Mathbox for Norm Megill < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  lfladdcl Structured version   Visualization version   GIF version

 Description: Closure of addition of two functionals. (Contributed by NM, 19-Oct-2014.)
Hypotheses
Ref Expression
Assertion
Ref Expression
lfladdcl (𝜑 → (𝐺f + 𝐻) ∈ 𝐹)

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lfladdcl.w . . . . 5 (𝜑𝑊 ∈ LMod)
21adantr 485 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → 𝑊 ∈ LMod)
3 simprl 771 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → 𝑥 ∈ (Base‘𝑅))
4 simprr 773 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → 𝑦 ∈ (Base‘𝑅))
5 lfladdcl.r . . . . 5 𝑅 = (Scalar‘𝑊)
6 eqid 2759 . . . . 5 (Base‘𝑅) = (Base‘𝑅)
7 lfladdcl.p . . . . 5 + = (+g𝑅)
85, 6, 7lmodacl 19706 . . . 4 ((𝑊 ∈ LMod ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥 + 𝑦) ∈ (Base‘𝑅))
92, 3, 4, 8syl3anc 1369 . . 3 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (𝑥 + 𝑦) ∈ (Base‘𝑅))
10 lfladdcl.g . . . 4 (𝜑𝐺𝐹)
11 eqid 2759 . . . . 5 (Base‘𝑊) = (Base‘𝑊)
12 lfladdcl.f . . . . 5 𝐹 = (LFnl‘𝑊)
135, 6, 11, 12lflf 36632 . . . 4 ((𝑊 ∈ LMod ∧ 𝐺𝐹) → 𝐺:(Base‘𝑊)⟶(Base‘𝑅))
141, 10, 13syl2anc 588 . . 3 (𝜑𝐺:(Base‘𝑊)⟶(Base‘𝑅))
15 lfladdcl.h . . . 4 (𝜑𝐻𝐹)
165, 6, 11, 12lflf 36632 . . . 4 ((𝑊 ∈ LMod ∧ 𝐻𝐹) → 𝐻:(Base‘𝑊)⟶(Base‘𝑅))
171, 15, 16syl2anc 588 . . 3 (𝜑𝐻:(Base‘𝑊)⟶(Base‘𝑅))
18 fvexd 6674 . . 3 (𝜑 → (Base‘𝑊) ∈ V)
19 inidm 4124 . . 3 ((Base‘𝑊) ∩ (Base‘𝑊)) = (Base‘𝑊)
209, 14, 17, 18, 18, 19off 7423 . 2 (𝜑 → (𝐺f + 𝐻):(Base‘𝑊)⟶(Base‘𝑅))
211adantr 485 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑊 ∈ LMod)
22 simpr1 1192 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑥 ∈ (Base‘𝑅))
23 simpr2 1193 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑦 ∈ (Base‘𝑊))
24 eqid 2759 . . . . . . . 8 ( ·𝑠𝑊) = ( ·𝑠𝑊)
2511, 5, 24, 6lmodvscl 19712 . . . . . . 7 ((𝑊 ∈ LMod ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊)) → (𝑥( ·𝑠𝑊)𝑦) ∈ (Base‘𝑊))
2621, 22, 23, 25syl3anc 1369 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → (𝑥( ·𝑠𝑊)𝑦) ∈ (Base‘𝑊))
27 simpr3 1194 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑧 ∈ (Base‘𝑊))
28 eqid 2759 . . . . . . 7 (+g𝑊) = (+g𝑊)
2911, 28lmodvacl 19709 . . . . . 6 ((𝑊 ∈ LMod ∧ (𝑥( ·𝑠𝑊)𝑦) ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊)) → ((𝑥( ·𝑠𝑊)𝑦)(+g𝑊)𝑧) ∈ (Base‘𝑊))
3021, 26, 27, 29syl3anc 1369 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → ((𝑥( ·𝑠𝑊)𝑦)(+g𝑊)𝑧) ∈ (Base‘𝑊))
3114ffnd 6500 . . . . . 6 (𝜑𝐺 Fn (Base‘𝑊))
3217ffnd 6500 . . . . . 6 (𝜑𝐻 Fn (Base‘𝑊))
33 eqidd 2760 . . . . . 6 ((𝜑 ∧ ((𝑥( ·𝑠𝑊)𝑦)(+g𝑊)𝑧) ∈ (Base‘𝑊)) → (𝐺‘((𝑥( ·𝑠𝑊)𝑦)(+g𝑊)𝑧)) = (𝐺‘((𝑥( ·𝑠𝑊)𝑦)(+g𝑊)𝑧)))
34 eqidd 2760 . . . . . 6 ((𝜑 ∧ ((𝑥( ·𝑠𝑊)𝑦)(+g𝑊)𝑧) ∈ (Base‘𝑊)) → (𝐻‘((𝑥( ·𝑠𝑊)𝑦)(+g𝑊)𝑧)) = (𝐻‘((𝑥( ·𝑠𝑊)𝑦)(+g𝑊)𝑧)))
3531, 32, 18, 18, 19, 33, 34ofval 7416 . . . . 5 ((𝜑 ∧ ((𝑥( ·𝑠𝑊)𝑦)(+g𝑊)𝑧) ∈ (Base‘𝑊)) → ((𝐺f + 𝐻)‘((𝑥( ·𝑠𝑊)𝑦)(+g𝑊)𝑧)) = ((𝐺‘((𝑥( ·𝑠𝑊)𝑦)(+g𝑊)𝑧)) + (𝐻‘((𝑥( ·𝑠𝑊)𝑦)(+g𝑊)𝑧))))
3630, 35syldan 595 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → ((𝐺f + 𝐻)‘((𝑥( ·𝑠𝑊)𝑦)(+g𝑊)𝑧)) = ((𝐺‘((𝑥( ·𝑠𝑊)𝑦)(+g𝑊)𝑧)) + (𝐻‘((𝑥( ·𝑠𝑊)𝑦)(+g𝑊)𝑧))))
37 eqidd 2760 . . . . . . . . 9 ((𝜑𝑦 ∈ (Base‘𝑊)) → (𝐺𝑦) = (𝐺𝑦))
38 eqidd 2760 . . . . . . . . 9 ((𝜑𝑦 ∈ (Base‘𝑊)) → (𝐻𝑦) = (𝐻𝑦))
3931, 32, 18, 18, 19, 37, 38ofval 7416 . . . . . . . 8 ((𝜑𝑦 ∈ (Base‘𝑊)) → ((𝐺f + 𝐻)‘𝑦) = ((𝐺𝑦) + (𝐻𝑦)))
4023, 39syldan 595 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → ((𝐺f + 𝐻)‘𝑦) = ((𝐺𝑦) + (𝐻𝑦)))
4140oveq2d 7167 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → (𝑥(.r𝑅)((𝐺f + 𝐻)‘𝑦)) = (𝑥(.r𝑅)((𝐺𝑦) + (𝐻𝑦))))
42 eqidd 2760 . . . . . . . 8 ((𝜑𝑧 ∈ (Base‘𝑊)) → (𝐺𝑧) = (𝐺𝑧))
43 eqidd 2760 . . . . . . . 8 ((𝜑𝑧 ∈ (Base‘𝑊)) → (𝐻𝑧) = (𝐻𝑧))
4431, 32, 18, 18, 19, 42, 43ofval 7416 . . . . . . 7 ((𝜑𝑧 ∈ (Base‘𝑊)) → ((𝐺f + 𝐻)‘𝑧) = ((𝐺𝑧) + (𝐻𝑧)))
4527, 44syldan 595 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → ((𝐺f + 𝐻)‘𝑧) = ((𝐺𝑧) + (𝐻𝑧)))
4641, 45oveq12d 7169 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → ((𝑥(.r𝑅)((𝐺f + 𝐻)‘𝑦)) + ((𝐺f + 𝐻)‘𝑧)) = ((𝑥(.r𝑅)((𝐺𝑦) + (𝐻𝑦))) + ((𝐺𝑧) + (𝐻𝑧))))
4710adantr 485 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝐺𝐹)
485, 7, 11, 28, 12lfladd 36635 . . . . . . . 8 ((𝑊 ∈ LMod ∧ 𝐺𝐹 ∧ ((𝑥( ·𝑠𝑊)𝑦) ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → (𝐺‘((𝑥( ·𝑠𝑊)𝑦)(+g𝑊)𝑧)) = ((𝐺‘(𝑥( ·𝑠𝑊)𝑦)) + (𝐺𝑧)))
4921, 47, 26, 27, 48syl112anc 1372 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → (𝐺‘((𝑥( ·𝑠𝑊)𝑦)(+g𝑊)𝑧)) = ((𝐺‘(𝑥( ·𝑠𝑊)𝑦)) + (𝐺𝑧)))
5015adantr 485 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝐻𝐹)
515, 7, 11, 28, 12lfladd 36635 . . . . . . . 8 ((𝑊 ∈ LMod ∧ 𝐻𝐹 ∧ ((𝑥( ·𝑠𝑊)𝑦) ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → (𝐻‘((𝑥( ·𝑠𝑊)𝑦)(+g𝑊)𝑧)) = ((𝐻‘(𝑥( ·𝑠𝑊)𝑦)) + (𝐻𝑧)))
5221, 50, 26, 27, 51syl112anc 1372 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → (𝐻‘((𝑥( ·𝑠𝑊)𝑦)(+g𝑊)𝑧)) = ((𝐻‘(𝑥( ·𝑠𝑊)𝑦)) + (𝐻𝑧)))
5349, 52oveq12d 7169 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → ((𝐺‘((𝑥( ·𝑠𝑊)𝑦)(+g𝑊)𝑧)) + (𝐻‘((𝑥( ·𝑠𝑊)𝑦)(+g𝑊)𝑧))) = (((𝐺‘(𝑥( ·𝑠𝑊)𝑦)) + (𝐺𝑧)) + ((𝐻‘(𝑥( ·𝑠𝑊)𝑦)) + (𝐻𝑧))))
545lmodring 19703 . . . . . . . . 9 (𝑊 ∈ LMod → 𝑅 ∈ Ring)
5521, 54syl 17 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑅 ∈ Ring)
56 ringcmn 19395 . . . . . . . 8 (𝑅 ∈ Ring → 𝑅 ∈ CMnd)
5755, 56syl 17 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑅 ∈ CMnd)
585, 6, 11, 12lflcl 36633 . . . . . . . 8 ((𝑊 ∈ LMod ∧ 𝐺𝐹 ∧ (𝑥( ·𝑠𝑊)𝑦) ∈ (Base‘𝑊)) → (𝐺‘(𝑥( ·𝑠𝑊)𝑦)) ∈ (Base‘𝑅))
5921, 47, 26, 58syl3anc 1369 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → (𝐺‘(𝑥( ·𝑠𝑊)𝑦)) ∈ (Base‘𝑅))
605, 6, 11, 12lflcl 36633 . . . . . . . 8 ((𝑊 ∈ LMod ∧ 𝐺𝐹𝑧 ∈ (Base‘𝑊)) → (𝐺𝑧) ∈ (Base‘𝑅))
6121, 47, 27, 60syl3anc 1369 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → (𝐺𝑧) ∈ (Base‘𝑅))
625, 6, 11, 12lflcl 36633 . . . . . . . 8 ((𝑊 ∈ LMod ∧ 𝐻𝐹 ∧ (𝑥( ·𝑠𝑊)𝑦) ∈ (Base‘𝑊)) → (𝐻‘(𝑥( ·𝑠𝑊)𝑦)) ∈ (Base‘𝑅))
6321, 50, 26, 62syl3anc 1369 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → (𝐻‘(𝑥( ·𝑠𝑊)𝑦)) ∈ (Base‘𝑅))
645, 6, 11, 12lflcl 36633 . . . . . . . 8 ((𝑊 ∈ LMod ∧ 𝐻𝐹𝑧 ∈ (Base‘𝑊)) → (𝐻𝑧) ∈ (Base‘𝑅))
6521, 50, 27, 64syl3anc 1369 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → (𝐻𝑧) ∈ (Base‘𝑅))
666, 7cmn4 18986 . . . . . . 7 ((𝑅 ∈ CMnd ∧ ((𝐺‘(𝑥( ·𝑠𝑊)𝑦)) ∈ (Base‘𝑅) ∧ (𝐺𝑧) ∈ (Base‘𝑅)) ∧ ((𝐻‘(𝑥( ·𝑠𝑊)𝑦)) ∈ (Base‘𝑅) ∧ (𝐻𝑧) ∈ (Base‘𝑅))) → (((𝐺‘(𝑥( ·𝑠𝑊)𝑦)) + (𝐺𝑧)) + ((𝐻‘(𝑥( ·𝑠𝑊)𝑦)) + (𝐻𝑧))) = (((𝐺‘(𝑥( ·𝑠𝑊)𝑦)) + (𝐻‘(𝑥( ·𝑠𝑊)𝑦))) + ((𝐺𝑧) + (𝐻𝑧))))
6757, 59, 61, 63, 65, 66syl122anc 1377 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → (((𝐺‘(𝑥( ·𝑠𝑊)𝑦)) + (𝐺𝑧)) + ((𝐻‘(𝑥( ·𝑠𝑊)𝑦)) + (𝐻𝑧))) = (((𝐺‘(𝑥( ·𝑠𝑊)𝑦)) + (𝐻‘(𝑥( ·𝑠𝑊)𝑦))) + ((𝐺𝑧) + (𝐻𝑧))))
68 eqid 2759 . . . . . . . . . . 11 (.r𝑅) = (.r𝑅)
695, 6, 68, 11, 24, 12lflmul 36637 . . . . . . . . . 10 ((𝑊 ∈ LMod ∧ 𝐺𝐹 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊))) → (𝐺‘(𝑥( ·𝑠𝑊)𝑦)) = (𝑥(.r𝑅)(𝐺𝑦)))
7021, 47, 22, 23, 69syl112anc 1372 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → (𝐺‘(𝑥( ·𝑠𝑊)𝑦)) = (𝑥(.r𝑅)(𝐺𝑦)))
715, 6, 68, 11, 24, 12lflmul 36637 . . . . . . . . . 10 ((𝑊 ∈ LMod ∧ 𝐻𝐹 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊))) → (𝐻‘(𝑥( ·𝑠𝑊)𝑦)) = (𝑥(.r𝑅)(𝐻𝑦)))
7221, 50, 22, 23, 71syl112anc 1372 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → (𝐻‘(𝑥( ·𝑠𝑊)𝑦)) = (𝑥(.r𝑅)(𝐻𝑦)))
7370, 72oveq12d 7169 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → ((𝐺‘(𝑥( ·𝑠𝑊)𝑦)) + (𝐻‘(𝑥( ·𝑠𝑊)𝑦))) = ((𝑥(.r𝑅)(𝐺𝑦)) + (𝑥(.r𝑅)(𝐻𝑦))))
745, 6, 11, 12lflcl 36633 . . . . . . . . . 10 ((𝑊 ∈ LMod ∧ 𝐺𝐹𝑦 ∈ (Base‘𝑊)) → (𝐺𝑦) ∈ (Base‘𝑅))
7521, 47, 23, 74syl3anc 1369 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → (𝐺𝑦) ∈ (Base‘𝑅))
765, 6, 11, 12lflcl 36633 . . . . . . . . . 10 ((𝑊 ∈ LMod ∧ 𝐻𝐹𝑦 ∈ (Base‘𝑊)) → (𝐻𝑦) ∈ (Base‘𝑅))
7721, 50, 23, 76syl3anc 1369 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → (𝐻𝑦) ∈ (Base‘𝑅))
786, 7, 68ringdi 19380 . . . . . . . . 9 ((𝑅 ∈ Ring ∧ (𝑥 ∈ (Base‘𝑅) ∧ (𝐺𝑦) ∈ (Base‘𝑅) ∧ (𝐻𝑦) ∈ (Base‘𝑅))) → (𝑥(.r𝑅)((𝐺𝑦) + (𝐻𝑦))) = ((𝑥(.r𝑅)(𝐺𝑦)) + (𝑥(.r𝑅)(𝐻𝑦))))
7955, 22, 75, 77, 78syl13anc 1370 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → (𝑥(.r𝑅)((𝐺𝑦) + (𝐻𝑦))) = ((𝑥(.r𝑅)(𝐺𝑦)) + (𝑥(.r𝑅)(𝐻𝑦))))
8073, 79eqtr4d 2797 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → ((𝐺‘(𝑥( ·𝑠𝑊)𝑦)) + (𝐻‘(𝑥( ·𝑠𝑊)𝑦))) = (𝑥(.r𝑅)((𝐺𝑦) + (𝐻𝑦))))
8180oveq1d 7166 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → (((𝐺‘(𝑥( ·𝑠𝑊)𝑦)) + (𝐻‘(𝑥( ·𝑠𝑊)𝑦))) + ((𝐺𝑧) + (𝐻𝑧))) = ((𝑥(.r𝑅)((𝐺𝑦) + (𝐻𝑦))) + ((𝐺𝑧) + (𝐻𝑧))))
8253, 67, 813eqtrd 2798 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → ((𝐺‘((𝑥( ·𝑠𝑊)𝑦)(+g𝑊)𝑧)) + (𝐻‘((𝑥( ·𝑠𝑊)𝑦)(+g𝑊)𝑧))) = ((𝑥(.r𝑅)((𝐺𝑦) + (𝐻𝑦))) + ((𝐺𝑧) + (𝐻𝑧))))
8346, 82eqtr4d 2797 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → ((𝑥(.r𝑅)((𝐺f + 𝐻)‘𝑦)) + ((𝐺f + 𝐻)‘𝑧)) = ((𝐺‘((𝑥( ·𝑠𝑊)𝑦)(+g𝑊)𝑧)) + (𝐻‘((𝑥( ·𝑠𝑊)𝑦)(+g𝑊)𝑧))))
8436, 83eqtr4d 2797 . . 3 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → ((𝐺f + 𝐻)‘((𝑥( ·𝑠𝑊)𝑦)(+g𝑊)𝑧)) = ((𝑥(.r𝑅)((𝐺f + 𝐻)‘𝑦)) + ((𝐺f + 𝐻)‘𝑧)))
8584ralrimivvva 3122 . 2 (𝜑 → ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑊)∀𝑧 ∈ (Base‘𝑊)((𝐺f + 𝐻)‘((𝑥( ·𝑠𝑊)𝑦)(+g𝑊)𝑧)) = ((𝑥(.r𝑅)((𝐺f + 𝐻)‘𝑦)) + ((𝐺f + 𝐻)‘𝑧)))
8611, 28, 5, 24, 6, 7, 68, 12islfl 36629 . . 3 (𝑊 ∈ LMod → ((𝐺f + 𝐻) ∈ 𝐹 ↔ ((𝐺f + 𝐻):(Base‘𝑊)⟶(Base‘𝑅) ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑊)∀𝑧 ∈ (Base‘𝑊)((𝐺f + 𝐻)‘((𝑥( ·𝑠𝑊)𝑦)(+g𝑊)𝑧)) = ((𝑥(.r𝑅)((𝐺f + 𝐻)‘𝑦)) + ((𝐺f + 𝐻)‘𝑧)))))
871, 86syl 17 . 2 (𝜑 → ((𝐺f + 𝐻) ∈ 𝐹 ↔ ((𝐺f + 𝐻):(Base‘𝑊)⟶(Base‘𝑅) ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑊)∀𝑧 ∈ (Base‘𝑊)((𝐺f + 𝐻)‘((𝑥( ·𝑠𝑊)𝑦)(+g𝑊)𝑧)) = ((𝑥(.r𝑅)((𝐺f + 𝐻)‘𝑦)) + ((𝐺f + 𝐻)‘𝑧)))))
8820, 85, 87mpbir2and 713 1 (𝜑 → (𝐺f + 𝐻) ∈ 𝐹)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1085   = wceq 1539   ∈ wcel 2112  ∀wral 3071  Vcvv 3410  ⟶wf 6332  ‘cfv 6336  (class class class)co 7151   ∘f cof 7404  Basecbs 16534  +gcplusg 16616  .rcmulr 16617  Scalarcsca 16619   ·𝑠 cvsca 16620  CMndccmn 18966  Ringcrg 19358  LModclmod 19695  LFnlclfn 36626 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 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5235  ax-pr 5299  ax-un 7460  ax-cnex 10624  ax-resscn 10625  ax-1cn 10626  ax-icn 10627  ax-addcl 10628  ax-addrcl 10629  ax-mulcl 10630  ax-mulrcl 10631  ax-mulcom 10632  ax-addass 10633  ax-mulass 10634  ax-distr 10635  ax-i2m1 10636  ax-1ne0 10637  ax-1rid 10638  ax-rnegex 10639  ax-rrecex 10640  ax-cnre 10641  ax-pre-lttri 10642  ax-pre-lttrn 10643  ax-pre-ltadd 10644  ax-pre-mulgt0 10645 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-nel 3057  df-ral 3076  df-rex 3077  df-reu 3078  df-rmo 3079  df-rab 3080  df-v 3412  df-sbc 3698  df-csb 3807  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-pss 3878  df-nul 4227  df-if 4422  df-pw 4497  df-sn 4524  df-pr 4526  df-tp 4528  df-op 4530  df-uni 4800  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5431  df-eprel 5436  df-po 5444  df-so 5445  df-fr 5484  df-we 5486  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-pred 6127  df-ord 6173  df-on 6174  df-lim 6175  df-suc 6176  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-f1 6341  df-fo 6342  df-f1o 6343  df-fv 6344  df-riota 7109  df-ov 7154  df-oprab 7155  df-mpo 7156  df-of 7406  df-om 7581  df-1st 7694  df-2nd 7695  df-wrecs 7958  df-recs 8019  df-rdg 8057  df-er 8300  df-map 8419  df-en 8529  df-dom 8530  df-sdom 8531  df-pnf 10708  df-mnf 10709  df-xr 10710  df-ltxr 10711  df-le 10712  df-sub 10903  df-neg 10904  df-nn 11668  df-2 11730  df-ndx 16537  df-slot 16538  df-base 16540  df-sets 16541  df-plusg 16629  df-0g 16766  df-mgm 17911  df-sgrp 17960  df-mnd 17971  df-grp 18165  df-minusg 18166  df-sbg 18167  df-cmn 18968  df-abl 18969  df-mgp 19301  df-ur 19313  df-ring 19360  df-lmod 19697  df-lfl 36627 This theorem is referenced by:  ldualvaddcl  36699
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