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Theorem prdslmodd 20445
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 2734 . 2 (πœ‘ β†’ (Baseβ€˜π‘Œ) = (Baseβ€˜π‘Œ))
2 eqidd 2734 . 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 7178 . . 3 (πœ‘ β†’ 𝑅 ∈ V)
83, 4, 7prdssca 17343 . 2 (πœ‘ β†’ 𝑆 = (Scalarβ€˜π‘Œ))
9 eqidd 2734 . 2 (πœ‘ β†’ ( ·𝑠 β€˜π‘Œ) = ( ·𝑠 β€˜π‘Œ))
10 eqidd 2734 . 2 (πœ‘ β†’ (Baseβ€˜π‘†) = (Baseβ€˜π‘†))
11 eqidd 2734 . 2 (πœ‘ β†’ (+gβ€˜π‘†) = (+gβ€˜π‘†))
12 eqidd 2734 . 2 (πœ‘ β†’ (.rβ€˜π‘†) = (.rβ€˜π‘†))
13 eqidd 2734 . 2 (πœ‘ β†’ (1rβ€˜π‘†) = (1rβ€˜π‘†))
14 lmodgrp 20343 . . . . 5 (π‘Ž ∈ LMod β†’ π‘Ž ∈ Grp)
1514ssriv 3949 . . . 4 LMod βŠ† Grp
16 fss 6686 . . . 4 ((𝑅:𝐼⟢LMod ∧ LMod βŠ† Grp) β†’ 𝑅:𝐼⟢Grp)
175, 15, 16sylancl 587 . . 3 (πœ‘ β†’ 𝑅:𝐼⟢Grp)
183, 6, 4, 17prdsgrpd 18862 . 2 (πœ‘ β†’ π‘Œ ∈ Grp)
19 eqid 2733 . . . 4 (Baseβ€˜π‘Œ) = (Baseβ€˜π‘Œ)
20 eqid 2733 . . . 4 ( ·𝑠 β€˜π‘Œ) = ( ·𝑠 β€˜π‘Œ)
21 eqid 2733 . . . 4 (Baseβ€˜π‘†) = (Baseβ€˜π‘†)
224adantr 482 . . . 4 ((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘Œ))) β†’ 𝑆 ∈ Ring)
236elexd 3464 . . . . 5 (πœ‘ β†’ 𝐼 ∈ V)
2423adantr 482 . . . 4 ((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘Œ))) β†’ 𝐼 ∈ V)
255adantr 482 . . . 4 ((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘Œ))) β†’ 𝑅:𝐼⟢LMod)
26 simprl 770 . . . 4 ((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘Œ))) β†’ π‘Ž ∈ (Baseβ€˜π‘†))
27 simprr 772 . . . 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 714 . . . 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 20444 . . 3 ((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘Œ))) β†’ (π‘Ž( ·𝑠 β€˜π‘Œ)𝑏) ∈ (Baseβ€˜π‘Œ))
31303impb 1116 . 2 ((πœ‘ ∧ π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘Œ)) β†’ (π‘Ž( ·𝑠 β€˜π‘Œ)𝑏) ∈ (Baseβ€˜π‘Œ))
325ffvelcdmda 7036 . . . . . . 7 ((πœ‘ ∧ 𝑦 ∈ 𝐼) β†’ (π‘…β€˜π‘¦) ∈ LMod)
3332adantlr 714 . . . . . 6 (((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘Œ) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ))) ∧ 𝑦 ∈ 𝐼) β†’ (π‘…β€˜π‘¦) ∈ LMod)
34 simplr1 1216 . . . . . . 7 (((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘Œ) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ))) ∧ 𝑦 ∈ 𝐼) β†’ π‘Ž ∈ (Baseβ€˜π‘†))
3528fveq2d 6847 . . . . . . . 8 ((πœ‘ ∧ 𝑦 ∈ 𝐼) β†’ (Baseβ€˜(Scalarβ€˜(π‘…β€˜π‘¦))) = (Baseβ€˜π‘†))
3635adantlr 714 . . . . . . 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 2837 . . . . . 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 725 . . . . . . 7 (((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘Œ) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ))) ∧ 𝑦 ∈ 𝐼) β†’ 𝑆 ∈ Ring)
3923ad2antrr 725 . . . . . . 7 (((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘Œ) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ))) ∧ 𝑦 ∈ 𝐼) β†’ 𝐼 ∈ V)
405ffnd 6670 . . . . . . . 8 (πœ‘ β†’ 𝑅 Fn 𝐼)
4140ad2antrr 725 . . . . . . 7 (((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘Œ) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ))) ∧ 𝑦 ∈ 𝐼) β†’ 𝑅 Fn 𝐼)
42 simplr2 1217 . . . . . . 7 (((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘Œ) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ))) ∧ 𝑦 ∈ 𝐼) β†’ 𝑏 ∈ (Baseβ€˜π‘Œ))
43 simpr 486 . . . . . . 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 17359 . . . . . 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 1218 . . . . . . 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 17359 . . . . . 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 2733 . . . . . . 7 (Baseβ€˜(π‘…β€˜π‘¦)) = (Baseβ€˜(π‘…β€˜π‘¦))
48 eqid 2733 . . . . . . 7 (+gβ€˜(π‘…β€˜π‘¦)) = (+gβ€˜(π‘…β€˜π‘¦))
49 eqid 2733 . . . . . . 7 (Scalarβ€˜(π‘…β€˜π‘¦)) = (Scalarβ€˜(π‘…β€˜π‘¦))
50 eqid 2733 . . . . . . 7 ( ·𝑠 β€˜(π‘…β€˜π‘¦)) = ( ·𝑠 β€˜(π‘…β€˜π‘¦))
51 eqid 2733 . . . . . . 7 (Baseβ€˜(Scalarβ€˜(π‘…β€˜π‘¦))) = (Baseβ€˜(Scalarβ€˜(π‘…β€˜π‘¦)))
5247, 48, 49, 50, 51lmodvsdi 20360 . . . . . 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 1373 . . . . 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 2733 . . . . . . 7 (+gβ€˜π‘Œ) = (+gβ€˜π‘Œ)
553, 19, 38, 39, 41, 42, 45, 54, 43prdsplusgfval 17361 . . . . . 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 7374 . . . . 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 17367 . . . . . 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 17367 . . . . . 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 7376 . . . . 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 2783 . . . 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 5206 . . 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 482 . . . 4 ((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘Œ) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ))) β†’ 𝑆 ∈ Ring)
6323adantr 482 . . . 4 ((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘Œ) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ))) β†’ 𝐼 ∈ V)
6440adantr 482 . . . 4 ((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘Œ) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ))) β†’ 𝑅 Fn 𝐼)
65 simpr1 1195 . . . 4 ((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘Œ) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ))) β†’ π‘Ž ∈ (Baseβ€˜π‘†))
6618adantr 482 . . . . 5 ((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘Œ) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ))) β†’ π‘Œ ∈ Grp)
67 simpr2 1196 . . . . 5 ((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘Œ) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ))) β†’ 𝑏 ∈ (Baseβ€˜π‘Œ))
68 simpr3 1197 . . . . 5 ((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘Œ) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ))) β†’ 𝑐 ∈ (Baseβ€˜π‘Œ))
6919, 54grpcl 18761 . . . . 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 1372 . . . 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 17366 . . 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 1172 . . . 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 482 . . . . . 6 ((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ))) β†’ 𝑆 ∈ Ring)
7423adantr 482 . . . . . 6 ((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ))) β†’ 𝐼 ∈ V)
755adantr 482 . . . . . 6 ((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ))) β†’ 𝑅:𝐼⟢LMod)
76 simprl 770 . . . . . 6 ((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ))) β†’ π‘Ž ∈ (Baseβ€˜π‘†))
77 simprr 772 . . . . . 6 ((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ))) β†’ 𝑐 ∈ (Baseβ€˜π‘Œ))
7828adantlr 714 . . . . . 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 20444 . . . . 5 ((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ))) β†’ (π‘Ž( ·𝑠 β€˜π‘Œ)𝑐) ∈ (Baseβ€˜π‘Œ))
80793adantr2 1171 . . . 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 17360 . . 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 2783 . 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 725 . . . . . . 7 (((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘†) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ))) ∧ 𝑦 ∈ 𝐼) β†’ 𝑆 ∈ Ring)
8423ad2antrr 725 . . . . . . 7 (((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘†) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ))) ∧ 𝑦 ∈ 𝐼) β†’ 𝐼 ∈ V)
8540ad2antrr 725 . . . . . . 7 (((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘†) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ))) ∧ 𝑦 ∈ 𝐼) β†’ 𝑅 Fn 𝐼)
86 simplr1 1216 . . . . . . 7 (((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘†) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ))) ∧ 𝑦 ∈ 𝐼) β†’ π‘Ž ∈ (Baseβ€˜π‘†))
87 simplr3 1218 . . . . . . 7 (((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘†) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ))) ∧ 𝑦 ∈ 𝐼) β†’ 𝑐 ∈ (Baseβ€˜π‘Œ))
88 simpr 486 . . . . . . 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 17367 . . . . . 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 1217 . . . . . . 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 17367 . . . . . 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 7376 . . . . 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 714 . . . . . 6 (((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘†) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ))) ∧ 𝑦 ∈ 𝐼) β†’ (π‘…β€˜π‘¦) ∈ LMod)
9435adantlr 714 . . . . . . 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 2837 . . . . . 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 2837 . . . . . 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 17359 . . . . . 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 2733 . . . . . . 7 (+gβ€˜(Scalarβ€˜(π‘…β€˜π‘¦))) = (+gβ€˜(Scalarβ€˜(π‘…β€˜π‘¦)))
9947, 48, 49, 50, 51, 98lmodvsdir 20361 . . . . . 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 1373 . . . . 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 714 . . . . . . . 8 (((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘†) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ))) ∧ 𝑦 ∈ 𝐼) β†’ (Scalarβ€˜(π‘…β€˜π‘¦)) = 𝑆)
102101fveq2d 6847 . . . . . . 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 7375 . . . . . 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 7373 . . . . 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 2780 . . . 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 5206 . . 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 482 . . . 4 ((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘†) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ))) β†’ 𝑆 ∈ Ring)
10823adantr 482 . . . 4 ((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘†) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ))) β†’ 𝐼 ∈ V)
10940adantr 482 . . . 4 ((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘†) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ))) β†’ 𝑅 Fn 𝐼)
110 simpr1 1195 . . . . 5 ((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘†) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ))) β†’ π‘Ž ∈ (Baseβ€˜π‘†))
111 simpr2 1196 . . . . 5 ((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘†) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ))) β†’ 𝑏 ∈ (Baseβ€˜π‘†))
112 eqid 2733 . . . . . 6 (+gβ€˜π‘†) = (+gβ€˜π‘†)
11321, 112ringacl 20004 . . . . 5 ((𝑆 ∈ Ring ∧ π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘†)) β†’ (π‘Ž(+gβ€˜π‘†)𝑏) ∈ (Baseβ€˜π‘†))
114107, 110, 111, 113syl3anc 1372 . . . 4 ((πœ‘ ∧ (π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘†) ∧ 𝑐 ∈ (Baseβ€˜π‘Œ))) β†’ (π‘Ž(+gβ€˜π‘†)𝑏) ∈ (Baseβ€˜π‘†))
115 simpr3 1197 . . . 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 17366 . . 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 1171 . . . 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 482 . . . . 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 20444 . . . 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 17360 . . 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 2783 . 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 7374 . . . . 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 2733 . . . . . . 7 (.rβ€˜(Scalarβ€˜(π‘…β€˜π‘¦))) = (.rβ€˜(Scalarβ€˜(π‘…β€˜π‘¦)))
12447, 49, 50, 51, 123lmodvsass 20362 . . . . . 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 1373 . . . . 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 6847 . . . . . . 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 7375 . . . . . 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 7373 . . . . 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 2780 . . . 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 5206 . . 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 2733 . . . . . 6 (.rβ€˜π‘†) = (.rβ€˜π‘†)
13221, 131ringcl 19986 . . . . 5 ((𝑆 ∈ Ring ∧ π‘Ž ∈ (Baseβ€˜π‘†) ∧ 𝑏 ∈ (Baseβ€˜π‘†)) β†’ (π‘Ž(.rβ€˜π‘†)𝑏) ∈ (Baseβ€˜π‘†))
133107, 110, 111, 132syl3anc 1372 . . . 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 17366 . . 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 17366 . . 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 2783 . 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 6847 . . . . . . 7 ((πœ‘ ∧ 𝑦 ∈ 𝐼) β†’ (1rβ€˜(Scalarβ€˜(π‘…β€˜π‘¦))) = (1rβ€˜π‘†))
138137adantlr 714 . . . . . 6 (((πœ‘ ∧ π‘Ž ∈ (Baseβ€˜π‘Œ)) ∧ 𝑦 ∈ 𝐼) β†’ (1rβ€˜(Scalarβ€˜(π‘…β€˜π‘¦))) = (1rβ€˜π‘†))
139138oveq1d 7373 . . . . 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 714 . . . . . 6 (((πœ‘ ∧ π‘Ž ∈ (Baseβ€˜π‘Œ)) ∧ 𝑦 ∈ 𝐼) β†’ (π‘…β€˜π‘¦) ∈ LMod)
1414ad2antrr 725 . . . . . . 7 (((πœ‘ ∧ π‘Ž ∈ (Baseβ€˜π‘Œ)) ∧ 𝑦 ∈ 𝐼) β†’ 𝑆 ∈ Ring)
14223ad2antrr 725 . . . . . . 7 (((πœ‘ ∧ π‘Ž ∈ (Baseβ€˜π‘Œ)) ∧ 𝑦 ∈ 𝐼) β†’ 𝐼 ∈ V)
14340ad2antrr 725 . . . . . . 7 (((πœ‘ ∧ π‘Ž ∈ (Baseβ€˜π‘Œ)) ∧ 𝑦 ∈ 𝐼) β†’ 𝑅 Fn 𝐼)
144 simplr 768 . . . . . . 7 (((πœ‘ ∧ π‘Ž ∈ (Baseβ€˜π‘Œ)) ∧ 𝑦 ∈ 𝐼) β†’ π‘Ž ∈ (Baseβ€˜π‘Œ))
145 simpr 486 . . . . . . 7 (((πœ‘ ∧ π‘Ž ∈ (Baseβ€˜π‘Œ)) ∧ 𝑦 ∈ 𝐼) β†’ 𝑦 ∈ 𝐼)
1463, 19, 141, 142, 143, 144, 145prdsbasprj 17359 . . . . . 6 (((πœ‘ ∧ π‘Ž ∈ (Baseβ€˜π‘Œ)) ∧ 𝑦 ∈ 𝐼) β†’ (π‘Žβ€˜π‘¦) ∈ (Baseβ€˜(π‘…β€˜π‘¦)))
147 eqid 2733 . . . . . . 7 (1rβ€˜(Scalarβ€˜(π‘…β€˜π‘¦))) = (1rβ€˜(Scalarβ€˜(π‘…β€˜π‘¦)))
14847, 49, 50, 147lmodvs1 20365 . . . . . 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 585 . . . . 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 2775 . . . 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 5206 . . 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 482 . . . 4 ((πœ‘ ∧ π‘Ž ∈ (Baseβ€˜π‘Œ)) β†’ 𝑆 ∈ Ring)
15323adantr 482 . . . 4 ((πœ‘ ∧ π‘Ž ∈ (Baseβ€˜π‘Œ)) β†’ 𝐼 ∈ V)
15440adantr 482 . . . 4 ((πœ‘ ∧ π‘Ž ∈ (Baseβ€˜π‘Œ)) β†’ 𝑅 Fn 𝐼)
155 eqid 2733 . . . . . . 7 (1rβ€˜π‘†) = (1rβ€˜π‘†)
15621, 155ringidcl 19994 . . . . . 6 (𝑆 ∈ Ring β†’ (1rβ€˜π‘†) ∈ (Baseβ€˜π‘†))
1574, 156syl 17 . . . . 5 (πœ‘ β†’ (1rβ€˜π‘†) ∈ (Baseβ€˜π‘†))
158157adantr 482 . . . 4 ((πœ‘ ∧ π‘Ž ∈ (Baseβ€˜π‘Œ)) β†’ (1rβ€˜π‘†) ∈ (Baseβ€˜π‘†))
159 simpr 486 . . . 4 ((πœ‘ ∧ π‘Ž ∈ (Baseβ€˜π‘Œ)) β†’ π‘Ž ∈ (Baseβ€˜π‘Œ))
1603, 19, 20, 21, 152, 153, 154, 158, 159prdsvscaval 17366 . . 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 17358 . . . 4 ((πœ‘ ∧ π‘Ž ∈ (Baseβ€˜π‘Œ)) β†’ π‘Ž Fn 𝐼)
162 dffn5 6902 . . . 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 2783 . 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 20342 1 (πœ‘ β†’ π‘Œ ∈ LMod)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  Vcvv 3444   βŠ† wss 3911   ↦ cmpt 5189   Fn wfn 6492  βŸΆwf 6493  β€˜cfv 6497  (class class class)co 7358  Basecbs 17088  +gcplusg 17138  .rcmulr 17139  Scalarcsca 17141   ·𝑠 cvsca 17142  Xscprds 17332  Grpcgrp 18753  1rcur 19918  Ringcrg 19969  LModclmod 20336
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-cnex 11112  ax-resscn 11113  ax-1cn 11114  ax-icn 11115  ax-addcl 11116  ax-addrcl 11117  ax-mulcl 11118  ax-mulrcl 11119  ax-mulcom 11120  ax-addass 11121  ax-mulass 11122  ax-distr 11123  ax-i2m1 11124  ax-1ne0 11125  ax-1rid 11126  ax-rnegex 11127  ax-rrecex 11128  ax-cnre 11129  ax-pre-lttri 11130  ax-pre-lttrn 11131  ax-pre-ltadd 11132  ax-pre-mulgt0 11133
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7804  df-1st 7922  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-1o 8413  df-er 8651  df-map 8770  df-ixp 8839  df-en 8887  df-dom 8888  df-sdom 8889  df-fin 8890  df-sup 9383  df-pnf 11196  df-mnf 11197  df-xr 11198  df-ltxr 11199  df-le 11200  df-sub 11392  df-neg 11393  df-nn 12159  df-2 12221  df-3 12222  df-4 12223  df-5 12224  df-6 12225  df-7 12226  df-8 12227  df-9 12228  df-n0 12419  df-z 12505  df-dec 12624  df-uz 12769  df-fz 13431  df-struct 17024  df-sets 17041  df-slot 17059  df-ndx 17071  df-base 17089  df-plusg 17151  df-mulr 17152  df-sca 17154  df-vsca 17155  df-ip 17156  df-tset 17157  df-ple 17158  df-ds 17160  df-hom 17162  df-cco 17163  df-0g 17328  df-prds 17334  df-mgm 18502  df-sgrp 18551  df-mnd 18562  df-grp 18756  df-minusg 18757  df-mgp 19902  df-ur 19919  df-ring 19971  df-lmod 20338
This theorem is referenced by:  pwslmod  20446  dsmmlss  21166  dsmmlmod  21167
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