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Theorem lnolin 28446
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 = (BaseSet‘𝑈)
lnoval.2 𝑌 = (BaseSet‘𝑊)
lnoval.3 𝐺 = ( +𝑣𝑈)
lnoval.4 𝐻 = ( +𝑣𝑊)
lnoval.5 𝑅 = ( ·𝑠OLD𝑈)
lnoval.6 𝑆 = ( ·𝑠OLD𝑊)
lnoval.7 𝐿 = (𝑈 LnOp 𝑊)
Assertion
Ref Expression
lnolin (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇𝐿) ∧ (𝐴 ∈ ℂ ∧ 𝐵𝑋𝐶𝑋)) → (𝑇‘((𝐴𝑅𝐵)𝐺𝐶)) = ((𝐴𝑆(𝑇𝐵))𝐻(𝑇𝐶)))

Proof of Theorem lnolin
Dummy variables 𝑢 𝑡 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lnoval.1 . . . . 5 𝑋 = (BaseSet‘𝑈)
2 lnoval.2 . . . . 5 𝑌 = (BaseSet‘𝑊)
3 lnoval.3 . . . . 5 𝐺 = ( +𝑣𝑈)
4 lnoval.4 . . . . 5 𝐻 = ( +𝑣𝑊)
5 lnoval.5 . . . . 5 𝑅 = ( ·𝑠OLD𝑈)
6 lnoval.6 . . . . 5 𝑆 = ( ·𝑠OLD𝑊)
7 lnoval.7 . . . . 5 𝐿 = (𝑈 LnOp 𝑊)
81, 2, 3, 4, 5, 6, 7islno 28445 . . . 4 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑇𝐿 ↔ (𝑇:𝑋𝑌 ∧ ∀𝑢 ∈ ℂ ∀𝑤𝑋𝑡𝑋 (𝑇‘((𝑢𝑅𝑤)𝐺𝑡)) = ((𝑢𝑆(𝑇𝑤))𝐻(𝑇𝑡)))))
98biimp3a 1462 . . 3 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇𝐿) → (𝑇:𝑋𝑌 ∧ ∀𝑢 ∈ ℂ ∀𝑤𝑋𝑡𝑋 (𝑇‘((𝑢𝑅𝑤)𝐺𝑡)) = ((𝑢𝑆(𝑇𝑤))𝐻(𝑇𝑡))))
109simprd 496 . 2 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇𝐿) → ∀𝑢 ∈ ℂ ∀𝑤𝑋𝑡𝑋 (𝑇‘((𝑢𝑅𝑤)𝐺𝑡)) = ((𝑢𝑆(𝑇𝑤))𝐻(𝑇𝑡)))
11 oveq1 7158 . . . . 5 (𝑢 = 𝐴 → (𝑢𝑅𝑤) = (𝐴𝑅𝑤))
1211fvoveq1d 7173 . . . 4 (𝑢 = 𝐴 → (𝑇‘((𝑢𝑅𝑤)𝐺𝑡)) = (𝑇‘((𝐴𝑅𝑤)𝐺𝑡)))
13 oveq1 7158 . . . . 5 (𝑢 = 𝐴 → (𝑢𝑆(𝑇𝑤)) = (𝐴𝑆(𝑇𝑤)))
1413oveq1d 7166 . . . 4 (𝑢 = 𝐴 → ((𝑢𝑆(𝑇𝑤))𝐻(𝑇𝑡)) = ((𝐴𝑆(𝑇𝑤))𝐻(𝑇𝑡)))
1512, 14eqeq12d 2840 . . 3 (𝑢 = 𝐴 → ((𝑇‘((𝑢𝑅𝑤)𝐺𝑡)) = ((𝑢𝑆(𝑇𝑤))𝐻(𝑇𝑡)) ↔ (𝑇‘((𝐴𝑅𝑤)𝐺𝑡)) = ((𝐴𝑆(𝑇𝑤))𝐻(𝑇𝑡))))
16 oveq2 7159 . . . . 5 (𝑤 = 𝐵 → (𝐴𝑅𝑤) = (𝐴𝑅𝐵))
1716fvoveq1d 7173 . . . 4 (𝑤 = 𝐵 → (𝑇‘((𝐴𝑅𝑤)𝐺𝑡)) = (𝑇‘((𝐴𝑅𝐵)𝐺𝑡)))
18 fveq2 6666 . . . . . 6 (𝑤 = 𝐵 → (𝑇𝑤) = (𝑇𝐵))
1918oveq2d 7167 . . . . 5 (𝑤 = 𝐵 → (𝐴𝑆(𝑇𝑤)) = (𝐴𝑆(𝑇𝐵)))
2019oveq1d 7166 . . . 4 (𝑤 = 𝐵 → ((𝐴𝑆(𝑇𝑤))𝐻(𝑇𝑡)) = ((𝐴𝑆(𝑇𝐵))𝐻(𝑇𝑡)))
2117, 20eqeq12d 2840 . . 3 (𝑤 = 𝐵 → ((𝑇‘((𝐴𝑅𝑤)𝐺𝑡)) = ((𝐴𝑆(𝑇𝑤))𝐻(𝑇𝑡)) ↔ (𝑇‘((𝐴𝑅𝐵)𝐺𝑡)) = ((𝐴𝑆(𝑇𝐵))𝐻(𝑇𝑡))))
22 oveq2 7159 . . . . 5 (𝑡 = 𝐶 → ((𝐴𝑅𝐵)𝐺𝑡) = ((𝐴𝑅𝐵)𝐺𝐶))
2322fveq2d 6670 . . . 4 (𝑡 = 𝐶 → (𝑇‘((𝐴𝑅𝐵)𝐺𝑡)) = (𝑇‘((𝐴𝑅𝐵)𝐺𝐶)))
24 fveq2 6666 . . . . 5 (𝑡 = 𝐶 → (𝑇𝑡) = (𝑇𝐶))
2524oveq2d 7167 . . . 4 (𝑡 = 𝐶 → ((𝐴𝑆(𝑇𝐵))𝐻(𝑇𝑡)) = ((𝐴𝑆(𝑇𝐵))𝐻(𝑇𝐶)))
2623, 25eqeq12d 2840 . . 3 (𝑡 = 𝐶 → ((𝑇‘((𝐴𝑅𝐵)𝐺𝑡)) = ((𝐴𝑆(𝑇𝐵))𝐻(𝑇𝑡)) ↔ (𝑇‘((𝐴𝑅𝐵)𝐺𝐶)) = ((𝐴𝑆(𝑇𝐵))𝐻(𝑇𝐶))))
2715, 21, 26rspc3v 3639 . 2 ((𝐴 ∈ ℂ ∧ 𝐵𝑋𝐶𝑋) → (∀𝑢 ∈ ℂ ∀𝑤𝑋𝑡𝑋 (𝑇‘((𝑢𝑅𝑤)𝐺𝑡)) = ((𝑢𝑆(𝑇𝑤))𝐻(𝑇𝑡)) → (𝑇‘((𝐴𝑅𝐵)𝐺𝐶)) = ((𝐴𝑆(𝑇𝐵))𝐻(𝑇𝐶))))
2810, 27mpan9 507 1 (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇𝐿) ∧ (𝐴 ∈ ℂ ∧ 𝐵𝑋𝐶𝑋)) → (𝑇‘((𝐴𝑅𝐵)𝐺𝐶)) = ((𝐴𝑆(𝑇𝐵))𝐻(𝑇𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1081   = wceq 1530  wcel 2106  wral 3142  wf 6347  cfv 6351  (class class class)co 7151  cc 10527  NrmCVeccnv 28276   +𝑣 cpv 28277  BaseSetcba 28278   ·𝑠OLD cns 28279   LnOp clno 28432
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 1904  ax-6 1963  ax-7 2008  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2152  ax-12 2167  ax-ext 2796  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2615  df-eu 2649  df-clab 2803  df-cleq 2817  df-clel 2897  df-nfc 2967  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-br 5063  df-opab 5125  df-id 5458  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-fv 6359  df-ov 7154  df-oprab 7155  df-mpo 7156  df-map 8401  df-lno 28436
This theorem is referenced by:  lno0  28448  lnocoi  28449  lnoadd  28450  lnosub  28451  lnomul  28452
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