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Theorem lnfnaddi 31291
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 ∈ β„‹ ∧ 𝐡 ∈ β„‹) β†’ (π‘‡β€˜(𝐴 +β„Ž 𝐡)) = ((π‘‡β€˜π΄) + (π‘‡β€˜π΅)))
Proof of Theorem lnfnaddi
StepHypRef Expression
1 ax-1cn 11167 . . 3 1 ∈ β„‚
2 lnfnl.1 . . . 4 𝑇 ∈ LinFn
32lnfnli 31288 . . 3 ((1 ∈ β„‚ ∧ 𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹) β†’ (π‘‡β€˜((1 Β·β„Ž 𝐴) +β„Ž 𝐡)) = ((1 Β· (π‘‡β€˜π΄)) + (π‘‡β€˜π΅)))
41, 3mp3an1 1448 . 2 ((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹) β†’ (π‘‡β€˜((1 Β·β„Ž 𝐴) +β„Ž 𝐡)) = ((1 Β· (π‘‡β€˜π΄)) + (π‘‡β€˜π΅)))
5 ax-hvmulid 30254 . . . 4 (𝐴 ∈ β„‹ β†’ (1 Β·β„Ž 𝐴) = 𝐴)
65fvoveq1d 7430 . . 3 (𝐴 ∈ β„‹ β†’ (π‘‡β€˜((1 Β·β„Ž 𝐴) +β„Ž 𝐡)) = (π‘‡β€˜(𝐴 +β„Ž 𝐡)))
76adantr 481 . 2 ((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹) β†’ (π‘‡β€˜((1 Β·β„Ž 𝐴) +β„Ž 𝐡)) = (π‘‡β€˜(𝐴 +β„Ž 𝐡)))
82lnfnfi 31289 . . . . . 6 𝑇: β„‹βŸΆβ„‚
98ffvelcdmi 7085 . . . . 5 (𝐴 ∈ β„‹ β†’ (π‘‡β€˜π΄) ∈ β„‚)
109mullidd 11231 . . . 4 (𝐴 ∈ β„‹ β†’ (1 Β· (π‘‡β€˜π΄)) = (π‘‡β€˜π΄))
1110adantr 481 . . 3 ((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹) β†’ (1 Β· (π‘‡β€˜π΄)) = (π‘‡β€˜π΄))
1211oveq1d 7423 . 2 ((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹) β†’ ((1 Β· (π‘‡β€˜π΄)) + (π‘‡β€˜π΅)) = ((π‘‡β€˜π΄) + (π‘‡β€˜π΅)))
134, 7, 123eqtr3d 2780 1 ((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹) β†’ (π‘‡β€˜(𝐴 +β„Ž 𝐡)) = ((π‘‡β€˜π΄) + (π‘‡β€˜π΅)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  β€˜cfv 6543  (class class class)co 7408  β„‚cc 11107  1c1 11110   + caddc 11112   Β· cmul 11114   β„‹chba 30167   +β„Ž cva 30168   Β·β„Ž csm 30169  LinFnclf 30202
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724  ax-cnex 11165  ax-resscn 11166  ax-1cn 11167  ax-icn 11168  ax-addcl 11169  ax-mulcl 11171  ax-mulcom 11173  ax-mulass 11175  ax-distr 11176  ax-1rid 11179  ax-cnre 11182  ax-hilex 30247  ax-hvmulid 30254
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-ov 7411  df-oprab 7412  df-mpo 7413  df-map 8821  df-lnfn 31096
This theorem is referenced by:  lnfnaddmuli  31293  nlelshi  31308
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