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Theorem prdslmodd 19017
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 2652 . 2 (𝜑 → (Base‘𝑌) = (Base‘𝑌))
2 eqidd 2652 . 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 6530 . . . 4 ((𝑅:𝐼⟶LMod ∧ 𝐼𝑉) → 𝑅 ∈ V)
85, 6, 7syl2anc 694 . . 3 (𝜑𝑅 ∈ V)
93, 4, 8prdssca 16163 . 2 (𝜑𝑆 = (Scalar‘𝑌))
10 eqidd 2652 . 2 (𝜑 → ( ·𝑠𝑌) = ( ·𝑠𝑌))
11 eqidd 2652 . 2 (𝜑 → (Base‘𝑆) = (Base‘𝑆))
12 eqidd 2652 . 2 (𝜑 → (+g𝑆) = (+g𝑆))
13 eqidd 2652 . 2 (𝜑 → (.r𝑆) = (.r𝑆))
14 eqidd 2652 . 2 (𝜑 → (1r𝑆) = (1r𝑆))
15 lmodgrp 18918 . . . . 5 (𝑎 ∈ LMod → 𝑎 ∈ Grp)
1615ssriv 3640 . . . 4 LMod ⊆ Grp
17 fss 6094 . . . 4 ((𝑅:𝐼⟶LMod ∧ LMod ⊆ Grp) → 𝑅:𝐼⟶Grp)
185, 16, 17sylancl 695 . . 3 (𝜑𝑅:𝐼⟶Grp)
193, 6, 4, 18prdsgrpd 17572 . 2 (𝜑𝑌 ∈ Grp)
20 eqid 2651 . . . 4 (Base‘𝑌) = (Base‘𝑌)
21 eqid 2651 . . . 4 ( ·𝑠𝑌) = ( ·𝑠𝑌)
22 eqid 2651 . . . 4 (Base‘𝑆) = (Base‘𝑆)
234adantr 480 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌))) → 𝑆 ∈ Ring)
24 elex 3243 . . . . . 6 (𝐼𝑉𝐼 ∈ V)
256, 24syl 17 . . . . 5 (𝜑𝐼 ∈ V)
2625adantr 480 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌))) → 𝐼 ∈ V)
275adantr 480 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌))) → 𝑅:𝐼⟶LMod)
28 simprl 809 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌))) → 𝑎 ∈ (Base‘𝑆))
29 simprr 811 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌))) → 𝑏 ∈ (Base‘𝑌))
30 prdslmodd.rs . . . . 5 ((𝜑𝑦𝐼) → (Scalar‘(𝑅𝑦)) = 𝑆)
3130adantlr 751 . . . 4 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → (Scalar‘(𝑅𝑦)) = 𝑆)
323, 20, 21, 22, 23, 26, 27, 28, 29, 31prdsvscacl 19016 . . 3 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌))) → (𝑎( ·𝑠𝑌)𝑏) ∈ (Base‘𝑌))
33323impb 1279 . 2 ((𝜑𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌)) → (𝑎( ·𝑠𝑌)𝑏) ∈ (Base‘𝑌))
345ffvelrnda 6399 . . . . . . 7 ((𝜑𝑦𝐼) → (𝑅𝑦) ∈ LMod)
3534adantlr 751 . . . . . 6 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → (𝑅𝑦) ∈ LMod)
36 simplr1 1123 . . . . . . 7 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → 𝑎 ∈ (Base‘𝑆))
3730fveq2d 6233 . . . . . . . 8 ((𝜑𝑦𝐼) → (Base‘(Scalar‘(𝑅𝑦))) = (Base‘𝑆))
3837adantlr 751 . . . . . . 7 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → (Base‘(Scalar‘(𝑅𝑦))) = (Base‘𝑆))
3936, 38eleqtrrd 2733 . . . . . 6 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → 𝑎 ∈ (Base‘(Scalar‘(𝑅𝑦))))
404ad2antrr 762 . . . . . . 7 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → 𝑆 ∈ Ring)
4125ad2antrr 762 . . . . . . 7 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → 𝐼 ∈ V)
42 ffn 6083 . . . . . . . . 9 (𝑅:𝐼⟶LMod → 𝑅 Fn 𝐼)
435, 42syl 17 . . . . . . . 8 (𝜑𝑅 Fn 𝐼)
4443ad2antrr 762 . . . . . . 7 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → 𝑅 Fn 𝐼)
45 simplr2 1124 . . . . . . 7 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → 𝑏 ∈ (Base‘𝑌))
46 simpr 476 . . . . . . 7 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → 𝑦𝐼)
473, 20, 40, 41, 44, 45, 46prdsbasprj 16179 . . . . . 6 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → (𝑏𝑦) ∈ (Base‘(𝑅𝑦)))
48 simplr3 1125 . . . . . . 7 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → 𝑐 ∈ (Base‘𝑌))
493, 20, 40, 41, 44, 48, 46prdsbasprj 16179 . . . . . 6 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → (𝑐𝑦) ∈ (Base‘(𝑅𝑦)))
50 eqid 2651 . . . . . . 7 (Base‘(𝑅𝑦)) = (Base‘(𝑅𝑦))
51 eqid 2651 . . . . . . 7 (+g‘(𝑅𝑦)) = (+g‘(𝑅𝑦))
52 eqid 2651 . . . . . . 7 (Scalar‘(𝑅𝑦)) = (Scalar‘(𝑅𝑦))
53 eqid 2651 . . . . . . 7 ( ·𝑠 ‘(𝑅𝑦)) = ( ·𝑠 ‘(𝑅𝑦))
54 eqid 2651 . . . . . . 7 (Base‘(Scalar‘(𝑅𝑦))) = (Base‘(Scalar‘(𝑅𝑦)))
5550, 51, 52, 53, 54lmodvsdi 18934 . . . . . 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))))
5635, 39, 47, 49, 55syl13anc 1368 . . . . 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))))
57 eqid 2651 . . . . . . 7 (+g𝑌) = (+g𝑌)
583, 20, 40, 41, 44, 45, 48, 57, 46prdsplusgfval 16181 . . . . . 6 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → ((𝑏(+g𝑌)𝑐)‘𝑦) = ((𝑏𝑦)(+g‘(𝑅𝑦))(𝑐𝑦)))
5958oveq2d 6706 . . . . 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))))
603, 20, 21, 22, 40, 41, 44, 36, 45, 46prdsvscafval 16187 . . . . . 6 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → ((𝑎( ·𝑠𝑌)𝑏)‘𝑦) = (𝑎( ·𝑠 ‘(𝑅𝑦))(𝑏𝑦)))
613, 20, 21, 22, 40, 41, 44, 36, 48, 46prdsvscafval 16187 . . . . . 6 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → ((𝑎( ·𝑠𝑌)𝑐)‘𝑦) = (𝑎( ·𝑠 ‘(𝑅𝑦))(𝑐𝑦)))
6260, 61oveq12d 6708 . . . . 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))))
6356, 59, 623eqtr4d 2695 . . . 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)))
6463mpteq2dva 4777 . . 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))))
654adantr 480 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑆 ∈ Ring)
6625adantr 480 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝐼 ∈ V)
6743adantr 480 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑅 Fn 𝐼)
68 simpr1 1087 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑎 ∈ (Base‘𝑆))
6919adantr 480 . . . . 5 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑌 ∈ Grp)
70 simpr2 1088 . . . . 5 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑏 ∈ (Base‘𝑌))
71 simpr3 1089 . . . . 5 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑐 ∈ (Base‘𝑌))
7220, 57grpcl 17477 . . . . 5 ((𝑌 ∈ Grp ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌)) → (𝑏(+g𝑌)𝑐) ∈ (Base‘𝑌))
7369, 70, 71, 72syl3anc 1366 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → (𝑏(+g𝑌)𝑐) ∈ (Base‘𝑌))
743, 20, 21, 22, 65, 66, 67, 68, 73prdsvscaval 16186 . . 3 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → (𝑎( ·𝑠𝑌)(𝑏(+g𝑌)𝑐)) = (𝑦𝐼 ↦ (𝑎( ·𝑠 ‘(𝑅𝑦))((𝑏(+g𝑌)𝑐)‘𝑦))))
75323adantr3 1242 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → (𝑎( ·𝑠𝑌)𝑏) ∈ (Base‘𝑌))
764adantr 480 . . . . . 6 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑆 ∈ Ring)
7725adantr 480 . . . . . 6 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝐼 ∈ V)
785adantr 480 . . . . . 6 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑅:𝐼⟶LMod)
79 simprl 809 . . . . . 6 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑎 ∈ (Base‘𝑆))
80 simprr 811 . . . . . 6 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑐 ∈ (Base‘𝑌))
8130adantlr 751 . . . . . 6 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → (Scalar‘(𝑅𝑦)) = 𝑆)
823, 20, 21, 22, 76, 77, 78, 79, 80, 81prdsvscacl 19016 . . . . 5 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → (𝑎( ·𝑠𝑌)𝑐) ∈ (Base‘𝑌))
83823adantr2 1241 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → (𝑎( ·𝑠𝑌)𝑐) ∈ (Base‘𝑌))
843, 20, 65, 66, 67, 75, 83, 57prdsplusgval 16180 . . 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))))
8564, 74, 843eqtr4d 2695 . 2 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → (𝑎( ·𝑠𝑌)(𝑏(+g𝑌)𝑐)) = ((𝑎( ·𝑠𝑌)𝑏)(+g𝑌)(𝑎( ·𝑠𝑌)𝑐)))
864ad2antrr 762 . . . . . . 7 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → 𝑆 ∈ Ring)
8725ad2antrr 762 . . . . . . 7 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → 𝐼 ∈ V)
8843ad2antrr 762 . . . . . . 7 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → 𝑅 Fn 𝐼)
89 simplr1 1123 . . . . . . 7 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → 𝑎 ∈ (Base‘𝑆))
90 simplr3 1125 . . . . . . 7 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → 𝑐 ∈ (Base‘𝑌))
91 simpr 476 . . . . . . 7 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → 𝑦𝐼)
923, 20, 21, 22, 86, 87, 88, 89, 90, 91prdsvscafval 16187 . . . . . 6 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → ((𝑎( ·𝑠𝑌)𝑐)‘𝑦) = (𝑎( ·𝑠 ‘(𝑅𝑦))(𝑐𝑦)))
93 simplr2 1124 . . . . . . 7 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → 𝑏 ∈ (Base‘𝑆))
943, 20, 21, 22, 86, 87, 88, 93, 90, 91prdsvscafval 16187 . . . . . 6 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → ((𝑏( ·𝑠𝑌)𝑐)‘𝑦) = (𝑏( ·𝑠 ‘(𝑅𝑦))(𝑐𝑦)))
9592, 94oveq12d 6708 . . . . 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))))
9634adantlr 751 . . . . . 6 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → (𝑅𝑦) ∈ LMod)
9737adantlr 751 . . . . . . 7 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → (Base‘(Scalar‘(𝑅𝑦))) = (Base‘𝑆))
9889, 97eleqtrrd 2733 . . . . . 6 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → 𝑎 ∈ (Base‘(Scalar‘(𝑅𝑦))))
9993, 97eleqtrrd 2733 . . . . . 6 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → 𝑏 ∈ (Base‘(Scalar‘(𝑅𝑦))))
1003, 20, 86, 87, 88, 90, 91prdsbasprj 16179 . . . . . 6 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → (𝑐𝑦) ∈ (Base‘(𝑅𝑦)))
101 eqid 2651 . . . . . . 7 (+g‘(Scalar‘(𝑅𝑦))) = (+g‘(Scalar‘(𝑅𝑦)))
10250, 51, 52, 53, 54, 101lmodvsdir 18935 . . . . . 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))))
10396, 98, 99, 100, 102syl13anc 1368 . . . . 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))))
10430adantlr 751 . . . . . . . 8 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → (Scalar‘(𝑅𝑦)) = 𝑆)
105104fveq2d 6233 . . . . . . 7 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → (+g‘(Scalar‘(𝑅𝑦))) = (+g𝑆))
106105oveqd 6707 . . . . . 6 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → (𝑎(+g‘(Scalar‘(𝑅𝑦)))𝑏) = (𝑎(+g𝑆)𝑏))
107106oveq1d 6705 . . . . 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)))
10895, 103, 1073eqtr2rd 2692 . . . 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)))
109108mpteq2dva 4777 . . 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))))
1104adantr 480 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑆 ∈ Ring)
11125adantr 480 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝐼 ∈ V)
11243adantr 480 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑅 Fn 𝐼)
113 simpr1 1087 . . . . 5 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑎 ∈ (Base‘𝑆))
114 simpr2 1088 . . . . 5 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑏 ∈ (Base‘𝑆))
115 eqid 2651 . . . . . 6 (+g𝑆) = (+g𝑆)
11622, 115ringacl 18624 . . . . 5 ((𝑆 ∈ Ring ∧ 𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆)) → (𝑎(+g𝑆)𝑏) ∈ (Base‘𝑆))
117110, 113, 114, 116syl3anc 1366 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → (𝑎(+g𝑆)𝑏) ∈ (Base‘𝑆))
118 simpr3 1089 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑐 ∈ (Base‘𝑌))
1193, 20, 21, 22, 110, 111, 112, 117, 118prdsvscaval 16186 . . 3 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → ((𝑎(+g𝑆)𝑏)( ·𝑠𝑌)𝑐) = (𝑦𝐼 ↦ ((𝑎(+g𝑆)𝑏)( ·𝑠 ‘(𝑅𝑦))(𝑐𝑦))))
120823adantr2 1241 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → (𝑎( ·𝑠𝑌)𝑐) ∈ (Base‘𝑌))
1215adantr 480 . . . . 5 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑅:𝐼⟶LMod)
1223, 20, 21, 22, 110, 111, 121, 114, 118, 104prdsvscacl 19016 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → (𝑏( ·𝑠𝑌)𝑐) ∈ (Base‘𝑌))
1233, 20, 110, 111, 112, 120, 122, 57prdsplusgval 16180 . . 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))))
124109, 119, 1233eqtr4d 2695 . 2 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → ((𝑎(+g𝑆)𝑏)( ·𝑠𝑌)𝑐) = ((𝑎( ·𝑠𝑌)𝑐)(+g𝑌)(𝑏( ·𝑠𝑌)𝑐)))
12594oveq2d 6706 . . . . 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))))
126 eqid 2651 . . . . . . 7 (.r‘(Scalar‘(𝑅𝑦))) = (.r‘(Scalar‘(𝑅𝑦)))
12750, 52, 53, 54, 126lmodvsass 18936 . . . . . 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))))
12896, 98, 99, 100, 127syl13anc 1368 . . . . 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))))
129104fveq2d 6233 . . . . . . 7 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → (.r‘(Scalar‘(𝑅𝑦))) = (.r𝑆))
130129oveqd 6707 . . . . . 6 (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦𝐼) → (𝑎(.r‘(Scalar‘(𝑅𝑦)))𝑏) = (𝑎(.r𝑆)𝑏))
131130oveq1d 6705 . . . . 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)))
132125, 128, 1313eqtr2rd 2692 . . . 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)))
133132mpteq2dva 4777 . . 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))))
134 eqid 2651 . . . . . 6 (.r𝑆) = (.r𝑆)
13522, 134ringcl 18607 . . . . 5 ((𝑆 ∈ Ring ∧ 𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆)) → (𝑎(.r𝑆)𝑏) ∈ (Base‘𝑆))
136110, 113, 114, 135syl3anc 1366 . . . 4 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → (𝑎(.r𝑆)𝑏) ∈ (Base‘𝑆))
1373, 20, 21, 22, 110, 111, 112, 136, 118prdsvscaval 16186 . . 3 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → ((𝑎(.r𝑆)𝑏)( ·𝑠𝑌)𝑐) = (𝑦𝐼 ↦ ((𝑎(.r𝑆)𝑏)( ·𝑠 ‘(𝑅𝑦))(𝑐𝑦))))
1383, 20, 21, 22, 110, 111, 112, 113, 122prdsvscaval 16186 . . 3 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → (𝑎( ·𝑠𝑌)(𝑏( ·𝑠𝑌)𝑐)) = (𝑦𝐼 ↦ (𝑎( ·𝑠 ‘(𝑅𝑦))((𝑏( ·𝑠𝑌)𝑐)‘𝑦))))
139133, 137, 1383eqtr4d 2695 . 2 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → ((𝑎(.r𝑆)𝑏)( ·𝑠𝑌)𝑐) = (𝑎( ·𝑠𝑌)(𝑏( ·𝑠𝑌)𝑐)))
14030fveq2d 6233 . . . . . . 7 ((𝜑𝑦𝐼) → (1r‘(Scalar‘(𝑅𝑦))) = (1r𝑆))
141140adantlr 751 . . . . . 6 (((𝜑𝑎 ∈ (Base‘𝑌)) ∧ 𝑦𝐼) → (1r‘(Scalar‘(𝑅𝑦))) = (1r𝑆))
142141oveq1d 6705 . . . . 5 (((𝜑𝑎 ∈ (Base‘𝑌)) ∧ 𝑦𝐼) → ((1r‘(Scalar‘(𝑅𝑦)))( ·𝑠 ‘(𝑅𝑦))(𝑎𝑦)) = ((1r𝑆)( ·𝑠 ‘(𝑅𝑦))(𝑎𝑦)))
14334adantlr 751 . . . . . 6 (((𝜑𝑎 ∈ (Base‘𝑌)) ∧ 𝑦𝐼) → (𝑅𝑦) ∈ LMod)
1444ad2antrr 762 . . . . . . 7 (((𝜑𝑎 ∈ (Base‘𝑌)) ∧ 𝑦𝐼) → 𝑆 ∈ Ring)
14525ad2antrr 762 . . . . . . 7 (((𝜑𝑎 ∈ (Base‘𝑌)) ∧ 𝑦𝐼) → 𝐼 ∈ V)
14643ad2antrr 762 . . . . . . 7 (((𝜑𝑎 ∈ (Base‘𝑌)) ∧ 𝑦𝐼) → 𝑅 Fn 𝐼)
147 simplr 807 . . . . . . 7 (((𝜑𝑎 ∈ (Base‘𝑌)) ∧ 𝑦𝐼) → 𝑎 ∈ (Base‘𝑌))
148 simpr 476 . . . . . . 7 (((𝜑𝑎 ∈ (Base‘𝑌)) ∧ 𝑦𝐼) → 𝑦𝐼)
1493, 20, 144, 145, 146, 147, 148prdsbasprj 16179 . . . . . 6 (((𝜑𝑎 ∈ (Base‘𝑌)) ∧ 𝑦𝐼) → (𝑎𝑦) ∈ (Base‘(𝑅𝑦)))
150 eqid 2651 . . . . . . 7 (1r‘(Scalar‘(𝑅𝑦))) = (1r‘(Scalar‘(𝑅𝑦)))
15150, 52, 53, 150lmodvs1 18939 . . . . . 6 (((𝑅𝑦) ∈ LMod ∧ (𝑎𝑦) ∈ (Base‘(𝑅𝑦))) → ((1r‘(Scalar‘(𝑅𝑦)))( ·𝑠 ‘(𝑅𝑦))(𝑎𝑦)) = (𝑎𝑦))
152143, 149, 151syl2anc 694 . . . . 5 (((𝜑𝑎 ∈ (Base‘𝑌)) ∧ 𝑦𝐼) → ((1r‘(Scalar‘(𝑅𝑦)))( ·𝑠 ‘(𝑅𝑦))(𝑎𝑦)) = (𝑎𝑦))
153142, 152eqtr3d 2687 . . . 4 (((𝜑𝑎 ∈ (Base‘𝑌)) ∧ 𝑦𝐼) → ((1r𝑆)( ·𝑠 ‘(𝑅𝑦))(𝑎𝑦)) = (𝑎𝑦))
154153mpteq2dva 4777 . . 3 ((𝜑𝑎 ∈ (Base‘𝑌)) → (𝑦𝐼 ↦ ((1r𝑆)( ·𝑠 ‘(𝑅𝑦))(𝑎𝑦))) = (𝑦𝐼 ↦ (𝑎𝑦)))
1554adantr 480 . . . 4 ((𝜑𝑎 ∈ (Base‘𝑌)) → 𝑆 ∈ Ring)
15625adantr 480 . . . 4 ((𝜑𝑎 ∈ (Base‘𝑌)) → 𝐼 ∈ V)
15743adantr 480 . . . 4 ((𝜑𝑎 ∈ (Base‘𝑌)) → 𝑅 Fn 𝐼)
158 eqid 2651 . . . . . . 7 (1r𝑆) = (1r𝑆)
15922, 158ringidcl 18614 . . . . . 6 (𝑆 ∈ Ring → (1r𝑆) ∈ (Base‘𝑆))
1604, 159syl 17 . . . . 5 (𝜑 → (1r𝑆) ∈ (Base‘𝑆))
161160adantr 480 . . . 4 ((𝜑𝑎 ∈ (Base‘𝑌)) → (1r𝑆) ∈ (Base‘𝑆))
162 simpr 476 . . . 4 ((𝜑𝑎 ∈ (Base‘𝑌)) → 𝑎 ∈ (Base‘𝑌))
1633, 20, 21, 22, 155, 156, 157, 161, 162prdsvscaval 16186 . . 3 ((𝜑𝑎 ∈ (Base‘𝑌)) → ((1r𝑆)( ·𝑠𝑌)𝑎) = (𝑦𝐼 ↦ ((1r𝑆)( ·𝑠 ‘(𝑅𝑦))(𝑎𝑦))))
1643, 20, 155, 156, 157, 162prdsbasfn 16178 . . . 4 ((𝜑𝑎 ∈ (Base‘𝑌)) → 𝑎 Fn 𝐼)
165 dffn5 6280 . . . 4 (𝑎 Fn 𝐼𝑎 = (𝑦𝐼 ↦ (𝑎𝑦)))
166164, 165sylib 208 . . 3 ((𝜑𝑎 ∈ (Base‘𝑌)) → 𝑎 = (𝑦𝐼 ↦ (𝑎𝑦)))
167154, 163, 1663eqtr4d 2695 . 2 ((𝜑𝑎 ∈ (Base‘𝑌)) → ((1r𝑆)( ·𝑠𝑌)𝑎) = 𝑎)
1681, 2, 9, 10, 11, 12, 13, 14, 4, 19, 33, 85, 124, 139, 167islmodd 18917 1 (𝜑𝑌 ∈ LMod)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1054   = wceq 1523  wcel 2030  Vcvv 3231  wss 3607  cmpt 4762   Fn wfn 5921  wf 5922  cfv 5926  (class class class)co 6690  Basecbs 15904  +gcplusg 15988  .rcmulr 15989  Scalarcsca 15991   ·𝑠 cvsca 15992  Xscprds 16153  Grpcgrp 17469  1rcur 18547  Ringcrg 18593  LModclmod 18911
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-map 7901  df-ixp 7951  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-sup 8389  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-3 11118  df-4 11119  df-5 11120  df-6 11121  df-7 11122  df-8 11123  df-9 11124  df-n0 11331  df-z 11416  df-dec 11532  df-uz 11726  df-fz 12365  df-struct 15906  df-ndx 15907  df-slot 15908  df-base 15910  df-sets 15911  df-plusg 16001  df-mulr 16002  df-sca 16004  df-vsca 16005  df-ip 16006  df-tset 16007  df-ple 16008  df-ds 16011  df-hom 16013  df-cco 16014  df-0g 16149  df-prds 16155  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-grp 17472  df-minusg 17473  df-mgp 18536  df-ur 18548  df-ring 18595  df-lmod 18913
This theorem is referenced by:  pwslmod  19018  dsmmlss  20136  dsmmlmod  20137
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