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Theorem lfladdcl 36199
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 483 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → 𝑊 ∈ LMod)
3 simprl 769 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → 𝑥 ∈ (Base‘𝑅))
4 simprr 771 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → 𝑦 ∈ (Base‘𝑅))
5 lfladdcl.r . . . . 5 𝑅 = (Scalar‘𝑊)
6 eqid 2819 . . . . 5 (Base‘𝑅) = (Base‘𝑅)
7 lfladdcl.p . . . . 5 + = (+g𝑅)
85, 6, 7lmodacl 19637 . . . 4 ((𝑊 ∈ LMod ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥 + 𝑦) ∈ (Base‘𝑅))
92, 3, 4, 8syl3anc 1366 . . 3 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (𝑥 + 𝑦) ∈ (Base‘𝑅))
10 lfladdcl.g . . . 4 (𝜑𝐺𝐹)
11 eqid 2819 . . . . 5 (Base‘𝑊) = (Base‘𝑊)
12 lfladdcl.f . . . . 5 𝐹 = (LFnl‘𝑊)
135, 6, 11, 12lflf 36191 . . . 4 ((𝑊 ∈ LMod ∧ 𝐺𝐹) → 𝐺:(Base‘𝑊)⟶(Base‘𝑅))
141, 10, 13syl2anc 586 . . 3 (𝜑𝐺:(Base‘𝑊)⟶(Base‘𝑅))
15 lfladdcl.h . . . 4 (𝜑𝐻𝐹)
165, 6, 11, 12lflf 36191 . . . 4 ((𝑊 ∈ LMod ∧ 𝐻𝐹) → 𝐻:(Base‘𝑊)⟶(Base‘𝑅))
171, 15, 16syl2anc 586 . . 3 (𝜑𝐻:(Base‘𝑊)⟶(Base‘𝑅))
18 fvexd 6678 . . 3 (𝜑 → (Base‘𝑊) ∈ V)
19 inidm 4193 . . 3 ((Base‘𝑊) ∩ (Base‘𝑊)) = (Base‘𝑊)
209, 14, 17, 18, 18, 19off 7416 . 2 (𝜑 → (𝐺f + 𝐻):(Base‘𝑊)⟶(Base‘𝑅))
211adantr 483 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑊 ∈ LMod)
22 simpr1 1189 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑥 ∈ (Base‘𝑅))
23 simpr2 1190 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑦 ∈ (Base‘𝑊))
24 eqid 2819 . . . . . . . 8 ( ·𝑠𝑊) = ( ·𝑠𝑊)
2511, 5, 24, 6lmodvscl 19643 . . . . . . 7 ((𝑊 ∈ LMod ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊)) → (𝑥( ·𝑠𝑊)𝑦) ∈ (Base‘𝑊))
2621, 22, 23, 25syl3anc 1366 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → (𝑥( ·𝑠𝑊)𝑦) ∈ (Base‘𝑊))
27 simpr3 1191 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑧 ∈ (Base‘𝑊))
28 eqid 2819 . . . . . . 7 (+g𝑊) = (+g𝑊)
2911, 28lmodvacl 19640 . . . . . 6 ((𝑊 ∈ LMod ∧ (𝑥( ·𝑠𝑊)𝑦) ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊)) → ((𝑥( ·𝑠𝑊)𝑦)(+g𝑊)𝑧) ∈ (Base‘𝑊))
3021, 26, 27, 29syl3anc 1366 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → ((𝑥( ·𝑠𝑊)𝑦)(+g𝑊)𝑧) ∈ (Base‘𝑊))
3114ffnd 6508 . . . . . 6 (𝜑𝐺 Fn (Base‘𝑊))
3217ffnd 6508 . . . . . 6 (𝜑𝐻 Fn (Base‘𝑊))
33 eqidd 2820 . . . . . 6 ((𝜑 ∧ ((𝑥( ·𝑠𝑊)𝑦)(+g𝑊)𝑧) ∈ (Base‘𝑊)) → (𝐺‘((𝑥( ·𝑠𝑊)𝑦)(+g𝑊)𝑧)) = (𝐺‘((𝑥( ·𝑠𝑊)𝑦)(+g𝑊)𝑧)))
34 eqidd 2820 . . . . . 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 7410 . . . . 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 593 . . . 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 2820 . . . . . . . . 9 ((𝜑𝑦 ∈ (Base‘𝑊)) → (𝐺𝑦) = (𝐺𝑦))
38 eqidd 2820 . . . . . . . . 9 ((𝜑𝑦 ∈ (Base‘𝑊)) → (𝐻𝑦) = (𝐻𝑦))
3931, 32, 18, 18, 19, 37, 38ofval 7410 . . . . . . . 8 ((𝜑𝑦 ∈ (Base‘𝑊)) → ((𝐺f + 𝐻)‘𝑦) = ((𝐺𝑦) + (𝐻𝑦)))
4023, 39syldan 593 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → ((𝐺f + 𝐻)‘𝑦) = ((𝐺𝑦) + (𝐻𝑦)))
4140oveq2d 7164 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → (𝑥(.r𝑅)((𝐺f + 𝐻)‘𝑦)) = (𝑥(.r𝑅)((𝐺𝑦) + (𝐻𝑦))))
42 eqidd 2820 . . . . . . . 8 ((𝜑𝑧 ∈ (Base‘𝑊)) → (𝐺𝑧) = (𝐺𝑧))
43 eqidd 2820 . . . . . . . 8 ((𝜑𝑧 ∈ (Base‘𝑊)) → (𝐻𝑧) = (𝐻𝑧))
4431, 32, 18, 18, 19, 42, 43ofval 7410 . . . . . . 7 ((𝜑𝑧 ∈ (Base‘𝑊)) → ((𝐺f + 𝐻)‘𝑧) = ((𝐺𝑧) + (𝐻𝑧)))
4527, 44syldan 593 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → ((𝐺f + 𝐻)‘𝑧) = ((𝐺𝑧) + (𝐻𝑧)))
4641, 45oveq12d 7166 . . . . 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 483 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝐺𝐹)
485, 7, 11, 28, 12lfladd 36194 . . . . . . . 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 1369 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → (𝐺‘((𝑥( ·𝑠𝑊)𝑦)(+g𝑊)𝑧)) = ((𝐺‘(𝑥( ·𝑠𝑊)𝑦)) + (𝐺𝑧)))
5015adantr 483 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝐻𝐹)
515, 7, 11, 28, 12lfladd 36194 . . . . . . . 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 1369 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → (𝐻‘((𝑥( ·𝑠𝑊)𝑦)(+g𝑊)𝑧)) = ((𝐻‘(𝑥( ·𝑠𝑊)𝑦)) + (𝐻𝑧)))
5349, 52oveq12d 7166 . . . . . 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 19634 . . . . . . . . 9 (𝑊 ∈ LMod → 𝑅 ∈ Ring)
5521, 54syl 17 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑅 ∈ Ring)
56 ringcmn 19323 . . . . . . . 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 36192 . . . . . . . 8 ((𝑊 ∈ LMod ∧ 𝐺𝐹 ∧ (𝑥( ·𝑠𝑊)𝑦) ∈ (Base‘𝑊)) → (𝐺‘(𝑥( ·𝑠𝑊)𝑦)) ∈ (Base‘𝑅))
5921, 47, 26, 58syl3anc 1366 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → (𝐺‘(𝑥( ·𝑠𝑊)𝑦)) ∈ (Base‘𝑅))
605, 6, 11, 12lflcl 36192 . . . . . . . 8 ((𝑊 ∈ LMod ∧ 𝐺𝐹𝑧 ∈ (Base‘𝑊)) → (𝐺𝑧) ∈ (Base‘𝑅))
6121, 47, 27, 60syl3anc 1366 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → (𝐺𝑧) ∈ (Base‘𝑅))
625, 6, 11, 12lflcl 36192 . . . . . . . 8 ((𝑊 ∈ LMod ∧ 𝐻𝐹 ∧ (𝑥( ·𝑠𝑊)𝑦) ∈ (Base‘𝑊)) → (𝐻‘(𝑥( ·𝑠𝑊)𝑦)) ∈ (Base‘𝑅))
6321, 50, 26, 62syl3anc 1366 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → (𝐻‘(𝑥( ·𝑠𝑊)𝑦)) ∈ (Base‘𝑅))
645, 6, 11, 12lflcl 36192 . . . . . . . 8 ((𝑊 ∈ LMod ∧ 𝐻𝐹𝑧 ∈ (Base‘𝑊)) → (𝐻𝑧) ∈ (Base‘𝑅))
6521, 50, 27, 64syl3anc 1366 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → (𝐻𝑧) ∈ (Base‘𝑅))
666, 7cmn4 18918 . . . . . . 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 1374 . . . . . 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 2819 . . . . . . . . . . 11 (.r𝑅) = (.r𝑅)
695, 6, 68, 11, 24, 12lflmul 36196 . . . . . . . . . 10 ((𝑊 ∈ LMod ∧ 𝐺𝐹 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊))) → (𝐺‘(𝑥( ·𝑠𝑊)𝑦)) = (𝑥(.r𝑅)(𝐺𝑦)))
7021, 47, 22, 23, 69syl112anc 1369 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → (𝐺‘(𝑥( ·𝑠𝑊)𝑦)) = (𝑥(.r𝑅)(𝐺𝑦)))
715, 6, 68, 11, 24, 12lflmul 36196 . . . . . . . . . 10 ((𝑊 ∈ LMod ∧ 𝐻𝐹 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊))) → (𝐻‘(𝑥( ·𝑠𝑊)𝑦)) = (𝑥(.r𝑅)(𝐻𝑦)))
7221, 50, 22, 23, 71syl112anc 1369 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → (𝐻‘(𝑥( ·𝑠𝑊)𝑦)) = (𝑥(.r𝑅)(𝐻𝑦)))
7370, 72oveq12d 7166 . . . . . . . 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 36192 . . . . . . . . . 10 ((𝑊 ∈ LMod ∧ 𝐺𝐹𝑦 ∈ (Base‘𝑊)) → (𝐺𝑦) ∈ (Base‘𝑅))
7521, 47, 23, 74syl3anc 1366 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → (𝐺𝑦) ∈ (Base‘𝑅))
765, 6, 11, 12lflcl 36192 . . . . . . . . . 10 ((𝑊 ∈ LMod ∧ 𝐻𝐹𝑦 ∈ (Base‘𝑊)) → (𝐻𝑦) ∈ (Base‘𝑅))
7721, 50, 23, 76syl3anc 1366 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → (𝐻𝑦) ∈ (Base‘𝑅))
786, 7, 68ringdi 19308 . . . . . . . . 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 1367 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → (𝑥(.r𝑅)((𝐺𝑦) + (𝐻𝑦))) = ((𝑥(.r𝑅)(𝐺𝑦)) + (𝑥(.r𝑅)(𝐻𝑦))))
8073, 79eqtr4d 2857 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → ((𝐺‘(𝑥( ·𝑠𝑊)𝑦)) + (𝐻‘(𝑥( ·𝑠𝑊)𝑦))) = (𝑥(.r𝑅)((𝐺𝑦) + (𝐻𝑦))))
8180oveq1d 7163 . . . . . 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 2858 . . . . 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 2857 . . . 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 2857 . . 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 3190 . 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 36188 . . 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 711 1 (𝜑 → (𝐺f + 𝐻) ∈ 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  w3a 1082   = wceq 1531  wcel 2108  wral 3136  Vcvv 3493  wf 6344  cfv 6348  (class class class)co 7148  f cof 7399  Basecbs 16475  +gcplusg 16557  .rcmulr 16558  Scalarcsca 16560   ·𝑠 cvsca 16561  CMndccmn 18898  Ringcrg 19289  LModclmod 19626  LFnlclfn 36185
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1083  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-nel 3122  df-ral 3141  df-rex 3142  df-reu 3143  df-rmo 3144  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-of 7401  df-om 7573  df-1st 7681  df-2nd 7682  df-wrecs 7939  df-recs 8000  df-rdg 8038  df-er 8281  df-map 8400  df-en 8502  df-dom 8503  df-sdom 8504  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-nn 11631  df-2 11692  df-ndx 16478  df-slot 16479  df-base 16481  df-sets 16482  df-plusg 16570  df-0g 16707  df-mgm 17844  df-sgrp 17893  df-mnd 17904  df-grp 18098  df-minusg 18099  df-sbg 18100  df-cmn 18900  df-abl 18901  df-mgp 19232  df-ur 19244  df-ring 19291  df-lmod 19628  df-lfl 36186
This theorem is referenced by:  ldualvaddcl  36258
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