Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  prdslmodd Structured version   Visualization version   GIF version

Theorem prdslmodd 19733
 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 2820 . 2 (𝜑 → (Base‘𝑌) = (Base‘𝑌))
2 eqidd 2820 . 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)
7 fex 6981 . . . 4 ((𝑅:𝐼⟶LMod ∧ 𝐼𝑉) → 𝑅 ∈ V)
85, 6, 7syl2anc 586 . . 3 (𝜑𝑅 ∈ V)
93, 4, 8prdssca 16721 . 2 (𝜑𝑆 = (Scalar‘𝑌))
10 eqidd 2820 . 2 (𝜑 → ( ·𝑠𝑌) = ( ·𝑠𝑌))
11 eqidd 2820 . 2 (𝜑 → (Base‘𝑆) = (Base‘𝑆))
12 eqidd 2820 . 2 (𝜑 → (+g𝑆) = (+g𝑆))
13 eqidd 2820 . 2 (𝜑 → (.r𝑆) = (.r𝑆))
14 eqidd 2820 . 2 (𝜑 → (1r𝑆) = (1r𝑆))
15 lmodgrp 19633 . . . . 5 (𝑎 ∈ LMod → 𝑎 ∈ Grp)
1615ssriv 3969 . . . 4 LMod ⊆ Grp
17 fss 6520 . . . 4 ((𝑅:𝐼⟶LMod ∧ LMod ⊆ Grp) → 𝑅:𝐼⟶Grp)
185, 16, 17sylancl 588 . . 3 (𝜑𝑅:𝐼⟶Grp)
193, 6, 4, 18prdsgrpd 18201 . 2 (𝜑𝑌 ∈ Grp)
20 eqid 2819 . . . 4 (Base‘𝑌) = (Base‘𝑌)
21 eqid 2819 . . . 4 ( ·𝑠𝑌) = ( ·𝑠𝑌)
22 eqid 2819 . . . 4 (Base‘𝑆) = (Base‘𝑆)
234adantr 483 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌))) → 𝑆 ∈ Ring)
246elexd 3513 . . . . 5 (𝜑𝐼 ∈ V)
2524adantr 483 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌))) → 𝐼 ∈ V)
265adantr 483 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌))) → 𝑅:𝐼⟶LMod)
27 simprl 769 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌))) → 𝑎 ∈ (Base‘𝑆))
28 simprr 771 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌))) → 𝑏 ∈ (Base‘𝑌))
29 prdslmodd.rs . . . . 5 ((𝜑𝑦𝐼) → (Scalar‘(𝑅𝑦)) = 𝑆)
3029adantlr 713 . . . 4 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → (Scalar‘(𝑅𝑦)) = 𝑆)
313, 20, 21, 22, 23, 25, 26, 27, 28, 30prdsvscacl 19732 . . 3 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌))) → (𝑎( ·𝑠𝑌)𝑏) ∈ (Base‘𝑌))
32313impb 1109 . 2 ((𝜑𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌)) → (𝑎( ·𝑠𝑌)𝑏) ∈ (Base‘𝑌))
335ffvelrnda 6844 . . . . . . 7 ((𝜑𝑦𝐼) → (𝑅𝑦) ∈ LMod)
3433adantlr 713 . . . . . 6 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → (𝑅𝑦) ∈ LMod)
35 simplr1 1209 . . . . . . 7 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → 𝑎 ∈ (Base‘𝑆))
3629fveq2d 6667 . . . . . . . 8 ((𝜑𝑦𝐼) → (Base‘(Scalar‘(𝑅𝑦))) = (Base‘𝑆))
3736adantlr 713 . . . . . . 7 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → (Base‘(Scalar‘(𝑅𝑦))) = (Base‘𝑆))
3835, 37eleqtrrd 2914 . . . . . 6 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → 𝑎 ∈ (Base‘(Scalar‘(𝑅𝑦))))
394ad2antrr 724 . . . . . . 7 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → 𝑆 ∈ Ring)
4024ad2antrr 724 . . . . . . 7 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → 𝐼 ∈ V)
415ffnd 6508 . . . . . . . 8 (𝜑𝑅 Fn 𝐼)
4241ad2antrr 724 . . . . . . 7 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → 𝑅 Fn 𝐼)
43 simplr2 1210 . . . . . . 7 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → 𝑏 ∈ (Base‘𝑌))
44 simpr 487 . . . . . . 7 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → 𝑦𝐼)
453, 20, 39, 40, 42, 43, 44prdsbasprj 16737 . . . . . 6 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → (𝑏𝑦) ∈ (Base‘(𝑅𝑦)))
46 simplr3 1211 . . . . . . 7 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → 𝑐 ∈ (Base‘𝑌))
473, 20, 39, 40, 42, 46, 44prdsbasprj 16737 . . . . . 6 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → (𝑐𝑦) ∈ (Base‘(𝑅𝑦)))
48 eqid 2819 . . . . . . 7 (Base‘(𝑅𝑦)) = (Base‘(𝑅𝑦))
49 eqid 2819 . . . . . . 7 (+g‘(𝑅𝑦)) = (+g‘(𝑅𝑦))
50 eqid 2819 . . . . . . 7 (Scalar‘(𝑅𝑦)) = (Scalar‘(𝑅𝑦))
51 eqid 2819 . . . . . . 7 ( ·𝑠 ‘(𝑅𝑦)) = ( ·𝑠 ‘(𝑅𝑦))
52 eqid 2819 . . . . . . 7 (Base‘(Scalar‘(𝑅𝑦))) = (Base‘(Scalar‘(𝑅𝑦)))
5348, 49, 50, 51, 52lmodvsdi 19649 . . . . . 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))))
5434, 38, 45, 47, 53syl13anc 1366 . . . . 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))))
55 eqid 2819 . . . . . . 7 (+g𝑌) = (+g𝑌)
563, 20, 39, 40, 42, 43, 46, 55, 44prdsplusgfval 16739 . . . . . 6 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → ((𝑏(+g𝑌)𝑐)‘𝑦) = ((𝑏𝑦)(+g‘(𝑅𝑦))(𝑐𝑦)))
5756oveq2d 7164 . . . . 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))))
583, 20, 21, 22, 39, 40, 42, 35, 43, 44prdsvscafval 16745 . . . . . 6 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → ((𝑎( ·𝑠𝑌)𝑏)‘𝑦) = (𝑎( ·𝑠 ‘(𝑅𝑦))(𝑏𝑦)))
593, 20, 21, 22, 39, 40, 42, 35, 46, 44prdsvscafval 16745 . . . . . 6 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → ((𝑎( ·𝑠𝑌)𝑐)‘𝑦) = (𝑎( ·𝑠 ‘(𝑅𝑦))(𝑐𝑦)))
6058, 59oveq12d 7166 . . . . 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))))
6154, 57, 603eqtr4d 2864 . . . 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)))
6261mpteq2dva 5152 . . 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))))
634adantr 483 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑆 ∈ Ring)
6424adantr 483 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝐼 ∈ V)
6541adantr 483 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑅 Fn 𝐼)
66 simpr1 1188 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑎 ∈ (Base‘𝑆))
6719adantr 483 . . . . 5 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑌 ∈ Grp)
68 simpr2 1189 . . . . 5 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑏 ∈ (Base‘𝑌))
69 simpr3 1190 . . . . 5 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑐 ∈ (Base‘𝑌))
7020, 55grpcl 18103 . . . . 5 ((𝑌 ∈ Grp ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌)) → (𝑏(+g𝑌)𝑐) ∈ (Base‘𝑌))
7167, 68, 69, 70syl3anc 1365 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → (𝑏(+g𝑌)𝑐) ∈ (Base‘𝑌))
723, 20, 21, 22, 63, 64, 65, 66, 71prdsvscaval 16744 . . 3 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → (𝑎( ·𝑠𝑌)(𝑏(+g𝑌)𝑐)) = (𝑦𝐼 ↦ (𝑎( ·𝑠 ‘(𝑅𝑦))((𝑏(+g𝑌)𝑐)‘𝑦))))
73313adantr3 1165 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → (𝑎( ·𝑠𝑌)𝑏) ∈ (Base‘𝑌))
744adantr 483 . . . . . 6 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑆 ∈ Ring)
7524adantr 483 . . . . . 6 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝐼 ∈ V)
765adantr 483 . . . . . 6 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑅:𝐼⟶LMod)
77 simprl 769 . . . . . 6 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑎 ∈ (Base‘𝑆))
78 simprr 771 . . . . . 6 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑐 ∈ (Base‘𝑌))
7929adantlr 713 . . . . . 6 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → (Scalar‘(𝑅𝑦)) = 𝑆)
803, 20, 21, 22, 74, 75, 76, 77, 78, 79prdsvscacl 19732 . . . . 5 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → (𝑎( ·𝑠𝑌)𝑐) ∈ (Base‘𝑌))
81803adantr2 1164 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → (𝑎( ·𝑠𝑌)𝑐) ∈ (Base‘𝑌))
823, 20, 63, 64, 65, 73, 81, 55prdsplusgval 16738 . . 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))))
8362, 72, 823eqtr4d 2864 . 2 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → (𝑎( ·𝑠𝑌)(𝑏(+g𝑌)𝑐)) = ((𝑎( ·𝑠𝑌)𝑏)(+g𝑌)(𝑎( ·𝑠𝑌)𝑐)))
844ad2antrr 724 . . . . . . 7 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → 𝑆 ∈ Ring)
8524ad2antrr 724 . . . . . . 7 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → 𝐼 ∈ V)
8641ad2antrr 724 . . . . . . 7 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → 𝑅 Fn 𝐼)
87 simplr1 1209 . . . . . . 7 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → 𝑎 ∈ (Base‘𝑆))
88 simplr3 1211 . . . . . . 7 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → 𝑐 ∈ (Base‘𝑌))
89 simpr 487 . . . . . . 7 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → 𝑦𝐼)
903, 20, 21, 22, 84, 85, 86, 87, 88, 89prdsvscafval 16745 . . . . . 6 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → ((𝑎( ·𝑠𝑌)𝑐)‘𝑦) = (𝑎( ·𝑠 ‘(𝑅𝑦))(𝑐𝑦)))
91 simplr2 1210 . . . . . . 7 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → 𝑏 ∈ (Base‘𝑆))
923, 20, 21, 22, 84, 85, 86, 91, 88, 89prdsvscafval 16745 . . . . . 6 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → ((𝑏( ·𝑠𝑌)𝑐)‘𝑦) = (𝑏( ·𝑠 ‘(𝑅𝑦))(𝑐𝑦)))
9390, 92oveq12d 7166 . . . . 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))))
9433adantlr 713 . . . . . 6 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → (𝑅𝑦) ∈ LMod)
9536adantlr 713 . . . . . . 7 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → (Base‘(Scalar‘(𝑅𝑦))) = (Base‘𝑆))
9687, 95eleqtrrd 2914 . . . . . 6 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → 𝑎 ∈ (Base‘(Scalar‘(𝑅𝑦))))
9791, 95eleqtrrd 2914 . . . . . 6 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → 𝑏 ∈ (Base‘(Scalar‘(𝑅𝑦))))
983, 20, 84, 85, 86, 88, 89prdsbasprj 16737 . . . . . 6 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → (𝑐𝑦) ∈ (Base‘(𝑅𝑦)))
99 eqid 2819 . . . . . . 7 (+g‘(Scalar‘(𝑅𝑦))) = (+g‘(Scalar‘(𝑅𝑦)))
10048, 49, 50, 51, 52, 99lmodvsdir 19650 . . . . . 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))))
10194, 96, 97, 98, 100syl13anc 1366 . . . . 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))))
10229adantlr 713 . . . . . . . 8 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → (Scalar‘(𝑅𝑦)) = 𝑆)
103102fveq2d 6667 . . . . . . 7 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → (+g‘(Scalar‘(𝑅𝑦))) = (+g𝑆))
104103oveqd 7165 . . . . . 6 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → (𝑎(+g‘(Scalar‘(𝑅𝑦)))𝑏) = (𝑎(+g𝑆)𝑏))
105104oveq1d 7163 . . . . 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)))
10693, 101, 1053eqtr2rd 2861 . . . 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)))
107106mpteq2dva 5152 . . 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))))
1084adantr 483 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑆 ∈ Ring)
10924adantr 483 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝐼 ∈ V)
11041adantr 483 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑅 Fn 𝐼)
111 simpr1 1188 . . . . 5 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑎 ∈ (Base‘𝑆))
112 simpr2 1189 . . . . 5 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑏 ∈ (Base‘𝑆))
113 eqid 2819 . . . . . 6 (+g𝑆) = (+g𝑆)
11422, 113ringacl 19320 . . . . 5 ((𝑆 ∈ Ring ∧ 𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆)) → (𝑎(+g𝑆)𝑏) ∈ (Base‘𝑆))
115108, 111, 112, 114syl3anc 1365 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → (𝑎(+g𝑆)𝑏) ∈ (Base‘𝑆))
116 simpr3 1190 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑐 ∈ (Base‘𝑌))
1173, 20, 21, 22, 108, 109, 110, 115, 116prdsvscaval 16744 . . 3 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → ((𝑎(+g𝑆)𝑏)( ·𝑠𝑌)𝑐) = (𝑦𝐼 ↦ ((𝑎(+g𝑆)𝑏)( ·𝑠 ‘(𝑅𝑦))(𝑐𝑦))))
118803adantr2 1164 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → (𝑎( ·𝑠𝑌)𝑐) ∈ (Base‘𝑌))
1195adantr 483 . . . . 5 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑅:𝐼⟶LMod)
1203, 20, 21, 22, 108, 109, 119, 112, 116, 102prdsvscacl 19732 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → (𝑏( ·𝑠𝑌)𝑐) ∈ (Base‘𝑌))
1213, 20, 108, 109, 110, 118, 120, 55prdsplusgval 16738 . . 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))))
122107, 117, 1213eqtr4d 2864 . 2 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → ((𝑎(+g𝑆)𝑏)( ·𝑠𝑌)𝑐) = ((𝑎( ·𝑠𝑌)𝑐)(+g𝑌)(𝑏( ·𝑠𝑌)𝑐)))
12392oveq2d 7164 . . . . 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))))
124 eqid 2819 . . . . . . 7 (.r‘(Scalar‘(𝑅𝑦))) = (.r‘(Scalar‘(𝑅𝑦)))
12548, 50, 51, 52, 124lmodvsass 19651 . . . . . 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))))
12694, 96, 97, 98, 125syl13anc 1366 . . . . 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))))
127102fveq2d 6667 . . . . . . 7 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → (.r‘(Scalar‘(𝑅𝑦))) = (.r𝑆))
128127oveqd 7165 . . . . . 6 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → (𝑎(.r‘(Scalar‘(𝑅𝑦)))𝑏) = (𝑎(.r𝑆)𝑏))
129128oveq1d 7163 . . . . 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)))
130123, 126, 1293eqtr2rd 2861 . . . 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)))
131130mpteq2dva 5152 . . 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))))
132 eqid 2819 . . . . . 6 (.r𝑆) = (.r𝑆)
13322, 132ringcl 19303 . . . . 5 ((𝑆 ∈ Ring ∧ 𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆)) → (𝑎(.r𝑆)𝑏) ∈ (Base‘𝑆))
134108, 111, 112, 133syl3anc 1365 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → (𝑎(.r𝑆)𝑏) ∈ (Base‘𝑆))
1353, 20, 21, 22, 108, 109, 110, 134, 116prdsvscaval 16744 . . 3 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → ((𝑎(.r𝑆)𝑏)( ·𝑠𝑌)𝑐) = (𝑦𝐼 ↦ ((𝑎(.r𝑆)𝑏)( ·𝑠 ‘(𝑅𝑦))(𝑐𝑦))))
1363, 20, 21, 22, 108, 109, 110, 111, 120prdsvscaval 16744 . . 3 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → (𝑎( ·𝑠𝑌)(𝑏( ·𝑠𝑌)𝑐)) = (𝑦𝐼 ↦ (𝑎( ·𝑠 ‘(𝑅𝑦))((𝑏( ·𝑠𝑌)𝑐)‘𝑦))))
137131, 135, 1363eqtr4d 2864 . 2 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → ((𝑎(.r𝑆)𝑏)( ·𝑠𝑌)𝑐) = (𝑎( ·𝑠𝑌)(𝑏( ·𝑠𝑌)𝑐)))
13829fveq2d 6667 . . . . . . 7 ((𝜑𝑦𝐼) → (1r‘(Scalar‘(𝑅𝑦))) = (1r𝑆))
139138adantlr 713 . . . . . 6 (((𝜑𝑎 ∈ (Base‘𝑌)) ∧ 𝑦𝐼) → (1r‘(Scalar‘(𝑅𝑦))) = (1r𝑆))
140139oveq1d 7163 . . . . 5 (((𝜑𝑎 ∈ (Base‘𝑌)) ∧ 𝑦𝐼) → ((1r‘(Scalar‘(𝑅𝑦)))( ·𝑠 ‘(𝑅𝑦))(𝑎𝑦)) = ((1r𝑆)( ·𝑠 ‘(𝑅𝑦))(𝑎𝑦)))
14133adantlr 713 . . . . . 6 (((𝜑𝑎 ∈ (Base‘𝑌)) ∧ 𝑦𝐼) → (𝑅𝑦) ∈ LMod)
1424ad2antrr 724 . . . . . . 7 (((𝜑𝑎 ∈ (Base‘𝑌)) ∧ 𝑦𝐼) → 𝑆 ∈ Ring)
14324ad2antrr 724 . . . . . . 7 (((𝜑𝑎 ∈ (Base‘𝑌)) ∧ 𝑦𝐼) → 𝐼 ∈ V)
14441ad2antrr 724 . . . . . . 7 (((𝜑𝑎 ∈ (Base‘𝑌)) ∧ 𝑦𝐼) → 𝑅 Fn 𝐼)
145 simplr 767 . . . . . . 7 (((𝜑𝑎 ∈ (Base‘𝑌)) ∧ 𝑦𝐼) → 𝑎 ∈ (Base‘𝑌))
146 simpr 487 . . . . . . 7 (((𝜑𝑎 ∈ (Base‘𝑌)) ∧ 𝑦𝐼) → 𝑦𝐼)
1473, 20, 142, 143, 144, 145, 146prdsbasprj 16737 . . . . . 6 (((𝜑𝑎 ∈ (Base‘𝑌)) ∧ 𝑦𝐼) → (𝑎𝑦) ∈ (Base‘(𝑅𝑦)))
148 eqid 2819 . . . . . . 7 (1r‘(Scalar‘(𝑅𝑦))) = (1r‘(Scalar‘(𝑅𝑦)))
14948, 50, 51, 148lmodvs1 19654 . . . . . 6 (((𝑅𝑦) ∈ LMod ∧ (𝑎𝑦) ∈ (Base‘(𝑅𝑦))) → ((1r‘(Scalar‘(𝑅𝑦)))( ·𝑠 ‘(𝑅𝑦))(𝑎𝑦)) = (𝑎𝑦))
150141, 147, 149syl2anc 586 . . . . 5 (((𝜑𝑎 ∈ (Base‘𝑌)) ∧ 𝑦𝐼) → ((1r‘(Scalar‘(𝑅𝑦)))( ·𝑠 ‘(𝑅𝑦))(𝑎𝑦)) = (𝑎𝑦))
151140, 150eqtr3d 2856 . . . 4 (((𝜑𝑎 ∈ (Base‘𝑌)) ∧ 𝑦𝐼) → ((1r𝑆)( ·𝑠 ‘(𝑅𝑦))(𝑎𝑦)) = (𝑎𝑦))
152151mpteq2dva 5152 . . 3 ((𝜑𝑎 ∈ (Base‘𝑌)) → (𝑦𝐼 ↦ ((1r𝑆)( ·𝑠 ‘(𝑅𝑦))(𝑎𝑦))) = (𝑦𝐼 ↦ (𝑎𝑦)))
1534adantr 483 . . . 4 ((𝜑𝑎 ∈ (Base‘𝑌)) → 𝑆 ∈ Ring)
15424adantr 483 . . . 4 ((𝜑𝑎 ∈ (Base‘𝑌)) → 𝐼 ∈ V)
15541adantr 483 . . . 4 ((𝜑𝑎 ∈ (Base‘𝑌)) → 𝑅 Fn 𝐼)
156 eqid 2819 . . . . . . 7 (1r𝑆) = (1r𝑆)
15722, 156ringidcl 19310 . . . . . 6 (𝑆 ∈ Ring → (1r𝑆) ∈ (Base‘𝑆))
1584, 157syl 17 . . . . 5 (𝜑 → (1r𝑆) ∈ (Base‘𝑆))
159158adantr 483 . . . 4 ((𝜑𝑎 ∈ (Base‘𝑌)) → (1r𝑆) ∈ (Base‘𝑆))
160 simpr 487 . . . 4 ((𝜑𝑎 ∈ (Base‘𝑌)) → 𝑎 ∈ (Base‘𝑌))
1613, 20, 21, 22, 153, 154, 155, 159, 160prdsvscaval 16744 . . 3 ((𝜑𝑎 ∈ (Base‘𝑌)) → ((1r𝑆)( ·𝑠𝑌)𝑎) = (𝑦𝐼 ↦ ((1r𝑆)( ·𝑠 ‘(𝑅𝑦))(𝑎𝑦))))
1623, 20, 153, 154, 155, 160prdsbasfn 16736 . . . 4 ((𝜑𝑎 ∈ (Base‘𝑌)) → 𝑎 Fn 𝐼)
163 dffn5 6717 . . . 4 (𝑎 Fn 𝐼𝑎 = (𝑦𝐼 ↦ (𝑎𝑦)))
164162, 163sylib 220 . . 3 ((𝜑𝑎 ∈ (Base‘𝑌)) → 𝑎 = (𝑦𝐼 ↦ (𝑎𝑦)))
165152, 161, 1643eqtr4d 2864 . 2 ((𝜑𝑎 ∈ (Base‘𝑌)) → ((1r𝑆)( ·𝑠𝑌)𝑎) = 𝑎)
1661, 2, 9, 10, 11, 12, 13, 14, 4, 19, 32, 83, 122, 137, 165islmodd 19632 1 (𝜑𝑌 ∈ LMod)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1081   = wceq 1530   ∈ wcel 2107  Vcvv 3493   ⊆ wss 3934   ↦ cmpt 5137   Fn wfn 6343  ⟶wf 6344  ‘cfv 6348  (class class class)co 7148  Basecbs 16475  +gcplusg 16557  .rcmulr 16558  Scalarcsca 16560   ·𝑠 cvsca 16561  Xscprds 16711  Grpcgrp 18095  1rcur 19243  Ringcrg 19289  LModclmod 19626 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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-nel 3122  df-ral 3141  df-rex 3142  df-reu 3143  df-rmo 3144  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7573  df-1st 7681  df-2nd 7682  df-wrecs 7939  df-recs 8000  df-rdg 8038  df-1o 8094  df-oadd 8098  df-er 8281  df-map 8400  df-ixp 8454  df-en 8502  df-dom 8503  df-sdom 8504  df-fin 8505  df-sup 8898  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-nn 11631  df-2 11692  df-3 11693  df-4 11694  df-5 11695  df-6 11696  df-7 11697  df-8 11698  df-9 11699  df-n0 11890  df-z 11974  df-dec 12091  df-uz 12236  df-fz 12885  df-struct 16477  df-ndx 16478  df-slot 16479  df-base 16481  df-sets 16482  df-plusg 16570  df-mulr 16571  df-sca 16573  df-vsca 16574  df-ip 16575  df-tset 16576  df-ple 16577  df-ds 16579  df-hom 16581  df-cco 16582  df-0g 16707  df-prds 16713  df-mgm 17844  df-sgrp 17893  df-mnd 17904  df-grp 18098  df-minusg 18099  df-mgp 19232  df-ur 19244  df-ring 19291  df-lmod 19628 This theorem is referenced by:  pwslmod  19734  dsmmlss  20880  dsmmlmod  20881
 Copyright terms: Public domain W3C validator