Theorem prdslmodd 20900

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	⊢ (𝜑 → 𝑆 ∈ Ring)
prdslmodd.i	⊢ (𝜑 → 𝐼 ∈ 𝑉)
prdslmodd.rm	⊢ (𝜑 → 𝑅:𝐼⟶LMod)
prdslmodd.rs	⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → (Scalar‘(𝑅‘𝑦)) = 𝑆)

Assertion

Ref	Expression
prdslmodd	⊢ (𝜑 → 𝑌 ∈ 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.

Step	Hyp	Ref	Expression
1		eqidd 2732	. 2 ⊢ (𝜑 → (Base‘𝑌) = (Base‘𝑌))
2		eqidd 2732	. 2 ⊢ (𝜑 → (+g‘𝑌) = (+g‘𝑌))
3		prdslmodd.y	. . 3 ⊢ 𝑌 = (𝑆Xs𝑅)
4		prdslmodd.s	. . 3 ⊢ (𝜑 → 𝑆 ∈ Ring)
5		prdslmodd.rm	. . . 4 ⊢ (𝜑 → 𝑅:𝐼⟶LMod)
6		prdslmodd.i	. . . 4 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
7	5, 6	fexd 7161	. . 3 ⊢ (𝜑 → 𝑅 ∈ V)
8	3, 4, 7	prdssca 17357	. 2 ⊢ (𝜑 → 𝑆 = (Scalar‘𝑌))
9		eqidd 2732	. 2 ⊢ (𝜑 → ( ·𝑠 ‘𝑌) = ( ·𝑠 ‘𝑌))
10		eqidd 2732	. 2 ⊢ (𝜑 → (Base‘𝑆) = (Base‘𝑆))
11		eqidd 2732	. 2 ⊢ (𝜑 → (+g‘𝑆) = (+g‘𝑆))
12		eqidd 2732	. 2 ⊢ (𝜑 → (.r‘𝑆) = (.r‘𝑆))
13		eqidd 2732	. 2 ⊢ (𝜑 → (1r‘𝑆) = (1r‘𝑆))
14		lmodgrp 20798	. . . . 5 ⊢ (𝑎 ∈ LMod → 𝑎 ∈ Grp)
15	14	ssriv 3938	. . . 4 ⊢ LMod ⊆ Grp
16		fss 6667	. . . 4 ⊢ ((𝑅:𝐼⟶LMod ∧ LMod ⊆ Grp) → 𝑅:𝐼⟶Grp)
17	5, 15, 16	sylancl 586	. . 3 ⊢ (𝜑 → 𝑅:𝐼⟶Grp)
18	3, 6, 4, 17	prdsgrpd 18960	. 2 ⊢ (𝜑 → 𝑌 ∈ Grp)
19		eqid 2731	. . . 4 ⊢ (Base‘𝑌) = (Base‘𝑌)
20		eqid 2731	. . . 4 ⊢ ( ·𝑠 ‘𝑌) = ( ·𝑠 ‘𝑌)
21		eqid 2731	. . . 4 ⊢ (Base‘𝑆) = (Base‘𝑆)
22	4	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌))) → 𝑆 ∈ Ring)
23	6	elexd 3460	. . . . 5 ⊢ (𝜑 → 𝐼 ∈ V)
24	23	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌))) → 𝐼 ∈ V)
25	5	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌))) → 𝑅:𝐼⟶LMod)
26		simprl 770	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌))) → 𝑎 ∈ (Base‘𝑆))
27		simprr 772	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌))) → 𝑏 ∈ (Base‘𝑌))
28		prdslmodd.rs	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → (Scalar‘(𝑅‘𝑦)) = 𝑆)
29	28	adantlr 715	. . . 4 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → (Scalar‘(𝑅‘𝑦)) = 𝑆)
30	3, 19, 20, 21, 22, 24, 25, 26, 27, 29	prdsvscacl 20899	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌))) → (𝑎( ·𝑠 ‘𝑌)𝑏) ∈ (Base‘𝑌))
31	30	3impb 1114	. 2 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌)) → (𝑎( ·𝑠 ‘𝑌)𝑏) ∈ (Base‘𝑌))
32	5	ffvelcdmda 7017	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → (𝑅‘𝑦) ∈ LMod)
33	32	adantlr 715	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → (𝑅‘𝑦) ∈ LMod)
34		simplr1 1216	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → 𝑎 ∈ (Base‘𝑆))
35	28	fveq2d 6826	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → (Base‘(Scalar‘(𝑅‘𝑦))) = (Base‘𝑆))
36	35	adantlr 715	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → (Base‘(Scalar‘(𝑅‘𝑦))) = (Base‘𝑆))
37	34, 36	eleqtrrd 2834	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → 𝑎 ∈ (Base‘(Scalar‘(𝑅‘𝑦))))
38	4	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → 𝑆 ∈ Ring)
39	23	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → 𝐼 ∈ V)
40	5	ffnd 6652	. . . . . . . 8 ⊢ (𝜑 → 𝑅 Fn 𝐼)
41	40	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → 𝑅 Fn 𝐼)
42		simplr2 1217	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → 𝑏 ∈ (Base‘𝑌))
43		simpr 484	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → 𝑦 ∈ 𝐼)
44	3, 19, 38, 39, 41, 42, 43	prdsbasprj 17373	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → (𝑏‘𝑦) ∈ (Base‘(𝑅‘𝑦)))
45		simplr3 1218	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → 𝑐 ∈ (Base‘𝑌))
46	3, 19, 38, 39, 41, 45, 43	prdsbasprj 17373	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → (𝑐‘𝑦) ∈ (Base‘(𝑅‘𝑦)))
47		eqid 2731	. . . . . . 7 ⊢ (Base‘(𝑅‘𝑦)) = (Base‘(𝑅‘𝑦))
48		eqid 2731	. . . . . . 7 ⊢ (+g‘(𝑅‘𝑦)) = (+g‘(𝑅‘𝑦))
49		eqid 2731	. . . . . . 7 ⊢ (Scalar‘(𝑅‘𝑦)) = (Scalar‘(𝑅‘𝑦))
50		eqid 2731	. . . . . . 7 ⊢ ( ·𝑠 ‘(𝑅‘𝑦)) = ( ·𝑠 ‘(𝑅‘𝑦))
51		eqid 2731	. . . . . . 7 ⊢ (Base‘(Scalar‘(𝑅‘𝑦))) = (Base‘(Scalar‘(𝑅‘𝑦)))
52	47, 48, 49, 50, 51	lmodvsdi 20816	. . . . . 6 ⊢ (((𝑅‘𝑦) ∈ LMod ∧ (𝑎 ∈ (Base‘(Scalar‘(𝑅‘𝑦))) ∧ (𝑏‘𝑦) ∈ (Base‘(𝑅‘𝑦)) ∧ (𝑐‘𝑦) ∈ (Base‘(𝑅‘𝑦)))) → (𝑎( ·𝑠 ‘(𝑅‘𝑦))((𝑏‘𝑦)(+g‘(𝑅‘𝑦))(𝑐‘𝑦))) = ((𝑎( ·𝑠 ‘(𝑅‘𝑦))(𝑏‘𝑦))(+g‘(𝑅‘𝑦))(𝑎( ·𝑠 ‘(𝑅‘𝑦))(𝑐‘𝑦))))
53	33, 37, 44, 46, 52	syl13anc 1374	. . . . 5 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → (𝑎( ·𝑠 ‘(𝑅‘𝑦))((𝑏‘𝑦)(+g‘(𝑅‘𝑦))(𝑐‘𝑦))) = ((𝑎( ·𝑠 ‘(𝑅‘𝑦))(𝑏‘𝑦))(+g‘(𝑅‘𝑦))(𝑎( ·𝑠 ‘(𝑅‘𝑦))(𝑐‘𝑦))))
54		eqid 2731	. . . . . . 7 ⊢ (+g‘𝑌) = (+g‘𝑌)
55	3, 19, 38, 39, 41, 42, 45, 54, 43	prdsplusgfval 17375	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → ((𝑏(+g‘𝑌)𝑐)‘𝑦) = ((𝑏‘𝑦)(+g‘(𝑅‘𝑦))(𝑐‘𝑦)))
56	55	oveq2d 7362	. . . . 5 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → (𝑎( ·𝑠 ‘(𝑅‘𝑦))((𝑏(+g‘𝑌)𝑐)‘𝑦)) = (𝑎( ·𝑠 ‘(𝑅‘𝑦))((𝑏‘𝑦)(+g‘(𝑅‘𝑦))(𝑐‘𝑦))))
57	3, 19, 20, 21, 38, 39, 41, 34, 42, 43	prdsvscafval 17381	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → ((𝑎( ·𝑠 ‘𝑌)𝑏)‘𝑦) = (𝑎( ·𝑠 ‘(𝑅‘𝑦))(𝑏‘𝑦)))
58	3, 19, 20, 21, 38, 39, 41, 34, 45, 43	prdsvscafval 17381	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → ((𝑎( ·𝑠 ‘𝑌)𝑐)‘𝑦) = (𝑎( ·𝑠 ‘(𝑅‘𝑦))(𝑐‘𝑦)))
59	57, 58	oveq12d 7364	. . . . 5 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → (((𝑎( ·𝑠 ‘𝑌)𝑏)‘𝑦)(+g‘(𝑅‘𝑦))((𝑎( ·𝑠 ‘𝑌)𝑐)‘𝑦)) = ((𝑎( ·𝑠 ‘(𝑅‘𝑦))(𝑏‘𝑦))(+g‘(𝑅‘𝑦))(𝑎( ·𝑠 ‘(𝑅‘𝑦))(𝑐‘𝑦))))
60	53, 56, 59	3eqtr4d 2776	. . . 4 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → (𝑎( ·𝑠 ‘(𝑅‘𝑦))((𝑏(+g‘𝑌)𝑐)‘𝑦)) = (((𝑎( ·𝑠 ‘𝑌)𝑏)‘𝑦)(+g‘(𝑅‘𝑦))((𝑎( ·𝑠 ‘𝑌)𝑐)‘𝑦)))
61	60	mpteq2dva 5184	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → (𝑦 ∈ 𝐼 ↦ (𝑎( ·𝑠 ‘(𝑅‘𝑦))((𝑏(+g‘𝑌)𝑐)‘𝑦))) = (𝑦 ∈ 𝐼 ↦ (((𝑎( ·𝑠 ‘𝑌)𝑏)‘𝑦)(+g‘(𝑅‘𝑦))((𝑎( ·𝑠 ‘𝑌)𝑐)‘𝑦))))
62	4	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑆 ∈ Ring)
63	23	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝐼 ∈ V)
64	40	adantr 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))
66	18	adantr 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))
69	19, 54	grpcl 18851	. . . . 5 ⊢ ((𝑌 ∈ Grp ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌)) → (𝑏(+g‘𝑌)𝑐) ∈ (Base‘𝑌))
70	66, 67, 68, 69	syl3anc 1373	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → (𝑏(+g‘𝑌)𝑐) ∈ (Base‘𝑌))
71	3, 19, 20, 21, 62, 63, 64, 65, 70	prdsvscaval 17380	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → (𝑎( ·𝑠 ‘𝑌)(𝑏(+g‘𝑌)𝑐)) = (𝑦 ∈ 𝐼 ↦ (𝑎( ·𝑠 ‘(𝑅‘𝑦))((𝑏(+g‘𝑌)𝑐)‘𝑦))))
72	30	3adantr3 1172	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → (𝑎( ·𝑠 ‘𝑌)𝑏) ∈ (Base‘𝑌))
73	4	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑆 ∈ Ring)
74	23	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝐼 ∈ V)
75	5	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑅:𝐼⟶LMod)
76		simprl 770	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑎 ∈ (Base‘𝑆))
77		simprr 772	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑐 ∈ (Base‘𝑌))
78	28	adantlr 715	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → (Scalar‘(𝑅‘𝑦)) = 𝑆)
79	3, 19, 20, 21, 73, 74, 75, 76, 77, 78	prdsvscacl 20899	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → (𝑎( ·𝑠 ‘𝑌)𝑐) ∈ (Base‘𝑌))
80	79	3adantr2 1171	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → (𝑎( ·𝑠 ‘𝑌)𝑐) ∈ (Base‘𝑌))
81	3, 19, 62, 63, 64, 72, 80, 54	prdsplusgval 17374	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → ((𝑎( ·𝑠 ‘𝑌)𝑏)(+g‘𝑌)(𝑎( ·𝑠 ‘𝑌)𝑐)) = (𝑦 ∈ 𝐼 ↦ (((𝑎( ·𝑠 ‘𝑌)𝑏)‘𝑦)(+g‘(𝑅‘𝑦))((𝑎( ·𝑠 ‘𝑌)𝑐)‘𝑦))))
82	61, 71, 81	3eqtr4d 2776	. 2 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → (𝑎( ·𝑠 ‘𝑌)(𝑏(+g‘𝑌)𝑐)) = ((𝑎( ·𝑠 ‘𝑌)𝑏)(+g‘𝑌)(𝑎( ·𝑠 ‘𝑌)𝑐)))
83	4	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → 𝑆 ∈ Ring)
84	23	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → 𝐼 ∈ V)
85	40	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → 𝑅 Fn 𝐼)
86		simplr1 1216	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → 𝑎 ∈ (Base‘𝑆))
87		simplr3 1218	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → 𝑐 ∈ (Base‘𝑌))
88		simpr 484	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → 𝑦 ∈ 𝐼)
89	3, 19, 20, 21, 83, 84, 85, 86, 87, 88	prdsvscafval 17381	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → ((𝑎( ·𝑠 ‘𝑌)𝑐)‘𝑦) = (𝑎( ·𝑠 ‘(𝑅‘𝑦))(𝑐‘𝑦)))
90		simplr2 1217	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → 𝑏 ∈ (Base‘𝑆))
91	3, 19, 20, 21, 83, 84, 85, 90, 87, 88	prdsvscafval 17381	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → ((𝑏( ·𝑠 ‘𝑌)𝑐)‘𝑦) = (𝑏( ·𝑠 ‘(𝑅‘𝑦))(𝑐‘𝑦)))
92	89, 91	oveq12d 7364	. . . . 5 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → (((𝑎( ·𝑠 ‘𝑌)𝑐)‘𝑦)(+g‘(𝑅‘𝑦))((𝑏( ·𝑠 ‘𝑌)𝑐)‘𝑦)) = ((𝑎( ·𝑠 ‘(𝑅‘𝑦))(𝑐‘𝑦))(+g‘(𝑅‘𝑦))(𝑏( ·𝑠 ‘(𝑅‘𝑦))(𝑐‘𝑦))))
93	32	adantlr 715	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → (𝑅‘𝑦) ∈ LMod)
94	35	adantlr 715	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → (Base‘(Scalar‘(𝑅‘𝑦))) = (Base‘𝑆))
95	86, 94	eleqtrrd 2834	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → 𝑎 ∈ (Base‘(Scalar‘(𝑅‘𝑦))))
96	90, 94	eleqtrrd 2834	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → 𝑏 ∈ (Base‘(Scalar‘(𝑅‘𝑦))))
97	3, 19, 83, 84, 85, 87, 88	prdsbasprj 17373	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → (𝑐‘𝑦) ∈ (Base‘(𝑅‘𝑦)))
98		eqid 2731	. . . . . . 7 ⊢ (+g‘(Scalar‘(𝑅‘𝑦))) = (+g‘(Scalar‘(𝑅‘𝑦)))
99	47, 48, 49, 50, 51, 98	lmodvsdir 20817	. . . . . 6 ⊢ (((𝑅‘𝑦) ∈ LMod ∧ (𝑎 ∈ (Base‘(Scalar‘(𝑅‘𝑦))) ∧ 𝑏 ∈ (Base‘(Scalar‘(𝑅‘𝑦))) ∧ (𝑐‘𝑦) ∈ (Base‘(𝑅‘𝑦)))) → ((𝑎(+g‘(Scalar‘(𝑅‘𝑦)))𝑏)( ·𝑠 ‘(𝑅‘𝑦))(𝑐‘𝑦)) = ((𝑎( ·𝑠 ‘(𝑅‘𝑦))(𝑐‘𝑦))(+g‘(𝑅‘𝑦))(𝑏( ·𝑠 ‘(𝑅‘𝑦))(𝑐‘𝑦))))
100	93, 95, 96, 97, 99	syl13anc 1374	. . . . 5 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → ((𝑎(+g‘(Scalar‘(𝑅‘𝑦)))𝑏)( ·𝑠 ‘(𝑅‘𝑦))(𝑐‘𝑦)) = ((𝑎( ·𝑠 ‘(𝑅‘𝑦))(𝑐‘𝑦))(+g‘(𝑅‘𝑦))(𝑏( ·𝑠 ‘(𝑅‘𝑦))(𝑐‘𝑦))))
101	28	adantlr 715	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → (Scalar‘(𝑅‘𝑦)) = 𝑆)
102	101	fveq2d 6826	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → (+g‘(Scalar‘(𝑅‘𝑦))) = (+g‘𝑆))
103	102	oveqd 7363	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → (𝑎(+g‘(Scalar‘(𝑅‘𝑦)))𝑏) = (𝑎(+g‘𝑆)𝑏))
104	103	oveq1d 7361	. . . . 5 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → ((𝑎(+g‘(Scalar‘(𝑅‘𝑦)))𝑏)( ·𝑠 ‘(𝑅‘𝑦))(𝑐‘𝑦)) = ((𝑎(+g‘𝑆)𝑏)( ·𝑠 ‘(𝑅‘𝑦))(𝑐‘𝑦)))
105	92, 100, 104	3eqtr2rd 2773	. . . 4 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → ((𝑎(+g‘𝑆)𝑏)( ·𝑠 ‘(𝑅‘𝑦))(𝑐‘𝑦)) = (((𝑎( ·𝑠 ‘𝑌)𝑐)‘𝑦)(+g‘(𝑅‘𝑦))((𝑏( ·𝑠 ‘𝑌)𝑐)‘𝑦)))
106	105	mpteq2dva 5184	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → (𝑦 ∈ 𝐼 ↦ ((𝑎(+g‘𝑆)𝑏)( ·𝑠 ‘(𝑅‘𝑦))(𝑐‘𝑦))) = (𝑦 ∈ 𝐼 ↦ (((𝑎( ·𝑠 ‘𝑌)𝑐)‘𝑦)(+g‘(𝑅‘𝑦))((𝑏( ·𝑠 ‘𝑌)𝑐)‘𝑦))))
107	4	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑆 ∈ Ring)
108	23	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝐼 ∈ V)
109	40	adantr 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 2731	. . . . . 6 ⊢ (+g‘𝑆) = (+g‘𝑆)
113	21, 112	ringacl 20194	. . . . 5 ⊢ ((𝑆 ∈ Ring ∧ 𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆)) → (𝑎(+g‘𝑆)𝑏) ∈ (Base‘𝑆))
114	107, 110, 111, 113	syl3anc 1373	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → (𝑎(+g‘𝑆)𝑏) ∈ (Base‘𝑆))
115		simpr3 1197	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑐 ∈ (Base‘𝑌))
116	3, 19, 20, 21, 107, 108, 109, 114, 115	prdsvscaval 17380	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → ((𝑎(+g‘𝑆)𝑏)( ·𝑠 ‘𝑌)𝑐) = (𝑦 ∈ 𝐼 ↦ ((𝑎(+g‘𝑆)𝑏)( ·𝑠 ‘(𝑅‘𝑦))(𝑐‘𝑦))))
117	79	3adantr2 1171	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → (𝑎( ·𝑠 ‘𝑌)𝑐) ∈ (Base‘𝑌))
118	5	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑅:𝐼⟶LMod)
119	3, 19, 20, 21, 107, 108, 118, 111, 115, 101	prdsvscacl 20899	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → (𝑏( ·𝑠 ‘𝑌)𝑐) ∈ (Base‘𝑌))
120	3, 19, 107, 108, 109, 117, 119, 54	prdsplusgval 17374	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → ((𝑎( ·𝑠 ‘𝑌)𝑐)(+g‘𝑌)(𝑏( ·𝑠 ‘𝑌)𝑐)) = (𝑦 ∈ 𝐼 ↦ (((𝑎( ·𝑠 ‘𝑌)𝑐)‘𝑦)(+g‘(𝑅‘𝑦))((𝑏( ·𝑠 ‘𝑌)𝑐)‘𝑦))))
121	106, 116, 120	3eqtr4d 2776	. 2 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → ((𝑎(+g‘𝑆)𝑏)( ·𝑠 ‘𝑌)𝑐) = ((𝑎( ·𝑠 ‘𝑌)𝑐)(+g‘𝑌)(𝑏( ·𝑠 ‘𝑌)𝑐)))
122	91	oveq2d 7362	. . . . 5 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → (𝑎( ·𝑠 ‘(𝑅‘𝑦))((𝑏( ·𝑠 ‘𝑌)𝑐)‘𝑦)) = (𝑎( ·𝑠 ‘(𝑅‘𝑦))(𝑏( ·𝑠 ‘(𝑅‘𝑦))(𝑐‘𝑦))))
123		eqid 2731	. . . . . . 7 ⊢ (.r‘(Scalar‘(𝑅‘𝑦))) = (.r‘(Scalar‘(𝑅‘𝑦)))
124	47, 49, 50, 51, 123	lmodvsass 20818	. . . . . 6 ⊢ (((𝑅‘𝑦) ∈ LMod ∧ (𝑎 ∈ (Base‘(Scalar‘(𝑅‘𝑦))) ∧ 𝑏 ∈ (Base‘(Scalar‘(𝑅‘𝑦))) ∧ (𝑐‘𝑦) ∈ (Base‘(𝑅‘𝑦)))) → ((𝑎(.r‘(Scalar‘(𝑅‘𝑦)))𝑏)( ·𝑠 ‘(𝑅‘𝑦))(𝑐‘𝑦)) = (𝑎( ·𝑠 ‘(𝑅‘𝑦))(𝑏( ·𝑠 ‘(𝑅‘𝑦))(𝑐‘𝑦))))
125	93, 95, 96, 97, 124	syl13anc 1374	. . . . 5 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → ((𝑎(.r‘(Scalar‘(𝑅‘𝑦)))𝑏)( ·𝑠 ‘(𝑅‘𝑦))(𝑐‘𝑦)) = (𝑎( ·𝑠 ‘(𝑅‘𝑦))(𝑏( ·𝑠 ‘(𝑅‘𝑦))(𝑐‘𝑦))))
126	101	fveq2d 6826	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → (.r‘(Scalar‘(𝑅‘𝑦))) = (.r‘𝑆))
127	126	oveqd 7363	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → (𝑎(.r‘(Scalar‘(𝑅‘𝑦)))𝑏) = (𝑎(.r‘𝑆)𝑏))
128	127	oveq1d 7361	. . . . 5 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → ((𝑎(.r‘(Scalar‘(𝑅‘𝑦)))𝑏)( ·𝑠 ‘(𝑅‘𝑦))(𝑐‘𝑦)) = ((𝑎(.r‘𝑆)𝑏)( ·𝑠 ‘(𝑅‘𝑦))(𝑐‘𝑦)))
129	122, 125, 128	3eqtr2rd 2773	. . . 4 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → ((𝑎(.r‘𝑆)𝑏)( ·𝑠 ‘(𝑅‘𝑦))(𝑐‘𝑦)) = (𝑎( ·𝑠 ‘(𝑅‘𝑦))((𝑏( ·𝑠 ‘𝑌)𝑐)‘𝑦)))
130	129	mpteq2dva 5184	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → (𝑦 ∈ 𝐼 ↦ ((𝑎(.r‘𝑆)𝑏)( ·𝑠 ‘(𝑅‘𝑦))(𝑐‘𝑦))) = (𝑦 ∈ 𝐼 ↦ (𝑎( ·𝑠 ‘(𝑅‘𝑦))((𝑏( ·𝑠 ‘𝑌)𝑐)‘𝑦))))
131		eqid 2731	. . . . . 6 ⊢ (.r‘𝑆) = (.r‘𝑆)
132	21, 131	ringcl 20166	. . . . 5 ⊢ ((𝑆 ∈ Ring ∧ 𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆)) → (𝑎(.r‘𝑆)𝑏) ∈ (Base‘𝑆))
133	107, 110, 111, 132	syl3anc 1373	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → (𝑎(.r‘𝑆)𝑏) ∈ (Base‘𝑆))
134	3, 19, 20, 21, 107, 108, 109, 133, 115	prdsvscaval 17380	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → ((𝑎(.r‘𝑆)𝑏)( ·𝑠 ‘𝑌)𝑐) = (𝑦 ∈ 𝐼 ↦ ((𝑎(.r‘𝑆)𝑏)( ·𝑠 ‘(𝑅‘𝑦))(𝑐‘𝑦))))
135	3, 19, 20, 21, 107, 108, 109, 110, 119	prdsvscaval 17380	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → (𝑎( ·𝑠 ‘𝑌)(𝑏( ·𝑠 ‘𝑌)𝑐)) = (𝑦 ∈ 𝐼 ↦ (𝑎( ·𝑠 ‘(𝑅‘𝑦))((𝑏( ·𝑠 ‘𝑌)𝑐)‘𝑦))))
136	130, 134, 135	3eqtr4d 2776	. 2 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → ((𝑎(.r‘𝑆)𝑏)( ·𝑠 ‘𝑌)𝑐) = (𝑎( ·𝑠 ‘𝑌)(𝑏( ·𝑠 ‘𝑌)𝑐)))
137	28	fveq2d 6826	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → (1r‘(Scalar‘(𝑅‘𝑦))) = (1r‘𝑆))
138	137	adantlr 715	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) ∧ 𝑦 ∈ 𝐼) → (1r‘(Scalar‘(𝑅‘𝑦))) = (1r‘𝑆))
139	138	oveq1d 7361	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) ∧ 𝑦 ∈ 𝐼) → ((1r‘(Scalar‘(𝑅‘𝑦)))( ·𝑠 ‘(𝑅‘𝑦))(𝑎‘𝑦)) = ((1r‘𝑆)( ·𝑠 ‘(𝑅‘𝑦))(𝑎‘𝑦)))
140	32	adantlr 715	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) ∧ 𝑦 ∈ 𝐼) → (𝑅‘𝑦) ∈ LMod)
141	4	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) ∧ 𝑦 ∈ 𝐼) → 𝑆 ∈ Ring)
142	23	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) ∧ 𝑦 ∈ 𝐼) → 𝐼 ∈ V)
143	40	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) ∧ 𝑦 ∈ 𝐼) → 𝑅 Fn 𝐼)
144		simplr 768	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) ∧ 𝑦 ∈ 𝐼) → 𝑎 ∈ (Base‘𝑌))
145		simpr 484	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) ∧ 𝑦 ∈ 𝐼) → 𝑦 ∈ 𝐼)
146	3, 19, 141, 142, 143, 144, 145	prdsbasprj 17373	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) ∧ 𝑦 ∈ 𝐼) → (𝑎‘𝑦) ∈ (Base‘(𝑅‘𝑦)))
147		eqid 2731	. . . . . . 7 ⊢ (1r‘(Scalar‘(𝑅‘𝑦))) = (1r‘(Scalar‘(𝑅‘𝑦)))
148	47, 49, 50, 147	lmodvs1 20821	. . . . . 6 ⊢ (((𝑅‘𝑦) ∈ LMod ∧ (𝑎‘𝑦) ∈ (Base‘(𝑅‘𝑦))) → ((1r‘(Scalar‘(𝑅‘𝑦)))( ·𝑠 ‘(𝑅‘𝑦))(𝑎‘𝑦)) = (𝑎‘𝑦))
149	140, 146, 148	syl2anc 584	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) ∧ 𝑦 ∈ 𝐼) → ((1r‘(Scalar‘(𝑅‘𝑦)))( ·𝑠 ‘(𝑅‘𝑦))(𝑎‘𝑦)) = (𝑎‘𝑦))
150	139, 149	eqtr3d 2768	. . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) ∧ 𝑦 ∈ 𝐼) → ((1r‘𝑆)( ·𝑠 ‘(𝑅‘𝑦))(𝑎‘𝑦)) = (𝑎‘𝑦))
151	150	mpteq2dva 5184	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) → (𝑦 ∈ 𝐼 ↦ ((1r‘𝑆)( ·𝑠 ‘(𝑅‘𝑦))(𝑎‘𝑦))) = (𝑦 ∈ 𝐼 ↦ (𝑎‘𝑦)))
152	4	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) → 𝑆 ∈ Ring)
153	23	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) → 𝐼 ∈ V)
154	40	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) → 𝑅 Fn 𝐼)
155		eqid 2731	. . . . . . 7 ⊢ (1r‘𝑆) = (1r‘𝑆)
156	21, 155	ringidcl 20181	. . . . . 6 ⊢ (𝑆 ∈ Ring → (1r‘𝑆) ∈ (Base‘𝑆))
157	4, 156	syl 17	. . . . 5 ⊢ (𝜑 → (1r‘𝑆) ∈ (Base‘𝑆))
158	157	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) → (1r‘𝑆) ∈ (Base‘𝑆))
159		simpr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) → 𝑎 ∈ (Base‘𝑌))
160	3, 19, 20, 21, 152, 153, 154, 158, 159	prdsvscaval 17380	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) → ((1r‘𝑆)( ·𝑠 ‘𝑌)𝑎) = (𝑦 ∈ 𝐼 ↦ ((1r‘𝑆)( ·𝑠 ‘(𝑅‘𝑦))(𝑎‘𝑦))))
161	3, 19, 152, 153, 154, 159	prdsbasfn 17372	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) → 𝑎 Fn 𝐼)
162		dffn5 6880	. . . 4 ⊢ (𝑎 Fn 𝐼 ↔ 𝑎 = (𝑦 ∈ 𝐼 ↦ (𝑎‘𝑦)))
163	161, 162	sylib 218	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) → 𝑎 = (𝑦 ∈ 𝐼 ↦ (𝑎‘𝑦)))
164	151, 160, 163	3eqtr4d 2776	. 2 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) → ((1r‘𝑆)( ·𝑠 ‘𝑌)𝑎) = 𝑎)
165	1, 2, 8, 9, 10, 11, 12, 13, 4, 18, 31, 82, 121, 136, 164	islmodd 20797	1 ⊢ (𝜑 → 𝑌 ∈ LMod)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111  Vcvv 3436   ⊆ wss 3902   ↦ cmpt 5172   Fn wfn 6476  ⟶wf 6477  ‘cfv 6481  (class class class)co 7346  Basecbs 17117  +_gcplusg 17158  ._rcmulr 17159  Scalarcsca 17161   ·_𝑠 cvsca 17162  X_scprds 17346  Grpcgrp 18843  1_rcur 20097  Ringcrg 20149  LModclmod 20791

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5217  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668  ax-cnex 11059  ax-resscn 11060  ax-1cn 11061  ax-icn 11062  ax-addcl 11063  ax-addrcl 11064  ax-mulcl 11065  ax-mulrcl 11066  ax-mulcom 11067  ax-addass 11068  ax-mulass 11069  ax-distr 11070  ax-i2m1 11071  ax-1ne0 11072  ax-1rid 11073  ax-rnegex 11074  ax-rrecex 11075  ax-cnre 11076  ax-pre-lttri 11077  ax-pre-lttrn 11078  ax-pre-ltadd 11079  ax-pre-mulgt0 11080

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-tp 4581  df-op 4583  df-uni 4860  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-tr 5199  df-id 5511  df-eprel 5516  df-po 5524  df-so 5525  df-fr 5569  df-we 5571  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-1st 7921  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-1o 8385  df-er 8622  df-map 8752  df-ixp 8822  df-en 8870  df-dom 8871  df-sdom 8872  df-fin 8873  df-sup 9326  df-pnf 11145  df-mnf 11146  df-xr 11147  df-ltxr 11148  df-le 11149  df-sub 11343  df-neg 11344  df-nn 12123  df-2 12185  df-3 12186  df-4 12187  df-5 12188  df-6 12189  df-7 12190  df-8 12191  df-9 12192  df-n0 12379  df-z 12466  df-dec 12586  df-uz 12730  df-fz 13405  df-struct 17055  df-sets 17072  df-slot 17090  df-ndx 17102  df-base 17118  df-plusg 17171  df-mulr 17172  df-sca 17174  df-vsca 17175  df-ip 17176  df-tset 17177  df-ple 17178  df-ds 17180  df-hom 17182  df-cco 17183  df-0g 17342  df-prds 17348  df-mgm 18545  df-sgrp 18624  df-mnd 18640  df-grp 18846  df-minusg 18847  df-mgp 20057  df-ur 20098  df-ring 20151  df-lmod 20793

This theorem is referenced by:  pwslmod  20901  dsmmlss  21679  dsmmlmod  21680