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Theorem prdslmodd 20806
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 2725 . 2 (𝜑 → (Base‘𝑌) = (Base‘𝑌))
2 eqidd 2725 . 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 7220 . . 3 (𝜑𝑅 ∈ V)
83, 4, 7prdssca 17401 . 2 (𝜑𝑆 = (Scalar‘𝑌))
9 eqidd 2725 . 2 (𝜑 → ( ·𝑠𝑌) = ( ·𝑠𝑌))
10 eqidd 2725 . 2 (𝜑 → (Base‘𝑆) = (Base‘𝑆))
11 eqidd 2725 . 2 (𝜑 → (+g𝑆) = (+g𝑆))
12 eqidd 2725 . 2 (𝜑 → (.r𝑆) = (.r𝑆))
13 eqidd 2725 . 2 (𝜑 → (1r𝑆) = (1r𝑆))
14 lmodgrp 20703 . . . . 5 (𝑎 ∈ LMod → 𝑎 ∈ Grp)
1514ssriv 3978 . . . 4 LMod ⊆ Grp
16 fss 6724 . . . 4 ((𝑅:𝐼⟶LMod ∧ LMod ⊆ Grp) → 𝑅:𝐼⟶Grp)
175, 15, 16sylancl 585 . . 3 (𝜑𝑅:𝐼⟶Grp)
183, 6, 4, 17prdsgrpd 18968 . 2 (𝜑𝑌 ∈ Grp)
19 eqid 2724 . . . 4 (Base‘𝑌) = (Base‘𝑌)
20 eqid 2724 . . . 4 ( ·𝑠𝑌) = ( ·𝑠𝑌)
21 eqid 2724 . . . 4 (Base‘𝑆) = (Base‘𝑆)
224adantr 480 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌))) → 𝑆 ∈ Ring)
236elexd 3487 . . . . 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 768 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌))) → 𝑎 ∈ (Base‘𝑆))
27 simprr 770 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌))) → 𝑏 ∈ (Base‘𝑌))
28 prdslmodd.rs . . . . 5 ((𝜑𝑦𝐼) → (Scalar‘(𝑅𝑦)) = 𝑆)
2928adantlr 712 . . . 4 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → (Scalar‘(𝑅𝑦)) = 𝑆)
303, 19, 20, 21, 22, 24, 25, 26, 27, 29prdsvscacl 20805 . . 3 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌))) → (𝑎( ·𝑠𝑌)𝑏) ∈ (Base‘𝑌))
31303impb 1112 . 2 ((𝜑𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌)) → (𝑎( ·𝑠𝑌)𝑏) ∈ (Base‘𝑌))
325ffvelcdmda 7076 . . . . . . 7 ((𝜑𝑦𝐼) → (𝑅𝑦) ∈ LMod)
3332adantlr 712 . . . . . 6 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → (𝑅𝑦) ∈ LMod)
34 simplr1 1212 . . . . . . 7 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → 𝑎 ∈ (Base‘𝑆))
3528fveq2d 6885 . . . . . . . 8 ((𝜑𝑦𝐼) → (Base‘(Scalar‘(𝑅𝑦))) = (Base‘𝑆))
3635adantlr 712 . . . . . . 7 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → (Base‘(Scalar‘(𝑅𝑦))) = (Base‘𝑆))
3734, 36eleqtrrd 2828 . . . . . 6 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → 𝑎 ∈ (Base‘(Scalar‘(𝑅𝑦))))
384ad2antrr 723 . . . . . . 7 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → 𝑆 ∈ Ring)
3923ad2antrr 723 . . . . . . 7 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → 𝐼 ∈ V)
405ffnd 6708 . . . . . . . 8 (𝜑𝑅 Fn 𝐼)
4140ad2antrr 723 . . . . . . 7 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → 𝑅 Fn 𝐼)
42 simplr2 1213 . . . . . . 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 17417 . . . . . 6 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → (𝑏𝑦) ∈ (Base‘(𝑅𝑦)))
45 simplr3 1214 . . . . . . 7 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → 𝑐 ∈ (Base‘𝑌))
463, 19, 38, 39, 41, 45, 43prdsbasprj 17417 . . . . . 6 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → (𝑐𝑦) ∈ (Base‘(𝑅𝑦)))
47 eqid 2724 . . . . . . 7 (Base‘(𝑅𝑦)) = (Base‘(𝑅𝑦))
48 eqid 2724 . . . . . . 7 (+g‘(𝑅𝑦)) = (+g‘(𝑅𝑦))
49 eqid 2724 . . . . . . 7 (Scalar‘(𝑅𝑦)) = (Scalar‘(𝑅𝑦))
50 eqid 2724 . . . . . . 7 ( ·𝑠 ‘(𝑅𝑦)) = ( ·𝑠 ‘(𝑅𝑦))
51 eqid 2724 . . . . . . 7 (Base‘(Scalar‘(𝑅𝑦))) = (Base‘(Scalar‘(𝑅𝑦)))
5247, 48, 49, 50, 51lmodvsdi 20721 . . . . . 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 1369 . . . . 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 2724 . . . . . . 7 (+g𝑌) = (+g𝑌)
553, 19, 38, 39, 41, 42, 45, 54, 43prdsplusgfval 17419 . . . . . 6 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → ((𝑏(+g𝑌)𝑐)‘𝑦) = ((𝑏𝑦)(+g‘(𝑅𝑦))(𝑐𝑦)))
5655oveq2d 7417 . . . . 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 17425 . . . . . 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 17425 . . . . . 6 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → ((𝑎( ·𝑠𝑌)𝑐)‘𝑦) = (𝑎( ·𝑠 ‘(𝑅𝑦))(𝑐𝑦)))
5957, 58oveq12d 7419 . . . . 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 2774 . . . 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 5238 . . 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 1191 . . . 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 1192 . . . . 5 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑏 ∈ (Base‘𝑌))
68 simpr3 1193 . . . . 5 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑐 ∈ (Base‘𝑌))
6919, 54grpcl 18861 . . . . 5 ((𝑌 ∈ Grp ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌)) → (𝑏(+g𝑌)𝑐) ∈ (Base‘𝑌))
7066, 67, 68, 69syl3anc 1368 . . . 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 17424 . . 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 1168 . . . 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 768 . . . . . 6 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑎 ∈ (Base‘𝑆))
77 simprr 770 . . . . . 6 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑐 ∈ (Base‘𝑌))
7828adantlr 712 . . . . . 6 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → (Scalar‘(𝑅𝑦)) = 𝑆)
793, 19, 20, 21, 73, 74, 75, 76, 77, 78prdsvscacl 20805 . . . . 5 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → (𝑎( ·𝑠𝑌)𝑐) ∈ (Base‘𝑌))
80793adantr2 1167 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → (𝑎( ·𝑠𝑌)𝑐) ∈ (Base‘𝑌))
813, 19, 62, 63, 64, 72, 80, 54prdsplusgval 17418 . . 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 2774 . 2 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → (𝑎( ·𝑠𝑌)(𝑏(+g𝑌)𝑐)) = ((𝑎( ·𝑠𝑌)𝑏)(+g𝑌)(𝑎( ·𝑠𝑌)𝑐)))
834ad2antrr 723 . . . . . . 7 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → 𝑆 ∈ Ring)
8423ad2antrr 723 . . . . . . 7 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → 𝐼 ∈ V)
8540ad2antrr 723 . . . . . . 7 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → 𝑅 Fn 𝐼)
86 simplr1 1212 . . . . . . 7 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → 𝑎 ∈ (Base‘𝑆))
87 simplr3 1214 . . . . . . 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 17425 . . . . . 6 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → ((𝑎( ·𝑠𝑌)𝑐)‘𝑦) = (𝑎( ·𝑠 ‘(𝑅𝑦))(𝑐𝑦)))
90 simplr2 1213 . . . . . . 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 17425 . . . . . 6 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → ((𝑏( ·𝑠𝑌)𝑐)‘𝑦) = (𝑏( ·𝑠 ‘(𝑅𝑦))(𝑐𝑦)))
9289, 91oveq12d 7419 . . . . 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 712 . . . . . 6 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → (𝑅𝑦) ∈ LMod)
9435adantlr 712 . . . . . . 7 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → (Base‘(Scalar‘(𝑅𝑦))) = (Base‘𝑆))
9586, 94eleqtrrd 2828 . . . . . 6 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → 𝑎 ∈ (Base‘(Scalar‘(𝑅𝑦))))
9690, 94eleqtrrd 2828 . . . . . 6 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → 𝑏 ∈ (Base‘(Scalar‘(𝑅𝑦))))
973, 19, 83, 84, 85, 87, 88prdsbasprj 17417 . . . . . 6 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → (𝑐𝑦) ∈ (Base‘(𝑅𝑦)))
98 eqid 2724 . . . . . . 7 (+g‘(Scalar‘(𝑅𝑦))) = (+g‘(Scalar‘(𝑅𝑦)))
9947, 48, 49, 50, 51, 98lmodvsdir 20722 . . . . . 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 1369 . . . . 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 712 . . . . . . . 8 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → (Scalar‘(𝑅𝑦)) = 𝑆)
102101fveq2d 6885 . . . . . . 7 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → (+g‘(Scalar‘(𝑅𝑦))) = (+g𝑆))
103102oveqd 7418 . . . . . 6 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → (𝑎(+g‘(Scalar‘(𝑅𝑦)))𝑏) = (𝑎(+g𝑆)𝑏))
104103oveq1d 7416 . . . . 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 2771 . . . 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 5238 . . 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 1191 . . . . 5 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑎 ∈ (Base‘𝑆))
111 simpr2 1192 . . . . 5 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑏 ∈ (Base‘𝑆))
112 eqid 2724 . . . . . 6 (+g𝑆) = (+g𝑆)
11321, 112ringacl 20167 . . . . 5 ((𝑆 ∈ Ring ∧ 𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆)) → (𝑎(+g𝑆)𝑏) ∈ (Base‘𝑆))
114107, 110, 111, 113syl3anc 1368 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → (𝑎(+g𝑆)𝑏) ∈ (Base‘𝑆))
115 simpr3 1193 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑐 ∈ (Base‘𝑌))
1163, 19, 20, 21, 107, 108, 109, 114, 115prdsvscaval 17424 . . 3 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → ((𝑎(+g𝑆)𝑏)( ·𝑠𝑌)𝑐) = (𝑦𝐼 ↦ ((𝑎(+g𝑆)𝑏)( ·𝑠 ‘(𝑅𝑦))(𝑐𝑦))))
117793adantr2 1167 . . . 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 20805 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → (𝑏( ·𝑠𝑌)𝑐) ∈ (Base‘𝑌))
1203, 19, 107, 108, 109, 117, 119, 54prdsplusgval 17418 . . 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 2774 . 2 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → ((𝑎(+g𝑆)𝑏)( ·𝑠𝑌)𝑐) = ((𝑎( ·𝑠𝑌)𝑐)(+g𝑌)(𝑏( ·𝑠𝑌)𝑐)))
12291oveq2d 7417 . . . . 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 2724 . . . . . . 7 (.r‘(Scalar‘(𝑅𝑦))) = (.r‘(Scalar‘(𝑅𝑦)))
12447, 49, 50, 51, 123lmodvsass 20723 . . . . . 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 1369 . . . . 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 6885 . . . . . . 7 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → (.r‘(Scalar‘(𝑅𝑦))) = (.r𝑆))
127126oveqd 7418 . . . . . 6 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → (𝑎(.r‘(Scalar‘(𝑅𝑦)))𝑏) = (𝑎(.r𝑆)𝑏))
128127oveq1d 7416 . . . . 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 2771 . . . 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 5238 . . 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 2724 . . . . . 6 (.r𝑆) = (.r𝑆)
13221, 131ringcl 20145 . . . . 5 ((𝑆 ∈ Ring ∧ 𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆)) → (𝑎(.r𝑆)𝑏) ∈ (Base‘𝑆))
133107, 110, 111, 132syl3anc 1368 . . . 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 17424 . . 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 17424 . . 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 2774 . 2 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → ((𝑎(.r𝑆)𝑏)( ·𝑠𝑌)𝑐) = (𝑎( ·𝑠𝑌)(𝑏( ·𝑠𝑌)𝑐)))
13728fveq2d 6885 . . . . . . 7 ((𝜑𝑦𝐼) → (1r‘(Scalar‘(𝑅𝑦))) = (1r𝑆))
138137adantlr 712 . . . . . 6 (((𝜑𝑎 ∈ (Base‘𝑌)) ∧ 𝑦𝐼) → (1r‘(Scalar‘(𝑅𝑦))) = (1r𝑆))
139138oveq1d 7416 . . . . 5 (((𝜑𝑎 ∈ (Base‘𝑌)) ∧ 𝑦𝐼) → ((1r‘(Scalar‘(𝑅𝑦)))( ·𝑠 ‘(𝑅𝑦))(𝑎𝑦)) = ((1r𝑆)( ·𝑠 ‘(𝑅𝑦))(𝑎𝑦)))
14032adantlr 712 . . . . . 6 (((𝜑𝑎 ∈ (Base‘𝑌)) ∧ 𝑦𝐼) → (𝑅𝑦) ∈ LMod)
1414ad2antrr 723 . . . . . . 7 (((𝜑𝑎 ∈ (Base‘𝑌)) ∧ 𝑦𝐼) → 𝑆 ∈ Ring)
14223ad2antrr 723 . . . . . . 7 (((𝜑𝑎 ∈ (Base‘𝑌)) ∧ 𝑦𝐼) → 𝐼 ∈ V)
14340ad2antrr 723 . . . . . . 7 (((𝜑𝑎 ∈ (Base‘𝑌)) ∧ 𝑦𝐼) → 𝑅 Fn 𝐼)
144 simplr 766 . . . . . . 7 (((𝜑𝑎 ∈ (Base‘𝑌)) ∧ 𝑦𝐼) → 𝑎 ∈ (Base‘𝑌))
145 simpr 484 . . . . . . 7 (((𝜑𝑎 ∈ (Base‘𝑌)) ∧ 𝑦𝐼) → 𝑦𝐼)
1463, 19, 141, 142, 143, 144, 145prdsbasprj 17417 . . . . . 6 (((𝜑𝑎 ∈ (Base‘𝑌)) ∧ 𝑦𝐼) → (𝑎𝑦) ∈ (Base‘(𝑅𝑦)))
147 eqid 2724 . . . . . . 7 (1r‘(Scalar‘(𝑅𝑦))) = (1r‘(Scalar‘(𝑅𝑦)))
14847, 49, 50, 147lmodvs1 20726 . . . . . 6 (((𝑅𝑦) ∈ LMod ∧ (𝑎𝑦) ∈ (Base‘(𝑅𝑦))) → ((1r‘(Scalar‘(𝑅𝑦)))( ·𝑠 ‘(𝑅𝑦))(𝑎𝑦)) = (𝑎𝑦))
149140, 146, 148syl2anc 583 . . . . 5 (((𝜑𝑎 ∈ (Base‘𝑌)) ∧ 𝑦𝐼) → ((1r‘(Scalar‘(𝑅𝑦)))( ·𝑠 ‘(𝑅𝑦))(𝑎𝑦)) = (𝑎𝑦))
150139, 149eqtr3d 2766 . . . 4 (((𝜑𝑎 ∈ (Base‘𝑌)) ∧ 𝑦𝐼) → ((1r𝑆)( ·𝑠 ‘(𝑅𝑦))(𝑎𝑦)) = (𝑎𝑦))
151150mpteq2dva 5238 . . 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 2724 . . . . . . 7 (1r𝑆) = (1r𝑆)
15621, 155ringidcl 20155 . . . . . 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 17424 . . 3 ((𝜑𝑎 ∈ (Base‘𝑌)) → ((1r𝑆)( ·𝑠𝑌)𝑎) = (𝑦𝐼 ↦ ((1r𝑆)( ·𝑠 ‘(𝑅𝑦))(𝑎𝑦))))
1613, 19, 152, 153, 154, 159prdsbasfn 17416 . . . 4 ((𝜑𝑎 ∈ (Base‘𝑌)) → 𝑎 Fn 𝐼)
162 dffn5 6940 . . . 4 (𝑎 Fn 𝐼𝑎 = (𝑦𝐼 ↦ (𝑎𝑦)))
163161, 162sylib 217 . . 3 ((𝜑𝑎 ∈ (Base‘𝑌)) → 𝑎 = (𝑦𝐼 ↦ (𝑎𝑦)))
164151, 160, 1633eqtr4d 2774 . 2 ((𝜑𝑎 ∈ (Base‘𝑌)) → ((1r𝑆)( ·𝑠𝑌)𝑎) = 𝑎)
1651, 2, 8, 9, 10, 11, 12, 13, 4, 18, 31, 82, 121, 136, 164islmodd 20702 1 (𝜑𝑌 ∈ LMod)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1084   = wceq 1533  wcel 2098  Vcvv 3466  wss 3940  cmpt 5221   Fn wfn 6528  wf 6529  cfv 6533  (class class class)co 7401  Basecbs 17143  +gcplusg 17196  .rcmulr 17197  Scalarcsca 17199   ·𝑠 cvsca 17200  Xscprds 17390  Grpcgrp 18853  1rcur 20076  Ringcrg 20128  LModclmod 20696
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 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183
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 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-nel 3039  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3959  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-tp 4625  df-op 4627  df-uni 4900  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-tr 5256  df-id 5564  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-we 5623  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6290  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7357  df-ov 7404  df-oprab 7405  df-mpo 7406  df-om 7849  df-1st 7968  df-2nd 7969  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-1o 8461  df-er 8699  df-map 8818  df-ixp 8888  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-sup 9433  df-pnf 11247  df-mnf 11248  df-xr 11249  df-ltxr 11250  df-le 11251  df-sub 11443  df-neg 11444  df-nn 12210  df-2 12272  df-3 12273  df-4 12274  df-5 12275  df-6 12276  df-7 12277  df-8 12278  df-9 12279  df-n0 12470  df-z 12556  df-dec 12675  df-uz 12820  df-fz 13482  df-struct 17079  df-sets 17096  df-slot 17114  df-ndx 17126  df-base 17144  df-plusg 17209  df-mulr 17210  df-sca 17212  df-vsca 17213  df-ip 17214  df-tset 17215  df-ple 17216  df-ds 17218  df-hom 17220  df-cco 17221  df-0g 17386  df-prds 17392  df-mgm 18563  df-sgrp 18642  df-mnd 18658  df-grp 18856  df-minusg 18857  df-mgp 20030  df-ur 20077  df-ring 20130  df-lmod 20698
This theorem is referenced by:  pwslmod  20807  dsmmlss  21607  dsmmlmod  21608
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