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Theorem lnolin 27593
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 27592 . . . 4 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑇𝐿 ↔ (𝑇:𝑋𝑌 ∧ ∀𝑢 ∈ ℂ ∀𝑤𝑋𝑡𝑋 (𝑇‘((𝑢𝑅𝑤)𝐺𝑡)) = ((𝑢𝑆(𝑇𝑤))𝐻(𝑇𝑡)))))
98biimp3a 1431 . . 3 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇𝐿) → (𝑇:𝑋𝑌 ∧ ∀𝑢 ∈ ℂ ∀𝑤𝑋𝑡𝑋 (𝑇‘((𝑢𝑅𝑤)𝐺𝑡)) = ((𝑢𝑆(𝑇𝑤))𝐻(𝑇𝑡))))
109simprd 479 . 2 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇𝐿) → ∀𝑢 ∈ ℂ ∀𝑤𝑋𝑡𝑋 (𝑇‘((𝑢𝑅𝑤)𝐺𝑡)) = ((𝑢𝑆(𝑇𝑤))𝐻(𝑇𝑡)))
11 oveq1 6654 . . . . . 6 (𝑢 = 𝐴 → (𝑢𝑅𝑤) = (𝐴𝑅𝑤))
1211oveq1d 6662 . . . . 5 (𝑢 = 𝐴 → ((𝑢𝑅𝑤)𝐺𝑡) = ((𝐴𝑅𝑤)𝐺𝑡))
1312fveq2d 6193 . . . 4 (𝑢 = 𝐴 → (𝑇‘((𝑢𝑅𝑤)𝐺𝑡)) = (𝑇‘((𝐴𝑅𝑤)𝐺𝑡)))
14 oveq1 6654 . . . . 5 (𝑢 = 𝐴 → (𝑢𝑆(𝑇𝑤)) = (𝐴𝑆(𝑇𝑤)))
1514oveq1d 6662 . . . 4 (𝑢 = 𝐴 → ((𝑢𝑆(𝑇𝑤))𝐻(𝑇𝑡)) = ((𝐴𝑆(𝑇𝑤))𝐻(𝑇𝑡)))
1613, 15eqeq12d 2636 . . 3 (𝑢 = 𝐴 → ((𝑇‘((𝑢𝑅𝑤)𝐺𝑡)) = ((𝑢𝑆(𝑇𝑤))𝐻(𝑇𝑡)) ↔ (𝑇‘((𝐴𝑅𝑤)𝐺𝑡)) = ((𝐴𝑆(𝑇𝑤))𝐻(𝑇𝑡))))
17 oveq2 6655 . . . . . 6 (𝑤 = 𝐵 → (𝐴𝑅𝑤) = (𝐴𝑅𝐵))
1817oveq1d 6662 . . . . 5 (𝑤 = 𝐵 → ((𝐴𝑅𝑤)𝐺𝑡) = ((𝐴𝑅𝐵)𝐺𝑡))
1918fveq2d 6193 . . . 4 (𝑤 = 𝐵 → (𝑇‘((𝐴𝑅𝑤)𝐺𝑡)) = (𝑇‘((𝐴𝑅𝐵)𝐺𝑡)))
20 fveq2 6189 . . . . . 6 (𝑤 = 𝐵 → (𝑇𝑤) = (𝑇𝐵))
2120oveq2d 6663 . . . . 5 (𝑤 = 𝐵 → (𝐴𝑆(𝑇𝑤)) = (𝐴𝑆(𝑇𝐵)))
2221oveq1d 6662 . . . 4 (𝑤 = 𝐵 → ((𝐴𝑆(𝑇𝑤))𝐻(𝑇𝑡)) = ((𝐴𝑆(𝑇𝐵))𝐻(𝑇𝑡)))
2319, 22eqeq12d 2636 . . 3 (𝑤 = 𝐵 → ((𝑇‘((𝐴𝑅𝑤)𝐺𝑡)) = ((𝐴𝑆(𝑇𝑤))𝐻(𝑇𝑡)) ↔ (𝑇‘((𝐴𝑅𝐵)𝐺𝑡)) = ((𝐴𝑆(𝑇𝐵))𝐻(𝑇𝑡))))
24 oveq2 6655 . . . . 5 (𝑡 = 𝐶 → ((𝐴𝑅𝐵)𝐺𝑡) = ((𝐴𝑅𝐵)𝐺𝐶))
2524fveq2d 6193 . . . 4 (𝑡 = 𝐶 → (𝑇‘((𝐴𝑅𝐵)𝐺𝑡)) = (𝑇‘((𝐴𝑅𝐵)𝐺𝐶)))
26 fveq2 6189 . . . . 5 (𝑡 = 𝐶 → (𝑇𝑡) = (𝑇𝐶))
2726oveq2d 6663 . . . 4 (𝑡 = 𝐶 → ((𝐴𝑆(𝑇𝐵))𝐻(𝑇𝑡)) = ((𝐴𝑆(𝑇𝐵))𝐻(𝑇𝐶)))
2825, 27eqeq12d 2636 . . 3 (𝑡 = 𝐶 → ((𝑇‘((𝐴𝑅𝐵)𝐺𝑡)) = ((𝐴𝑆(𝑇𝐵))𝐻(𝑇𝑡)) ↔ (𝑇‘((𝐴𝑅𝐵)𝐺𝐶)) = ((𝐴𝑆(𝑇𝐵))𝐻(𝑇𝐶))))
2916, 23, 28rspc3v 3323 . 2 ((𝐴 ∈ ℂ ∧ 𝐵𝑋𝐶𝑋) → (∀𝑢 ∈ ℂ ∀𝑤𝑋𝑡𝑋 (𝑇‘((𝑢𝑅𝑤)𝐺𝑡)) = ((𝑢𝑆(𝑇𝑤))𝐻(𝑇𝑡)) → (𝑇‘((𝐴𝑅𝐵)𝐺𝐶)) = ((𝐴𝑆(𝑇𝐵))𝐻(𝑇𝐶))))
3010, 29mpan9 486 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 384  w3a 1037   = wceq 1482  wcel 1989  wral 2911  wf 5882  cfv 5886  (class class class)co 6647  cc 9931  NrmCVeccnv 27423   +𝑣 cpv 27424  BaseSetcba 27425   ·𝑠OLD cns 27426   LnOp clno 27579
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-sbc 3434  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-br 4652  df-opab 4711  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-fv 5894  df-ov 6650  df-oprab 6651  df-mpt2 6652  df-map 7856  df-lno 27583
This theorem is referenced by:  lno0  27595  lnocoi  27596  lnoadd  27597  lnosub  27598  lnomul  27599
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