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Theorem lfladdcl 33877
Description: Closure of addition of two functionals. (Contributed by NM, 19-Oct-2014.)
Hypotheses
Ref Expression
lfladdcl.r 𝑅 = (Scalar‘𝑊)
lfladdcl.p + = (+g𝑅)
lfladdcl.f 𝐹 = (LFnl‘𝑊)
lfladdcl.w (𝜑𝑊 ∈ LMod)
lfladdcl.g (𝜑𝐺𝐹)
lfladdcl.h (𝜑𝐻𝐹)
Assertion
Ref Expression
lfladdcl (𝜑 → (𝐺𝑓 + 𝐻) ∈ 𝐹)

Proof of Theorem lfladdcl
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lfladdcl.w . . . . 5 (𝜑𝑊 ∈ LMod)
21adantr 481 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → 𝑊 ∈ LMod)
3 simprl 793 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → 𝑥 ∈ (Base‘𝑅))
4 simprr 795 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → 𝑦 ∈ (Base‘𝑅))
5 lfladdcl.r . . . . 5 𝑅 = (Scalar‘𝑊)
6 eqid 2621 . . . . 5 (Base‘𝑅) = (Base‘𝑅)
7 lfladdcl.p . . . . 5 + = (+g𝑅)
85, 6, 7lmodacl 18814 . . . 4 ((𝑊 ∈ LMod ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥 + 𝑦) ∈ (Base‘𝑅))
92, 3, 4, 8syl3anc 1323 . . 3 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (𝑥 + 𝑦) ∈ (Base‘𝑅))
10 lfladdcl.g . . . 4 (𝜑𝐺𝐹)
11 eqid 2621 . . . . 5 (Base‘𝑊) = (Base‘𝑊)
12 lfladdcl.f . . . . 5 𝐹 = (LFnl‘𝑊)
135, 6, 11, 12lflf 33869 . . . 4 ((𝑊 ∈ LMod ∧ 𝐺𝐹) → 𝐺:(Base‘𝑊)⟶(Base‘𝑅))
141, 10, 13syl2anc 692 . . 3 (𝜑𝐺:(Base‘𝑊)⟶(Base‘𝑅))
15 lfladdcl.h . . . 4 (𝜑𝐻𝐹)
165, 6, 11, 12lflf 33869 . . . 4 ((𝑊 ∈ LMod ∧ 𝐻𝐹) → 𝐻:(Base‘𝑊)⟶(Base‘𝑅))
171, 15, 16syl2anc 692 . . 3 (𝜑𝐻:(Base‘𝑊)⟶(Base‘𝑅))
18 fvexd 6170 . . 3 (𝜑 → (Base‘𝑊) ∈ V)
19 inidm 3806 . . 3 ((Base‘𝑊) ∩ (Base‘𝑊)) = (Base‘𝑊)
209, 14, 17, 18, 18, 19off 6877 . 2 (𝜑 → (𝐺𝑓 + 𝐻):(Base‘𝑊)⟶(Base‘𝑅))
211adantr 481 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑊 ∈ LMod)
22 simpr1 1065 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑥 ∈ (Base‘𝑅))
23 simpr2 1066 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑦 ∈ (Base‘𝑊))
24 eqid 2621 . . . . . . . 8 ( ·𝑠𝑊) = ( ·𝑠𝑊)
2511, 5, 24, 6lmodvscl 18820 . . . . . . 7 ((𝑊 ∈ LMod ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊)) → (𝑥( ·𝑠𝑊)𝑦) ∈ (Base‘𝑊))
2621, 22, 23, 25syl3anc 1323 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → (𝑥( ·𝑠𝑊)𝑦) ∈ (Base‘𝑊))
27 simpr3 1067 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑧 ∈ (Base‘𝑊))
28 eqid 2621 . . . . . . 7 (+g𝑊) = (+g𝑊)
2911, 28lmodvacl 18817 . . . . . 6 ((𝑊 ∈ LMod ∧ (𝑥( ·𝑠𝑊)𝑦) ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊)) → ((𝑥( ·𝑠𝑊)𝑦)(+g𝑊)𝑧) ∈ (Base‘𝑊))
3021, 26, 27, 29syl3anc 1323 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → ((𝑥( ·𝑠𝑊)𝑦)(+g𝑊)𝑧) ∈ (Base‘𝑊))
31 ffn 6012 . . . . . . 7 (𝐺:(Base‘𝑊)⟶(Base‘𝑅) → 𝐺 Fn (Base‘𝑊))
3214, 31syl 17 . . . . . 6 (𝜑𝐺 Fn (Base‘𝑊))
33 ffn 6012 . . . . . . 7 (𝐻:(Base‘𝑊)⟶(Base‘𝑅) → 𝐻 Fn (Base‘𝑊))
3417, 33syl 17 . . . . . 6 (𝜑𝐻 Fn (Base‘𝑊))
35 eqidd 2622 . . . . . 6 ((𝜑 ∧ ((𝑥( ·𝑠𝑊)𝑦)(+g𝑊)𝑧) ∈ (Base‘𝑊)) → (𝐺‘((𝑥( ·𝑠𝑊)𝑦)(+g𝑊)𝑧)) = (𝐺‘((𝑥( ·𝑠𝑊)𝑦)(+g𝑊)𝑧)))
36 eqidd 2622 . . . . . 6 ((𝜑 ∧ ((𝑥( ·𝑠𝑊)𝑦)(+g𝑊)𝑧) ∈ (Base‘𝑊)) → (𝐻‘((𝑥( ·𝑠𝑊)𝑦)(+g𝑊)𝑧)) = (𝐻‘((𝑥( ·𝑠𝑊)𝑦)(+g𝑊)𝑧)))
3732, 34, 18, 18, 19, 35, 36ofval 6871 . . . . 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))))
3830, 37syldan 487 . . . 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))))
39 eqidd 2622 . . . . . . . . 9 ((𝜑𝑦 ∈ (Base‘𝑊)) → (𝐺𝑦) = (𝐺𝑦))
40 eqidd 2622 . . . . . . . . 9 ((𝜑𝑦 ∈ (Base‘𝑊)) → (𝐻𝑦) = (𝐻𝑦))
4132, 34, 18, 18, 19, 39, 40ofval 6871 . . . . . . . 8 ((𝜑𝑦 ∈ (Base‘𝑊)) → ((𝐺𝑓 + 𝐻)‘𝑦) = ((𝐺𝑦) + (𝐻𝑦)))
4223, 41syldan 487 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → ((𝐺𝑓 + 𝐻)‘𝑦) = ((𝐺𝑦) + (𝐻𝑦)))
4342oveq2d 6631 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → (𝑥(.r𝑅)((𝐺𝑓 + 𝐻)‘𝑦)) = (𝑥(.r𝑅)((𝐺𝑦) + (𝐻𝑦))))
44 eqidd 2622 . . . . . . . 8 ((𝜑𝑧 ∈ (Base‘𝑊)) → (𝐺𝑧) = (𝐺𝑧))
45 eqidd 2622 . . . . . . . 8 ((𝜑𝑧 ∈ (Base‘𝑊)) → (𝐻𝑧) = (𝐻𝑧))
4632, 34, 18, 18, 19, 44, 45ofval 6871 . . . . . . 7 ((𝜑𝑧 ∈ (Base‘𝑊)) → ((𝐺𝑓 + 𝐻)‘𝑧) = ((𝐺𝑧) + (𝐻𝑧)))
4727, 46syldan 487 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → ((𝐺𝑓 + 𝐻)‘𝑧) = ((𝐺𝑧) + (𝐻𝑧)))
4843, 47oveq12d 6633 . . . . 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))))
4910adantr 481 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝐺𝐹)
505, 7, 11, 28, 12lfladd 33872 . . . . . . . 8 ((𝑊 ∈ LMod ∧ 𝐺𝐹 ∧ ((𝑥( ·𝑠𝑊)𝑦) ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → (𝐺‘((𝑥( ·𝑠𝑊)𝑦)(+g𝑊)𝑧)) = ((𝐺‘(𝑥( ·𝑠𝑊)𝑦)) + (𝐺𝑧)))
5121, 49, 26, 27, 50syl112anc 1327 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → (𝐺‘((𝑥( ·𝑠𝑊)𝑦)(+g𝑊)𝑧)) = ((𝐺‘(𝑥( ·𝑠𝑊)𝑦)) + (𝐺𝑧)))
5215adantr 481 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝐻𝐹)
535, 7, 11, 28, 12lfladd 33872 . . . . . . . 8 ((𝑊 ∈ LMod ∧ 𝐻𝐹 ∧ ((𝑥( ·𝑠𝑊)𝑦) ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → (𝐻‘((𝑥( ·𝑠𝑊)𝑦)(+g𝑊)𝑧)) = ((𝐻‘(𝑥( ·𝑠𝑊)𝑦)) + (𝐻𝑧)))
5421, 52, 26, 27, 53syl112anc 1327 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → (𝐻‘((𝑥( ·𝑠𝑊)𝑦)(+g𝑊)𝑧)) = ((𝐻‘(𝑥( ·𝑠𝑊)𝑦)) + (𝐻𝑧)))
5551, 54oveq12d 6633 . . . . . 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))))
565lmodring 18811 . . . . . . . . 9 (𝑊 ∈ LMod → 𝑅 ∈ Ring)
5721, 56syl 17 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑅 ∈ Ring)
58 ringcmn 18521 . . . . . . . 8 (𝑅 ∈ Ring → 𝑅 ∈ CMnd)
5957, 58syl 17 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑅 ∈ CMnd)
605, 6, 11, 12lflcl 33870 . . . . . . . 8 ((𝑊 ∈ LMod ∧ 𝐺𝐹 ∧ (𝑥( ·𝑠𝑊)𝑦) ∈ (Base‘𝑊)) → (𝐺‘(𝑥( ·𝑠𝑊)𝑦)) ∈ (Base‘𝑅))
6121, 49, 26, 60syl3anc 1323 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → (𝐺‘(𝑥( ·𝑠𝑊)𝑦)) ∈ (Base‘𝑅))
625, 6, 11, 12lflcl 33870 . . . . . . . 8 ((𝑊 ∈ LMod ∧ 𝐺𝐹𝑧 ∈ (Base‘𝑊)) → (𝐺𝑧) ∈ (Base‘𝑅))
6321, 49, 27, 62syl3anc 1323 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → (𝐺𝑧) ∈ (Base‘𝑅))
645, 6, 11, 12lflcl 33870 . . . . . . . 8 ((𝑊 ∈ LMod ∧ 𝐻𝐹 ∧ (𝑥( ·𝑠𝑊)𝑦) ∈ (Base‘𝑊)) → (𝐻‘(𝑥( ·𝑠𝑊)𝑦)) ∈ (Base‘𝑅))
6521, 52, 26, 64syl3anc 1323 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → (𝐻‘(𝑥( ·𝑠𝑊)𝑦)) ∈ (Base‘𝑅))
665, 6, 11, 12lflcl 33870 . . . . . . . 8 ((𝑊 ∈ LMod ∧ 𝐻𝐹𝑧 ∈ (Base‘𝑊)) → (𝐻𝑧) ∈ (Base‘𝑅))
6721, 52, 27, 66syl3anc 1323 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → (𝐻𝑧) ∈ (Base‘𝑅))
686, 7cmn4 18152 . . . . . . 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))))
6959, 61, 63, 65, 67, 68syl122anc 1332 . . . . . 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))))
70 eqid 2621 . . . . . . . . . . 11 (.r𝑅) = (.r𝑅)
715, 6, 70, 11, 24, 12lflmul 33874 . . . . . . . . . 10 ((𝑊 ∈ LMod ∧ 𝐺𝐹 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊))) → (𝐺‘(𝑥( ·𝑠𝑊)𝑦)) = (𝑥(.r𝑅)(𝐺𝑦)))
7221, 49, 22, 23, 71syl112anc 1327 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → (𝐺‘(𝑥( ·𝑠𝑊)𝑦)) = (𝑥(.r𝑅)(𝐺𝑦)))
735, 6, 70, 11, 24, 12lflmul 33874 . . . . . . . . . 10 ((𝑊 ∈ LMod ∧ 𝐻𝐹 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊))) → (𝐻‘(𝑥( ·𝑠𝑊)𝑦)) = (𝑥(.r𝑅)(𝐻𝑦)))
7421, 52, 22, 23, 73syl112anc 1327 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → (𝐻‘(𝑥( ·𝑠𝑊)𝑦)) = (𝑥(.r𝑅)(𝐻𝑦)))
7572, 74oveq12d 6633 . . . . . . . 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))))
765, 6, 11, 12lflcl 33870 . . . . . . . . . 10 ((𝑊 ∈ LMod ∧ 𝐺𝐹𝑦 ∈ (Base‘𝑊)) → (𝐺𝑦) ∈ (Base‘𝑅))
7721, 49, 23, 76syl3anc 1323 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → (𝐺𝑦) ∈ (Base‘𝑅))
785, 6, 11, 12lflcl 33870 . . . . . . . . . 10 ((𝑊 ∈ LMod ∧ 𝐻𝐹𝑦 ∈ (Base‘𝑊)) → (𝐻𝑦) ∈ (Base‘𝑅))
7921, 52, 23, 78syl3anc 1323 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → (𝐻𝑦) ∈ (Base‘𝑅))
806, 7, 70ringdi 18506 . . . . . . . . 9 ((𝑅 ∈ Ring ∧ (𝑥 ∈ (Base‘𝑅) ∧ (𝐺𝑦) ∈ (Base‘𝑅) ∧ (𝐻𝑦) ∈ (Base‘𝑅))) → (𝑥(.r𝑅)((𝐺𝑦) + (𝐻𝑦))) = ((𝑥(.r𝑅)(𝐺𝑦)) + (𝑥(.r𝑅)(𝐻𝑦))))
8157, 22, 77, 79, 80syl13anc 1325 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → (𝑥(.r𝑅)((𝐺𝑦) + (𝐻𝑦))) = ((𝑥(.r𝑅)(𝐺𝑦)) + (𝑥(.r𝑅)(𝐻𝑦))))
8275, 81eqtr4d 2658 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → ((𝐺‘(𝑥( ·𝑠𝑊)𝑦)) + (𝐻‘(𝑥( ·𝑠𝑊)𝑦))) = (𝑥(.r𝑅)((𝐺𝑦) + (𝐻𝑦))))
8382oveq1d 6630 . . . . . 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))))
8455, 69, 833eqtrd 2659 . . . . 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))))
8548, 84eqtr4d 2658 . . . 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))))
8638, 85eqtr4d 2658 . . 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)))
8786ralrimivvva 2968 . 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)))
8811, 28, 5, 24, 6, 7, 70, 12islfl 33866 . . 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)))))
891, 88syl 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)))))
9020, 87, 89mpbir2and 956 1 (𝜑 → (𝐺𝑓 + 𝐻) ∈ 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2908  Vcvv 3190   Fn wfn 5852  wf 5853  cfv 5857  (class class class)co 6615  𝑓 cof 6860  Basecbs 15800  +gcplusg 15881  .rcmulr 15882  Scalarcsca 15884   ·𝑠 cvsca 15885  CMndccmn 18133  Ringcrg 18487  LModclmod 18803  LFnlclfn 33863
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-cnex 9952  ax-resscn 9953  ax-1cn 9954  ax-icn 9955  ax-addcl 9956  ax-addrcl 9957  ax-mulcl 9958  ax-mulrcl 9959  ax-mulcom 9960  ax-addass 9961  ax-mulass 9962  ax-distr 9963  ax-i2m1 9964  ax-1ne0 9965  ax-1rid 9966  ax-rnegex 9967  ax-rrecex 9968  ax-cnre 9969  ax-pre-lttri 9970  ax-pre-lttrn 9971  ax-pre-ltadd 9972  ax-pre-mulgt0 9973
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-of 6862  df-om 7028  df-1st 7128  df-2nd 7129  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-er 7702  df-map 7819  df-en 7916  df-dom 7917  df-sdom 7918  df-pnf 10036  df-mnf 10037  df-xr 10038  df-ltxr 10039  df-le 10040  df-sub 10228  df-neg 10229  df-nn 10981  df-2 11039  df-ndx 15803  df-slot 15804  df-base 15805  df-sets 15806  df-plusg 15894  df-0g 16042  df-mgm 17182  df-sgrp 17224  df-mnd 17235  df-grp 17365  df-minusg 17366  df-sbg 17367  df-cmn 18135  df-abl 18136  df-mgp 18430  df-ur 18442  df-ring 18489  df-lmod 18805  df-lfl 33864
This theorem is referenced by:  ldualvaddcl  33936
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