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Theorem prdslmodd 20985
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 𝑌 = (𝑆Xs𝑅)
prdslmodd.s (𝜑𝑆 ∈ Ring)
prdslmodd.i (𝜑𝐼𝑉)
prdslmodd.rm (𝜑𝑅:𝐼⟶LMod)
prdslmodd.rs ((𝜑𝑦𝐼) → (Scalar‘(𝑅𝑦)) = 𝑆)
Assertion
Ref Expression
prdslmodd (𝜑𝑌 ∈ LMod)
Distinct variable groups:   𝑦,𝐼   𝜑,𝑦   𝑦,𝑅   𝑦,𝑆   𝑦,𝑌
Allowed substitution hint:   𝑉(𝑦)

Proof of Theorem prdslmodd
Dummy variables 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqidd 2736 . 2 (𝜑 → (Base‘𝑌) = (Base‘𝑌))
2 eqidd 2736 . 2 (𝜑 → (+g𝑌) = (+g𝑌))
3 prdslmodd.y . . 3 𝑌 = (𝑆Xs𝑅)
4 prdslmodd.s . . 3 (𝜑𝑆 ∈ Ring)
5 prdslmodd.rm . . . 4 (𝜑𝑅:𝐼⟶LMod)
6 prdslmodd.i . . . 4 (𝜑𝐼𝑉)
75, 6fexd 7247 . . 3 (𝜑𝑅 ∈ V)
83, 4, 7prdssca 17503 . 2 (𝜑𝑆 = (Scalar‘𝑌))
9 eqidd 2736 . 2 (𝜑 → ( ·𝑠𝑌) = ( ·𝑠𝑌))
10 eqidd 2736 . 2 (𝜑 → (Base‘𝑆) = (Base‘𝑆))
11 eqidd 2736 . 2 (𝜑 → (+g𝑆) = (+g𝑆))
12 eqidd 2736 . 2 (𝜑 → (.r𝑆) = (.r𝑆))
13 eqidd 2736 . 2 (𝜑 → (1r𝑆) = (1r𝑆))
14 lmodgrp 20882 . . . . 5 (𝑎 ∈ LMod → 𝑎 ∈ Grp)
1514ssriv 3999 . . . 4 LMod ⊆ Grp
16 fss 6753 . . . 4 ((𝑅:𝐼⟶LMod ∧ LMod ⊆ Grp) → 𝑅:𝐼⟶Grp)
175, 15, 16sylancl 586 . . 3 (𝜑𝑅:𝐼⟶Grp)
183, 6, 4, 17prdsgrpd 19081 . 2 (𝜑𝑌 ∈ Grp)
19 eqid 2735 . . . 4 (Base‘𝑌) = (Base‘𝑌)
20 eqid 2735 . . . 4 ( ·𝑠𝑌) = ( ·𝑠𝑌)
21 eqid 2735 . . . 4 (Base‘𝑆) = (Base‘𝑆)
224adantr 480 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌))) → 𝑆 ∈ Ring)
236elexd 3502 . . . . 5 (𝜑𝐼 ∈ V)
2423adantr 480 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌))) → 𝐼 ∈ V)
255adantr 480 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌))) → 𝑅:𝐼⟶LMod)
26 simprl 771 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌))) → 𝑎 ∈ (Base‘𝑆))
27 simprr 773 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌))) → 𝑏 ∈ (Base‘𝑌))
28 prdslmodd.rs . . . . 5 ((𝜑𝑦𝐼) → (Scalar‘(𝑅𝑦)) = 𝑆)
2928adantlr 715 . . . 4 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → (Scalar‘(𝑅𝑦)) = 𝑆)
303, 19, 20, 21, 22, 24, 25, 26, 27, 29prdsvscacl 20984 . . 3 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌))) → (𝑎( ·𝑠𝑌)𝑏) ∈ (Base‘𝑌))
31303impb 1114 . 2 ((𝜑𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌)) → (𝑎( ·𝑠𝑌)𝑏) ∈ (Base‘𝑌))
325ffvelcdmda 7104 . . . . . . 7 ((𝜑𝑦𝐼) → (𝑅𝑦) ∈ LMod)
3332adantlr 715 . . . . . 6 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → (𝑅𝑦) ∈ LMod)
34 simplr1 1214 . . . . . . 7 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → 𝑎 ∈ (Base‘𝑆))
3528fveq2d 6911 . . . . . . . 8 ((𝜑𝑦𝐼) → (Base‘(Scalar‘(𝑅𝑦))) = (Base‘𝑆))
3635adantlr 715 . . . . . . 7 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → (Base‘(Scalar‘(𝑅𝑦))) = (Base‘𝑆))
3734, 36eleqtrrd 2842 . . . . . 6 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → 𝑎 ∈ (Base‘(Scalar‘(𝑅𝑦))))
384ad2antrr 726 . . . . . . 7 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → 𝑆 ∈ Ring)
3923ad2antrr 726 . . . . . . 7 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → 𝐼 ∈ V)
405ffnd 6738 . . . . . . . 8 (𝜑𝑅 Fn 𝐼)
4140ad2antrr 726 . . . . . . 7 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → 𝑅 Fn 𝐼)
42 simplr2 1215 . . . . . . 7 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → 𝑏 ∈ (Base‘𝑌))
43 simpr 484 . . . . . . 7 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → 𝑦𝐼)
443, 19, 38, 39, 41, 42, 43prdsbasprj 17519 . . . . . 6 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → (𝑏𝑦) ∈ (Base‘(𝑅𝑦)))
45 simplr3 1216 . . . . . . 7 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → 𝑐 ∈ (Base‘𝑌))
463, 19, 38, 39, 41, 45, 43prdsbasprj 17519 . . . . . 6 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → (𝑐𝑦) ∈ (Base‘(𝑅𝑦)))
47 eqid 2735 . . . . . . 7 (Base‘(𝑅𝑦)) = (Base‘(𝑅𝑦))
48 eqid 2735 . . . . . . 7 (+g‘(𝑅𝑦)) = (+g‘(𝑅𝑦))
49 eqid 2735 . . . . . . 7 (Scalar‘(𝑅𝑦)) = (Scalar‘(𝑅𝑦))
50 eqid 2735 . . . . . . 7 ( ·𝑠 ‘(𝑅𝑦)) = ( ·𝑠 ‘(𝑅𝑦))
51 eqid 2735 . . . . . . 7 (Base‘(Scalar‘(𝑅𝑦))) = (Base‘(Scalar‘(𝑅𝑦)))
5247, 48, 49, 50, 51lmodvsdi 20900 . . . . . 6 (((𝑅𝑦) ∈ LMod ∧ (𝑎 ∈ (Base‘(Scalar‘(𝑅𝑦))) ∧ (𝑏𝑦) ∈ (Base‘(𝑅𝑦)) ∧ (𝑐𝑦) ∈ (Base‘(𝑅𝑦)))) → (𝑎( ·𝑠 ‘(𝑅𝑦))((𝑏𝑦)(+g‘(𝑅𝑦))(𝑐𝑦))) = ((𝑎( ·𝑠 ‘(𝑅𝑦))(𝑏𝑦))(+g‘(𝑅𝑦))(𝑎( ·𝑠 ‘(𝑅𝑦))(𝑐𝑦))))
5333, 37, 44, 46, 52syl13anc 1371 . . . . 5 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → (𝑎( ·𝑠 ‘(𝑅𝑦))((𝑏𝑦)(+g‘(𝑅𝑦))(𝑐𝑦))) = ((𝑎( ·𝑠 ‘(𝑅𝑦))(𝑏𝑦))(+g‘(𝑅𝑦))(𝑎( ·𝑠 ‘(𝑅𝑦))(𝑐𝑦))))
54 eqid 2735 . . . . . . 7 (+g𝑌) = (+g𝑌)
553, 19, 38, 39, 41, 42, 45, 54, 43prdsplusgfval 17521 . . . . . 6 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → ((𝑏(+g𝑌)𝑐)‘𝑦) = ((𝑏𝑦)(+g‘(𝑅𝑦))(𝑐𝑦)))
5655oveq2d 7447 . . . . 5 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → (𝑎( ·𝑠 ‘(𝑅𝑦))((𝑏(+g𝑌)𝑐)‘𝑦)) = (𝑎( ·𝑠 ‘(𝑅𝑦))((𝑏𝑦)(+g‘(𝑅𝑦))(𝑐𝑦))))
573, 19, 20, 21, 38, 39, 41, 34, 42, 43prdsvscafval 17527 . . . . . 6 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → ((𝑎( ·𝑠𝑌)𝑏)‘𝑦) = (𝑎( ·𝑠 ‘(𝑅𝑦))(𝑏𝑦)))
583, 19, 20, 21, 38, 39, 41, 34, 45, 43prdsvscafval 17527 . . . . . 6 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → ((𝑎( ·𝑠𝑌)𝑐)‘𝑦) = (𝑎( ·𝑠 ‘(𝑅𝑦))(𝑐𝑦)))
5957, 58oveq12d 7449 . . . . 5 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → (((𝑎( ·𝑠𝑌)𝑏)‘𝑦)(+g‘(𝑅𝑦))((𝑎( ·𝑠𝑌)𝑐)‘𝑦)) = ((𝑎( ·𝑠 ‘(𝑅𝑦))(𝑏𝑦))(+g‘(𝑅𝑦))(𝑎( ·𝑠 ‘(𝑅𝑦))(𝑐𝑦))))
6053, 56, 593eqtr4d 2785 . . . 4 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → (𝑎( ·𝑠 ‘(𝑅𝑦))((𝑏(+g𝑌)𝑐)‘𝑦)) = (((𝑎( ·𝑠𝑌)𝑏)‘𝑦)(+g‘(𝑅𝑦))((𝑎( ·𝑠𝑌)𝑐)‘𝑦)))
6160mpteq2dva 5248 . . 3 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → (𝑦𝐼 ↦ (𝑎( ·𝑠 ‘(𝑅𝑦))((𝑏(+g𝑌)𝑐)‘𝑦))) = (𝑦𝐼 ↦ (((𝑎( ·𝑠𝑌)𝑏)‘𝑦)(+g‘(𝑅𝑦))((𝑎( ·𝑠𝑌)𝑐)‘𝑦))))
624adantr 480 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑆 ∈ Ring)
6323adantr 480 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝐼 ∈ V)
6440adantr 480 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑅 Fn 𝐼)
65 simpr1 1193 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑎 ∈ (Base‘𝑆))
6618adantr 480 . . . . 5 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑌 ∈ Grp)
67 simpr2 1194 . . . . 5 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑏 ∈ (Base‘𝑌))
68 simpr3 1195 . . . . 5 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑐 ∈ (Base‘𝑌))
6919, 54grpcl 18972 . . . . 5 ((𝑌 ∈ Grp ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌)) → (𝑏(+g𝑌)𝑐) ∈ (Base‘𝑌))
7066, 67, 68, 69syl3anc 1370 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → (𝑏(+g𝑌)𝑐) ∈ (Base‘𝑌))
713, 19, 20, 21, 62, 63, 64, 65, 70prdsvscaval 17526 . . 3 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → (𝑎( ·𝑠𝑌)(𝑏(+g𝑌)𝑐)) = (𝑦𝐼 ↦ (𝑎( ·𝑠 ‘(𝑅𝑦))((𝑏(+g𝑌)𝑐)‘𝑦))))
72303adantr3 1170 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → (𝑎( ·𝑠𝑌)𝑏) ∈ (Base‘𝑌))
734adantr 480 . . . . . 6 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑆 ∈ Ring)
7423adantr 480 . . . . . 6 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝐼 ∈ V)
755adantr 480 . . . . . 6 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑅:𝐼⟶LMod)
76 simprl 771 . . . . . 6 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑎 ∈ (Base‘𝑆))
77 simprr 773 . . . . . 6 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑐 ∈ (Base‘𝑌))
7828adantlr 715 . . . . . 6 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → (Scalar‘(𝑅𝑦)) = 𝑆)
793, 19, 20, 21, 73, 74, 75, 76, 77, 78prdsvscacl 20984 . . . . 5 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → (𝑎( ·𝑠𝑌)𝑐) ∈ (Base‘𝑌))
80793adantr2 1169 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → (𝑎( ·𝑠𝑌)𝑐) ∈ (Base‘𝑌))
813, 19, 62, 63, 64, 72, 80, 54prdsplusgval 17520 . . 3 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → ((𝑎( ·𝑠𝑌)𝑏)(+g𝑌)(𝑎( ·𝑠𝑌)𝑐)) = (𝑦𝐼 ↦ (((𝑎( ·𝑠𝑌)𝑏)‘𝑦)(+g‘(𝑅𝑦))((𝑎( ·𝑠𝑌)𝑐)‘𝑦))))
8261, 71, 813eqtr4d 2785 . 2 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → (𝑎( ·𝑠𝑌)(𝑏(+g𝑌)𝑐)) = ((𝑎( ·𝑠𝑌)𝑏)(+g𝑌)(𝑎( ·𝑠𝑌)𝑐)))
834ad2antrr 726 . . . . . . 7 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → 𝑆 ∈ Ring)
8423ad2antrr 726 . . . . . . 7 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → 𝐼 ∈ V)
8540ad2antrr 726 . . . . . . 7 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → 𝑅 Fn 𝐼)
86 simplr1 1214 . . . . . . 7 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → 𝑎 ∈ (Base‘𝑆))
87 simplr3 1216 . . . . . . 7 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → 𝑐 ∈ (Base‘𝑌))
88 simpr 484 . . . . . . 7 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → 𝑦𝐼)
893, 19, 20, 21, 83, 84, 85, 86, 87, 88prdsvscafval 17527 . . . . . 6 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → ((𝑎( ·𝑠𝑌)𝑐)‘𝑦) = (𝑎( ·𝑠 ‘(𝑅𝑦))(𝑐𝑦)))
90 simplr2 1215 . . . . . . 7 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → 𝑏 ∈ (Base‘𝑆))
913, 19, 20, 21, 83, 84, 85, 90, 87, 88prdsvscafval 17527 . . . . . 6 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → ((𝑏( ·𝑠𝑌)𝑐)‘𝑦) = (𝑏( ·𝑠 ‘(𝑅𝑦))(𝑐𝑦)))
9289, 91oveq12d 7449 . . . . 5 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → (((𝑎( ·𝑠𝑌)𝑐)‘𝑦)(+g‘(𝑅𝑦))((𝑏( ·𝑠𝑌)𝑐)‘𝑦)) = ((𝑎( ·𝑠 ‘(𝑅𝑦))(𝑐𝑦))(+g‘(𝑅𝑦))(𝑏( ·𝑠 ‘(𝑅𝑦))(𝑐𝑦))))
9332adantlr 715 . . . . . 6 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → (𝑅𝑦) ∈ LMod)
9435adantlr 715 . . . . . . 7 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → (Base‘(Scalar‘(𝑅𝑦))) = (Base‘𝑆))
9586, 94eleqtrrd 2842 . . . . . 6 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → 𝑎 ∈ (Base‘(Scalar‘(𝑅𝑦))))
9690, 94eleqtrrd 2842 . . . . . 6 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → 𝑏 ∈ (Base‘(Scalar‘(𝑅𝑦))))
973, 19, 83, 84, 85, 87, 88prdsbasprj 17519 . . . . . 6 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → (𝑐𝑦) ∈ (Base‘(𝑅𝑦)))
98 eqid 2735 . . . . . . 7 (+g‘(Scalar‘(𝑅𝑦))) = (+g‘(Scalar‘(𝑅𝑦)))
9947, 48, 49, 50, 51, 98lmodvsdir 20901 . . . . . 6 (((𝑅𝑦) ∈ LMod ∧ (𝑎 ∈ (Base‘(Scalar‘(𝑅𝑦))) ∧ 𝑏 ∈ (Base‘(Scalar‘(𝑅𝑦))) ∧ (𝑐𝑦) ∈ (Base‘(𝑅𝑦)))) → ((𝑎(+g‘(Scalar‘(𝑅𝑦)))𝑏)( ·𝑠 ‘(𝑅𝑦))(𝑐𝑦)) = ((𝑎( ·𝑠 ‘(𝑅𝑦))(𝑐𝑦))(+g‘(𝑅𝑦))(𝑏( ·𝑠 ‘(𝑅𝑦))(𝑐𝑦))))
10093, 95, 96, 97, 99syl13anc 1371 . . . . 5 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → ((𝑎(+g‘(Scalar‘(𝑅𝑦)))𝑏)( ·𝑠 ‘(𝑅𝑦))(𝑐𝑦)) = ((𝑎( ·𝑠 ‘(𝑅𝑦))(𝑐𝑦))(+g‘(𝑅𝑦))(𝑏( ·𝑠 ‘(𝑅𝑦))(𝑐𝑦))))
10128adantlr 715 . . . . . . . 8 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → (Scalar‘(𝑅𝑦)) = 𝑆)
102101fveq2d 6911 . . . . . . 7 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → (+g‘(Scalar‘(𝑅𝑦))) = (+g𝑆))
103102oveqd 7448 . . . . . 6 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → (𝑎(+g‘(Scalar‘(𝑅𝑦)))𝑏) = (𝑎(+g𝑆)𝑏))
104103oveq1d 7446 . . . . 5 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → ((𝑎(+g‘(Scalar‘(𝑅𝑦)))𝑏)( ·𝑠 ‘(𝑅𝑦))(𝑐𝑦)) = ((𝑎(+g𝑆)𝑏)( ·𝑠 ‘(𝑅𝑦))(𝑐𝑦)))
10592, 100, 1043eqtr2rd 2782 . . . 4 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → ((𝑎(+g𝑆)𝑏)( ·𝑠 ‘(𝑅𝑦))(𝑐𝑦)) = (((𝑎( ·𝑠𝑌)𝑐)‘𝑦)(+g‘(𝑅𝑦))((𝑏( ·𝑠𝑌)𝑐)‘𝑦)))
106105mpteq2dva 5248 . . 3 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → (𝑦𝐼 ↦ ((𝑎(+g𝑆)𝑏)( ·𝑠 ‘(𝑅𝑦))(𝑐𝑦))) = (𝑦𝐼 ↦ (((𝑎( ·𝑠𝑌)𝑐)‘𝑦)(+g‘(𝑅𝑦))((𝑏( ·𝑠𝑌)𝑐)‘𝑦))))
1074adantr 480 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑆 ∈ Ring)
10823adantr 480 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝐼 ∈ V)
10940adantr 480 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑅 Fn 𝐼)
110 simpr1 1193 . . . . 5 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑎 ∈ (Base‘𝑆))
111 simpr2 1194 . . . . 5 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑏 ∈ (Base‘𝑆))
112 eqid 2735 . . . . . 6 (+g𝑆) = (+g𝑆)
11321, 112ringacl 20292 . . . . 5 ((𝑆 ∈ Ring ∧ 𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆)) → (𝑎(+g𝑆)𝑏) ∈ (Base‘𝑆))
114107, 110, 111, 113syl3anc 1370 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → (𝑎(+g𝑆)𝑏) ∈ (Base‘𝑆))
115 simpr3 1195 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑐 ∈ (Base‘𝑌))
1163, 19, 20, 21, 107, 108, 109, 114, 115prdsvscaval 17526 . . 3 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → ((𝑎(+g𝑆)𝑏)( ·𝑠𝑌)𝑐) = (𝑦𝐼 ↦ ((𝑎(+g𝑆)𝑏)( ·𝑠 ‘(𝑅𝑦))(𝑐𝑦))))
117793adantr2 1169 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → (𝑎( ·𝑠𝑌)𝑐) ∈ (Base‘𝑌))
1185adantr 480 . . . . 5 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑅:𝐼⟶LMod)
1193, 19, 20, 21, 107, 108, 118, 111, 115, 101prdsvscacl 20984 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → (𝑏( ·𝑠𝑌)𝑐) ∈ (Base‘𝑌))
1203, 19, 107, 108, 109, 117, 119, 54prdsplusgval 17520 . . 3 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → ((𝑎( ·𝑠𝑌)𝑐)(+g𝑌)(𝑏( ·𝑠𝑌)𝑐)) = (𝑦𝐼 ↦ (((𝑎( ·𝑠𝑌)𝑐)‘𝑦)(+g‘(𝑅𝑦))((𝑏( ·𝑠𝑌)𝑐)‘𝑦))))
121106, 116, 1203eqtr4d 2785 . 2 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → ((𝑎(+g𝑆)𝑏)( ·𝑠𝑌)𝑐) = ((𝑎( ·𝑠𝑌)𝑐)(+g𝑌)(𝑏( ·𝑠𝑌)𝑐)))
12291oveq2d 7447 . . . . 5 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → (𝑎( ·𝑠 ‘(𝑅𝑦))((𝑏( ·𝑠𝑌)𝑐)‘𝑦)) = (𝑎( ·𝑠 ‘(𝑅𝑦))(𝑏( ·𝑠 ‘(𝑅𝑦))(𝑐𝑦))))
123 eqid 2735 . . . . . . 7 (.r‘(Scalar‘(𝑅𝑦))) = (.r‘(Scalar‘(𝑅𝑦)))
12447, 49, 50, 51, 123lmodvsass 20902 . . . . . 6 (((𝑅𝑦) ∈ LMod ∧ (𝑎 ∈ (Base‘(Scalar‘(𝑅𝑦))) ∧ 𝑏 ∈ (Base‘(Scalar‘(𝑅𝑦))) ∧ (𝑐𝑦) ∈ (Base‘(𝑅𝑦)))) → ((𝑎(.r‘(Scalar‘(𝑅𝑦)))𝑏)( ·𝑠 ‘(𝑅𝑦))(𝑐𝑦)) = (𝑎( ·𝑠 ‘(𝑅𝑦))(𝑏( ·𝑠 ‘(𝑅𝑦))(𝑐𝑦))))
12593, 95, 96, 97, 124syl13anc 1371 . . . . 5 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → ((𝑎(.r‘(Scalar‘(𝑅𝑦)))𝑏)( ·𝑠 ‘(𝑅𝑦))(𝑐𝑦)) = (𝑎( ·𝑠 ‘(𝑅𝑦))(𝑏( ·𝑠 ‘(𝑅𝑦))(𝑐𝑦))))
126101fveq2d 6911 . . . . . . 7 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → (.r‘(Scalar‘(𝑅𝑦))) = (.r𝑆))
127126oveqd 7448 . . . . . 6 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → (𝑎(.r‘(Scalar‘(𝑅𝑦)))𝑏) = (𝑎(.r𝑆)𝑏))
128127oveq1d 7446 . . . . 5 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → ((𝑎(.r‘(Scalar‘(𝑅𝑦)))𝑏)( ·𝑠 ‘(𝑅𝑦))(𝑐𝑦)) = ((𝑎(.r𝑆)𝑏)( ·𝑠 ‘(𝑅𝑦))(𝑐𝑦)))
129122, 125, 1283eqtr2rd 2782 . . . 4 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → ((𝑎(.r𝑆)𝑏)( ·𝑠 ‘(𝑅𝑦))(𝑐𝑦)) = (𝑎( ·𝑠 ‘(𝑅𝑦))((𝑏( ·𝑠𝑌)𝑐)‘𝑦)))
130129mpteq2dva 5248 . . 3 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → (𝑦𝐼 ↦ ((𝑎(.r𝑆)𝑏)( ·𝑠 ‘(𝑅𝑦))(𝑐𝑦))) = (𝑦𝐼 ↦ (𝑎( ·𝑠 ‘(𝑅𝑦))((𝑏( ·𝑠𝑌)𝑐)‘𝑦))))
131 eqid 2735 . . . . . 6 (.r𝑆) = (.r𝑆)
13221, 131ringcl 20268 . . . . 5 ((𝑆 ∈ Ring ∧ 𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆)) → (𝑎(.r𝑆)𝑏) ∈ (Base‘𝑆))
133107, 110, 111, 132syl3anc 1370 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → (𝑎(.r𝑆)𝑏) ∈ (Base‘𝑆))
1343, 19, 20, 21, 107, 108, 109, 133, 115prdsvscaval 17526 . . 3 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → ((𝑎(.r𝑆)𝑏)( ·𝑠𝑌)𝑐) = (𝑦𝐼 ↦ ((𝑎(.r𝑆)𝑏)( ·𝑠 ‘(𝑅𝑦))(𝑐𝑦))))
1353, 19, 20, 21, 107, 108, 109, 110, 119prdsvscaval 17526 . . 3 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → (𝑎( ·𝑠𝑌)(𝑏( ·𝑠𝑌)𝑐)) = (𝑦𝐼 ↦ (𝑎( ·𝑠 ‘(𝑅𝑦))((𝑏( ·𝑠𝑌)𝑐)‘𝑦))))
136130, 134, 1353eqtr4d 2785 . 2 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → ((𝑎(.r𝑆)𝑏)( ·𝑠𝑌)𝑐) = (𝑎( ·𝑠𝑌)(𝑏( ·𝑠𝑌)𝑐)))
13728fveq2d 6911 . . . . . . 7 ((𝜑𝑦𝐼) → (1r‘(Scalar‘(𝑅𝑦))) = (1r𝑆))
138137adantlr 715 . . . . . 6 (((𝜑𝑎 ∈ (Base‘𝑌)) ∧ 𝑦𝐼) → (1r‘(Scalar‘(𝑅𝑦))) = (1r𝑆))
139138oveq1d 7446 . . . . 5 (((𝜑𝑎 ∈ (Base‘𝑌)) ∧ 𝑦𝐼) → ((1r‘(Scalar‘(𝑅𝑦)))( ·𝑠 ‘(𝑅𝑦))(𝑎𝑦)) = ((1r𝑆)( ·𝑠 ‘(𝑅𝑦))(𝑎𝑦)))
14032adantlr 715 . . . . . 6 (((𝜑𝑎 ∈ (Base‘𝑌)) ∧ 𝑦𝐼) → (𝑅𝑦) ∈ LMod)
1414ad2antrr 726 . . . . . . 7 (((𝜑𝑎 ∈ (Base‘𝑌)) ∧ 𝑦𝐼) → 𝑆 ∈ Ring)
14223ad2antrr 726 . . . . . . 7 (((𝜑𝑎 ∈ (Base‘𝑌)) ∧ 𝑦𝐼) → 𝐼 ∈ V)
14340ad2antrr 726 . . . . . . 7 (((𝜑𝑎 ∈ (Base‘𝑌)) ∧ 𝑦𝐼) → 𝑅 Fn 𝐼)
144 simplr 769 . . . . . . 7 (((𝜑𝑎 ∈ (Base‘𝑌)) ∧ 𝑦𝐼) → 𝑎 ∈ (Base‘𝑌))
145 simpr 484 . . . . . . 7 (((𝜑𝑎 ∈ (Base‘𝑌)) ∧ 𝑦𝐼) → 𝑦𝐼)
1463, 19, 141, 142, 143, 144, 145prdsbasprj 17519 . . . . . 6 (((𝜑𝑎 ∈ (Base‘𝑌)) ∧ 𝑦𝐼) → (𝑎𝑦) ∈ (Base‘(𝑅𝑦)))
147 eqid 2735 . . . . . . 7 (1r‘(Scalar‘(𝑅𝑦))) = (1r‘(Scalar‘(𝑅𝑦)))
14847, 49, 50, 147lmodvs1 20905 . . . . . 6 (((𝑅𝑦) ∈ LMod ∧ (𝑎𝑦) ∈ (Base‘(𝑅𝑦))) → ((1r‘(Scalar‘(𝑅𝑦)))( ·𝑠 ‘(𝑅𝑦))(𝑎𝑦)) = (𝑎𝑦))
149140, 146, 148syl2anc 584 . . . . 5 (((𝜑𝑎 ∈ (Base‘𝑌)) ∧ 𝑦𝐼) → ((1r‘(Scalar‘(𝑅𝑦)))( ·𝑠 ‘(𝑅𝑦))(𝑎𝑦)) = (𝑎𝑦))
150139, 149eqtr3d 2777 . . . 4 (((𝜑𝑎 ∈ (Base‘𝑌)) ∧ 𝑦𝐼) → ((1r𝑆)( ·𝑠 ‘(𝑅𝑦))(𝑎𝑦)) = (𝑎𝑦))
151150mpteq2dva 5248 . . 3 ((𝜑𝑎 ∈ (Base‘𝑌)) → (𝑦𝐼 ↦ ((1r𝑆)( ·𝑠 ‘(𝑅𝑦))(𝑎𝑦))) = (𝑦𝐼 ↦ (𝑎𝑦)))
1524adantr 480 . . . 4 ((𝜑𝑎 ∈ (Base‘𝑌)) → 𝑆 ∈ Ring)
15323adantr 480 . . . 4 ((𝜑𝑎 ∈ (Base‘𝑌)) → 𝐼 ∈ V)
15440adantr 480 . . . 4 ((𝜑𝑎 ∈ (Base‘𝑌)) → 𝑅 Fn 𝐼)
155 eqid 2735 . . . . . . 7 (1r𝑆) = (1r𝑆)
15621, 155ringidcl 20280 . . . . . 6 (𝑆 ∈ Ring → (1r𝑆) ∈ (Base‘𝑆))
1574, 156syl 17 . . . . 5 (𝜑 → (1r𝑆) ∈ (Base‘𝑆))
158157adantr 480 . . . 4 ((𝜑𝑎 ∈ (Base‘𝑌)) → (1r𝑆) ∈ (Base‘𝑆))
159 simpr 484 . . . 4 ((𝜑𝑎 ∈ (Base‘𝑌)) → 𝑎 ∈ (Base‘𝑌))
1603, 19, 20, 21, 152, 153, 154, 158, 159prdsvscaval 17526 . . 3 ((𝜑𝑎 ∈ (Base‘𝑌)) → ((1r𝑆)( ·𝑠𝑌)𝑎) = (𝑦𝐼 ↦ ((1r𝑆)( ·𝑠 ‘(𝑅𝑦))(𝑎𝑦))))
1613, 19, 152, 153, 154, 159prdsbasfn 17518 . . . 4 ((𝜑𝑎 ∈ (Base‘𝑌)) → 𝑎 Fn 𝐼)
162 dffn5 6967 . . . 4 (𝑎 Fn 𝐼𝑎 = (𝑦𝐼 ↦ (𝑎𝑦)))
163161, 162sylib 218 . . 3 ((𝜑𝑎 ∈ (Base‘𝑌)) → 𝑎 = (𝑦𝐼 ↦ (𝑎𝑦)))
164151, 160, 1633eqtr4d 2785 . 2 ((𝜑𝑎 ∈ (Base‘𝑌)) → ((1r𝑆)( ·𝑠𝑌)𝑎) = 𝑎)
1651, 2, 8, 9, 10, 11, 12, 13, 4, 18, 31, 82, 121, 136, 164islmodd 20881 1 (𝜑𝑌 ∈ LMod)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1537  wcel 2106  Vcvv 3478  wss 3963  cmpt 5231   Fn wfn 6558  wf 6559  cfv 6563  (class class class)co 7431  Basecbs 17245  +gcplusg 17298  .rcmulr 17299  Scalarcsca 17301   ·𝑠 cvsca 17302  Xscprds 17492  Grpcgrp 18964  1rcur 20199  Ringcrg 20251  LModclmod 20875
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-er 8744  df-map 8867  df-ixp 8937  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-sup 9480  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12525  df-z 12612  df-dec 12732  df-uz 12877  df-fz 13545  df-struct 17181  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-plusg 17311  df-mulr 17312  df-sca 17314  df-vsca 17315  df-ip 17316  df-tset 17317  df-ple 17318  df-ds 17320  df-hom 17322  df-cco 17323  df-0g 17488  df-prds 17494  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-grp 18967  df-minusg 18968  df-mgp 20153  df-ur 20200  df-ring 20253  df-lmod 20877
This theorem is referenced by:  pwslmod  20986  dsmmlss  21782  dsmmlmod  21783
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