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Theorem lnfnaddi 31034
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 11117 . . 3 1 ∈ β„‚
2 lnfnl.1 . . . 4 𝑇 ∈ LinFn
32lnfnli 31031 . . 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 1449 . 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 29997 . . . 4 (𝐴 ∈ β„‹ β†’ (1 Β·β„Ž 𝐴) = 𝐴)
65fvoveq1d 7383 . . 3 (𝐴 ∈ β„‹ β†’ (π‘‡β€˜((1 Β·β„Ž 𝐴) +β„Ž 𝐡)) = (π‘‡β€˜(𝐴 +β„Ž 𝐡)))
76adantr 482 . 2 ((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹) β†’ (π‘‡β€˜((1 Β·β„Ž 𝐴) +β„Ž 𝐡)) = (π‘‡β€˜(𝐴 +β„Ž 𝐡)))
82lnfnfi 31032 . . . . . 6 𝑇: β„‹βŸΆβ„‚
98ffvelcdmi 7038 . . . . 5 (𝐴 ∈ β„‹ β†’ (π‘‡β€˜π΄) ∈ β„‚)
109mullidd 11181 . . . 4 (𝐴 ∈ β„‹ β†’ (1 Β· (π‘‡β€˜π΄)) = (π‘‡β€˜π΄))
1110adantr 482 . . 3 ((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹) β†’ (1 Β· (π‘‡β€˜π΄)) = (π‘‡β€˜π΄))
1211oveq1d 7376 . 2 ((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹) β†’ ((1 Β· (π‘‡β€˜π΄)) + (π‘‡β€˜π΅)) = ((π‘‡β€˜π΄) + (π‘‡β€˜π΅)))
134, 7, 123eqtr3d 2781 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 397   = wceq 1542   ∈ wcel 2107  β€˜cfv 6500  (class class class)co 7361  β„‚cc 11057  1c1 11060   + caddc 11062   Β· cmul 11064   β„‹chba 29910   +β„Ž cva 29911   Β·β„Ž csm 29912  LinFnclf 29945
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676  ax-cnex 11115  ax-resscn 11116  ax-1cn 11117  ax-icn 11118  ax-addcl 11119  ax-mulcl 11121  ax-mulcom 11123  ax-mulass 11125  ax-distr 11126  ax-1rid 11129  ax-cnre 11132  ax-hilex 29990  ax-hvmulid 29997
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-sbc 3744  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-fv 6508  df-ov 7364  df-oprab 7365  df-mpo 7366  df-map 8773  df-lnfn 30839
This theorem is referenced by:  lnfnaddmuli  31036  nlelshi  31051
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