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Theorem prdslmodd 20857

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 = (𝑆X_s𝑅)
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 2726	. 2 ⊢ (𝜑 → (Base‘𝑌) = (Base‘𝑌))
2		eqidd 2726	. 2 ⊢ (𝜑 → (+_g‘𝑌) = (+_g‘𝑌))
3		prdslmodd.y	. . 3 ⊢ 𝑌 = (𝑆X_s𝑅)
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 7235	. . 3 ⊢ (𝜑 → 𝑅 ∈ V)
8	3, 4, 7	prdssca 17437	. 2 ⊢ (𝜑 → 𝑆 = (Scalar‘𝑌))
9		eqidd 2726	. 2 ⊢ (𝜑 → ( ·_𝑠 ‘𝑌) = ( ·_𝑠 ‘𝑌))
10		eqidd 2726	. 2 ⊢ (𝜑 → (Base‘𝑆) = (Base‘𝑆))
11		eqidd 2726	. 2 ⊢ (𝜑 → (+_g‘𝑆) = (+_g‘𝑆))
12		eqidd 2726	. 2 ⊢ (𝜑 → (._r‘𝑆) = (._r‘𝑆))
13		eqidd 2726	. 2 ⊢ (𝜑 → (1_r‘𝑆) = (1_r‘𝑆))
14		lmodgrp 20754	. . . . 5 ⊢ (𝑎 ∈ LMod → 𝑎 ∈ Grp)
15	14	ssriv 3976	. . . 4 ⊢ LMod ⊆ Grp
16		fss 6734	. . . 4 ⊢ ((𝑅:𝐼⟶LMod ∧ LMod ⊆ Grp) → 𝑅:𝐼⟶Grp)
17	5, 15, 16	sylancl 584	. . 3 ⊢ (𝜑 → 𝑅:𝐼⟶Grp)
18	3, 6, 4, 17	prdsgrpd 19010	. 2 ⊢ (𝜑 → 𝑌 ∈ Grp)
19		eqid 2725	. . . 4 ⊢ (Base‘𝑌) = (Base‘𝑌)
20		eqid 2725	. . . 4 ⊢ ( ·_𝑠 ‘𝑌) = ( ·_𝑠 ‘𝑌)
21		eqid 2725	. . . 4 ⊢ (Base‘𝑆) = (Base‘𝑆)
22	4	adantr 479	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌))) → 𝑆 ∈ Ring)
23	6	elexd 3484	. . . . 5 ⊢ (𝜑 → 𝐼 ∈ V)
24	23	adantr 479	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌))) → 𝐼 ∈ V)
25	5	adantr 479	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌))) → 𝑅:𝐼⟶LMod)
26		simprl 769	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌))) → 𝑎 ∈ (Base‘𝑆))
27		simprr 771	. . . 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 713	. . . 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 20856	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌))) → (𝑎( ·_𝑠 ‘𝑌)𝑏) ∈ (Base‘𝑌))
31	30	3impb 1112	. 2 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌)) → (𝑎( ·_𝑠 ‘𝑌)𝑏) ∈ (Base‘𝑌))
32	5	ffvelcdmda 7089	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → (𝑅‘𝑦) ∈ LMod)
33	32	adantlr 713	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → (𝑅‘𝑦) ∈ LMod)
34		simplr1 1212	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → 𝑎 ∈ (Base‘𝑆))
35	28	fveq2d 6896	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → (Base‘(Scalar‘(𝑅‘𝑦))) = (Base‘𝑆))
36	35	adantlr 713	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → (Base‘(Scalar‘(𝑅‘𝑦))) = (Base‘𝑆))
37	34, 36	eleqtrrd 2828	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → 𝑎 ∈ (Base‘(Scalar‘(𝑅‘𝑦))))
38	4	ad2antrr 724	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → 𝑆 ∈ Ring)
39	23	ad2antrr 724	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → 𝐼 ∈ V)
40	5	ffnd 6718	. . . . . . . 8 ⊢ (𝜑 → 𝑅 Fn 𝐼)
41	40	ad2antrr 724	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → 𝑅 Fn 𝐼)
42		simplr2 1213	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → 𝑏 ∈ (Base‘𝑌))
43		simpr 483	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → 𝑦 ∈ 𝐼)
44	3, 19, 38, 39, 41, 42, 43	prdsbasprj 17453	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → (𝑏‘𝑦) ∈ (Base‘(𝑅‘𝑦)))
45		simplr3 1214	. . . . . . 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 17453	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → (𝑐‘𝑦) ∈ (Base‘(𝑅‘𝑦)))
47		eqid 2725	. . . . . . 7 ⊢ (Base‘(𝑅‘𝑦)) = (Base‘(𝑅‘𝑦))
48		eqid 2725	. . . . . . 7 ⊢ (+_g‘(𝑅‘𝑦)) = (+_g‘(𝑅‘𝑦))
49		eqid 2725	. . . . . . 7 ⊢ (Scalar‘(𝑅‘𝑦)) = (Scalar‘(𝑅‘𝑦))
50		eqid 2725	. . . . . . 7 ⊢ ( ·_𝑠 ‘(𝑅‘𝑦)) = ( ·_𝑠 ‘(𝑅‘𝑦))
51		eqid 2725	. . . . . . 7 ⊢ (Base‘(Scalar‘(𝑅‘𝑦))) = (Base‘(Scalar‘(𝑅‘𝑦)))
52	47, 48, 49, 50, 51	lmodvsdi 20772	. . . . . 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 1369	. . . . 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 2725	. . . . . . 7 ⊢ (+_g‘𝑌) = (+_g‘𝑌)
55	3, 19, 38, 39, 41, 42, 45, 54, 43	prdsplusgfval 17455	. . . . . 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 7432	. . . . 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 17461	. . . . . 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 17461	. . . . . 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 7434	. . . . 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 2775	. . . 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 5243	. . 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 479	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑆 ∈ Ring)
63	23	adantr 479	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝐼 ∈ V)
64	40	adantr 479	. . . 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))
66	18	adantr 479	. . . . 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))
69	19, 54	grpcl 18902	. . . . 5 ⊢ ((𝑌 ∈ Grp ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌)) → (𝑏(+_g‘𝑌)𝑐) ∈ (Base‘𝑌))
70	66, 67, 68, 69	syl3anc 1368	. . . 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 17460	. . 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 1168	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑌) ∧ 𝑐 ∈ (Base‘𝑌))) → (𝑎( ·_𝑠 ‘𝑌)𝑏) ∈ (Base‘𝑌))
73	4	adantr 479	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑆 ∈ Ring)
74	23	adantr 479	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝐼 ∈ V)
75	5	adantr 479	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑅:𝐼⟶LMod)
76		simprl 769	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑎 ∈ (Base‘𝑆))
77		simprr 771	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑐 ∈ (Base‘𝑌))
78	28	adantlr 713	. . . . . 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 20856	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → (𝑎( ·_𝑠 ‘𝑌)𝑐) ∈ (Base‘𝑌))
80	79	3adantr2 1167	. . . 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 17454	. . 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 2775	. 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 724	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → 𝑆 ∈ Ring)
84	23	ad2antrr 724	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → 𝐼 ∈ V)
85	40	ad2antrr 724	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → 𝑅 Fn 𝐼)
86		simplr1 1212	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → 𝑎 ∈ (Base‘𝑆))
87		simplr3 1214	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → 𝑐 ∈ (Base‘𝑌))
88		simpr 483	. . . . . . 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 17461	. . . . . 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 1213	. . . . . . 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 17461	. . . . . 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 7434	. . . . 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 713	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → (𝑅‘𝑦) ∈ LMod)
94	35	adantlr 713	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → (Base‘(Scalar‘(𝑅‘𝑦))) = (Base‘𝑆))
95	86, 94	eleqtrrd 2828	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → 𝑎 ∈ (Base‘(Scalar‘(𝑅‘𝑦))))
96	90, 94	eleqtrrd 2828	. . . . . 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 17453	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → (𝑐‘𝑦) ∈ (Base‘(𝑅‘𝑦)))
98		eqid 2725	. . . . . . 7 ⊢ (+_g‘(Scalar‘(𝑅‘𝑦))) = (+_g‘(Scalar‘(𝑅‘𝑦)))
99	47, 48, 49, 50, 51, 98	lmodvsdir 20773	. . . . . 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 1369	. . . . 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 713	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → (Scalar‘(𝑅‘𝑦)) = 𝑆)
102	101	fveq2d 6896	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → (+_g‘(Scalar‘(𝑅‘𝑦))) = (+_g‘𝑆))
103	102	oveqd 7433	. . . . . 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 7431	. . . . 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 2772	. . . 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 5243	. . 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 479	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝑆 ∈ Ring)
108	23	adantr 479	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → 𝐼 ∈ V)
109	40	adantr 479	. . . 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 2725	. . . . . 6 ⊢ (+_g‘𝑆) = (+_g‘𝑆)
113	21, 112	ringacl 20218	. . . . 5 ⊢ ((𝑆 ∈ Ring ∧ 𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆)) → (𝑎(+_g‘𝑆)𝑏) ∈ (Base‘𝑆))
114	107, 110, 111, 113	syl3anc 1368	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → (𝑎(+_g‘𝑆)𝑏) ∈ (Base‘𝑆))
115		simpr3 1193	. . . 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 17460	. . 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 1167	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) → (𝑎( ·_𝑠 ‘𝑌)𝑐) ∈ (Base‘𝑌))
118	5	adantr 479	. . . . 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 20856	. . . 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 17454	. . 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 2775	. 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 7432	. . . . 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 2725	. . . . . . 7 ⊢ (._r‘(Scalar‘(𝑅‘𝑦))) = (._r‘(Scalar‘(𝑅‘𝑦)))
124	47, 49, 50, 51, 123	lmodvsass 20774	. . . . . 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 1369	. . . . 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 6896	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆) ∧ 𝑐 ∈ (Base‘𝑌))) ∧ 𝑦 ∈ 𝐼) → (._r‘(Scalar‘(𝑅‘𝑦))) = (._r‘𝑆))
127	126	oveqd 7433	. . . . . 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 7431	. . . . 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 2772	. . . 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 5243	. . 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 2725	. . . . . 6 ⊢ (._r‘𝑆) = (._r‘𝑆)
132	21, 131	ringcl 20194	. . . . 5 ⊢ ((𝑆 ∈ Ring ∧ 𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑆)) → (𝑎(._r‘𝑆)𝑏) ∈ (Base‘𝑆))
133	107, 110, 111, 132	syl3anc 1368	. . . 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 17460	. . 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 17460	. . 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 2775	. 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 6896	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → (1_r‘(Scalar‘(𝑅‘𝑦))) = (1_r‘𝑆))
138	137	adantlr 713	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) ∧ 𝑦 ∈ 𝐼) → (1_r‘(Scalar‘(𝑅‘𝑦))) = (1_r‘𝑆))
139	138	oveq1d 7431	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) ∧ 𝑦 ∈ 𝐼) → ((1_r‘(Scalar‘(𝑅‘𝑦)))( ·_𝑠 ‘(𝑅‘𝑦))(𝑎‘𝑦)) = ((1_r‘𝑆)( ·_𝑠 ‘(𝑅‘𝑦))(𝑎‘𝑦)))
140	32	adantlr 713	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) ∧ 𝑦 ∈ 𝐼) → (𝑅‘𝑦) ∈ LMod)
141	4	ad2antrr 724	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) ∧ 𝑦 ∈ 𝐼) → 𝑆 ∈ Ring)
142	23	ad2antrr 724	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) ∧ 𝑦 ∈ 𝐼) → 𝐼 ∈ V)
143	40	ad2antrr 724	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) ∧ 𝑦 ∈ 𝐼) → 𝑅 Fn 𝐼)
144		simplr 767	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) ∧ 𝑦 ∈ 𝐼) → 𝑎 ∈ (Base‘𝑌))
145		simpr 483	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) ∧ 𝑦 ∈ 𝐼) → 𝑦 ∈ 𝐼)
146	3, 19, 141, 142, 143, 144, 145	prdsbasprj 17453	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) ∧ 𝑦 ∈ 𝐼) → (𝑎‘𝑦) ∈ (Base‘(𝑅‘𝑦)))
147		eqid 2725	. . . . . . 7 ⊢ (1_r‘(Scalar‘(𝑅‘𝑦))) = (1_r‘(Scalar‘(𝑅‘𝑦)))
148	47, 49, 50, 147	lmodvs1 20777	. . . . . 6 ⊢ (((𝑅‘𝑦) ∈ LMod ∧ (𝑎‘𝑦) ∈ (Base‘(𝑅‘𝑦))) → ((1_r‘(Scalar‘(𝑅‘𝑦)))( ·_𝑠 ‘(𝑅‘𝑦))(𝑎‘𝑦)) = (𝑎‘𝑦))
149	140, 146, 148	syl2anc 582	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) ∧ 𝑦 ∈ 𝐼) → ((1_r‘(Scalar‘(𝑅‘𝑦)))( ·_𝑠 ‘(𝑅‘𝑦))(𝑎‘𝑦)) = (𝑎‘𝑦))
150	139, 149	eqtr3d 2767	. . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) ∧ 𝑦 ∈ 𝐼) → ((1_r‘𝑆)( ·_𝑠 ‘(𝑅‘𝑦))(𝑎‘𝑦)) = (𝑎‘𝑦))
151	150	mpteq2dva 5243	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) → (𝑦 ∈ 𝐼 ↦ ((1_r‘𝑆)( ·_𝑠 ‘(𝑅‘𝑦))(𝑎‘𝑦))) = (𝑦 ∈ 𝐼 ↦ (𝑎‘𝑦)))
152	4	adantr 479	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) → 𝑆 ∈ Ring)
153	23	adantr 479	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) → 𝐼 ∈ V)
154	40	adantr 479	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) → 𝑅 Fn 𝐼)
155		eqid 2725	. . . . . . 7 ⊢ (1_r‘𝑆) = (1_r‘𝑆)
156	21, 155	ringidcl 20206	. . . . . 6 ⊢ (𝑆 ∈ Ring → (1_r‘𝑆) ∈ (Base‘𝑆))
157	4, 156	syl 17	. . . . 5 ⊢ (𝜑 → (1_r‘𝑆) ∈ (Base‘𝑆))
158	157	adantr 479	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) → (1_r‘𝑆) ∈ (Base‘𝑆))
159		simpr 483	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) → 𝑎 ∈ (Base‘𝑌))
160	3, 19, 20, 21, 152, 153, 154, 158, 159	prdsvscaval 17460	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) → ((1_r‘𝑆)( ·_𝑠 ‘𝑌)𝑎) = (𝑦 ∈ 𝐼 ↦ ((1_r‘𝑆)( ·_𝑠 ‘(𝑅‘𝑦))(𝑎‘𝑦))))
161	3, 19, 152, 153, 154, 159	prdsbasfn 17452	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) → 𝑎 Fn 𝐼)
162		dffn5 6952	. . . 4 ⊢ (𝑎 Fn 𝐼 ↔ 𝑎 = (𝑦 ∈ 𝐼 ↦ (𝑎‘𝑦)))
163	161, 162	sylib 217	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) → 𝑎 = (𝑦 ∈ 𝐼 ↦ (𝑎‘𝑦)))
164	151, 160, 163	3eqtr4d 2775	. 2 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) → ((1_r‘𝑆)( ·_𝑠 ‘𝑌)𝑎) = 𝑎)
165	1, 2, 8, 9, 10, 11, 12, 13, 4, 18, 31, 82, 121, 136, 164	islmodd 20753	1 ⊢ (𝜑 → 𝑌 ∈ LMod)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 Vcvv 3463 ⊆ wss 3939 ↦ cmpt 5226 Fn wfn 6538 ⟶wf 6539 ‘cfv 6543 (class class class)co 7416 Basecbs 17179 +_gcplusg 17232 ._rcmulr 17233 Scalarcsca 17235 ·_𝑠 cvsca 17236 X_scprds 17426 Grpcgrp 18894 1_rcur 20125 Ringcrg 20177 LModclmod 20747

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 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7991 df-2nd 7992 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-er 8723 df-map 8845 df-ixp 8915 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 df-sup 9465 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-5 12308 df-6 12309 df-7 12310 df-8 12311 df-9 12312 df-n0 12503 df-z 12589 df-dec 12708 df-uz 12853 df-fz 13517 df-struct 17115 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-plusg 17245 df-mulr 17246 df-sca 17248 df-vsca 17249 df-ip 17250 df-tset 17251 df-ple 17252 df-ds 17254 df-hom 17256 df-cco 17257 df-0g 17422 df-prds 17428 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-grp 18897 df-minusg 18898 df-mgp 20079 df-ur 20126 df-ring 20179 df-lmod 20749

This theorem is referenced by: pwslmod 20858 dsmmlss 21682 dsmmlmod 21683

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