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 7446 | . 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((1 · (𝑇‘𝐴)) + (𝑇‘𝐵)) = ((𝑇‘𝐴) + (𝑇‘𝐵))) | 
| 13 | 4, 7, 12 | 3eqtr3d 2785 | 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 6561 (class class class)co 7431 ℂcc 11153 1c1 11156 + caddc 11158 · cmul 11160 ℋchba 30938 +ℎ cva 30939 ·ℎ csm 30940 LinFnclf 30973 | 
| 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 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-mulcl 11217 ax-mulcom 11219 ax-mulass 11221 ax-distr 11222 ax-1rid 11225 ax-cnre 11228 ax-hilex 31018 ax-hvmulid 31025 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-map 8868 df-lnfn 31867 | 
| This theorem is referenced by: lnfnaddmuli 32064 nlelshi 32079 | 
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