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Theorem ellnfn 28972
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 6303 . . . . . 6 (𝑡 = 𝑇 → (𝑡‘((𝑥 · 𝑦) + 𝑧)) = (𝑇‘((𝑥 · 𝑦) + 𝑧)))
2 fveq1 6303 . . . . . . . 8 (𝑡 = 𝑇 → (𝑡𝑦) = (𝑇𝑦))
32oveq2d 6781 . . . . . . 7 (𝑡 = 𝑇 → (𝑥 · (𝑡𝑦)) = (𝑥 · (𝑇𝑦)))
4 fveq1 6303 . . . . . . 7 (𝑡 = 𝑇 → (𝑡𝑧) = (𝑇𝑧))
53, 4oveq12d 6783 . . . . . 6 (𝑡 = 𝑇 → ((𝑥 · (𝑡𝑦)) + (𝑡𝑧)) = ((𝑥 · (𝑇𝑦)) + (𝑇𝑧)))
61, 5eqeq12d 2739 . . . . 5 (𝑡 = 𝑇 → ((𝑡‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 · (𝑡𝑦)) + (𝑡𝑧)) ↔ (𝑇‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 · (𝑇𝑦)) + (𝑇𝑧))))
76ralbidv 3088 . . . 4 (𝑡 = 𝑇 → (∀𝑧 ∈ ℋ (𝑡‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 · (𝑡𝑦)) + (𝑡𝑧)) ↔ ∀𝑧 ∈ ℋ (𝑇‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 · (𝑇𝑦)) + (𝑇𝑧))))
872ralbidv 3091 . . 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 28937 . . 3 LinFn = {𝑡 ∈ (ℂ ↑𝑚 ℋ) ∣ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ (𝑡‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 · (𝑡𝑦)) + (𝑡𝑧))}
108, 9elrab2 3472 . 2 (𝑇 ∈ LinFn ↔ (𝑇 ∈ (ℂ ↑𝑚 ℋ) ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ (𝑇‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 · (𝑇𝑦)) + (𝑇𝑧))))
11 cnex 10130 . . . 4 ℂ ∈ V
12 ax-hilex 28086 . . . 4 ℋ ∈ V
1311, 12elmap 8003 . . 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 1596  wcel 2103  wral 3014  wf 5997  cfv 6001  (class class class)co 6765  𝑚 cmap 7974  cc 10047   + caddc 10052   · cmul 10054  chil 28006   + cva 28007   · csm 28008  LinFnclf 28041
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066  ax-cnex 10105  ax-hilex 28086
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ral 3019  df-rex 3020  df-rab 3023  df-v 3306  df-sbc 3542  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-op 4292  df-uni 4545  df-br 4761  df-opab 4821  df-id 5128  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-fv 6009  df-ov 6768  df-oprab 6769  df-mpt2 6770  df-map 7976  df-lnfn 28937
This theorem is referenced by:  lnfnf  28973  lnfnl  29020  bralnfn  29037  0lnfn  29074  cnlnadjlem2  29157
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