Theorem prdslmodd 21127

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 2767	. 2 ⊢ (𝜑 → (Base‘𝑌) = (Base‘𝑌))
2		eqidd 2767	. 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 7232	. . 3 ⊢ (𝜑 → 𝑅 ∈ V)
8	3, 4, 7	prdssca 17534	. 2 ⊢ (𝜑 → 𝑆 = (Scalar‘𝑌))
9		eqidd 2767	. 2 ⊢ (𝜑 → ( ·𝑠 ‘𝑌) = ( ·𝑠 ‘𝑌))
10		eqidd 2767	. 2 ⊢ (𝜑 → (Base‘𝑆) = (Base‘𝑆))
11		eqidd 2767	. 2 ⊢ (𝜑 → (+g‘𝑆) = (+g‘𝑆))
12		eqidd 2767	. 2 ⊢ (𝜑 → (.r‘𝑆) = (.r‘𝑆))
13		eqidd 2767	. 2 ⊢ (𝜑 → (1r‘𝑆) = (1r‘𝑆))
14		lmodgrp 21025	. . . . 5 ⊢ (𝑎 ∈ LMod → 𝑎 ∈ Grp)
15	14	ssriv 3944	. . . 4 ⊢ LMod ⊆ Grp
16		fss 6729	. . . 4 ⊢ ((𝑅:𝐼⟶LMod ∧ LMod ⊆ Grp) → 𝑅:𝐼⟶Grp)
17	5, 15, 16	sylancl 598	. . 3 ⊢ (𝜑 → 𝑅:𝐼⟶Grp)
18	3, 6, 4, 17	prdsgrpd 19147	. 2 ⊢ (𝜑 → 𝑌 ∈ Grp)
19		eqid 2766	. . . 4 ⊢ (Base‘𝑌) = (Base‘𝑌)
20		eqid 2766	. . . 4 ⊢ ( ·𝑠 ‘𝑌) = ( ·𝑠 ‘𝑌)
21		eqid 2766	. . . 4 ⊢ (Base‘𝑆) = (Base‘𝑆)
22	4	adantr 486	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌))) → 𝑆 ∈ Ring)
23	6	elexd 3481	. . . . 5 ⊢ (𝜑 → 𝐼 ∈ V)
24	23	adantr 486	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌))) → 𝐼 ∈ V)
25	5	adantr 486	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌))) → 𝑅:𝐼⟶LMod)
26		simprl 783	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌))) → 𝑎 ∈ (Base‘𝑆))
27		simprr 785	. . . 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 728	. . . 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 21126	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌))) → (𝑎( ·𝑠 ‘𝑌)𝑏) ∈ (Base‘𝑌))
31	30	3impb 1132	. 2 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌)) → (𝑎( ·𝑠 ‘𝑌)𝑏) ∈ (Base‘𝑌))
32	5	ffvelcdmda 7086	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → (𝑅‘𝑦) ∈ LMod)
33	32	adantlr 728	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → (𝑅‘𝑦) ∈ LMod)
34		simplr1 1234	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → 𝑎 ∈ (Base‘𝑆))
35	28	fveq2d 6892	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → (Base‘(Scalar‘(𝑅‘𝑦))) = (Base‘𝑆))
36	35	adantlr 728	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → (Base‘(Scalar‘(𝑅‘𝑦))) = (Base‘𝑆))
37	34, 36	eleqtrrd 2869	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → 𝑎 ∈ (Base‘(Scalar‘(𝑅‘𝑦))))
38	4	ad2antrr 739	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → 𝑆 ∈ Ring)
39	23	ad2antrr 739	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → 𝐼 ∈ V)
40	5	ffnd 6713	. . . . . . . 8 ⊢ (𝜑 → 𝑅 Fn 𝐼)
41	40	ad2antrr 739	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → 𝑅 Fn 𝐼)
42		simplr2 1235	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → 𝑏 ∈ (Base‘𝑌))
43		simpr 490	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → 𝑦 ∈ 𝐼)
44	3, 19, 38, 39, 41, 42, 43	prdsbasprj 17550	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → (𝑏‘𝑦) ∈ (Base‘(𝑅‘𝑦)))
45		simplr3 1236	. . . . . . 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 17550	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → (𝑐‘𝑦) ∈ (Base‘(𝑅‘𝑦)))
47		eqid 2766	. . . . . . 7 ⊢ (Base‘(𝑅‘𝑦)) = (Base‘(𝑅‘𝑦))
48		eqid 2766	. . . . . . 7 ⊢ (+g‘(𝑅‘𝑦)) = (+g‘(𝑅‘𝑦))
49		eqid 2766	. . . . . . 7 ⊢ (Scalar‘(𝑅‘𝑦)) = (Scalar‘(𝑅‘𝑦))
50		eqid 2766	. . . . . . 7 ⊢ ( ·𝑠 ‘(𝑅‘𝑦)) = ( ·𝑠 ‘(𝑅‘𝑦))
51		eqid 2766	. . . . . . 7 ⊢ (Base‘(Scalar‘(𝑅‘𝑦))) = (Base‘(Scalar‘(𝑅‘𝑦)))
52	47, 48, 49, 50, 51	lmodvsdi 21043	. . . . . 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 1399	. . . . 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 2766	. . . . . . 7 ⊢ (+g‘𝑌) = (+g‘𝑌)
55	3, 19, 38, 39, 41, 42, 45, 54, 43	prdsplusgfval 17552	. . . . . 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 7439	. . . . 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 17558	. . . . . 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 17558	. . . . . 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 7441	. . . . 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 2811	. . . 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 5209	. . 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 486	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑆 ∈ Ring)
63	23	adantr 486	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝐼 ∈ V)
64	40	adantr 486	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑅 Fn 𝐼)
65		simpr1 1213	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑎 ∈ (Base‘𝑆))
66	18	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑌 ∈ Grp)
67		simpr2 1214	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑏 ∈ (Base‘𝑌))
68		simpr3 1215	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑐 ∈ (Base‘𝑌))
69	19, 54	grpcl 19039	. . . . 5 ⊢ ((𝑌 ∈ Grp ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌)) → (𝑏(+g‘𝑌)𝑐) ∈ (Base‘𝑌))
70	66, 67, 68, 69	syl3anc 1398	. . . 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 17557	. . 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 1190	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → (𝑎( ·𝑠 ‘𝑌)𝑏) ∈ (Base‘𝑌))
73	4	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑆 ∈ Ring)
74	23	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝐼 ∈ V)
75	5	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑅:𝐼⟶LMod)
76		simprl 783	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑎 ∈ (Base‘𝑆))
77		simprr 785	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑐 ∈ (Base‘𝑌))
78	28	adantlr 728	. . . . . 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 21126	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → (𝑎( ·𝑠 ‘𝑌)𝑐) ∈ (Base‘𝑌))
80	79	3adantr2 1189	. . . 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 17551	. . 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 2811	. 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 739	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → 𝑆 ∈ Ring)
84	23	ad2antrr 739	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → 𝐼 ∈ V)
85	40	ad2antrr 739	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → 𝑅 Fn 𝐼)
86		simplr1 1234	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → 𝑎 ∈ (Base‘𝑆))
87		simplr3 1236	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → 𝑐 ∈ (Base‘𝑌))
88		simpr 490	. . . . . . 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 17558	. . . . . 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 1235	. . . . . . 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 17558	. . . . . 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 7441	. . . . 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 728	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → (𝑅‘𝑦) ∈ LMod)
94	35	adantlr 728	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → (Base‘(Scalar‘(𝑅‘𝑦))) = (Base‘𝑆))
95	86, 94	eleqtrrd 2869	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → 𝑎 ∈ (Base‘(Scalar‘(𝑅‘𝑦))))
96	90, 94	eleqtrrd 2869	. . . . . 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 17550	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → (𝑐‘𝑦) ∈ (Base‘(𝑅‘𝑦)))
98		eqid 2766	. . . . . . 7 ⊢ (+g‘(Scalar‘(𝑅‘𝑦))) = (+g‘(Scalar‘(𝑅‘𝑦)))
99	47, 48, 49, 50, 51, 98	lmodvsdir 21044	. . . . . 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 1399	. . . . 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 728	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → (Scalar‘(𝑅‘𝑦)) = 𝑆)
102	101	fveq2d 6892	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → (+g‘(Scalar‘(𝑅‘𝑦))) = (+g‘𝑆))
103	102	oveqd 7440	. . . . . 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 7438	. . . . 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 2808	. . . 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 5209	. . 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 486	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑆 ∈ Ring)
108	23	adantr 486	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝐼 ∈ V)
109	40	adantr 486	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑅 Fn 𝐼)
110		simpr1 1213	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑎 ∈ (Base‘𝑆))
111		simpr2 1214	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑏 ∈ (Base‘𝑆))
112		eqid 2766	. . . . . 6 ⊢ (+g‘𝑆) = (+g‘𝑆)
113	21, 112	ringacl 20393	. . . . 5 ⊢ ((𝑆 ∈ Ring ∧ 𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆)) → (𝑎(+g‘𝑆)𝑏) ∈ (Base‘𝑆))
114	107, 110, 111, 113	syl3anc 1398	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → (𝑎(+g‘𝑆)𝑏) ∈ (Base‘𝑆))
115		simpr3 1215	. . . 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 17557	. . 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 1189	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → (𝑎( ·𝑠 ‘𝑌)𝑐) ∈ (Base‘𝑌))
118	5	adantr 486	. . . . 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 21126	. . . 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 17551	. . 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 2811	. 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 7439	. . . . 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 2766	. . . . . . 7 ⊢ (.r‘(Scalar‘(𝑅‘𝑦))) = (.r‘(Scalar‘(𝑅‘𝑦)))
124	47, 49, 50, 51, 123	lmodvsass 21045	. . . . . 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 1399	. . . . 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 6892	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → (.r‘(Scalar‘(𝑅‘𝑦))) = (.r‘𝑆))
127	126	oveqd 7440	. . . . . 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 7438	. . . . 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 2808	. . . 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 5209	. . 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 2766	. . . . . 6 ⊢ (.r‘𝑆) = (.r‘𝑆)
132	21, 131	ringcl 20363	. . . . 5 ⊢ ((𝑆 ∈ Ring ∧ 𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆)) → (𝑎(.r‘𝑆)𝑏) ∈ (Base‘𝑆))
133	107, 110, 111, 132	syl3anc 1398	. . . 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 17557	. . 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 17557	. . 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 2811	. 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 6892	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → (1r‘(Scalar‘(𝑅‘𝑦))) = (1r‘𝑆))
138	137	adantlr 728	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) ∧ 𝑦 ∈ 𝐼) → (1r‘(Scalar‘(𝑅‘𝑦))) = (1r‘𝑆))
139	138	oveq1d 7438	. . . . 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 728	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) ∧ 𝑦 ∈ 𝐼) → (𝑅‘𝑦) ∈ LMod)
141	4	ad2antrr 739	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) ∧ 𝑦 ∈ 𝐼) → 𝑆 ∈ Ring)
142	23	ad2antrr 739	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) ∧ 𝑦 ∈ 𝐼) → 𝐼 ∈ V)
143	40	ad2antrr 739	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) ∧ 𝑦 ∈ 𝐼) → 𝑅 Fn 𝐼)
144		simplr 781	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) ∧ 𝑦 ∈ 𝐼) → 𝑎 ∈ (Base‘𝑌))
145		simpr 490	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) ∧ 𝑦 ∈ 𝐼) → 𝑦 ∈ 𝐼)
146	3, 19, 141, 142, 143, 144, 145	prdsbasprj 17550	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) ∧ 𝑦 ∈ 𝐼) → (𝑎‘𝑦) ∈ (Base‘(𝑅‘𝑦)))
147		eqid 2766	. . . . . . 7 ⊢ (1r‘(Scalar‘(𝑅‘𝑦))) = (1r‘(Scalar‘(𝑅‘𝑦)))
148	47, 49, 50, 147	lmodvs1 21048	. . . . . 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 596	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) ∧ 𝑦 ∈ 𝐼) → ((1r‘(Scalar‘(𝑅‘𝑦)))( ·𝑠 ‘(𝑅‘𝑦))(𝑎‘𝑦)) = (𝑎‘𝑦))
150	139, 149	eqtr3d 2803	. . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) ∧ 𝑦 ∈ 𝐼) → ((1r‘𝑆)( ·𝑠 ‘(𝑅‘𝑦))(𝑎‘𝑦)) = (𝑎‘𝑦))
151	150	mpteq2dva 5209	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) → (𝑦 ∈ 𝐼 ↦ ((1r‘𝑆)( ·𝑠 ‘(𝑅‘𝑦))(𝑎‘𝑦))) = (𝑦 ∈ 𝐼 ↦ (𝑎‘𝑦)))
152	4	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) → 𝑆 ∈ Ring)
153	23	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) → 𝐼 ∈ V)
154	40	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) → 𝑅 Fn 𝐼)
155		eqid 2766	. . . . . . 7 ⊢ (1r‘𝑆) = (1r‘𝑆)
156	21, 155	ringidcl 20380	. . . . . 6 ⊢ (𝑆 ∈ Ring → (1r‘𝑆) ∈ (Base‘𝑆))
157	4, 156	syl 18	. . . . 5 ⊢ (𝜑 → (1r‘𝑆) ∈ (Base‘𝑆))
158	157	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) → (1r‘𝑆) ∈ (Base‘𝑆))
159		simpr 490	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) → 𝑎 ∈ (Base‘𝑌))
160	3, 19, 20, 21, 152, 153, 154, 158, 159	prdsvscaval 17557	. . 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 17549	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) → 𝑎 Fn 𝐼)
162		dffn5 6946	. . . 4 ⊢ (𝑎 Fn 𝐼 ↔ 𝑎 = (𝑦 ∈ 𝐼 ↦ (𝑎‘𝑦)))
163	161, 162	sylib 221	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) → 𝑎 = (𝑦 ∈ 𝐼 ↦ (𝑎‘𝑦)))
164	151, 160, 163	3eqtr4d 2811	. 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 21024	1 ⊢ (𝜑 → 𝑌 ∈ LMod)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146  Vcvv 3458   ⊆ wss 3908   ↦ cmpt 5197   Fn wfn 6538  ⟶wf 6539  ‘cfv 6543  (class class class)co 7423  Basecbs 17294  +_gcplusg 17335  ._rcmulr 17336  Scalarcsca 17338   ·_𝑠 cvsca 17339  X_scprds 17523  Grpcgrp 19031  1_rcur 20294  Ringcrg 20346  LModclmod 21018

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-rep 5243  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409  ax-un 7745  ax-cnex 11174  ax-resscn 11175  ax-1cn 11176  ax-icn 11177  ax-addcl 11178  ax-addrcl 11179  ax-mulcl 11180  ax-mulrcl 11181  ax-mulcom 11182  ax-addass 11183  ax-mulass 11184  ax-distr 11185  ax-i2m1 11186  ax-1ne0 11187  ax-1rid 11188  ax-rnegex 11189  ax-rrecex 11190  ax-cnre 11191  ax-pre-lttri 11192  ax-pre-lttrn 11193  ax-pre-ltadd 11194  ax-pre-mulgt0 11195

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-nel 3068  df-ral 3083  df-rex 3093  df-rmo 3372  df-reu 3373  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3928  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-tp 4599  df-op 4601  df-uni 4878  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-tr 5224  df-id 5561  df-eprel 5566  df-po 5574  df-so 5575  df-fr 5619  df-we 5621  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 6309  df-ord 6370  df-on 6371  df-lim 6372  df-suc 6373  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7380  df-ov 7426  df-oprab 7427  df-mpo 7428  df-om 7872  df-1st 7995  df-2nd 7996  df-frecs 8287  df-wrecs 8318  df-recs 8367  df-rdg 8406  df-1o 8462  df-er 8703  df-map 8835  df-ixp 8905  df-en 8953  df-dom 8954  df-sdom 8955  df-fin 8956  df-sup 9412  df-pnf 11263  df-mnf 11264  df-xr 11265  df-ltxr 11266  df-le 11267  df-sub 11461  df-neg 11462  df-nn 12252  df-2 12321  df-3 12322  df-4 12323  df-5 12324  df-6 12325  df-7 12326  df-8 12327  df-9 12328  df-n0 12523  df-z 12610  df-dec 12730  df-uz 12881  df-fz 13554  df-struct 17232  df-sets 17249  df-slot 17267  df-ndx 17279  df-base 17295  df-plusg 17348  df-mulr 17349  df-sca 17351  df-vsca 17352  df-ip 17353  df-tset 17354  df-ple 17355  df-ds 17357  df-hom 17359  df-cco 17360  df-0g 17519  df-prds 17525  df-mgm 18723  df-sgrp 18806  df-mnd 18822  df-grp 19034  df-minusg 19035  df-mgp 20248  df-ur 20295  df-ring 20348  df-lmod 21020

This theorem is used by:  pwslmod  21128  dsmmlss  21931  dsmmlmod  21932