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Theorem lnoadd 29742
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 11114 . . 3 1 ∈ β„‚
2 lnoadd.1 . . . 4 𝑋 = (BaseSetβ€˜π‘ˆ)
3 eqid 2733 . . . 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 2733 . . . 4 ( ·𝑠OLD β€˜π‘ˆ) = ( ·𝑠OLD β€˜π‘ˆ)
7 eqid 2733 . . . 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 29738 . . 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 1459 . 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 1137 . . . 4 ((π‘ˆ ∈ NrmCVec ∧ π‘Š ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) β†’ π‘ˆ ∈ NrmCVec)
12 simpl 484 . . . 4 ((𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ 𝐴 ∈ 𝑋)
132, 6nvsid 29611 . . . 4 ((π‘ˆ ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) β†’ (1( ·𝑠OLD β€˜π‘ˆ)𝐴) = 𝐴)
1411, 12, 13syl2an 597 . . 3 (((π‘ˆ ∈ NrmCVec ∧ π‘Š ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) β†’ (1( ·𝑠OLD β€˜π‘ˆ)𝐴) = 𝐴)
1514fvoveq1d 7380 . 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 1193 . . . 4 (((π‘ˆ ∈ NrmCVec ∧ π‘Š ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) β†’ π‘Š ∈ NrmCVec)
172, 3, 8lnof 29739 . . . . 5 ((π‘ˆ ∈ NrmCVec ∧ π‘Š ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) β†’ 𝑇:π‘‹βŸΆ(BaseSetβ€˜π‘Š))
18 ffvelcdm 7033 . . . . 5 ((𝑇:π‘‹βŸΆ(BaseSetβ€˜π‘Š) ∧ 𝐴 ∈ 𝑋) β†’ (π‘‡β€˜π΄) ∈ (BaseSetβ€˜π‘Š))
1917, 12, 18syl2an 597 . . . 4 (((π‘ˆ ∈ NrmCVec ∧ π‘Š ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) β†’ (π‘‡β€˜π΄) ∈ (BaseSetβ€˜π‘Š))
203, 7nvsid 29611 . . . 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 585 . . 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 7373 . 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 2781 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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βŸΆwf 6493  β€˜cfv 6497  (class class class)co 7358  β„‚cc 11054  1c1 11057  NrmCVeccnv 29568   +𝑣 cpv 29569  BaseSetcba 29570   ·𝑠OLD cns 29571   LnOp clno 29724
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-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-1cn 11114
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-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7922  df-2nd 7923  df-map 8770  df-vc 29543  df-nv 29576  df-va 29579  df-ba 29580  df-sm 29581  df-0v 29582  df-nmcv 29584  df-lno 29728
This theorem is referenced by: (None)
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