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Theorem lnoadd 29998
Description: Addition property of a linear operator. (Contributed by NM, 7-Dec-2007.) (Revised by Mario Carneiro, 19-Nov-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
lnoadd.1 𝑋 = (BaseSetβ€˜π‘ˆ)
lnoadd.5 𝐺 = ( +𝑣 β€˜π‘ˆ)
lnoadd.6 𝐻 = ( +𝑣 β€˜π‘Š)
lnoadd.7 𝐿 = (π‘ˆ LnOp π‘Š)
Assertion
Ref Expression
lnoadd (((π‘ˆ ∈ NrmCVec ∧ π‘Š ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) β†’ (π‘‡β€˜(𝐴𝐺𝐡)) = ((π‘‡β€˜π΄)𝐻(π‘‡β€˜π΅)))
Proof of Theorem lnoadd
StepHypRef Expression
1 ax-1cn 11164 . . 3 1 ∈ β„‚
2 lnoadd.1 . . . 4 𝑋 = (BaseSetβ€˜π‘ˆ)
3 eqid 2732 . . . 4 (BaseSetβ€˜π‘Š) = (BaseSetβ€˜π‘Š)
4 lnoadd.5 . . . 4 𝐺 = ( +𝑣 β€˜π‘ˆ)
5 lnoadd.6 . . . 4 𝐻 = ( +𝑣 β€˜π‘Š)
6 eqid 2732 . . . 4 ( ·𝑠OLD β€˜π‘ˆ) = ( ·𝑠OLD β€˜π‘ˆ)
7 eqid 2732 . . . 4 ( ·𝑠OLD β€˜π‘Š) = ( ·𝑠OLD β€˜π‘Š)
8 lnoadd.7 . . . 4 𝐿 = (π‘ˆ LnOp π‘Š)
92, 3, 4, 5, 6, 7, 8lnolin 29994 . . 3 (((π‘ˆ ∈ NrmCVec ∧ π‘Š ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) ∧ (1 ∈ β„‚ ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) β†’ (π‘‡β€˜((1( ·𝑠OLD β€˜π‘ˆ)𝐴)𝐺𝐡)) = ((1( ·𝑠OLD β€˜π‘Š)(π‘‡β€˜π΄))𝐻(π‘‡β€˜π΅)))
101, 9mp3anr1 1458 . 2 (((π‘ˆ ∈ NrmCVec ∧ π‘Š ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) β†’ (π‘‡β€˜((1( ·𝑠OLD β€˜π‘ˆ)𝐴)𝐺𝐡)) = ((1( ·𝑠OLD β€˜π‘Š)(π‘‡β€˜π΄))𝐻(π‘‡β€˜π΅)))
11 simp1 1136 . . . 4 ((π‘ˆ ∈ NrmCVec ∧ π‘Š ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) β†’ π‘ˆ ∈ NrmCVec)
12 simpl 483 . . . 4 ((𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ 𝐴 ∈ 𝑋)
132, 6nvsid 29867 . . . 4 ((π‘ˆ ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) β†’ (1( ·𝑠OLD β€˜π‘ˆ)𝐴) = 𝐴)
1411, 12, 13syl2an 596 . . 3 (((π‘ˆ ∈ NrmCVec ∧ π‘Š ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) β†’ (1( ·𝑠OLD β€˜π‘ˆ)𝐴) = 𝐴)
1514fvoveq1d 7427 . 2 (((π‘ˆ ∈ NrmCVec ∧ π‘Š ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) β†’ (π‘‡β€˜((1( ·𝑠OLD β€˜π‘ˆ)𝐴)𝐺𝐡)) = (π‘‡β€˜(𝐴𝐺𝐡)))
16 simpl2 1192 . . . 4 (((π‘ˆ ∈ NrmCVec ∧ π‘Š ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) β†’ π‘Š ∈ NrmCVec)
172, 3, 8lnof 29995 . . . . 5 ((π‘ˆ ∈ NrmCVec ∧ π‘Š ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) β†’ 𝑇:π‘‹βŸΆ(BaseSetβ€˜π‘Š))
18 ffvelcdm 7080 . . . . 5 ((𝑇:π‘‹βŸΆ(BaseSetβ€˜π‘Š) ∧ 𝐴 ∈ 𝑋) β†’ (π‘‡β€˜π΄) ∈ (BaseSetβ€˜π‘Š))
1917, 12, 18syl2an 596 . . . 4 (((π‘ˆ ∈ NrmCVec ∧ π‘Š ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) β†’ (π‘‡β€˜π΄) ∈ (BaseSetβ€˜π‘Š))
203, 7nvsid 29867 . . . 4 ((π‘Š ∈ NrmCVec ∧ (π‘‡β€˜π΄) ∈ (BaseSetβ€˜π‘Š)) β†’ (1( ·𝑠OLD β€˜π‘Š)(π‘‡β€˜π΄)) = (π‘‡β€˜π΄))
2116, 19, 20syl2anc 584 . . 3 (((π‘ˆ ∈ NrmCVec ∧ π‘Š ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) β†’ (1( ·𝑠OLD β€˜π‘Š)(π‘‡β€˜π΄)) = (π‘‡β€˜π΄))
2221oveq1d 7420 . 2 (((π‘ˆ ∈ NrmCVec ∧ π‘Š ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) β†’ ((1( ·𝑠OLD β€˜π‘Š)(π‘‡β€˜π΄))𝐻(π‘‡β€˜π΅)) = ((π‘‡β€˜π΄)𝐻(π‘‡β€˜π΅)))
2310, 15, 223eqtr3d 2780 1 (((π‘ˆ ∈ NrmCVec ∧ π‘Š ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) β†’ (π‘‡β€˜(𝐴𝐺𝐡)) = ((π‘‡β€˜π΄)𝐻(π‘‡β€˜π΅)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  βŸΆwf 6536  β€˜cfv 6540  (class class class)co 7405  β„‚cc 11104  1c1 11107  NrmCVeccnv 29824   +𝑣 cpv 29825  BaseSetcba 29826   ·𝑠OLD cns 29827   LnOp clno 29980
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-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-1cn 11164
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-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7971  df-2nd 7972  df-map 8818  df-vc 29799  df-nv 29832  df-va 29835  df-ba 29836  df-sm 29837  df-0v 29838  df-nmcv 29840  df-lno 29984
This theorem is referenced by: (None)
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