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Theorem prdslmodd 20967
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 2738 . 2 (𝜑 → (Base‘𝑌) = (Base‘𝑌))
2 eqidd 2738 . 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 17501 . 2 (𝜑𝑆 = (Scalar‘𝑌))
9 eqidd 2738 . 2 (𝜑 → ( ·𝑠𝑌) = ( ·𝑠𝑌))
10 eqidd 2738 . 2 (𝜑 → (Base‘𝑆) = (Base‘𝑆))
11 eqidd 2738 . 2 (𝜑 → (+g𝑆) = (+g𝑆))
12 eqidd 2738 . 2 (𝜑 → (.r𝑆) = (.r𝑆))
13 eqidd 2738 . 2 (𝜑 → (1r𝑆) = (1r𝑆))
14 lmodgrp 20865 . . . . 5 (𝑎 ∈ LMod → 𝑎 ∈ Grp)
1514ssriv 3987 . . . 4 LMod ⊆ Grp
16 fss 6752 . . . 4 ((𝑅:𝐼⟶LMod ∧ LMod ⊆ Grp) → 𝑅:𝐼⟶Grp)
175, 15, 16sylancl 586 . . 3 (𝜑𝑅:𝐼⟶Grp)
183, 6, 4, 17prdsgrpd 19068 . 2 (𝜑𝑌 ∈ Grp)
19 eqid 2737 . . . 4 (Base‘𝑌) = (Base‘𝑌)
20 eqid 2737 . . . 4 ( ·𝑠𝑌) = ( ·𝑠𝑌)
21 eqid 2737 . . . 4 (Base‘𝑆) = (Base‘𝑆)
224adantr 480 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌))) → 𝑆 ∈ Ring)
236elexd 3504 . . . . 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 20966 . . 3 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌))) → (𝑎( ·𝑠𝑌)𝑏) ∈ (Base‘𝑌))
31303impb 1115 . 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 1216 . . . . . . 7 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → 𝑎 ∈ (Base‘𝑆))
3528fveq2d 6910 . . . . . . . 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 2844 . . . . . 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 6737 . . . . . . . 8 (𝜑𝑅 Fn 𝐼)
4140ad2antrr 726 . . . . . . 7 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → 𝑅 Fn 𝐼)
42 simplr2 1217 . . . . . . 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 17517 . . . . . 6 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → (𝑏𝑦) ∈ (Base‘(𝑅𝑦)))
45 simplr3 1218 . . . . . . 7 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → 𝑐 ∈ (Base‘𝑌))
463, 19, 38, 39, 41, 45, 43prdsbasprj 17517 . . . . . 6 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → (𝑐𝑦) ∈ (Base‘(𝑅𝑦)))
47 eqid 2737 . . . . . . 7 (Base‘(𝑅𝑦)) = (Base‘(𝑅𝑦))
48 eqid 2737 . . . . . . 7 (+g‘(𝑅𝑦)) = (+g‘(𝑅𝑦))
49 eqid 2737 . . . . . . 7 (Scalar‘(𝑅𝑦)) = (Scalar‘(𝑅𝑦))
50 eqid 2737 . . . . . . 7 ( ·𝑠 ‘(𝑅𝑦)) = ( ·𝑠 ‘(𝑅𝑦))
51 eqid 2737 . . . . . . 7 (Base‘(Scalar‘(𝑅𝑦))) = (Base‘(Scalar‘(𝑅𝑦)))
5247, 48, 49, 50, 51lmodvsdi 20883 . . . . . 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 1374 . . . . 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 2737 . . . . . . 7 (+g𝑌) = (+g𝑌)
553, 19, 38, 39, 41, 42, 45, 54, 43prdsplusgfval 17519 . . . . . 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 17525 . . . . . 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 17525 . . . . . 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 2787 . . . 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 5242 . . 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 1195 . . . 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 1196 . . . . 5 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑏 ∈ (Base‘𝑌))
68 simpr3 1197 . . . . 5 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑐 ∈ (Base‘𝑌))
6919, 54grpcl 18959 . . . . 5 ((𝑌 ∈ Grp ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌)) → (𝑏(+g𝑌)𝑐) ∈ (Base‘𝑌))
7066, 67, 68, 69syl3anc 1373 . . . 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 17524 . . 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 1172 . . . 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 20966 . . . . 5 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → (𝑎( ·𝑠𝑌)𝑐) ∈ (Base‘𝑌))
80793adantr2 1171 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → (𝑎( ·𝑠𝑌)𝑐) ∈ (Base‘𝑌))
813, 19, 62, 63, 64, 72, 80, 54prdsplusgval 17518 . . 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 2787 . 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 1216 . . . . . . 7 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → 𝑎 ∈ (Base‘𝑆))
87 simplr3 1218 . . . . . . 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 17525 . . . . . 6 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → ((𝑎( ·𝑠𝑌)𝑐)‘𝑦) = (𝑎( ·𝑠 ‘(𝑅𝑦))(𝑐𝑦)))
90 simplr2 1217 . . . . . . 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 17525 . . . . . 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 2844 . . . . . 6 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → 𝑎 ∈ (Base‘(Scalar‘(𝑅𝑦))))
9690, 94eleqtrrd 2844 . . . . . 6 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → 𝑏 ∈ (Base‘(Scalar‘(𝑅𝑦))))
973, 19, 83, 84, 85, 87, 88prdsbasprj 17517 . . . . . 6 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → (𝑐𝑦) ∈ (Base‘(𝑅𝑦)))
98 eqid 2737 . . . . . . 7 (+g‘(Scalar‘(𝑅𝑦))) = (+g‘(Scalar‘(𝑅𝑦)))
9947, 48, 49, 50, 51, 98lmodvsdir 20884 . . . . . 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 1374 . . . . 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 6910 . . . . . . 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 2784 . . . 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 5242 . . 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 1195 . . . . 5 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑎 ∈ (Base‘𝑆))
111 simpr2 1196 . . . . 5 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑏 ∈ (Base‘𝑆))
112 eqid 2737 . . . . . 6 (+g𝑆) = (+g𝑆)
11321, 112ringacl 20275 . . . . 5 ((𝑆 ∈ Ring ∧ 𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆)) → (𝑎(+g𝑆)𝑏) ∈ (Base‘𝑆))
114107, 110, 111, 113syl3anc 1373 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → (𝑎(+g𝑆)𝑏) ∈ (Base‘𝑆))
115 simpr3 1197 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑐 ∈ (Base‘𝑌))
1163, 19, 20, 21, 107, 108, 109, 114, 115prdsvscaval 17524 . . 3 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → ((𝑎(+g𝑆)𝑏)( ·𝑠𝑌)𝑐) = (𝑦𝐼 ↦ ((𝑎(+g𝑆)𝑏)( ·𝑠 ‘(𝑅𝑦))(𝑐𝑦))))
117793adantr2 1171 . . . 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 20966 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → (𝑏( ·𝑠𝑌)𝑐) ∈ (Base‘𝑌))
1203, 19, 107, 108, 109, 117, 119, 54prdsplusgval 17518 . . 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 2787 . 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 2737 . . . . . . 7 (.r‘(Scalar‘(𝑅𝑦))) = (.r‘(Scalar‘(𝑅𝑦)))
12447, 49, 50, 51, 123lmodvsass 20885 . . . . . 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 1374 . . . . 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 6910 . . . . . . 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 2784 . . . 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 5242 . . 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 2737 . . . . . 6 (.r𝑆) = (.r𝑆)
13221, 131ringcl 20247 . . . . 5 ((𝑆 ∈ Ring ∧ 𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆)) → (𝑎(.r𝑆)𝑏) ∈ (Base‘𝑆))
133107, 110, 111, 132syl3anc 1373 . . . 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 17524 . . 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 17524 . . 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 2787 . 2 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → ((𝑎(.r𝑆)𝑏)( ·𝑠𝑌)𝑐) = (𝑎( ·𝑠𝑌)(𝑏( ·𝑠𝑌)𝑐)))
13728fveq2d 6910 . . . . . . 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 17517 . . . . . 6 (((𝜑𝑎 ∈ (Base‘𝑌)) ∧ 𝑦𝐼) → (𝑎𝑦) ∈ (Base‘(𝑅𝑦)))
147 eqid 2737 . . . . . . 7 (1r‘(Scalar‘(𝑅𝑦))) = (1r‘(Scalar‘(𝑅𝑦)))
14847, 49, 50, 147lmodvs1 20888 . . . . . 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 2779 . . . 4 (((𝜑𝑎 ∈ (Base‘𝑌)) ∧ 𝑦𝐼) → ((1r𝑆)( ·𝑠 ‘(𝑅𝑦))(𝑎𝑦)) = (𝑎𝑦))
151150mpteq2dva 5242 . . 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 2737 . . . . . . 7 (1r𝑆) = (1r𝑆)
15621, 155ringidcl 20262 . . . . . 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 17524 . . 3 ((𝜑𝑎 ∈ (Base‘𝑌)) → ((1r𝑆)( ·𝑠𝑌)𝑎) = (𝑦𝐼 ↦ ((1r𝑆)( ·𝑠 ‘(𝑅𝑦))(𝑎𝑦))))
1613, 19, 152, 153, 154, 159prdsbasfn 17516 . . . 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 2787 . 2 ((𝜑𝑎 ∈ (Base‘𝑌)) → ((1r𝑆)( ·𝑠𝑌)𝑎) = 𝑎)
1651, 2, 8, 9, 10, 11, 12, 13, 4, 18, 31, 82, 121, 136, 164islmodd 20864 1 (𝜑𝑌 ∈ LMod)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1540  wcel 2108  Vcvv 3480  wss 3951  cmpt 5225   Fn wfn 6556  wf 6557  cfv 6561  (class class class)co 7431  Basecbs 17247  +gcplusg 17297  .rcmulr 17298  Scalarcsca 17300   ·𝑠 cvsca 17301  Xscprds 17490  Grpcgrp 18951  1rcur 20178  Ringcrg 20230  LModclmod 20858
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-map 8868  df-ixp 8938  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-sup 9482  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-z 12614  df-dec 12734  df-uz 12879  df-fz 13548  df-struct 17184  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-plusg 17310  df-mulr 17311  df-sca 17313  df-vsca 17314  df-ip 17315  df-tset 17316  df-ple 17317  df-ds 17319  df-hom 17321  df-cco 17322  df-0g 17486  df-prds 17492  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-grp 18954  df-minusg 18955  df-mgp 20138  df-ur 20179  df-ring 20232  df-lmod 20860
This theorem is referenced by:  pwslmod  20968  dsmmlss  21764  dsmmlmod  21765
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