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Theorem prdslmodd 20816
Description: The product of a family of left modules is a left module. (Contributed by Stefan O'Rear, 10-Jan-2015.)
Hypotheses
Ref Expression
prdslmodd.y π‘Œ = (𝑆Xs𝑅)
prdslmodd.s (πœ‘ β†’ 𝑆 ∈ Ring)
prdslmodd.i (πœ‘ β†’ 𝐼 ∈ 𝑉)
prdslmodd.rm (πœ‘ β†’ 𝑅:𝐼⟢LMod)
prdslmodd.rs ((πœ‘ ∧ 𝑦 ∈ 𝐼) β†’ (Scalarβ€˜(π‘…β€˜π‘¦)) = 𝑆)
Assertion
Ref Expression
prdslmodd (πœ‘ β†’ π‘Œ ∈ LMod)
Distinct variable groups:   𝑦,𝐼   πœ‘,𝑦   𝑦,𝑅   𝑦,𝑆   𝑦,π‘Œ
Allowed substitution hint:   𝑉(𝑦)

Proof of Theorem prdslmodd
Dummy variables π‘Ž 𝑏 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqidd 2727 . 2 (πœ‘ β†’ (Baseβ€˜π‘Œ) = (Baseβ€˜π‘Œ))
2 eqidd 2727 . 2 (πœ‘ β†’ (+gβ€˜π‘Œ) = (+gβ€˜π‘Œ))
3 prdslmodd.y . . 3 π‘Œ = (𝑆Xs𝑅)
4 prdslmodd.s . . 3 (πœ‘ β†’ 𝑆 ∈ Ring)
5 prdslmodd.rm . . . 4 (πœ‘ β†’ 𝑅:𝐼⟢LMod)
6 prdslmodd.i . . . 4 (πœ‘ β†’ 𝐼 ∈ 𝑉)
75, 6fexd 7224 . . 3 (πœ‘ β†’ 𝑅 ∈ V)
83, 4, 7prdssca 17411 . 2 (πœ‘ β†’ 𝑆 = (Scalarβ€˜π‘Œ))
9 eqidd 2727 . 2 (πœ‘ β†’ ( ·𝑠 β€˜π‘Œ) = ( ·𝑠 β€˜π‘Œ))
10 eqidd 2727 . 2 (πœ‘ β†’ (Baseβ€˜π‘†) = (Baseβ€˜π‘†))
11 eqidd 2727 . 2 (πœ‘ β†’ (+gβ€˜π‘†) = (+gβ€˜π‘†))
12 eqidd 2727 . 2 (πœ‘ β†’ (.rβ€˜π‘†) = (.rβ€˜π‘†))
13 eqidd 2727 . 2 (πœ‘ β†’ (1rβ€˜π‘†) = (1rβ€˜π‘†))
14 lmodgrp 20713 . . . . 5 (π‘Ž ∈ LMod β†’ π‘Ž ∈ Grp)
1514ssriv 3981 . . . 4 LMod βŠ† Grp
16 fss 6728 . . . 4 ((𝑅:𝐼⟢LMod ∧ LMod βŠ† Grp) β†’ 𝑅:𝐼⟢Grp)
175, 15, 16sylancl 585 . . 3 (πœ‘ β†’ 𝑅:𝐼⟢Grp)
183, 6, 4, 17prdsgrpd 18978 . 2 (πœ‘ β†’ π‘Œ ∈ Grp)
19 eqid 2726 . . . 4 (Baseβ€˜π‘Œ) = (Baseβ€˜π‘Œ)
20 eqid 2726 . . . 4 ( ·𝑠 β€˜π‘Œ) = ( ·𝑠 β€˜π‘Œ)
21 eqid 2726 . . . 4 (Baseβ€˜π‘†) = (Baseβ€˜π‘†)
224adantr 480 . . . 4 ((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘Œ))) β†’ 𝑆 ∈ Ring)
236elexd 3489 . . . . 5 (πœ‘ β†’ 𝐼 ∈ V)
2423adantr 480 . . . 4 ((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘Œ))) β†’ 𝐼 ∈ V)
255adantr 480 . . . 4 ((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘Œ))) β†’ 𝑅:𝐼⟢LMod)
26 simprl 768 . . . 4 ((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘Œ))) β†’ π‘Ž ∈ (Baseβ€˜π‘†))
27 simprr 770 . . . 4 ((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘Œ))) β†’ 𝑏 ∈ (Baseβ€˜π‘Œ))
28 prdslmodd.rs . . . . 5 ((πœ‘ ∧ 𝑦 ∈ 𝐼) β†’ (Scalarβ€˜(π‘…β€˜π‘¦)) = 𝑆)
2928adantlr 712 . . . 4 (((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘Œ))) ∧ 𝑦 ∈ 𝐼) β†’ (Scalarβ€˜(π‘…β€˜π‘¦)) = 𝑆)
303, 19, 20, 21, 22, 24, 25, 26, 27, 29prdsvscacl 20815 . . 3 ((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘Œ))) β†’ (π‘Ž( ·𝑠 β€˜π‘Œ)𝑏) ∈ (Baseβ€˜π‘Œ))
31303impb 1112 . 2 ((πœ‘ ∧ π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘Œ)) β†’ (π‘Ž( ·𝑠 β€˜π‘Œ)𝑏) ∈ (Baseβ€˜π‘Œ))
325ffvelcdmda 7080 . . . . . . 7 ((πœ‘ ∧ 𝑦 ∈ 𝐼) β†’ (π‘…β€˜π‘¦) ∈ LMod)
3332adantlr 712 . . . . . 6 (((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘Œ) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ))) ∧ 𝑦 ∈ 𝐼) β†’ (π‘…β€˜π‘¦) ∈ LMod)
34 simplr1 1212 . . . . . . 7 (((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘Œ) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ))) ∧ 𝑦 ∈ 𝐼) β†’ π‘Ž ∈ (Baseβ€˜π‘†))
3528fveq2d 6889 . . . . . . . 8 ((πœ‘ ∧ 𝑦 ∈ 𝐼) β†’ (Baseβ€˜(Scalarβ€˜(π‘…β€˜π‘¦))) = (Baseβ€˜π‘†))
3635adantlr 712 . . . . . . 7 (((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘Œ) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ))) ∧ 𝑦 ∈ 𝐼) β†’ (Baseβ€˜(Scalarβ€˜(π‘…β€˜π‘¦))) = (Baseβ€˜π‘†))
3734, 36eleqtrrd 2830 . . . . . 6 (((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘Œ) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ))) ∧ 𝑦 ∈ 𝐼) β†’ π‘Ž ∈ (Baseβ€˜(Scalarβ€˜(π‘…β€˜π‘¦))))
384ad2antrr 723 . . . . . . 7 (((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘Œ) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ))) ∧ 𝑦 ∈ 𝐼) β†’ 𝑆 ∈ Ring)
3923ad2antrr 723 . . . . . . 7 (((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘Œ) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ))) ∧ 𝑦 ∈ 𝐼) β†’ 𝐼 ∈ V)
405ffnd 6712 . . . . . . . 8 (πœ‘ β†’ 𝑅 Fn 𝐼)
4140ad2antrr 723 . . . . . . 7 (((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘Œ) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ))) ∧ 𝑦 ∈ 𝐼) β†’ 𝑅 Fn 𝐼)
42 simplr2 1213 . . . . . . 7 (((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘Œ) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ))) ∧ 𝑦 ∈ 𝐼) β†’ 𝑏 ∈ (Baseβ€˜π‘Œ))
43 simpr 484 . . . . . . 7 (((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘Œ) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ))) ∧ 𝑦 ∈ 𝐼) β†’ 𝑦 ∈ 𝐼)
443, 19, 38, 39, 41, 42, 43prdsbasprj 17427 . . . . . 6 (((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘Œ) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ))) ∧ 𝑦 ∈ 𝐼) β†’ (π‘β€˜π‘¦) ∈ (Baseβ€˜(π‘…β€˜π‘¦)))
45 simplr3 1214 . . . . . . 7 (((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘Œ) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ))) ∧ 𝑦 ∈ 𝐼) β†’ 𝑐 ∈ (Baseβ€˜π‘Œ))
463, 19, 38, 39, 41, 45, 43prdsbasprj 17427 . . . . . 6 (((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘Œ) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ))) ∧ 𝑦 ∈ 𝐼) β†’ (π‘β€˜π‘¦) ∈ (Baseβ€˜(π‘…β€˜π‘¦)))
47 eqid 2726 . . . . . . 7 (Baseβ€˜(π‘…β€˜π‘¦)) = (Baseβ€˜(π‘…β€˜π‘¦))
48 eqid 2726 . . . . . . 7 (+gβ€˜(π‘…β€˜π‘¦)) = (+gβ€˜(π‘…β€˜π‘¦))
49 eqid 2726 . . . . . . 7 (Scalarβ€˜(π‘…β€˜π‘¦)) = (Scalarβ€˜(π‘…β€˜π‘¦))
50 eqid 2726 . . . . . . 7 ( ·𝑠 β€˜(π‘…β€˜π‘¦)) = ( ·𝑠 β€˜(π‘…β€˜π‘¦))
51 eqid 2726 . . . . . . 7 (Baseβ€˜(Scalarβ€˜(π‘…β€˜π‘¦))) = (Baseβ€˜(Scalarβ€˜(π‘…β€˜π‘¦)))
5247, 48, 49, 50, 51lmodvsdi 20731 . . . . . 6 (((π‘…β€˜π‘¦) ∈ LMod ∧ (π‘Ž ∈ (Baseβ€˜(Scalarβ€˜(π‘…β€˜π‘¦))) ∧ (π‘β€˜π‘¦) ∈ (Baseβ€˜(π‘…β€˜π‘¦)) ∧ (π‘β€˜π‘¦) ∈ (Baseβ€˜(π‘…β€˜π‘¦)))) β†’ (π‘Ž( ·𝑠 β€˜(π‘…β€˜π‘¦))((π‘β€˜π‘¦)(+gβ€˜(π‘…β€˜π‘¦))(π‘β€˜π‘¦))) = ((π‘Ž( ·𝑠 β€˜(π‘…β€˜π‘¦))(π‘β€˜π‘¦))(+gβ€˜(π‘…β€˜π‘¦))(π‘Ž( ·𝑠 β€˜(π‘…β€˜π‘¦))(π‘β€˜π‘¦))))
5333, 37, 44, 46, 52syl13anc 1369 . . . . 5 (((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘Œ) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ))) ∧ 𝑦 ∈ 𝐼) β†’ (π‘Ž( ·𝑠 β€˜(π‘…β€˜π‘¦))((π‘β€˜π‘¦)(+gβ€˜(π‘…β€˜π‘¦))(π‘β€˜π‘¦))) = ((π‘Ž( ·𝑠 β€˜(π‘…β€˜π‘¦))(π‘β€˜π‘¦))(+gβ€˜(π‘…β€˜π‘¦))(π‘Ž( ·𝑠 β€˜(π‘…β€˜π‘¦))(π‘β€˜π‘¦))))
54 eqid 2726 . . . . . . 7 (+gβ€˜π‘Œ) = (+gβ€˜π‘Œ)
553, 19, 38, 39, 41, 42, 45, 54, 43prdsplusgfval 17429 . . . . . 6 (((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘Œ) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ))) ∧ 𝑦 ∈ 𝐼) β†’ ((𝑏(+gβ€˜π‘Œ)𝑐)β€˜π‘¦) = ((π‘β€˜π‘¦)(+gβ€˜(π‘…β€˜π‘¦))(π‘β€˜π‘¦)))
5655oveq2d 7421 . . . . 5 (((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘Œ) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ))) ∧ 𝑦 ∈ 𝐼) β†’ (π‘Ž( ·𝑠 β€˜(π‘…β€˜π‘¦))((𝑏(+gβ€˜π‘Œ)𝑐)β€˜π‘¦)) = (π‘Ž( ·𝑠 β€˜(π‘…β€˜π‘¦))((π‘β€˜π‘¦)(+gβ€˜(π‘…β€˜π‘¦))(π‘β€˜π‘¦))))
573, 19, 20, 21, 38, 39, 41, 34, 42, 43prdsvscafval 17435 . . . . . 6 (((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘Œ) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ))) ∧ 𝑦 ∈ 𝐼) β†’ ((π‘Ž( ·𝑠 β€˜π‘Œ)𝑏)β€˜π‘¦) = (π‘Ž( ·𝑠 β€˜(π‘…β€˜π‘¦))(π‘β€˜π‘¦)))
583, 19, 20, 21, 38, 39, 41, 34, 45, 43prdsvscafval 17435 . . . . . 6 (((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘Œ) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ))) ∧ 𝑦 ∈ 𝐼) β†’ ((π‘Ž( ·𝑠 β€˜π‘Œ)𝑐)β€˜π‘¦) = (π‘Ž( ·𝑠 β€˜(π‘…β€˜π‘¦))(π‘β€˜π‘¦)))
5957, 58oveq12d 7423 . . . . 5 (((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘Œ) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ))) ∧ 𝑦 ∈ 𝐼) β†’ (((π‘Ž( ·𝑠 β€˜π‘Œ)𝑏)β€˜π‘¦)(+gβ€˜(π‘…β€˜π‘¦))((π‘Ž( ·𝑠 β€˜π‘Œ)𝑐)β€˜π‘¦)) = ((π‘Ž( ·𝑠 β€˜(π‘…β€˜π‘¦))(π‘β€˜π‘¦))(+gβ€˜(π‘…β€˜π‘¦))(π‘Ž( ·𝑠 β€˜(π‘…β€˜π‘¦))(π‘β€˜π‘¦))))
6053, 56, 593eqtr4d 2776 . . . 4 (((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘Œ) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ))) ∧ 𝑦 ∈ 𝐼) β†’ (π‘Ž( ·𝑠 β€˜(π‘…β€˜π‘¦))((𝑏(+gβ€˜π‘Œ)𝑐)β€˜π‘¦)) = (((π‘Ž( ·𝑠 β€˜π‘Œ)𝑏)β€˜π‘¦)(+gβ€˜(π‘…β€˜π‘¦))((π‘Ž( ·𝑠 β€˜π‘Œ)𝑐)β€˜π‘¦)))
6160mpteq2dva 5241 . . 3 ((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘Œ) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ))) β†’ (𝑦 ∈ 𝐼 ↦ (π‘Ž( ·𝑠 β€˜(π‘…β€˜π‘¦))((𝑏(+gβ€˜π‘Œ)𝑐)β€˜π‘¦))) = (𝑦 ∈ 𝐼 ↦ (((π‘Ž( ·𝑠 β€˜π‘Œ)𝑏)β€˜π‘¦)(+gβ€˜(π‘…β€˜π‘¦))((π‘Ž( ·𝑠 β€˜π‘Œ)𝑐)β€˜π‘¦))))
624adantr 480 . . . 4 ((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘Œ) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ))) β†’ 𝑆 ∈ Ring)
6323adantr 480 . . . 4 ((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘Œ) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ))) β†’ 𝐼 ∈ V)
6440adantr 480 . . . 4 ((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘Œ) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ))) β†’ 𝑅 Fn 𝐼)
65 simpr1 1191 . . . 4 ((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘Œ) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ))) β†’ π‘Ž ∈ (Baseβ€˜π‘†))
6618adantr 480 . . . . 5 ((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘Œ) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ))) β†’ π‘Œ ∈ Grp)
67 simpr2 1192 . . . . 5 ((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘Œ) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ))) β†’ 𝑏 ∈ (Baseβ€˜π‘Œ))
68 simpr3 1193 . . . . 5 ((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘Œ) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ))) β†’ 𝑐 ∈ (Baseβ€˜π‘Œ))
6919, 54grpcl 18871 . . . . 5 ((π‘Œ ∈ Grp ∧ 𝑏 ∈ (Baseβ€˜π‘Œ) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ)) β†’ (𝑏(+gβ€˜π‘Œ)𝑐) ∈ (Baseβ€˜π‘Œ))
7066, 67, 68, 69syl3anc 1368 . . . 4 ((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘Œ) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ))) β†’ (𝑏(+gβ€˜π‘Œ)𝑐) ∈ (Baseβ€˜π‘Œ))
713, 19, 20, 21, 62, 63, 64, 65, 70prdsvscaval 17434 . . 3 ((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘Œ) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ))) β†’ (π‘Ž( ·𝑠 β€˜π‘Œ)(𝑏(+gβ€˜π‘Œ)𝑐)) = (𝑦 ∈ 𝐼 ↦ (π‘Ž( ·𝑠 β€˜(π‘…β€˜π‘¦))((𝑏(+gβ€˜π‘Œ)𝑐)β€˜π‘¦))))
72303adantr3 1168 . . . 4 ((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘Œ) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ))) β†’ (π‘Ž( ·𝑠 β€˜π‘Œ)𝑏) ∈ (Baseβ€˜π‘Œ))
734adantr 480 . . . . . 6 ((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ))) β†’ 𝑆 ∈ Ring)
7423adantr 480 . . . . . 6 ((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ))) β†’ 𝐼 ∈ V)
755adantr 480 . . . . . 6 ((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ))) β†’ 𝑅:𝐼⟢LMod)
76 simprl 768 . . . . . 6 ((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ))) β†’ π‘Ž ∈ (Baseβ€˜π‘†))
77 simprr 770 . . . . . 6 ((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ))) β†’ 𝑐 ∈ (Baseβ€˜π‘Œ))
7828adantlr 712 . . . . . 6 (((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ))) ∧ 𝑦 ∈ 𝐼) β†’ (Scalarβ€˜(π‘…β€˜π‘¦)) = 𝑆)
793, 19, 20, 21, 73, 74, 75, 76, 77, 78prdsvscacl 20815 . . . . 5 ((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ))) β†’ (π‘Ž( ·𝑠 β€˜π‘Œ)𝑐) ∈ (Baseβ€˜π‘Œ))
80793adantr2 1167 . . . 4 ((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘Œ) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ))) β†’ (π‘Ž( ·𝑠 β€˜π‘Œ)𝑐) ∈ (Baseβ€˜π‘Œ))
813, 19, 62, 63, 64, 72, 80, 54prdsplusgval 17428 . . 3 ((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘Œ) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ))) β†’ ((π‘Ž( ·𝑠 β€˜π‘Œ)𝑏)(+gβ€˜π‘Œ)(π‘Ž( ·𝑠 β€˜π‘Œ)𝑐)) = (𝑦 ∈ 𝐼 ↦ (((π‘Ž( ·𝑠 β€˜π‘Œ)𝑏)β€˜π‘¦)(+gβ€˜(π‘…β€˜π‘¦))((π‘Ž( ·𝑠 β€˜π‘Œ)𝑐)β€˜π‘¦))))
8261, 71, 813eqtr4d 2776 . 2 ((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘Œ) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ))) β†’ (π‘Ž( ·𝑠 β€˜π‘Œ)(𝑏(+gβ€˜π‘Œ)𝑐)) = ((π‘Ž( ·𝑠 β€˜π‘Œ)𝑏)(+gβ€˜π‘Œ)(π‘Ž( ·𝑠 β€˜π‘Œ)𝑐)))
834ad2antrr 723 . . . . . . 7 (((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘†) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ))) ∧ 𝑦 ∈ 𝐼) β†’ 𝑆 ∈ Ring)
8423ad2antrr 723 . . . . . . 7 (((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘†) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ))) ∧ 𝑦 ∈ 𝐼) β†’ 𝐼 ∈ V)
8540ad2antrr 723 . . . . . . 7 (((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘†) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ))) ∧ 𝑦 ∈ 𝐼) β†’ 𝑅 Fn 𝐼)
86 simplr1 1212 . . . . . . 7 (((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘†) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ))) ∧ 𝑦 ∈ 𝐼) β†’ π‘Ž ∈ (Baseβ€˜π‘†))
87 simplr3 1214 . . . . . . 7 (((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘†) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ))) ∧ 𝑦 ∈ 𝐼) β†’ 𝑐 ∈ (Baseβ€˜π‘Œ))
88 simpr 484 . . . . . . 7 (((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘†) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ))) ∧ 𝑦 ∈ 𝐼) β†’ 𝑦 ∈ 𝐼)
893, 19, 20, 21, 83, 84, 85, 86, 87, 88prdsvscafval 17435 . . . . . 6 (((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘†) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ))) ∧ 𝑦 ∈ 𝐼) β†’ ((π‘Ž( ·𝑠 β€˜π‘Œ)𝑐)β€˜π‘¦) = (π‘Ž( ·𝑠 β€˜(π‘…β€˜π‘¦))(π‘β€˜π‘¦)))
90 simplr2 1213 . . . . . . 7 (((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘†) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ))) ∧ 𝑦 ∈ 𝐼) β†’ 𝑏 ∈ (Baseβ€˜π‘†))
913, 19, 20, 21, 83, 84, 85, 90, 87, 88prdsvscafval 17435 . . . . . 6 (((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘†) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ))) ∧ 𝑦 ∈ 𝐼) β†’ ((𝑏( ·𝑠 β€˜π‘Œ)𝑐)β€˜π‘¦) = (𝑏( ·𝑠 β€˜(π‘…β€˜π‘¦))(π‘β€˜π‘¦)))
9289, 91oveq12d 7423 . . . . 5 (((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘†) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ))) ∧ 𝑦 ∈ 𝐼) β†’ (((π‘Ž( ·𝑠 β€˜π‘Œ)𝑐)β€˜π‘¦)(+gβ€˜(π‘…β€˜π‘¦))((𝑏( ·𝑠 β€˜π‘Œ)𝑐)β€˜π‘¦)) = ((π‘Ž( ·𝑠 β€˜(π‘…β€˜π‘¦))(π‘β€˜π‘¦))(+gβ€˜(π‘…β€˜π‘¦))(𝑏( ·𝑠 β€˜(π‘…β€˜π‘¦))(π‘β€˜π‘¦))))
9332adantlr 712 . . . . . 6 (((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘†) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ))) ∧ 𝑦 ∈ 𝐼) β†’ (π‘…β€˜π‘¦) ∈ LMod)
9435adantlr 712 . . . . . . 7 (((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘†) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ))) ∧ 𝑦 ∈ 𝐼) β†’ (Baseβ€˜(Scalarβ€˜(π‘…β€˜π‘¦))) = (Baseβ€˜π‘†))
9586, 94eleqtrrd 2830 . . . . . 6 (((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘†) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ))) ∧ 𝑦 ∈ 𝐼) β†’ π‘Ž ∈ (Baseβ€˜(Scalarβ€˜(π‘…β€˜π‘¦))))
9690, 94eleqtrrd 2830 . . . . . 6 (((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘†) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ))) ∧ 𝑦 ∈ 𝐼) β†’ 𝑏 ∈ (Baseβ€˜(Scalarβ€˜(π‘…β€˜π‘¦))))
973, 19, 83, 84, 85, 87, 88prdsbasprj 17427 . . . . . 6 (((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘†) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ))) ∧ 𝑦 ∈ 𝐼) β†’ (π‘β€˜π‘¦) ∈ (Baseβ€˜(π‘…β€˜π‘¦)))
98 eqid 2726 . . . . . . 7 (+gβ€˜(Scalarβ€˜(π‘…β€˜π‘¦))) = (+gβ€˜(Scalarβ€˜(π‘…β€˜π‘¦)))
9947, 48, 49, 50, 51, 98lmodvsdir 20732 . . . . . 6 (((π‘…β€˜π‘¦) ∈ LMod ∧ (π‘Ž ∈ (Baseβ€˜(Scalarβ€˜(π‘…β€˜π‘¦))) ∧ 𝑏 ∈ (Baseβ€˜(Scalarβ€˜(π‘…β€˜π‘¦))) ∧ (π‘β€˜π‘¦) ∈ (Baseβ€˜(π‘…β€˜π‘¦)))) β†’ ((π‘Ž(+gβ€˜(Scalarβ€˜(π‘…β€˜π‘¦)))𝑏)( ·𝑠 β€˜(π‘…β€˜π‘¦))(π‘β€˜π‘¦)) = ((π‘Ž( ·𝑠 β€˜(π‘…β€˜π‘¦))(π‘β€˜π‘¦))(+gβ€˜(π‘…β€˜π‘¦))(𝑏( ·𝑠 β€˜(π‘…β€˜π‘¦))(π‘β€˜π‘¦))))
10093, 95, 96, 97, 99syl13anc 1369 . . . . 5 (((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘†) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ))) ∧ 𝑦 ∈ 𝐼) β†’ ((π‘Ž(+gβ€˜(Scalarβ€˜(π‘…β€˜π‘¦)))𝑏)( ·𝑠 β€˜(π‘…β€˜π‘¦))(π‘β€˜π‘¦)) = ((π‘Ž( ·𝑠 β€˜(π‘…β€˜π‘¦))(π‘β€˜π‘¦))(+gβ€˜(π‘…β€˜π‘¦))(𝑏( ·𝑠 β€˜(π‘…β€˜π‘¦))(π‘β€˜π‘¦))))
10128adantlr 712 . . . . . . . 8 (((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘†) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ))) ∧ 𝑦 ∈ 𝐼) β†’ (Scalarβ€˜(π‘…β€˜π‘¦)) = 𝑆)
102101fveq2d 6889 . . . . . . 7 (((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘†) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ))) ∧ 𝑦 ∈ 𝐼) β†’ (+gβ€˜(Scalarβ€˜(π‘…β€˜π‘¦))) = (+gβ€˜π‘†))
103102oveqd 7422 . . . . . 6 (((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘†) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ))) ∧ 𝑦 ∈ 𝐼) β†’ (π‘Ž(+gβ€˜(Scalarβ€˜(π‘…β€˜π‘¦)))𝑏) = (π‘Ž(+gβ€˜π‘†)𝑏))
104103oveq1d 7420 . . . . 5 (((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘†) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ))) ∧ 𝑦 ∈ 𝐼) β†’ ((π‘Ž(+gβ€˜(Scalarβ€˜(π‘…β€˜π‘¦)))𝑏)( ·𝑠 β€˜(π‘…β€˜π‘¦))(π‘β€˜π‘¦)) = ((π‘Ž(+gβ€˜π‘†)𝑏)( ·𝑠 β€˜(π‘…β€˜π‘¦))(π‘β€˜π‘¦)))
10592, 100, 1043eqtr2rd 2773 . . . 4 (((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘†) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ))) ∧ 𝑦 ∈ 𝐼) β†’ ((π‘Ž(+gβ€˜π‘†)𝑏)( ·𝑠 β€˜(π‘…β€˜π‘¦))(π‘β€˜π‘¦)) = (((π‘Ž( ·𝑠 β€˜π‘Œ)𝑐)β€˜π‘¦)(+gβ€˜(π‘…β€˜π‘¦))((𝑏( ·𝑠 β€˜π‘Œ)𝑐)β€˜π‘¦)))
106105mpteq2dva 5241 . . 3 ((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘†) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ))) β†’ (𝑦 ∈ 𝐼 ↦ ((π‘Ž(+gβ€˜π‘†)𝑏)( ·𝑠 β€˜(π‘…β€˜π‘¦))(π‘β€˜π‘¦))) = (𝑦 ∈ 𝐼 ↦ (((π‘Ž( ·𝑠 β€˜π‘Œ)𝑐)β€˜π‘¦)(+gβ€˜(π‘…β€˜π‘¦))((𝑏( ·𝑠 β€˜π‘Œ)𝑐)β€˜π‘¦))))
1074adantr 480 . . . 4 ((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘†) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ))) β†’ 𝑆 ∈ Ring)
10823adantr 480 . . . 4 ((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘†) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ))) β†’ 𝐼 ∈ V)
10940adantr 480 . . . 4 ((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘†) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ))) β†’ 𝑅 Fn 𝐼)
110 simpr1 1191 . . . . 5 ((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘†) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ))) β†’ π‘Ž ∈ (Baseβ€˜π‘†))
111 simpr2 1192 . . . . 5 ((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘†) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ))) β†’ 𝑏 ∈ (Baseβ€˜π‘†))
112 eqid 2726 . . . . . 6 (+gβ€˜π‘†) = (+gβ€˜π‘†)
11321, 112ringacl 20177 . . . . 5 ((𝑆 ∈ Ring ∧ π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘†)) β†’ (π‘Ž(+gβ€˜π‘†)𝑏) ∈ (Baseβ€˜π‘†))
114107, 110, 111, 113syl3anc 1368 . . . 4 ((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘†) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ))) β†’ (π‘Ž(+gβ€˜π‘†)𝑏) ∈ (Baseβ€˜π‘†))
115 simpr3 1193 . . . 4 ((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘†) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ))) β†’ 𝑐 ∈ (Baseβ€˜π‘Œ))
1163, 19, 20, 21, 107, 108, 109, 114, 115prdsvscaval 17434 . . 3 ((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘†) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ))) β†’ ((π‘Ž(+gβ€˜π‘†)𝑏)( ·𝑠 β€˜π‘Œ)𝑐) = (𝑦 ∈ 𝐼 ↦ ((π‘Ž(+gβ€˜π‘†)𝑏)( ·𝑠 β€˜(π‘…β€˜π‘¦))(π‘β€˜π‘¦))))
117793adantr2 1167 . . . 4 ((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘†) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ))) β†’ (π‘Ž( ·𝑠 β€˜π‘Œ)𝑐) ∈ (Baseβ€˜π‘Œ))
1185adantr 480 . . . . 5 ((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘†) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ))) β†’ 𝑅:𝐼⟢LMod)
1193, 19, 20, 21, 107, 108, 118, 111, 115, 101prdsvscacl 20815 . . . 4 ((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘†) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ))) β†’ (𝑏( ·𝑠 β€˜π‘Œ)𝑐) ∈ (Baseβ€˜π‘Œ))
1203, 19, 107, 108, 109, 117, 119, 54prdsplusgval 17428 . . 3 ((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘†) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ))) β†’ ((π‘Ž( ·𝑠 β€˜π‘Œ)𝑐)(+gβ€˜π‘Œ)(𝑏( ·𝑠 β€˜π‘Œ)𝑐)) = (𝑦 ∈ 𝐼 ↦ (((π‘Ž( ·𝑠 β€˜π‘Œ)𝑐)β€˜π‘¦)(+gβ€˜(π‘…β€˜π‘¦))((𝑏( ·𝑠 β€˜π‘Œ)𝑐)β€˜π‘¦))))
121106, 116, 1203eqtr4d 2776 . 2 ((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘†) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ))) β†’ ((π‘Ž(+gβ€˜π‘†)𝑏)( ·𝑠 β€˜π‘Œ)𝑐) = ((π‘Ž( ·𝑠 β€˜π‘Œ)𝑐)(+gβ€˜π‘Œ)(𝑏( ·𝑠 β€˜π‘Œ)𝑐)))
12291oveq2d 7421 . . . . 5 (((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘†) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ))) ∧ 𝑦 ∈ 𝐼) β†’ (π‘Ž( ·𝑠 β€˜(π‘…β€˜π‘¦))((𝑏( ·𝑠 β€˜π‘Œ)𝑐)β€˜π‘¦)) = (π‘Ž( ·𝑠 β€˜(π‘…β€˜π‘¦))(𝑏( ·𝑠 β€˜(π‘…β€˜π‘¦))(π‘β€˜π‘¦))))
123 eqid 2726 . . . . . . 7 (.rβ€˜(Scalarβ€˜(π‘…β€˜π‘¦))) = (.rβ€˜(Scalarβ€˜(π‘…β€˜π‘¦)))
12447, 49, 50, 51, 123lmodvsass 20733 . . . . . 6 (((π‘…β€˜π‘¦) ∈ LMod ∧ (π‘Ž ∈ (Baseβ€˜(Scalarβ€˜(π‘…β€˜π‘¦))) ∧ 𝑏 ∈ (Baseβ€˜(Scalarβ€˜(π‘…β€˜π‘¦))) ∧ (π‘β€˜π‘¦) ∈ (Baseβ€˜(π‘…β€˜π‘¦)))) β†’ ((π‘Ž(.rβ€˜(Scalarβ€˜(π‘…β€˜π‘¦)))𝑏)( ·𝑠 β€˜(π‘…β€˜π‘¦))(π‘β€˜π‘¦)) = (π‘Ž( ·𝑠 β€˜(π‘…β€˜π‘¦))(𝑏( ·𝑠 β€˜(π‘…β€˜π‘¦))(π‘β€˜π‘¦))))
12593, 95, 96, 97, 124syl13anc 1369 . . . . 5 (((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘†) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ))) ∧ 𝑦 ∈ 𝐼) β†’ ((π‘Ž(.rβ€˜(Scalarβ€˜(π‘…β€˜π‘¦)))𝑏)( ·𝑠 β€˜(π‘…β€˜π‘¦))(π‘β€˜π‘¦)) = (π‘Ž( ·𝑠 β€˜(π‘…β€˜π‘¦))(𝑏( ·𝑠 β€˜(π‘…β€˜π‘¦))(π‘β€˜π‘¦))))
126101fveq2d 6889 . . . . . . 7 (((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘†) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ))) ∧ 𝑦 ∈ 𝐼) β†’ (.rβ€˜(Scalarβ€˜(π‘…β€˜π‘¦))) = (.rβ€˜π‘†))
127126oveqd 7422 . . . . . 6 (((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘†) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ))) ∧ 𝑦 ∈ 𝐼) β†’ (π‘Ž(.rβ€˜(Scalarβ€˜(π‘…β€˜π‘¦)))𝑏) = (π‘Ž(.rβ€˜π‘†)𝑏))
128127oveq1d 7420 . . . . 5 (((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘†) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ))) ∧ 𝑦 ∈ 𝐼) β†’ ((π‘Ž(.rβ€˜(Scalarβ€˜(π‘…β€˜π‘¦)))𝑏)( ·𝑠 β€˜(π‘…β€˜π‘¦))(π‘β€˜π‘¦)) = ((π‘Ž(.rβ€˜π‘†)𝑏)( ·𝑠 β€˜(π‘…β€˜π‘¦))(π‘β€˜π‘¦)))
129122, 125, 1283eqtr2rd 2773 . . . 4 (((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘†) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ))) ∧ 𝑦 ∈ 𝐼) β†’ ((π‘Ž(.rβ€˜π‘†)𝑏)( ·𝑠 β€˜(π‘…β€˜π‘¦))(π‘β€˜π‘¦)) = (π‘Ž( ·𝑠 β€˜(π‘…β€˜π‘¦))((𝑏( ·𝑠 β€˜π‘Œ)𝑐)β€˜π‘¦)))
130129mpteq2dva 5241 . . 3 ((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘†) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ))) β†’ (𝑦 ∈ 𝐼 ↦ ((π‘Ž(.rβ€˜π‘†)𝑏)( ·𝑠 β€˜(π‘…β€˜π‘¦))(π‘β€˜π‘¦))) = (𝑦 ∈ 𝐼 ↦ (π‘Ž( ·𝑠 β€˜(π‘…β€˜π‘¦))((𝑏( ·𝑠 β€˜π‘Œ)𝑐)β€˜π‘¦))))
131 eqid 2726 . . . . . 6 (.rβ€˜π‘†) = (.rβ€˜π‘†)
13221, 131ringcl 20155 . . . . 5 ((𝑆 ∈ Ring ∧ π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘†)) β†’ (π‘Ž(.rβ€˜π‘†)𝑏) ∈ (Baseβ€˜π‘†))
133107, 110, 111, 132syl3anc 1368 . . . 4 ((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘†) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ))) β†’ (π‘Ž(.rβ€˜π‘†)𝑏) ∈ (Baseβ€˜π‘†))
1343, 19, 20, 21, 107, 108, 109, 133, 115prdsvscaval 17434 . . 3 ((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘†) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ))) β†’ ((π‘Ž(.rβ€˜π‘†)𝑏)( ·𝑠 β€˜π‘Œ)𝑐) = (𝑦 ∈ 𝐼 ↦ ((π‘Ž(.rβ€˜π‘†)𝑏)( ·𝑠 β€˜(π‘…β€˜π‘¦))(π‘β€˜π‘¦))))
1353, 19, 20, 21, 107, 108, 109, 110, 119prdsvscaval 17434 . . 3 ((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘†) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ))) β†’ (π‘Ž( ·𝑠 β€˜π‘Œ)(𝑏( ·𝑠 β€˜π‘Œ)𝑐)) = (𝑦 ∈ 𝐼 ↦ (π‘Ž( ·𝑠 β€˜(π‘…β€˜π‘¦))((𝑏( ·𝑠 β€˜π‘Œ)𝑐)β€˜π‘¦))))
136130, 134, 1353eqtr4d 2776 . 2 ((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘†) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ))) β†’ ((π‘Ž(.rβ€˜π‘†)𝑏)( ·𝑠 β€˜π‘Œ)𝑐) = (π‘Ž( ·𝑠 β€˜π‘Œ)(𝑏( ·𝑠 β€˜π‘Œ)𝑐)))
13728fveq2d 6889 . . . . . . 7 ((πœ‘ ∧ 𝑦 ∈ 𝐼) β†’ (1rβ€˜(Scalarβ€˜(π‘…β€˜π‘¦))) = (1rβ€˜π‘†))
138137adantlr 712 . . . . . 6 (((πœ‘ ∧ π‘Ž ∈ (Baseβ€˜π‘Œ)) ∧ 𝑦 ∈ 𝐼) β†’ (1rβ€˜(Scalarβ€˜(π‘…β€˜π‘¦))) = (1rβ€˜π‘†))
139138oveq1d 7420 . . . . 5 (((πœ‘ ∧ π‘Ž ∈ (Baseβ€˜π‘Œ)) ∧ 𝑦 ∈ 𝐼) β†’ ((1rβ€˜(Scalarβ€˜(π‘…β€˜π‘¦)))( ·𝑠 β€˜(π‘…β€˜π‘¦))(π‘Žβ€˜π‘¦)) = ((1rβ€˜π‘†)( ·𝑠 β€˜(π‘…β€˜π‘¦))(π‘Žβ€˜π‘¦)))
14032adantlr 712 . . . . . 6 (((πœ‘ ∧ π‘Ž ∈ (Baseβ€˜π‘Œ)) ∧ 𝑦 ∈ 𝐼) β†’ (π‘…β€˜π‘¦) ∈ LMod)
1414ad2antrr 723 . . . . . . 7 (((πœ‘ ∧ π‘Ž ∈ (Baseβ€˜π‘Œ)) ∧ 𝑦 ∈ 𝐼) β†’ 𝑆 ∈ Ring)
14223ad2antrr 723 . . . . . . 7 (((πœ‘ ∧ π‘Ž ∈ (Baseβ€˜π‘Œ)) ∧ 𝑦 ∈ 𝐼) β†’ 𝐼 ∈ V)
14340ad2antrr 723 . . . . . . 7 (((πœ‘ ∧ π‘Ž ∈ (Baseβ€˜π‘Œ)) ∧ 𝑦 ∈ 𝐼) β†’ 𝑅 Fn 𝐼)
144 simplr 766 . . . . . . 7 (((πœ‘ ∧ π‘Ž ∈ (Baseβ€˜π‘Œ)) ∧ 𝑦 ∈ 𝐼) β†’ π‘Ž ∈ (Baseβ€˜π‘Œ))
145 simpr 484 . . . . . . 7 (((πœ‘ ∧ π‘Ž ∈ (Baseβ€˜π‘Œ)) ∧ 𝑦 ∈ 𝐼) β†’ 𝑦 ∈ 𝐼)
1463, 19, 141, 142, 143, 144, 145prdsbasprj 17427 . . . . . 6 (((πœ‘ ∧ π‘Ž ∈ (Baseβ€˜π‘Œ)) ∧ 𝑦 ∈ 𝐼) β†’ (π‘Žβ€˜π‘¦) ∈ (Baseβ€˜(π‘…β€˜π‘¦)))
147 eqid 2726 . . . . . . 7 (1rβ€˜(Scalarβ€˜(π‘…β€˜π‘¦))) = (1rβ€˜(Scalarβ€˜(π‘…β€˜π‘¦)))
14847, 49, 50, 147lmodvs1 20736 . . . . . 6 (((π‘…β€˜π‘¦) ∈ LMod ∧ (π‘Žβ€˜π‘¦) ∈ (Baseβ€˜(π‘…β€˜π‘¦))) β†’ ((1rβ€˜(Scalarβ€˜(π‘…β€˜π‘¦)))( ·𝑠 β€˜(π‘…β€˜π‘¦))(π‘Žβ€˜π‘¦)) = (π‘Žβ€˜π‘¦))
149140, 146, 148syl2anc 583 . . . . 5 (((πœ‘ ∧ π‘Ž ∈ (Baseβ€˜π‘Œ)) ∧ 𝑦 ∈ 𝐼) β†’ ((1rβ€˜(Scalarβ€˜(π‘…β€˜π‘¦)))( ·𝑠 β€˜(π‘…β€˜π‘¦))(π‘Žβ€˜π‘¦)) = (π‘Žβ€˜π‘¦))
150139, 149eqtr3d 2768 . . . 4 (((πœ‘ ∧ π‘Ž ∈ (Baseβ€˜π‘Œ)) ∧ 𝑦 ∈ 𝐼) β†’ ((1rβ€˜π‘†)( ·𝑠 β€˜(π‘…β€˜π‘¦))(π‘Žβ€˜π‘¦)) = (π‘Žβ€˜π‘¦))
151150mpteq2dva 5241 . . 3 ((πœ‘ ∧ π‘Ž ∈ (Baseβ€˜π‘Œ)) β†’ (𝑦 ∈ 𝐼 ↦ ((1rβ€˜π‘†)( ·𝑠 β€˜(π‘…β€˜π‘¦))(π‘Žβ€˜π‘¦))) = (𝑦 ∈ 𝐼 ↦ (π‘Žβ€˜π‘¦)))
1524adantr 480 . . . 4 ((πœ‘ ∧ π‘Ž ∈ (Baseβ€˜π‘Œ)) β†’ 𝑆 ∈ Ring)
15323adantr 480 . . . 4 ((πœ‘ ∧ π‘Ž ∈ (Baseβ€˜π‘Œ)) β†’ 𝐼 ∈ V)
15440adantr 480 . . . 4 ((πœ‘ ∧ π‘Ž ∈ (Baseβ€˜π‘Œ)) β†’ 𝑅 Fn 𝐼)
155 eqid 2726 . . . . . . 7 (1rβ€˜π‘†) = (1rβ€˜π‘†)
15621, 155ringidcl 20165 . . . . . 6 (𝑆 ∈ Ring β†’ (1rβ€˜π‘†) ∈ (Baseβ€˜π‘†))
1574, 156syl 17 . . . . 5 (πœ‘ β†’ (1rβ€˜π‘†) ∈ (Baseβ€˜π‘†))
158157adantr 480 . . . 4 ((πœ‘ ∧ π‘Ž ∈ (Baseβ€˜π‘Œ)) β†’ (1rβ€˜π‘†) ∈ (Baseβ€˜π‘†))
159 simpr 484 . . . 4 ((πœ‘ ∧ π‘Ž ∈ (Baseβ€˜π‘Œ)) β†’ π‘Ž ∈ (Baseβ€˜π‘Œ))
1603, 19, 20, 21, 152, 153, 154, 158, 159prdsvscaval 17434 . . 3 ((πœ‘ ∧ π‘Ž ∈ (Baseβ€˜π‘Œ)) β†’ ((1rβ€˜π‘†)( ·𝑠 β€˜π‘Œ)π‘Ž) = (𝑦 ∈ 𝐼 ↦ ((1rβ€˜π‘†)( ·𝑠 β€˜(π‘…β€˜π‘¦))(π‘Žβ€˜π‘¦))))
1613, 19, 152, 153, 154, 159prdsbasfn 17426 . . . 4 ((πœ‘ ∧ π‘Ž ∈ (Baseβ€˜π‘Œ)) β†’ π‘Ž Fn 𝐼)
162 dffn5 6944 . . . 4 (π‘Ž Fn 𝐼 ↔ π‘Ž = (𝑦 ∈ 𝐼 ↦ (π‘Žβ€˜π‘¦)))
163161, 162sylib 217 . . 3 ((πœ‘ ∧ π‘Ž ∈ (Baseβ€˜π‘Œ)) β†’ π‘Ž = (𝑦 ∈ 𝐼 ↦ (π‘Žβ€˜π‘¦)))
164151, 160, 1633eqtr4d 2776 . 2 ((πœ‘ ∧ π‘Ž ∈ (Baseβ€˜π‘Œ)) β†’ ((1rβ€˜π‘†)( ·𝑠 β€˜π‘Œ)π‘Ž) = π‘Ž)
1651, 2, 8, 9, 10, 11, 12, 13, 4, 18, 31, 82, 121, 136, 164islmodd 20712 1 (πœ‘ β†’ π‘Œ ∈ LMod)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  Vcvv 3468   βŠ† wss 3943   ↦ cmpt 5224   Fn wfn 6532  βŸΆwf 6533  β€˜cfv 6537  (class class class)co 7405  Basecbs 17153  +gcplusg 17206  .rcmulr 17207  Scalarcsca 17209   ·𝑠 cvsca 17210  Xscprds 17400  Grpcgrp 18863  1rcur 20086  Ringcrg 20138  LModclmod 20706
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-tp 4628  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7853  df-1st 7974  df-2nd 7975  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-rdg 8411  df-1o 8467  df-er 8705  df-map 8824  df-ixp 8894  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-sup 9439  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-2 12279  df-3 12280  df-4 12281  df-5 12282  df-6 12283  df-7 12284  df-8 12285  df-9 12286  df-n0 12477  df-z 12563  df-dec 12682  df-uz 12827  df-fz 13491  df-struct 17089  df-sets 17106  df-slot 17124  df-ndx 17136  df-base 17154  df-plusg 17219  df-mulr 17220  df-sca 17222  df-vsca 17223  df-ip 17224  df-tset 17225  df-ple 17226  df-ds 17228  df-hom 17230  df-cco 17231  df-0g 17396  df-prds 17402  df-mgm 18573  df-sgrp 18652  df-mnd 18668  df-grp 18866  df-minusg 18867  df-mgp 20040  df-ur 20087  df-ring 20140  df-lmod 20708
This theorem is referenced by:  pwslmod  20817  dsmmlss  21639  dsmmlmod  21640
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