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Theorem lnolin 29738
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 29737 . . . 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 1470 . . 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 497 . 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 7365 . . . . 5 (𝑒 = 𝐴 β†’ (𝑒𝑅𝑀) = (𝐴𝑅𝑀))
1211fvoveq1d 7380 . . . 4 (𝑒 = 𝐴 β†’ (π‘‡β€˜((𝑒𝑅𝑀)𝐺𝑑)) = (π‘‡β€˜((𝐴𝑅𝑀)𝐺𝑑)))
13 oveq1 7365 . . . . 5 (𝑒 = 𝐴 β†’ (𝑒𝑆(π‘‡β€˜π‘€)) = (𝐴𝑆(π‘‡β€˜π‘€)))
1413oveq1d 7373 . . . 4 (𝑒 = 𝐴 β†’ ((𝑒𝑆(π‘‡β€˜π‘€))𝐻(π‘‡β€˜π‘‘)) = ((𝐴𝑆(π‘‡β€˜π‘€))𝐻(π‘‡β€˜π‘‘)))
1512, 14eqeq12d 2749 . . 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 7366 . . . . 5 (𝑀 = 𝐡 β†’ (𝐴𝑅𝑀) = (𝐴𝑅𝐡))
1716fvoveq1d 7380 . . . 4 (𝑀 = 𝐡 β†’ (π‘‡β€˜((𝐴𝑅𝑀)𝐺𝑑)) = (π‘‡β€˜((𝐴𝑅𝐡)𝐺𝑑)))
18 fveq2 6843 . . . . . 6 (𝑀 = 𝐡 β†’ (π‘‡β€˜π‘€) = (π‘‡β€˜π΅))
1918oveq2d 7374 . . . . 5 (𝑀 = 𝐡 β†’ (𝐴𝑆(π‘‡β€˜π‘€)) = (𝐴𝑆(π‘‡β€˜π΅)))
2019oveq1d 7373 . . . 4 (𝑀 = 𝐡 β†’ ((𝐴𝑆(π‘‡β€˜π‘€))𝐻(π‘‡β€˜π‘‘)) = ((𝐴𝑆(π‘‡β€˜π΅))𝐻(π‘‡β€˜π‘‘)))
2117, 20eqeq12d 2749 . . 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 7366 . . . . 5 (𝑑 = 𝐢 β†’ ((𝐴𝑅𝐡)𝐺𝑑) = ((𝐴𝑅𝐡)𝐺𝐢))
2322fveq2d 6847 . . . 4 (𝑑 = 𝐢 β†’ (π‘‡β€˜((𝐴𝑅𝐡)𝐺𝑑)) = (π‘‡β€˜((𝐴𝑅𝐡)𝐺𝐢)))
24 fveq2 6843 . . . . 5 (𝑑 = 𝐢 β†’ (π‘‡β€˜π‘‘) = (π‘‡β€˜πΆ))
2524oveq2d 7374 . . . 4 (𝑑 = 𝐢 β†’ ((𝐴𝑆(π‘‡β€˜π΅))𝐻(π‘‡β€˜π‘‘)) = ((𝐴𝑆(π‘‡β€˜π΅))𝐻(π‘‡β€˜πΆ)))
2623, 25eqeq12d 2749 . . 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 3592 . 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 508 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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3061  βŸΆwf 6493  β€˜cfv 6497  (class class class)co 7358  β„‚cc 11054  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-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
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 3446  df-sbc 3741  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-br 5107  df-opab 5169  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363  df-map 8770  df-lno 29728
This theorem is referenced by:  lno0  29740  lnocoi  29741  lnoadd  29742  lnosub  29743  lnomul  29744
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