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Mirrors > Home > HSE Home > Th. List > ellnfn | Structured version Visualization version GIF version |
Description: Property defining a linear functional. (Contributed by NM, 11-Feb-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ellnfn | ⊢ (𝑇 ∈ LinFn ↔ (𝑇: ℋ⟶ℂ ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ (𝑇‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) = ((𝑥 · (𝑇‘𝑦)) + (𝑇‘𝑧)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq1 6663 | . . . . . 6 ⊢ (𝑡 = 𝑇 → (𝑡‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) = (𝑇‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧))) | |
2 | fveq1 6663 | . . . . . . . 8 ⊢ (𝑡 = 𝑇 → (𝑡‘𝑦) = (𝑇‘𝑦)) | |
3 | 2 | oveq2d 7166 | . . . . . . 7 ⊢ (𝑡 = 𝑇 → (𝑥 · (𝑡‘𝑦)) = (𝑥 · (𝑇‘𝑦))) |
4 | fveq1 6663 | . . . . . . 7 ⊢ (𝑡 = 𝑇 → (𝑡‘𝑧) = (𝑇‘𝑧)) | |
5 | 3, 4 | oveq12d 7168 | . . . . . 6 ⊢ (𝑡 = 𝑇 → ((𝑥 · (𝑡‘𝑦)) + (𝑡‘𝑧)) = ((𝑥 · (𝑇‘𝑦)) + (𝑇‘𝑧))) |
6 | 1, 5 | eqeq12d 2837 | . . . . 5 ⊢ (𝑡 = 𝑇 → ((𝑡‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) = ((𝑥 · (𝑡‘𝑦)) + (𝑡‘𝑧)) ↔ (𝑇‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) = ((𝑥 · (𝑇‘𝑦)) + (𝑇‘𝑧)))) |
7 | 6 | ralbidv 3197 | . . . 4 ⊢ (𝑡 = 𝑇 → (∀𝑧 ∈ ℋ (𝑡‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) = ((𝑥 · (𝑡‘𝑦)) + (𝑡‘𝑧)) ↔ ∀𝑧 ∈ ℋ (𝑇‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) = ((𝑥 · (𝑇‘𝑦)) + (𝑇‘𝑧)))) |
8 | 7 | 2ralbidv 3199 | . . 3 ⊢ (𝑡 = 𝑇 → (∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ (𝑡‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) = ((𝑥 · (𝑡‘𝑦)) + (𝑡‘𝑧)) ↔ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ (𝑇‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) = ((𝑥 · (𝑇‘𝑦)) + (𝑇‘𝑧)))) |
9 | df-lnfn 29619 | . . 3 ⊢ LinFn = {𝑡 ∈ (ℂ ↑m ℋ) ∣ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ (𝑡‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) = ((𝑥 · (𝑡‘𝑦)) + (𝑡‘𝑧))} | |
10 | 8, 9 | elrab2 3682 | . 2 ⊢ (𝑇 ∈ LinFn ↔ (𝑇 ∈ (ℂ ↑m ℋ) ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ (𝑇‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) = ((𝑥 · (𝑇‘𝑦)) + (𝑇‘𝑧)))) |
11 | cnex 10612 | . . . 4 ⊢ ℂ ∈ V | |
12 | ax-hilex 28770 | . . . 4 ⊢ ℋ ∈ V | |
13 | 11, 12 | elmap 8429 | . . 3 ⊢ (𝑇 ∈ (ℂ ↑m ℋ) ↔ 𝑇: ℋ⟶ℂ) |
14 | 13 | anbi1i 625 | . 2 ⊢ ((𝑇 ∈ (ℂ ↑m ℋ) ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ (𝑇‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) = ((𝑥 · (𝑇‘𝑦)) + (𝑇‘𝑧))) ↔ (𝑇: ℋ⟶ℂ ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ (𝑇‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) = ((𝑥 · (𝑇‘𝑦)) + (𝑇‘𝑧)))) |
15 | 10, 14 | bitri 277 | 1 ⊢ (𝑇 ∈ LinFn ↔ (𝑇: ℋ⟶ℂ ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ (𝑇‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) = ((𝑥 · (𝑇‘𝑦)) + (𝑇‘𝑧)))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∀wral 3138 ⟶wf 6345 ‘cfv 6349 (class class class)co 7150 ↑m cmap 8400 ℂcc 10529 + caddc 10534 · cmul 10536 ℋchba 28690 +ℎ cva 28691 ·ℎ csm 28692 LinFnclf 28725 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-hilex 28770 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-fv 6357 df-ov 7153 df-oprab 7154 df-mpo 7155 df-map 8402 df-lnfn 29619 |
This theorem is referenced by: lnfnf 29655 lnfnl 29702 bralnfn 29719 0lnfn 29756 cnlnadjlem2 29839 |
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