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Theorem lnolin 28547
 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 28546 . . . 4 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑇𝐿 ↔ (𝑇:𝑋𝑌 ∧ ∀𝑢 ∈ ℂ ∀𝑤𝑋𝑡𝑋 (𝑇‘((𝑢𝑅𝑤)𝐺𝑡)) = ((𝑢𝑆(𝑇𝑤))𝐻(𝑇𝑡)))))
98biimp3a 1466 . . 3 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇𝐿) → (𝑇:𝑋𝑌 ∧ ∀𝑢 ∈ ℂ ∀𝑤𝑋𝑡𝑋 (𝑇‘((𝑢𝑅𝑤)𝐺𝑡)) = ((𝑢𝑆(𝑇𝑤))𝐻(𝑇𝑡))))
109simprd 499 . 2 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇𝐿) → ∀𝑢 ∈ ℂ ∀𝑤𝑋𝑡𝑋 (𝑇‘((𝑢𝑅𝑤)𝐺𝑡)) = ((𝑢𝑆(𝑇𝑤))𝐻(𝑇𝑡)))
11 oveq1 7143 . . . . 5 (𝑢 = 𝐴 → (𝑢𝑅𝑤) = (𝐴𝑅𝑤))
1211fvoveq1d 7158 . . . 4 (𝑢 = 𝐴 → (𝑇‘((𝑢𝑅𝑤)𝐺𝑡)) = (𝑇‘((𝐴𝑅𝑤)𝐺𝑡)))
13 oveq1 7143 . . . . 5 (𝑢 = 𝐴 → (𝑢𝑆(𝑇𝑤)) = (𝐴𝑆(𝑇𝑤)))
1413oveq1d 7151 . . . 4 (𝑢 = 𝐴 → ((𝑢𝑆(𝑇𝑤))𝐻(𝑇𝑡)) = ((𝐴𝑆(𝑇𝑤))𝐻(𝑇𝑡)))
1512, 14eqeq12d 2814 . . 3 (𝑢 = 𝐴 → ((𝑇‘((𝑢𝑅𝑤)𝐺𝑡)) = ((𝑢𝑆(𝑇𝑤))𝐻(𝑇𝑡)) ↔ (𝑇‘((𝐴𝑅𝑤)𝐺𝑡)) = ((𝐴𝑆(𝑇𝑤))𝐻(𝑇𝑡))))
16 oveq2 7144 . . . . 5 (𝑤 = 𝐵 → (𝐴𝑅𝑤) = (𝐴𝑅𝐵))
1716fvoveq1d 7158 . . . 4 (𝑤 = 𝐵 → (𝑇‘((𝐴𝑅𝑤)𝐺𝑡)) = (𝑇‘((𝐴𝑅𝐵)𝐺𝑡)))
18 fveq2 6646 . . . . . 6 (𝑤 = 𝐵 → (𝑇𝑤) = (𝑇𝐵))
1918oveq2d 7152 . . . . 5 (𝑤 = 𝐵 → (𝐴𝑆(𝑇𝑤)) = (𝐴𝑆(𝑇𝐵)))
2019oveq1d 7151 . . . 4 (𝑤 = 𝐵 → ((𝐴𝑆(𝑇𝑤))𝐻(𝑇𝑡)) = ((𝐴𝑆(𝑇𝐵))𝐻(𝑇𝑡)))
2117, 20eqeq12d 2814 . . 3 (𝑤 = 𝐵 → ((𝑇‘((𝐴𝑅𝑤)𝐺𝑡)) = ((𝐴𝑆(𝑇𝑤))𝐻(𝑇𝑡)) ↔ (𝑇‘((𝐴𝑅𝐵)𝐺𝑡)) = ((𝐴𝑆(𝑇𝐵))𝐻(𝑇𝑡))))
22 oveq2 7144 . . . . 5 (𝑡 = 𝐶 → ((𝐴𝑅𝐵)𝐺𝑡) = ((𝐴𝑅𝐵)𝐺𝐶))
2322fveq2d 6650 . . . 4 (𝑡 = 𝐶 → (𝑇‘((𝐴𝑅𝐵)𝐺𝑡)) = (𝑇‘((𝐴𝑅𝐵)𝐺𝐶)))
24 fveq2 6646 . . . . 5 (𝑡 = 𝐶 → (𝑇𝑡) = (𝑇𝐶))
2524oveq2d 7152 . . . 4 (𝑡 = 𝐶 → ((𝐴𝑆(𝑇𝐵))𝐻(𝑇𝑡)) = ((𝐴𝑆(𝑇𝐵))𝐻(𝑇𝐶)))
2623, 25eqeq12d 2814 . . 3 (𝑡 = 𝐶 → ((𝑇‘((𝐴𝑅𝐵)𝐺𝑡)) = ((𝐴𝑆(𝑇𝐵))𝐻(𝑇𝑡)) ↔ (𝑇‘((𝐴𝑅𝐵)𝐺𝐶)) = ((𝐴𝑆(𝑇𝐵))𝐻(𝑇𝐶))))
2715, 21, 26rspc3v 3584 . 2 ((𝐴 ∈ ℂ ∧ 𝐵𝑋𝐶𝑋) → (∀𝑢 ∈ ℂ ∀𝑤𝑋𝑡𝑋 (𝑇‘((𝑢𝑅𝑤)𝐺𝑡)) = ((𝑢𝑆(𝑇𝑤))𝐻(𝑇𝑡)) → (𝑇‘((𝐴𝑅𝐵)𝐺𝐶)) = ((𝐴𝑆(𝑇𝐵))𝐻(𝑇𝐶))))
2810, 27mpan9 510 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 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∀wral 3106  ⟶wf 6321  ‘cfv 6325  (class class class)co 7136  ℂcc 10527  NrmCVeccnv 28377   +𝑣 cpv 28378  BaseSetcba 28379   ·𝑠OLD cns 28380   LnOp clno 28533 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7444 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5032  df-opab 5094  df-id 5426  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-iota 6284  df-fun 6327  df-fn 6328  df-f 6329  df-fv 6333  df-ov 7139  df-oprab 7140  df-mpo 7141  df-map 8394  df-lno 28537 This theorem is referenced by:  lno0  28549  lnocoi  28550  lnoadd  28551  lnosub  28552  lnomul  28553
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