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Theorem lnolin 30577
Description: Basic linearity property of a linear operator. (Contributed by NM, 4-Dec-2007.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
lnoval.1 𝑋 = (BaseSetβ€˜π‘ˆ)
lnoval.2 π‘Œ = (BaseSetβ€˜π‘Š)
lnoval.3 𝐺 = ( +𝑣 β€˜π‘ˆ)
lnoval.4 𝐻 = ( +𝑣 β€˜π‘Š)
lnoval.5 𝑅 = ( ·𝑠OLD β€˜π‘ˆ)
lnoval.6 𝑆 = ( ·𝑠OLD β€˜π‘Š)
lnoval.7 𝐿 = (π‘ˆ LnOp π‘Š)
Assertion
Ref Expression
lnolin (((π‘ˆ ∈ NrmCVec ∧ π‘Š ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) ∧ (𝐴 ∈ β„‚ ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ (π‘‡β€˜((𝐴𝑅𝐡)𝐺𝐢)) = ((𝐴𝑆(π‘‡β€˜π΅))𝐻(π‘‡β€˜πΆ)))

Proof of Theorem lnolin
Dummy variables 𝑒 𝑑 𝑀 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lnoval.1 . . . . 5 𝑋 = (BaseSetβ€˜π‘ˆ)
2 lnoval.2 . . . . 5 π‘Œ = (BaseSetβ€˜π‘Š)
3 lnoval.3 . . . . 5 𝐺 = ( +𝑣 β€˜π‘ˆ)
4 lnoval.4 . . . . 5 𝐻 = ( +𝑣 β€˜π‘Š)
5 lnoval.5 . . . . 5 𝑅 = ( ·𝑠OLD β€˜π‘ˆ)
6 lnoval.6 . . . . 5 𝑆 = ( ·𝑠OLD β€˜π‘Š)
7 lnoval.7 . . . . 5 𝐿 = (π‘ˆ LnOp π‘Š)
81, 2, 3, 4, 5, 6, 7islno 30576 . . . 4 ((π‘ˆ ∈ NrmCVec ∧ π‘Š ∈ NrmCVec) β†’ (𝑇 ∈ 𝐿 ↔ (𝑇:π‘‹βŸΆπ‘Œ ∧ βˆ€π‘’ ∈ β„‚ βˆ€π‘€ ∈ 𝑋 βˆ€π‘‘ ∈ 𝑋 (π‘‡β€˜((𝑒𝑅𝑀)𝐺𝑑)) = ((𝑒𝑆(π‘‡β€˜π‘€))𝐻(π‘‡β€˜π‘‘)))))
98biimp3a 1466 . . 3 ((π‘ˆ ∈ NrmCVec ∧ π‘Š ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) β†’ (𝑇:π‘‹βŸΆπ‘Œ ∧ βˆ€π‘’ ∈ β„‚ βˆ€π‘€ ∈ 𝑋 βˆ€π‘‘ ∈ 𝑋 (π‘‡β€˜((𝑒𝑅𝑀)𝐺𝑑)) = ((𝑒𝑆(π‘‡β€˜π‘€))𝐻(π‘‡β€˜π‘‘))))
109simprd 495 . 2 ((π‘ˆ ∈ NrmCVec ∧ π‘Š ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) β†’ βˆ€π‘’ ∈ β„‚ βˆ€π‘€ ∈ 𝑋 βˆ€π‘‘ ∈ 𝑋 (π‘‡β€˜((𝑒𝑅𝑀)𝐺𝑑)) = ((𝑒𝑆(π‘‡β€˜π‘€))𝐻(π‘‡β€˜π‘‘)))
11 oveq1 7427 . . . . 5 (𝑒 = 𝐴 β†’ (𝑒𝑅𝑀) = (𝐴𝑅𝑀))
1211fvoveq1d 7442 . . . 4 (𝑒 = 𝐴 β†’ (π‘‡β€˜((𝑒𝑅𝑀)𝐺𝑑)) = (π‘‡β€˜((𝐴𝑅𝑀)𝐺𝑑)))
13 oveq1 7427 . . . . 5 (𝑒 = 𝐴 β†’ (𝑒𝑆(π‘‡β€˜π‘€)) = (𝐴𝑆(π‘‡β€˜π‘€)))
1413oveq1d 7435 . . . 4 (𝑒 = 𝐴 β†’ ((𝑒𝑆(π‘‡β€˜π‘€))𝐻(π‘‡β€˜π‘‘)) = ((𝐴𝑆(π‘‡β€˜π‘€))𝐻(π‘‡β€˜π‘‘)))
1512, 14eqeq12d 2744 . . 3 (𝑒 = 𝐴 β†’ ((π‘‡β€˜((𝑒𝑅𝑀)𝐺𝑑)) = ((𝑒𝑆(π‘‡β€˜π‘€))𝐻(π‘‡β€˜π‘‘)) ↔ (π‘‡β€˜((𝐴𝑅𝑀)𝐺𝑑)) = ((𝐴𝑆(π‘‡β€˜π‘€))𝐻(π‘‡β€˜π‘‘))))
16 oveq2 7428 . . . . 5 (𝑀 = 𝐡 β†’ (𝐴𝑅𝑀) = (𝐴𝑅𝐡))
1716fvoveq1d 7442 . . . 4 (𝑀 = 𝐡 β†’ (π‘‡β€˜((𝐴𝑅𝑀)𝐺𝑑)) = (π‘‡β€˜((𝐴𝑅𝐡)𝐺𝑑)))
18 fveq2 6897 . . . . . 6 (𝑀 = 𝐡 β†’ (π‘‡β€˜π‘€) = (π‘‡β€˜π΅))
1918oveq2d 7436 . . . . 5 (𝑀 = 𝐡 β†’ (𝐴𝑆(π‘‡β€˜π‘€)) = (𝐴𝑆(π‘‡β€˜π΅)))
2019oveq1d 7435 . . . 4 (𝑀 = 𝐡 β†’ ((𝐴𝑆(π‘‡β€˜π‘€))𝐻(π‘‡β€˜π‘‘)) = ((𝐴𝑆(π‘‡β€˜π΅))𝐻(π‘‡β€˜π‘‘)))
2117, 20eqeq12d 2744 . . 3 (𝑀 = 𝐡 β†’ ((π‘‡β€˜((𝐴𝑅𝑀)𝐺𝑑)) = ((𝐴𝑆(π‘‡β€˜π‘€))𝐻(π‘‡β€˜π‘‘)) ↔ (π‘‡β€˜((𝐴𝑅𝐡)𝐺𝑑)) = ((𝐴𝑆(π‘‡β€˜π΅))𝐻(π‘‡β€˜π‘‘))))
22 oveq2 7428 . . . . 5 (𝑑 = 𝐢 β†’ ((𝐴𝑅𝐡)𝐺𝑑) = ((𝐴𝑅𝐡)𝐺𝐢))
2322fveq2d 6901 . . . 4 (𝑑 = 𝐢 β†’ (π‘‡β€˜((𝐴𝑅𝐡)𝐺𝑑)) = (π‘‡β€˜((𝐴𝑅𝐡)𝐺𝐢)))
24 fveq2 6897 . . . . 5 (𝑑 = 𝐢 β†’ (π‘‡β€˜π‘‘) = (π‘‡β€˜πΆ))
2524oveq2d 7436 . . . 4 (𝑑 = 𝐢 β†’ ((𝐴𝑆(π‘‡β€˜π΅))𝐻(π‘‡β€˜π‘‘)) = ((𝐴𝑆(π‘‡β€˜π΅))𝐻(π‘‡β€˜πΆ)))
2623, 25eqeq12d 2744 . . 3 (𝑑 = 𝐢 β†’ ((π‘‡β€˜((𝐴𝑅𝐡)𝐺𝑑)) = ((𝐴𝑆(π‘‡β€˜π΅))𝐻(π‘‡β€˜π‘‘)) ↔ (π‘‡β€˜((𝐴𝑅𝐡)𝐺𝐢)) = ((𝐴𝑆(π‘‡β€˜π΅))𝐻(π‘‡β€˜πΆ))))
2715, 21, 26rspc3v 3625 . 2 ((𝐴 ∈ β„‚ ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋) β†’ (βˆ€π‘’ ∈ β„‚ βˆ€π‘€ ∈ 𝑋 βˆ€π‘‘ ∈ 𝑋 (π‘‡β€˜((𝑒𝑅𝑀)𝐺𝑑)) = ((𝑒𝑆(π‘‡β€˜π‘€))𝐻(π‘‡β€˜π‘‘)) β†’ (π‘‡β€˜((𝐴𝑅𝐡)𝐺𝐢)) = ((𝐴𝑆(π‘‡β€˜π΅))𝐻(π‘‡β€˜πΆ))))
2810, 27mpan9 506 1 (((π‘ˆ ∈ NrmCVec ∧ π‘Š ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) ∧ (𝐴 ∈ β„‚ ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ (π‘‡β€˜((𝐴𝑅𝐡)𝐺𝐢)) = ((𝐴𝑆(π‘‡β€˜π΅))𝐻(π‘‡β€˜πΆ)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   ∧ w3a 1085   = wceq 1534   ∈ wcel 2099  βˆ€wral 3058  βŸΆwf 6544  β€˜cfv 6548  (class class class)co 7420  β„‚cc 11137  NrmCVeccnv 30407   +𝑣 cpv 30408  BaseSetcba 30409   ·𝑠OLD cns 30410   LnOp clno 30563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-fv 6556  df-ov 7423  df-oprab 7424  df-mpo 7425  df-map 8847  df-lno 30567
This theorem is referenced by:  lno0  30579  lnocoi  30580  lnoadd  30581  lnosub  30582  lnomul  30583
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