Theorem frlmsslsp 21776

Description: A subset of a free module obtained by restricting the support set is spanned by the relevant unit vectors. (Contributed by Stefan O'Rear, 6-Feb-2015.) (Revised by AV, 24-Jun-2019.)

Hypotheses

Ref	Expression
frlmsslsp.y	⊢ 𝑌 = (𝑅 freeLMod 𝐼)
frlmsslsp.u	⊢ 𝑈 = (𝑅 unitVec 𝐼)
frlmsslsp.k	⊢ 𝐾 = (LSpan‘𝑌)
frlmsslsp.b	⊢ 𝐵 = (Base‘𝑌)
frlmsslsp.z	⊢ 0 = (0g‘𝑅)
frlmsslsp.c	⊢ 𝐶 = {𝑥 ∈ 𝐵 ∣ (𝑥 supp 0 ) ⊆ 𝐽}

Assertion

Ref	Expression
frlmsslsp	⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → (𝐾‘(𝑈 “ 𝐽)) = 𝐶)

Distinct variable groups:   𝑥,𝑌   𝑥,𝑈   𝑥,𝐵   𝑥, 0   𝑥,𝑅   𝑥,𝐼   𝑥,𝑉   𝑥,𝐽   𝑥,𝐾

Allowed substitution hint:   𝐶(𝑥)

Proof of Theorem frlmsslsp

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		frlmsslsp.y	. . . . 5 ⊢ 𝑌 = (𝑅 freeLMod 𝐼)
2	1	frlmlmod 21729	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → 𝑌 ∈ LMod)
3	2	3adant3 1133	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → 𝑌 ∈ LMod)
4		eqid 2736	. . . 4 ⊢ (LSubSp‘𝑌) = (LSubSp‘𝑌)
5		frlmsslsp.b	. . . 4 ⊢ 𝐵 = (Base‘𝑌)
6		frlmsslsp.z	. . . 4 ⊢ 0 = (0g‘𝑅)
7		frlmsslsp.c	. . . 4 ⊢ 𝐶 = {𝑥 ∈ 𝐵 ∣ (𝑥 supp 0 ) ⊆ 𝐽}
8	1, 4, 5, 6, 7	frlmsslss2 21755	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → 𝐶 ∈ (LSubSp‘𝑌))
9		frlmsslsp.u	. . . . . . . . . 10 ⊢ 𝑈 = (𝑅 unitVec 𝐼)
10	9, 1, 5	uvcff 21771	. . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → 𝑈:𝐼⟶𝐵)
11	10	3adant3 1133	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → 𝑈:𝐼⟶𝐵)
12	11	adantr 480	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐽) → 𝑈:𝐼⟶𝐵)
13		simp3 1139	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → 𝐽 ⊆ 𝐼)
14	13	sselda 3921	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐽) → 𝑦 ∈ 𝐼)
15	12, 14	ffvelcdmd 7037	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐽) → (𝑈‘𝑦) ∈ 𝐵)
16		simpl2 1194	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐽) → 𝐼 ∈ 𝑉)
17		eqid 2736	. . . . . . . . 9 ⊢ (Base‘𝑅) = (Base‘𝑅)
18	1, 17, 5	frlmbasf 21740	. . . . . . . 8 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑈‘𝑦) ∈ 𝐵) → (𝑈‘𝑦):𝐼⟶(Base‘𝑅))
19	16, 15, 18	syl2anc 585	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐽) → (𝑈‘𝑦):𝐼⟶(Base‘𝑅))
20		simpll1 1214	. . . . . . . 8 ⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐽) ∧ 𝑥 ∈ (𝐼 ∖ 𝐽)) → 𝑅 ∈ Ring)
21		simpll2 1215	. . . . . . . 8 ⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐽) ∧ 𝑥 ∈ (𝐼 ∖ 𝐽)) → 𝐼 ∈ 𝑉)
22	14	adantr 480	. . . . . . . 8 ⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐽) ∧ 𝑥 ∈ (𝐼 ∖ 𝐽)) → 𝑦 ∈ 𝐼)
23		eldifi 4071	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐼 ∖ 𝐽) → 𝑥 ∈ 𝐼)
24	23	adantl 481	. . . . . . . 8 ⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐽) ∧ 𝑥 ∈ (𝐼 ∖ 𝐽)) → 𝑥 ∈ 𝐼)
25		elneeldif 3903	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝐽 ∧ 𝑥 ∈ (𝐼 ∖ 𝐽)) → 𝑦 ≠ 𝑥)
26	25	adantll 715	. . . . . . . 8 ⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐽) ∧ 𝑥 ∈ (𝐼 ∖ 𝐽)) → 𝑦 ≠ 𝑥)
27	9, 20, 21, 22, 24, 26, 6	uvcvv0 21770	. . . . . . 7 ⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐽) ∧ 𝑥 ∈ (𝐼 ∖ 𝐽)) → ((𝑈‘𝑦)‘𝑥) = 0 )
28	19, 27	suppss 8144	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐽) → ((𝑈‘𝑦) supp 0 ) ⊆ 𝐽)
29		oveq1 7374	. . . . . . . 8 ⊢ (𝑥 = (𝑈‘𝑦) → (𝑥 supp 0 ) = ((𝑈‘𝑦) supp 0 ))
30	29	sseq1d 3953	. . . . . . 7 ⊢ (𝑥 = (𝑈‘𝑦) → ((𝑥 supp 0 ) ⊆ 𝐽 ↔ ((𝑈‘𝑦) supp 0 ) ⊆ 𝐽))
31	30, 7	elrab2 3637	. . . . . 6 ⊢ ((𝑈‘𝑦) ∈ 𝐶 ↔ ((𝑈‘𝑦) ∈ 𝐵 ∧ ((𝑈‘𝑦) supp 0 ) ⊆ 𝐽))
32	15, 28, 31	sylanbrc 584	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐽) → (𝑈‘𝑦) ∈ 𝐶)
33	32	ralrimiva 3129	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → ∀𝑦 ∈ 𝐽 (𝑈‘𝑦) ∈ 𝐶)
34	11	ffund 6672	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → Fun 𝑈)
35	11	fdmd 6678	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → dom 𝑈 = 𝐼)
36	13, 35	sseqtrrd 3959	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → 𝐽 ⊆ dom 𝑈)
37		funimass4 6904	. . . . 5 ⊢ ((Fun 𝑈 ∧ 𝐽 ⊆ dom 𝑈) → ((𝑈 “ 𝐽) ⊆ 𝐶 ↔ ∀𝑦 ∈ 𝐽 (𝑈‘𝑦) ∈ 𝐶))
38	34, 36, 37	syl2anc 585	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → ((𝑈 “ 𝐽) ⊆ 𝐶 ↔ ∀𝑦 ∈ 𝐽 (𝑈‘𝑦) ∈ 𝐶))
39	33, 38	mpbird 257	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → (𝑈 “ 𝐽) ⊆ 𝐶)
40		frlmsslsp.k	. . . 4 ⊢ 𝐾 = (LSpan‘𝑌)
41	4, 40	lspssp 20983	. . 3 ⊢ ((𝑌 ∈ LMod ∧ 𝐶 ∈ (LSubSp‘𝑌) ∧ (𝑈 “ 𝐽) ⊆ 𝐶) → (𝐾‘(𝑈 “ 𝐽)) ⊆ 𝐶)
42	3, 8, 39, 41	syl3anc 1374	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → (𝐾‘(𝑈 “ 𝐽)) ⊆ 𝐶)
43		simpl1 1193	. . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) → 𝑅 ∈ Ring)
44		simpl2 1194	. . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) → 𝐼 ∈ 𝑉)
45	7	ssrab3 4022	. . . . . 6 ⊢ 𝐶 ⊆ 𝐵
46	45	a1i 11	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → 𝐶 ⊆ 𝐵)
47	46	sselda 3921	. . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) → 𝑦 ∈ 𝐵)
48		eqid 2736	. . . . 5 ⊢ ( ·𝑠 ‘𝑌) = ( ·𝑠 ‘𝑌)
49	9, 1, 5, 48	uvcresum 21773	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝑦 ∈ 𝐵) → 𝑦 = (𝑌 Σg (𝑦 ∘f ( ·𝑠 ‘𝑌)𝑈)))
50	43, 44, 47, 49	syl3anc 1374	. . 3 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) → 𝑦 = (𝑌 Σg (𝑦 ∘f ( ·𝑠 ‘𝑌)𝑈)))
51		eqid 2736	. . . 4 ⊢ (0g‘𝑌) = (0g‘𝑌)
52		lmodabl 20904	. . . . . 6 ⊢ (𝑌 ∈ LMod → 𝑌 ∈ Abel)
53	3, 52	syl 17	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → 𝑌 ∈ Abel)
54	53	adantr 480	. . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) → 𝑌 ∈ Abel)
55		imassrn 6036	. . . . . . . 8 ⊢ (𝑈 “ 𝐽) ⊆ ran 𝑈
56	11	frnd 6676	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → ran 𝑈 ⊆ 𝐵)
57	55, 56	sstrid 3933	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → (𝑈 “ 𝐽) ⊆ 𝐵)
58	5, 4, 40	lspcl 20971	. . . . . . 7 ⊢ ((𝑌 ∈ LMod ∧ (𝑈 “ 𝐽) ⊆ 𝐵) → (𝐾‘(𝑈 “ 𝐽)) ∈ (LSubSp‘𝑌))
59	3, 57, 58	syl2anc 585	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → (𝐾‘(𝑈 “ 𝐽)) ∈ (LSubSp‘𝑌))
60	4	lsssubg 20952	. . . . . 6 ⊢ ((𝑌 ∈ LMod ∧ (𝐾‘(𝑈 “ 𝐽)) ∈ (LSubSp‘𝑌)) → (𝐾‘(𝑈 “ 𝐽)) ∈ (SubGrp‘𝑌))
61	3, 59, 60	syl2anc 585	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → (𝐾‘(𝑈 “ 𝐽)) ∈ (SubGrp‘𝑌))
62	61	adantr 480	. . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) → (𝐾‘(𝑈 “ 𝐽)) ∈ (SubGrp‘𝑌))
63	1, 17, 5	frlmbasf 21740	. . . . . . . . 9 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑦 ∈ 𝐵) → 𝑦:𝐼⟶(Base‘𝑅))
64	63	3ad2antl2 1188	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐵) → 𝑦:𝐼⟶(Base‘𝑅))
65	64	ffnd 6669	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐵) → 𝑦 Fn 𝐼)
66	11	ffnd 6669	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → 𝑈 Fn 𝐼)
67	66	adantr 480	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐵) → 𝑈 Fn 𝐼)
68		simpl2 1194	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐵) → 𝐼 ∈ 𝑉)
69		inidm 4167	. . . . . . 7 ⊢ (𝐼 ∩ 𝐼) = 𝐼
70	65, 67, 68, 68, 69	offn 7644	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐵) → (𝑦 ∘f ( ·𝑠 ‘𝑌)𝑈) Fn 𝐼)
71	47, 70	syldan 592	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) → (𝑦 ∘f ( ·𝑠 ‘𝑌)𝑈) Fn 𝐼)
72	47, 65	syldan 592	. . . . . . . . . 10 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) → 𝑦 Fn 𝐼)
73	72	adantrr 718	. . . . . . . . 9 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐼)) → 𝑦 Fn 𝐼)
74	66	adantr 480	. . . . . . . . 9 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐼)) → 𝑈 Fn 𝐼)
75		simpl2 1194	. . . . . . . . 9 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐼)) → 𝐼 ∈ 𝑉)
76		simprr 773	. . . . . . . . 9 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐼)) → 𝑧 ∈ 𝐼)
77		fnfvof 7648	. . . . . . . . 9 ⊢ (((𝑦 Fn 𝐼 ∧ 𝑈 Fn 𝐼) ∧ (𝐼 ∈ 𝑉 ∧ 𝑧 ∈ 𝐼)) → ((𝑦 ∘f ( ·𝑠 ‘𝑌)𝑈)‘𝑧) = ((𝑦‘𝑧)( ·𝑠 ‘𝑌)(𝑈‘𝑧)))
78	73, 74, 75, 76, 77	syl22anc 839	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐼)) → ((𝑦 ∘f ( ·𝑠 ‘𝑌)𝑈)‘𝑧) = ((𝑦‘𝑧)( ·𝑠 ‘𝑌)(𝑈‘𝑧)))
79	3	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐽)) → 𝑌 ∈ LMod)
80	59	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐽)) → (𝐾‘(𝑈 “ 𝐽)) ∈ (LSubSp‘𝑌))
81	45	sseli 3917	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ 𝐶 → 𝑦 ∈ 𝐵)
82	81, 64	sylan2 594	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) → 𝑦:𝐼⟶(Base‘𝑅))
83	82	adantrr 718	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐽)) → 𝑦:𝐼⟶(Base‘𝑅))
84	13	sselda 3921	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑧 ∈ 𝐽) → 𝑧 ∈ 𝐼)
85	84	adantrl 717	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐽)) → 𝑧 ∈ 𝐼)
86	83, 85	ffvelcdmd 7037	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐽)) → (𝑦‘𝑧) ∈ (Base‘𝑅))
87	1	frlmsca 21733	. . . . . . . . . . . . . . . 16 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → 𝑅 = (Scalar‘𝑌))
88	87	3adant3 1133	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → 𝑅 = (Scalar‘𝑌))
89	88	fveq2d 6844	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → (Base‘𝑅) = (Base‘(Scalar‘𝑌)))
90	89	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐽)) → (Base‘𝑅) = (Base‘(Scalar‘𝑌)))
91	86, 90	eleqtrd 2838	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐽)) → (𝑦‘𝑧) ∈ (Base‘(Scalar‘𝑌)))
92	5, 40	lspssid 20980	. . . . . . . . . . . . . . 15 ⊢ ((𝑌 ∈ LMod ∧ (𝑈 “ 𝐽) ⊆ 𝐵) → (𝑈 “ 𝐽) ⊆ (𝐾‘(𝑈 “ 𝐽)))
93	3, 57, 92	syl2anc 585	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → (𝑈 “ 𝐽) ⊆ (𝐾‘(𝑈 “ 𝐽)))
94	93	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐽)) → (𝑈 “ 𝐽) ⊆ (𝐾‘(𝑈 “ 𝐽)))
95		funfvima2 7186	. . . . . . . . . . . . . . . 16 ⊢ ((Fun 𝑈 ∧ 𝐽 ⊆ dom 𝑈) → (𝑧 ∈ 𝐽 → (𝑈‘𝑧) ∈ (𝑈 “ 𝐽)))
96	34, 36, 95	syl2anc 585	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → (𝑧 ∈ 𝐽 → (𝑈‘𝑧) ∈ (𝑈 “ 𝐽)))
97	96	imp 406	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑧 ∈ 𝐽) → (𝑈‘𝑧) ∈ (𝑈 “ 𝐽))
98	97	adantrl 717	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐽)) → (𝑈‘𝑧) ∈ (𝑈 “ 𝐽))
99	94, 98	sseldd 3922	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐽)) → (𝑈‘𝑧) ∈ (𝐾‘(𝑈 “ 𝐽)))
100		eqid 2736	. . . . . . . . . . . . 13 ⊢ (Scalar‘𝑌) = (Scalar‘𝑌)
101		eqid 2736	. . . . . . . . . . . . 13 ⊢ (Base‘(Scalar‘𝑌)) = (Base‘(Scalar‘𝑌))
102	100, 48, 101, 4	lssvscl 20950	. . . . . . . . . . . 12 ⊢ (((𝑌 ∈ LMod ∧ (𝐾‘(𝑈 “ 𝐽)) ∈ (LSubSp‘𝑌)) ∧ ((𝑦‘𝑧) ∈ (Base‘(Scalar‘𝑌)) ∧ (𝑈‘𝑧) ∈ (𝐾‘(𝑈 “ 𝐽)))) → ((𝑦‘𝑧)( ·𝑠 ‘𝑌)(𝑈‘𝑧)) ∈ (𝐾‘(𝑈 “ 𝐽)))
103	79, 80, 91, 99, 102	syl22anc 839	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐽)) → ((𝑦‘𝑧)( ·𝑠 ‘𝑌)(𝑈‘𝑧)) ∈ (𝐾‘(𝑈 “ 𝐽)))
104	103	anassrs 467	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) ∧ 𝑧 ∈ 𝐽) → ((𝑦‘𝑧)( ·𝑠 ‘𝑌)(𝑈‘𝑧)) ∈ (𝐾‘(𝑈 “ 𝐽)))
105	104	adantlrr 722	. . . . . . . . 9 ⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐼)) ∧ 𝑧 ∈ 𝐽) → ((𝑦‘𝑧)( ·𝑠 ‘𝑌)(𝑈‘𝑧)) ∈ (𝐾‘(𝑈 “ 𝐽)))
106		id 22	. . . . . . . . . . . . . . . 16 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) → ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶))
107	106	adantrr 718	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐼)) → ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶))
108	107	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐼)) ∧ ¬ 𝑧 ∈ 𝐽) → ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶))
109		simplrr 778	. . . . . . . . . . . . . . 15 ⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐼)) ∧ ¬ 𝑧 ∈ 𝐽) → 𝑧 ∈ 𝐼)
110		simpr 484	. . . . . . . . . . . . . . 15 ⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐼)) ∧ ¬ 𝑧 ∈ 𝐽) → ¬ 𝑧 ∈ 𝐽)
111	109, 110	eldifd 3900	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐼)) ∧ ¬ 𝑧 ∈ 𝐽) → 𝑧 ∈ (𝐼 ∖ 𝐽))
112		oveq1 7374	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑦 → (𝑥 supp 0 ) = (𝑦 supp 0 ))
113	112	sseq1d 3953	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑦 → ((𝑥 supp 0 ) ⊆ 𝐽 ↔ (𝑦 supp 0 ) ⊆ 𝐽))
114	113, 7	elrab2 3637	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ 𝐶 ↔ (𝑦 ∈ 𝐵 ∧ (𝑦 supp 0 ) ⊆ 𝐽))
115	114	simprbi 497	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ 𝐶 → (𝑦 supp 0 ) ⊆ 𝐽)
116	115	adantl 481	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) → (𝑦 supp 0 ) ⊆ 𝐽)
117	6	fvexi 6854	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ V
118	117	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) → 0 ∈ V)
119	82, 116, 44, 118	suppssr 8145	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) ∧ 𝑧 ∈ (𝐼 ∖ 𝐽)) → (𝑦‘𝑧) = 0 )
120	108, 111, 119	syl2anc 585	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐼)) ∧ ¬ 𝑧 ∈ 𝐽) → (𝑦‘𝑧) = 0 )
121	88	fveq2d 6844	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → (0g‘𝑅) = (0g‘(Scalar‘𝑌)))
122	6, 121	eqtrid 2783	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → 0 = (0g‘(Scalar‘𝑌)))
123	122	ad2antrr 727	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐼)) ∧ ¬ 𝑧 ∈ 𝐽) → 0 = (0g‘(Scalar‘𝑌)))
124	120, 123	eqtrd 2771	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐼)) ∧ ¬ 𝑧 ∈ 𝐽) → (𝑦‘𝑧) = (0g‘(Scalar‘𝑌)))
125	124	oveq1d 7382	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐼)) ∧ ¬ 𝑧 ∈ 𝐽) → ((𝑦‘𝑧)( ·𝑠 ‘𝑌)(𝑈‘𝑧)) = ((0g‘(Scalar‘𝑌))( ·𝑠 ‘𝑌)(𝑈‘𝑧)))
126	3	ad2antrr 727	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐼)) ∧ ¬ 𝑧 ∈ 𝐽) → 𝑌 ∈ LMod)
127	11	ffvelcdmda 7036	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑧 ∈ 𝐼) → (𝑈‘𝑧) ∈ 𝐵)
128	127	adantrl 717	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐼)) → (𝑈‘𝑧) ∈ 𝐵)
129	128	adantr 480	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐼)) ∧ ¬ 𝑧 ∈ 𝐽) → (𝑈‘𝑧) ∈ 𝐵)
130		eqid 2736	. . . . . . . . . . . . 13 ⊢ (0g‘(Scalar‘𝑌)) = (0g‘(Scalar‘𝑌))
131	5, 100, 48, 130, 51	lmod0vs 20890	. . . . . . . . . . . 12 ⊢ ((𝑌 ∈ LMod ∧ (𝑈‘𝑧) ∈ 𝐵) → ((0g‘(Scalar‘𝑌))( ·𝑠 ‘𝑌)(𝑈‘𝑧)) = (0g‘𝑌))
132	126, 129, 131	syl2anc 585	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐼)) ∧ ¬ 𝑧 ∈ 𝐽) → ((0g‘(Scalar‘𝑌))( ·𝑠 ‘𝑌)(𝑈‘𝑧)) = (0g‘𝑌))
133	125, 132	eqtrd 2771	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐼)) ∧ ¬ 𝑧 ∈ 𝐽) → ((𝑦‘𝑧)( ·𝑠 ‘𝑌)(𝑈‘𝑧)) = (0g‘𝑌))
134	59	ad2antrr 727	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐼)) ∧ ¬ 𝑧 ∈ 𝐽) → (𝐾‘(𝑈 “ 𝐽)) ∈ (LSubSp‘𝑌))
135	51, 4	lss0cl 20942	. . . . . . . . . . 11 ⊢ ((𝑌 ∈ LMod ∧ (𝐾‘(𝑈 “ 𝐽)) ∈ (LSubSp‘𝑌)) → (0g‘𝑌) ∈ (𝐾‘(𝑈 “ 𝐽)))
136	126, 134, 135	syl2anc 585	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐼)) ∧ ¬ 𝑧 ∈ 𝐽) → (0g‘𝑌) ∈ (𝐾‘(𝑈 “ 𝐽)))
137	133, 136	eqeltrd 2836	. . . . . . . . 9 ⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐼)) ∧ ¬ 𝑧 ∈ 𝐽) → ((𝑦‘𝑧)( ·𝑠 ‘𝑌)(𝑈‘𝑧)) ∈ (𝐾‘(𝑈 “ 𝐽)))
138	105, 137	pm2.61dan 813	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐼)) → ((𝑦‘𝑧)( ·𝑠 ‘𝑌)(𝑈‘𝑧)) ∈ (𝐾‘(𝑈 “ 𝐽)))
139	78, 138	eqeltrd 2836	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐼)) → ((𝑦 ∘f ( ·𝑠 ‘𝑌)𝑈)‘𝑧) ∈ (𝐾‘(𝑈 “ 𝐽)))
140	139	expr 456	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) → (𝑧 ∈ 𝐼 → ((𝑦 ∘f ( ·𝑠 ‘𝑌)𝑈)‘𝑧) ∈ (𝐾‘(𝑈 “ 𝐽))))
141	140	ralrimiv 3128	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) → ∀𝑧 ∈ 𝐼 ((𝑦 ∘f ( ·𝑠 ‘𝑌)𝑈)‘𝑧) ∈ (𝐾‘(𝑈 “ 𝐽)))
142		ffnfv 7071	. . . . 5 ⊢ ((𝑦 ∘f ( ·𝑠 ‘𝑌)𝑈):𝐼⟶(𝐾‘(𝑈 “ 𝐽)) ↔ ((𝑦 ∘f ( ·𝑠 ‘𝑌)𝑈) Fn 𝐼 ∧ ∀𝑧 ∈ 𝐼 ((𝑦 ∘f ( ·𝑠 ‘𝑌)𝑈)‘𝑧) ∈ (𝐾‘(𝑈 “ 𝐽))))
143	71, 141, 142	sylanbrc 584	. . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) → (𝑦 ∘f ( ·𝑠 ‘𝑌)𝑈):𝐼⟶(𝐾‘(𝑈 “ 𝐽)))
144	1, 6, 5	frlmbasfsupp 21738	. . . . . . . 8 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑦 ∈ 𝐵) → 𝑦 finSupp 0 )
145	144	fsuppimpd 9282	. . . . . . 7 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑦 ∈ 𝐵) → (𝑦 supp 0 ) ∈ Fin)
146	44, 47, 145	syl2anc 585	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) → (𝑦 supp 0 ) ∈ Fin)
147		dffn2 6670	. . . . . . . . 9 ⊢ ((𝑦 ∘f ( ·𝑠 ‘𝑌)𝑈) Fn 𝐼 ↔ (𝑦 ∘f ( ·𝑠 ‘𝑌)𝑈):𝐼⟶V)
148	70, 147	sylib 218	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐵) → (𝑦 ∘f ( ·𝑠 ‘𝑌)𝑈):𝐼⟶V)
149	65	adantr 480	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 ∈ (𝐼 ∖ (𝑦 supp 0 ))) → 𝑦 Fn 𝐼)
150	66	ad2antrr 727	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 ∈ (𝐼 ∖ (𝑦 supp 0 ))) → 𝑈 Fn 𝐼)
151		simpll2 1215	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 ∈ (𝐼 ∖ (𝑦 supp 0 ))) → 𝐼 ∈ 𝑉)
152		eldifi 4071	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝐼 ∖ (𝑦 supp 0 )) → 𝑥 ∈ 𝐼)
153	152	adantl 481	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 ∈ (𝐼 ∖ (𝑦 supp 0 ))) → 𝑥 ∈ 𝐼)
154		fnfvof 7648	. . . . . . . . . 10 ⊢ (((𝑦 Fn 𝐼 ∧ 𝑈 Fn 𝐼) ∧ (𝐼 ∈ 𝑉 ∧ 𝑥 ∈ 𝐼)) → ((𝑦 ∘f ( ·𝑠 ‘𝑌)𝑈)‘𝑥) = ((𝑦‘𝑥)( ·𝑠 ‘𝑌)(𝑈‘𝑥)))
155	149, 150, 151, 153, 154	syl22anc 839	. . . . . . . . 9 ⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 ∈ (𝐼 ∖ (𝑦 supp 0 ))) → ((𝑦 ∘f ( ·𝑠 ‘𝑌)𝑈)‘𝑥) = ((𝑦‘𝑥)( ·𝑠 ‘𝑌)(𝑈‘𝑥)))
156		ssidd 3945	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐵) → (𝑦 supp 0 ) ⊆ (𝑦 supp 0 ))
157	117	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐵) → 0 ∈ V)
158	64, 156, 68, 157	suppssr 8145	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 ∈ (𝐼 ∖ (𝑦 supp 0 ))) → (𝑦‘𝑥) = 0 )
159	122	ad2antrr 727	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 ∈ (𝐼 ∖ (𝑦 supp 0 ))) → 0 = (0g‘(Scalar‘𝑌)))
160	158, 159	eqtrd 2771	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 ∈ (𝐼 ∖ (𝑦 supp 0 ))) → (𝑦‘𝑥) = (0g‘(Scalar‘𝑌)))
161	160	oveq1d 7382	. . . . . . . . 9 ⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 ∈ (𝐼 ∖ (𝑦 supp 0 ))) → ((𝑦‘𝑥)( ·𝑠 ‘𝑌)(𝑈‘𝑥)) = ((0g‘(Scalar‘𝑌))( ·𝑠 ‘𝑌)(𝑈‘𝑥)))
162	3	ad2antrr 727	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 ∈ (𝐼 ∖ (𝑦 supp 0 ))) → 𝑌 ∈ LMod)
163	11	adantr 480	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐵) → 𝑈:𝐼⟶𝐵)
164		ffvelcdm 7033	. . . . . . . . . . 11 ⊢ ((𝑈:𝐼⟶𝐵 ∧ 𝑥 ∈ 𝐼) → (𝑈‘𝑥) ∈ 𝐵)
165	163, 152, 164	syl2an 597	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 ∈ (𝐼 ∖ (𝑦 supp 0 ))) → (𝑈‘𝑥) ∈ 𝐵)
166	5, 100, 48, 130, 51	lmod0vs 20890	. . . . . . . . . 10 ⊢ ((𝑌 ∈ LMod ∧ (𝑈‘𝑥) ∈ 𝐵) → ((0g‘(Scalar‘𝑌))( ·𝑠 ‘𝑌)(𝑈‘𝑥)) = (0g‘𝑌))
167	162, 165, 166	syl2anc 585	. . . . . . . . 9 ⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 ∈ (𝐼 ∖ (𝑦 supp 0 ))) → ((0g‘(Scalar‘𝑌))( ·𝑠 ‘𝑌)(𝑈‘𝑥)) = (0g‘𝑌))
168	155, 161, 167	3eqtrd 2775	. . . . . . . 8 ⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 ∈ (𝐼 ∖ (𝑦 supp 0 ))) → ((𝑦 ∘f ( ·𝑠 ‘𝑌)𝑈)‘𝑥) = (0g‘𝑌))
169	148, 168	suppss 8144	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐵) → ((𝑦 ∘f ( ·𝑠 ‘𝑌)𝑈) supp (0g‘𝑌)) ⊆ (𝑦 supp 0 ))
170	47, 169	syldan 592	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) → ((𝑦 ∘f ( ·𝑠 ‘𝑌)𝑈) supp (0g‘𝑌)) ⊆ (𝑦 supp 0 ))
171	146, 170	ssfid 9179	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) → ((𝑦 ∘f ( ·𝑠 ‘𝑌)𝑈) supp (0g‘𝑌)) ∈ Fin)
172		simp2 1138	. . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → 𝐼 ∈ 𝑉)
173	1, 17, 5	frlmbasmap 21739	. . . . . . . . 9 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ ((Base‘𝑅) ↑m 𝐼))
174	172, 81, 173	syl2an 597	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) → 𝑦 ∈ ((Base‘𝑅) ↑m 𝐼))
175		elmapfn 8812	. . . . . . . 8 ⊢ (𝑦 ∈ ((Base‘𝑅) ↑m 𝐼) → 𝑦 Fn 𝐼)
176	174, 175	syl 17	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) → 𝑦 Fn 𝐼)
177	11	adantr 480	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) → 𝑈:𝐼⟶𝐵)
178	177	ffnd 6669	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) → 𝑈 Fn 𝐼)
179	176, 178, 44, 44	offun 7645	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) → Fun (𝑦 ∘f ( ·𝑠 ‘𝑌)𝑈))
180		ovexd 7402	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) → (𝑦 ∘f ( ·𝑠 ‘𝑌)𝑈) ∈ V)
181		fvexd 6855	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) → (0g‘𝑌) ∈ V)
182		funisfsupp 9280	. . . . . 6 ⊢ ((Fun (𝑦 ∘f ( ·𝑠 ‘𝑌)𝑈) ∧ (𝑦 ∘f ( ·𝑠 ‘𝑌)𝑈) ∈ V ∧ (0g‘𝑌) ∈ V) → ((𝑦 ∘f ( ·𝑠 ‘𝑌)𝑈) finSupp (0g‘𝑌) ↔ ((𝑦 ∘f ( ·𝑠 ‘𝑌)𝑈) supp (0g‘𝑌)) ∈ Fin))
183	179, 180, 181, 182	syl3anc 1374	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) → ((𝑦 ∘f ( ·𝑠 ‘𝑌)𝑈) finSupp (0g‘𝑌) ↔ ((𝑦 ∘f ( ·𝑠 ‘𝑌)𝑈) supp (0g‘𝑌)) ∈ Fin))
184	171, 183	mpbird 257	. . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) → (𝑦 ∘f ( ·𝑠 ‘𝑌)𝑈) finSupp (0g‘𝑌))
185	51, 54, 44, 62, 143, 184	gsumsubgcl 19895	. . 3 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) → (𝑌 Σg (𝑦 ∘f ( ·𝑠 ‘𝑌)𝑈)) ∈ (𝐾‘(𝑈 “ 𝐽)))
186	50, 185	eqeltrd 2836	. 2 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) ∧ 𝑦 ∈ 𝐶) → 𝑦 ∈ (𝐾‘(𝑈 “ 𝐽)))
187	42, 186	eqelssd 3943	1 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → (𝐾‘(𝑈 “ 𝐽)) = 𝐶)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2932  ∀wral 3051  {crab 3389  Vcvv 3429   ∖ cdif 3886   ⊆ wss 3889   class class class wbr 5085  dom cdm 5631  ran crn 5632   “ cima 5634  Fun wfun 6492   Fn wfn 6493  ⟶wf 6494  ‘cfv 6498  (class class class)co 7367   ∘_f cof 7629   supp csupp 8110   ↑_m cmap 8773  Fincfn 8893   finSupp cfsupp 9274  Basecbs 17179  Scalarcsca 17223   ·_𝑠 cvsca 17224  0_gc0g 17402   Σ_g cgsu 17403  SubGrpcsubg 19096  Abelcabl 19756  Ringcrg 20214  LModclmod 20855  LSubSpclss 20926  LSpanclspn 20966   freeLMod cfrlm 21726   unitVec cuvc 21762

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-iin 4936  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-isom 6507  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-of 7631  df-om 7818  df-1st 7942  df-2nd 7943  df-supp 8111  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-2o 8406  df-er 8643  df-map 8775  df-ixp 8846  df-en 8894  df-dom 8895  df-sdom 8896  df-fin 8897  df-fsupp 9275  df-sup 9355  df-oi 9425  df-card 9863  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-sub 11379  df-neg 11380  df-nn 12175  df-2 12244  df-3 12245  df-4 12246  df-5 12247  df-6 12248  df-7 12249  df-8 12250  df-9 12251  df-n0 12438  df-z 12525  df-dec 12645  df-uz 12789  df-fz 13462  df-fzo 13609  df-seq 13964  df-hash 14293  df-struct 17117  df-sets 17134  df-slot 17152  df-ndx 17164  df-base 17180  df-ress 17201  df-plusg 17233  df-mulr 17234  df-sca 17236  df-vsca 17237  df-ip 17238  df-tset 17239  df-ple 17240  df-ds 17242  df-hom 17244  df-cco 17245  df-0g 17404  df-gsum 17405  df-prds 17410  df-pws 17412  df-mre 17548  df-mrc 17549  df-acs 17551  df-mgm 18608  df-sgrp 18687  df-mnd 18703  df-mhm 18751  df-submnd 18752  df-grp 18912  df-minusg 18913  df-sbg 18914  df-mulg 19044  df-subg 19099  df-ghm 19188  df-cntz 19292  df-cmn 19757  df-abl 19758  df-mgp 20122  df-rng 20134  df-ur 20163  df-ring 20216  df-subrg 20547  df-lmod 20857  df-lss 20927  df-lsp 20967  df-lmhm 21017  df-sra 21168  df-rgmod 21169  df-dsmm 21712  df-frlm 21727  df-uvc 21763

This theorem is referenced by:  frlmlbs  21777