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Theorem lnoadd 30516
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 11167 . . 3 1 ∈ β„‚
2 lnoadd.1 . . . 4 𝑋 = (BaseSetβ€˜π‘ˆ)
3 eqid 2726 . . . 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 2726 . . . 4 ( ·𝑠OLD β€˜π‘ˆ) = ( ·𝑠OLD β€˜π‘ˆ)
7 eqid 2726 . . . 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 30512 . . 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 1454 . 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 1133 . . . 4 ((π‘ˆ ∈ NrmCVec ∧ π‘Š ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) β†’ π‘ˆ ∈ NrmCVec)
12 simpl 482 . . . 4 ((𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ 𝐴 ∈ 𝑋)
132, 6nvsid 30385 . . . 4 ((π‘ˆ ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) β†’ (1( ·𝑠OLD β€˜π‘ˆ)𝐴) = 𝐴)
1411, 12, 13syl2an 595 . . 3 (((π‘ˆ ∈ NrmCVec ∧ π‘Š ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) β†’ (1( ·𝑠OLD β€˜π‘ˆ)𝐴) = 𝐴)
1514fvoveq1d 7426 . 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 1189 . . . 4 (((π‘ˆ ∈ NrmCVec ∧ π‘Š ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) β†’ π‘Š ∈ NrmCVec)
172, 3, 8lnof 30513 . . . . 5 ((π‘ˆ ∈ NrmCVec ∧ π‘Š ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) β†’ 𝑇:π‘‹βŸΆ(BaseSetβ€˜π‘Š))
18 ffvelcdm 7076 . . . . 5 ((𝑇:π‘‹βŸΆ(BaseSetβ€˜π‘Š) ∧ 𝐴 ∈ 𝑋) β†’ (π‘‡β€˜π΄) ∈ (BaseSetβ€˜π‘Š))
1917, 12, 18syl2an 595 . . . 4 (((π‘ˆ ∈ NrmCVec ∧ π‘Š ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) β†’ (π‘‡β€˜π΄) ∈ (BaseSetβ€˜π‘Š))
203, 7nvsid 30385 . . . 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 583 . . 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 7419 . 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 2774 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 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  βŸΆwf 6532  β€˜cfv 6536  (class class class)co 7404  β„‚cc 11107  1c1 11110  NrmCVeccnv 30342   +𝑣 cpv 30343  BaseSetcba 30344   ·𝑠OLD cns 30345   LnOp clno 30498
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721  ax-1cn 11167
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7407  df-oprab 7408  df-mpo 7409  df-1st 7971  df-2nd 7972  df-map 8821  df-vc 30317  df-nv 30350  df-va 30353  df-ba 30354  df-sm 30355  df-0v 30356  df-nmcv 30358  df-lno 30502
This theorem is referenced by: (None)
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