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Theorem ellnfn 29072
 Description: Property defining a linear functional. (Contributed by NM, 11-Feb-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
Assertion
Ref Expression
ellnfn (𝑇 ∈ LinFn ↔ (𝑇: ℋ⟶ℂ ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ (𝑇‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 · (𝑇𝑦)) + (𝑇𝑧))))
Distinct variable group:   𝑥,𝑦,𝑧,𝑇

Proof of Theorem ellnfn
Dummy variable 𝑡 is distinct from all other variables.
StepHypRef Expression
1 fveq1 6352 . . . . . 6 (𝑡 = 𝑇 → (𝑡‘((𝑥 · 𝑦) + 𝑧)) = (𝑇‘((𝑥 · 𝑦) + 𝑧)))
2 fveq1 6352 . . . . . . . 8 (𝑡 = 𝑇 → (𝑡𝑦) = (𝑇𝑦))
32oveq2d 6830 . . . . . . 7 (𝑡 = 𝑇 → (𝑥 · (𝑡𝑦)) = (𝑥 · (𝑇𝑦)))
4 fveq1 6352 . . . . . . 7 (𝑡 = 𝑇 → (𝑡𝑧) = (𝑇𝑧))
53, 4oveq12d 6832 . . . . . 6 (𝑡 = 𝑇 → ((𝑥 · (𝑡𝑦)) + (𝑡𝑧)) = ((𝑥 · (𝑇𝑦)) + (𝑇𝑧)))
61, 5eqeq12d 2775 . . . . 5 (𝑡 = 𝑇 → ((𝑡‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 · (𝑡𝑦)) + (𝑡𝑧)) ↔ (𝑇‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 · (𝑇𝑦)) + (𝑇𝑧))))
76ralbidv 3124 . . . 4 (𝑡 = 𝑇 → (∀𝑧 ∈ ℋ (𝑡‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 · (𝑡𝑦)) + (𝑡𝑧)) ↔ ∀𝑧 ∈ ℋ (𝑇‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 · (𝑇𝑦)) + (𝑇𝑧))))
872ralbidv 3127 . . 3 (𝑡 = 𝑇 → (∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ (𝑡‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 · (𝑡𝑦)) + (𝑡𝑧)) ↔ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ (𝑇‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 · (𝑇𝑦)) + (𝑇𝑧))))
9 df-lnfn 29037 . . 3 LinFn = {𝑡 ∈ (ℂ ↑𝑚 ℋ) ∣ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ (𝑡‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 · (𝑡𝑦)) + (𝑡𝑧))}
108, 9elrab2 3507 . 2 (𝑇 ∈ LinFn ↔ (𝑇 ∈ (ℂ ↑𝑚 ℋ) ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ (𝑇‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 · (𝑇𝑦)) + (𝑇𝑧))))
11 cnex 10229 . . . 4 ℂ ∈ V
12 ax-hilex 28186 . . . 4 ℋ ∈ V
1311, 12elmap 8054 . . 3 (𝑇 ∈ (ℂ ↑𝑚 ℋ) ↔ 𝑇: ℋ⟶ℂ)
1413anbi1i 733 . 2 ((𝑇 ∈ (ℂ ↑𝑚 ℋ) ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ (𝑇‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 · (𝑇𝑦)) + (𝑇𝑧))) ↔ (𝑇: ℋ⟶ℂ ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ (𝑇‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 · (𝑇𝑦)) + (𝑇𝑧))))
1510, 14bitri 264 1 (𝑇 ∈ LinFn ↔ (𝑇: ℋ⟶ℂ ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ (𝑇‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 · (𝑇𝑦)) + (𝑇𝑧))))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 196   ∧ wa 383   = wceq 1632   ∈ wcel 2139  ∀wral 3050  ⟶wf 6045  ‘cfv 6049  (class class class)co 6814   ↑𝑚 cmap 8025  ℂcc 10146   + caddc 10151   · cmul 10153   ℋchil 28106   +ℎ cva 28107   ·ℎ csm 28108  LinFnclf 28141 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115  ax-cnex 10204  ax-hilex 28186 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-fv 6057  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-map 8027  df-lnfn 29037 This theorem is referenced by:  lnfnf  29073  lnfnl  29120  bralnfn  29137  0lnfn  29174  cnlnadjlem2  29257
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