Theorem resssra 33749

Description: The subring algebra of a restricted structure is the restriction of the subring algebra. (Contributed by Thierry Arnoux, 2-Apr-2025.)

Hypotheses

Ref	Expression
resssra.a	⊢ 𝐴 = (Base‘𝑅)
resssra.s	⊢ 𝑆 = (𝑅 ↾s 𝐵)
resssra.b	⊢ (𝜑 → 𝐵 ⊆ 𝐴)
resssra.c	⊢ (𝜑 → 𝐶 ⊆ 𝐵)
resssra.r	⊢ (𝜑 → 𝑅 ∈ 𝑉)

Assertion

Ref	Expression
resssra	⊢ (𝜑 → ((subringAlg ‘𝑆)‘𝐶) = (((subringAlg ‘𝑅)‘𝐶) ↾s 𝐵))

Proof of Theorem resssra

Step	Hyp	Ref	Expression
1		resssra.a	. . . . . . 7 ⊢ 𝐴 = (Base‘𝑅)
2		eqidd 2738	. . . . . . . 8 ⊢ (𝜑 → ((subringAlg ‘𝑅)‘𝐶) = ((subringAlg ‘𝑅)‘𝐶))
3		resssra.c	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶 ⊆ 𝐵)
4		resssra.b	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ⊆ 𝐴)
5	3, 4	sstrd 3933	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ⊆ 𝐴)
6	5, 1	sseqtrdi 3963	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ⊆ (Base‘𝑅))
7	2, 6	srabase 21167	. . . . . . 7 ⊢ (𝜑 → (Base‘𝑅) = (Base‘((subringAlg ‘𝑅)‘𝐶)))
8	1, 7	eqtrid 2784	. . . . . 6 ⊢ (𝜑 → 𝐴 = (Base‘((subringAlg ‘𝑅)‘𝐶)))
9	8	oveq2d 7377	. . . . 5 ⊢ (𝜑 → (((subringAlg ‘𝑅)‘𝐶) ↾s 𝐴) = (((subringAlg ‘𝑅)‘𝐶) ↾s (Base‘((subringAlg ‘𝑅)‘𝐶))))
10	9	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ⊆ 𝐵) → (((subringAlg ‘𝑅)‘𝐶) ↾s 𝐴) = (((subringAlg ‘𝑅)‘𝐶) ↾s (Base‘((subringAlg ‘𝑅)‘𝐶))))
11		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ⊆ 𝐵) → 𝐴 ⊆ 𝐵)
12	4	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ⊆ 𝐵) → 𝐵 ⊆ 𝐴)
13	11, 12	eqssd 3940	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ⊆ 𝐵) → 𝐴 = 𝐵)
14	13	oveq2d 7377	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ⊆ 𝐵) → (((subringAlg ‘𝑅)‘𝐶) ↾s 𝐴) = (((subringAlg ‘𝑅)‘𝐶) ↾s 𝐵))
15		fvex 6848	. . . . 5 ⊢ ((subringAlg ‘𝑅)‘𝐶) ∈ V
16		eqid 2737	. . . . . 6 ⊢ (Base‘((subringAlg ‘𝑅)‘𝐶)) = (Base‘((subringAlg ‘𝑅)‘𝐶))
17	16	ressid 17208	. . . . 5 ⊢ (((subringAlg ‘𝑅)‘𝐶) ∈ V → (((subringAlg ‘𝑅)‘𝐶) ↾s (Base‘((subringAlg ‘𝑅)‘𝐶))) = ((subringAlg ‘𝑅)‘𝐶))
18	15, 17	mp1i 13	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ⊆ 𝐵) → (((subringAlg ‘𝑅)‘𝐶) ↾s (Base‘((subringAlg ‘𝑅)‘𝐶))) = ((subringAlg ‘𝑅)‘𝐶))
19	10, 14, 18	3eqtr3d 2780	. . 3 ⊢ ((𝜑 ∧ 𝐴 ⊆ 𝐵) → (((subringAlg ‘𝑅)‘𝐶) ↾s 𝐵) = ((subringAlg ‘𝑅)‘𝐶))
20	1	oveq2i 7372	. . . . . . . 8 ⊢ (𝑅 ↾s 𝐴) = (𝑅 ↾s (Base‘𝑅))
21		resssra.r	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ 𝑉)
22	21	elexd 3454	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 ∈ V)
23		eqid 2737	. . . . . . . . . 10 ⊢ (Base‘𝑅) = (Base‘𝑅)
24	23	ressid 17208	. . . . . . . . 9 ⊢ (𝑅 ∈ V → (𝑅 ↾s (Base‘𝑅)) = 𝑅)
25	22, 24	syl 17	. . . . . . . 8 ⊢ (𝜑 → (𝑅 ↾s (Base‘𝑅)) = 𝑅)
26	20, 25	eqtrid 2784	. . . . . . 7 ⊢ (𝜑 → (𝑅 ↾s 𝐴) = 𝑅)
27	26	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ⊆ 𝐵) → (𝑅 ↾s 𝐴) = 𝑅)
28	13	oveq2d 7377	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ⊆ 𝐵) → (𝑅 ↾s 𝐴) = (𝑅 ↾s 𝐵))
29		resssra.s	. . . . . . 7 ⊢ 𝑆 = (𝑅 ↾s 𝐵)
30	28, 29	eqtr4di 2790	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ⊆ 𝐵) → (𝑅 ↾s 𝐴) = 𝑆)
31	27, 30	eqtr3d 2774	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ⊆ 𝐵) → 𝑅 = 𝑆)
32	31	fveq2d 6839	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ⊆ 𝐵) → (subringAlg ‘𝑅) = (subringAlg ‘𝑆))
33	32	fveq1d 6837	. . 3 ⊢ ((𝜑 ∧ 𝐴 ⊆ 𝐵) → ((subringAlg ‘𝑅)‘𝐶) = ((subringAlg ‘𝑆)‘𝐶))
34	19, 33	eqtr2d 2773	. 2 ⊢ ((𝜑 ∧ 𝐴 ⊆ 𝐵) → ((subringAlg ‘𝑆)‘𝐶) = (((subringAlg ‘𝑅)‘𝐶) ↾s 𝐵))
35		simpr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 𝐴 ⊆ 𝐵) → ¬ 𝐴 ⊆ 𝐵)
36	22	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 𝐴 ⊆ 𝐵) → 𝑅 ∈ V)
37	1	fvexi 6849	. . . . . . . . . . . . . 14 ⊢ 𝐴 ∈ V
38	37	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴 ∈ V)
39	38, 4	ssexd 5262	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 ∈ V)
40	39	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 𝐴 ⊆ 𝐵) → 𝐵 ∈ V)
41	29, 1	ressval2 17199	. . . . . . . . . . 11 ⊢ ((¬ 𝐴 ⊆ 𝐵 ∧ 𝑅 ∈ V ∧ 𝐵 ∈ V) → 𝑆 = (𝑅 sSet ⟨(Base‘ndx), (𝐵 ∩ 𝐴)⟩))
42	35, 36, 40, 41	syl3anc 1374	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝐴 ⊆ 𝐵) → 𝑆 = (𝑅 sSet ⟨(Base‘ndx), (𝐵 ∩ 𝐴)⟩))
43		dfss2 3908	. . . . . . . . . . . . . 14 ⊢ (𝐵 ⊆ 𝐴 ↔ (𝐵 ∩ 𝐴) = 𝐵)
44	4, 43	sylib 218	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐵 ∩ 𝐴) = 𝐵)
45	44	opeq2d 4824	. . . . . . . . . . . 12 ⊢ (𝜑 → ⟨(Base‘ndx), (𝐵 ∩ 𝐴)⟩ = ⟨(Base‘ndx), 𝐵⟩)
46	45	oveq2d 7377	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑅 sSet ⟨(Base‘ndx), (𝐵 ∩ 𝐴)⟩) = (𝑅 sSet ⟨(Base‘ndx), 𝐵⟩))
47	46	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝐴 ⊆ 𝐵) → (𝑅 sSet ⟨(Base‘ndx), (𝐵 ∩ 𝐴)⟩) = (𝑅 sSet ⟨(Base‘ndx), 𝐵⟩))
48	42, 47	eqtrd 2772	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐴 ⊆ 𝐵) → 𝑆 = (𝑅 sSet ⟨(Base‘ndx), 𝐵⟩))
49	29	oveq1i 7371	. . . . . . . . . . . 12 ⊢ (𝑆 ↾s 𝐶) = ((𝑅 ↾s 𝐵) ↾s 𝐶)
50		ressabs 17212	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ V ∧ 𝐶 ⊆ 𝐵) → ((𝑅 ↾s 𝐵) ↾s 𝐶) = (𝑅 ↾s 𝐶))
51	39, 3, 50	syl2anc 585	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑅 ↾s 𝐵) ↾s 𝐶) = (𝑅 ↾s 𝐶))
52	49, 51	eqtrid 2784	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑆 ↾s 𝐶) = (𝑅 ↾s 𝐶))
53	52	opeq2d 4824	. . . . . . . . . 10 ⊢ (𝜑 → ⟨(Scalar‘ndx), (𝑆 ↾s 𝐶)⟩ = ⟨(Scalar‘ndx), (𝑅 ↾s 𝐶)⟩)
54	53	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐴 ⊆ 𝐵) → ⟨(Scalar‘ndx), (𝑆 ↾s 𝐶)⟩ = ⟨(Scalar‘ndx), (𝑅 ↾s 𝐶)⟩)
55	48, 54	oveq12d 7379	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐴 ⊆ 𝐵) → (𝑆 sSet ⟨(Scalar‘ndx), (𝑆 ↾s 𝐶)⟩) = ((𝑅 sSet ⟨(Base‘ndx), 𝐵⟩) sSet ⟨(Scalar‘ndx), (𝑅 ↾s 𝐶)⟩))
56		scandxnbasendx 17273	. . . . . . . . . . 11 ⊢ (Scalar‘ndx) ≠ (Base‘ndx)
57	56	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → (Scalar‘ndx) ≠ (Base‘ndx))
58		ovexd 7396	. . . . . . . . . 10 ⊢ (𝜑 → (𝑅 ↾s 𝐶) ∈ V)
59		fvex 6848	. . . . . . . . . . 11 ⊢ (Scalar‘ndx) ∈ V
60		fvex 6848	. . . . . . . . . . 11 ⊢ (Base‘ndx) ∈ V
61	59, 60	setscom 17144	. . . . . . . . . 10 ⊢ (((𝑅 ∈ V ∧ (Scalar‘ndx) ≠ (Base‘ndx)) ∧ ((𝑅 ↾s 𝐶) ∈ V ∧ 𝐵 ∈ V)) → ((𝑅 sSet ⟨(Scalar‘ndx), (𝑅 ↾s 𝐶)⟩) sSet ⟨(Base‘ndx), 𝐵⟩) = ((𝑅 sSet ⟨(Base‘ndx), 𝐵⟩) sSet ⟨(Scalar‘ndx), (𝑅 ↾s 𝐶)⟩))
62	22, 57, 58, 39, 61	syl22anc 839	. . . . . . . . 9 ⊢ (𝜑 → ((𝑅 sSet ⟨(Scalar‘ndx), (𝑅 ↾s 𝐶)⟩) sSet ⟨(Base‘ndx), 𝐵⟩) = ((𝑅 sSet ⟨(Base‘ndx), 𝐵⟩) sSet ⟨(Scalar‘ndx), (𝑅 ↾s 𝐶)⟩))
63	62	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐴 ⊆ 𝐵) → ((𝑅 sSet ⟨(Scalar‘ndx), (𝑅 ↾s 𝐶)⟩) sSet ⟨(Base‘ndx), 𝐵⟩) = ((𝑅 sSet ⟨(Base‘ndx), 𝐵⟩) sSet ⟨(Scalar‘ndx), (𝑅 ↾s 𝐶)⟩))
64	55, 63	eqtr4d 2775	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝐴 ⊆ 𝐵) → (𝑆 sSet ⟨(Scalar‘ndx), (𝑆 ↾s 𝐶)⟩) = ((𝑅 sSet ⟨(Scalar‘ndx), (𝑅 ↾s 𝐶)⟩) sSet ⟨(Base‘ndx), 𝐵⟩))
65		eqid 2737	. . . . . . . . . . . 12 ⊢ (.r‘𝑅) = (.r‘𝑅)
66	29, 65	ressmulr 17264	. . . . . . . . . . 11 ⊢ (𝐵 ∈ V → (.r‘𝑅) = (.r‘𝑆))
67	39, 66	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (.r‘𝑅) = (.r‘𝑆))
68	67	eqcomd 2743	. . . . . . . . 9 ⊢ (𝜑 → (.r‘𝑆) = (.r‘𝑅))
69	68	opeq2d 4824	. . . . . . . 8 ⊢ (𝜑 → ⟨( ·𝑠 ‘ndx), (.r‘𝑆)⟩ = ⟨( ·𝑠 ‘ndx), (.r‘𝑅)⟩)
70	69	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝐴 ⊆ 𝐵) → ⟨( ·𝑠 ‘ndx), (.r‘𝑆)⟩ = ⟨( ·𝑠 ‘ndx), (.r‘𝑅)⟩)
71	64, 70	oveq12d 7379	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐴 ⊆ 𝐵) → ((𝑆 sSet ⟨(Scalar‘ndx), (𝑆 ↾s 𝐶)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r‘𝑆)⟩) = (((𝑅 sSet ⟨(Scalar‘ndx), (𝑅 ↾s 𝐶)⟩) sSet ⟨(Base‘ndx), 𝐵⟩) sSet ⟨( ·𝑠 ‘ndx), (.r‘𝑅)⟩))
72		ovexd 7396	. . . . . . . 8 ⊢ (𝜑 → (𝑅 sSet ⟨(Scalar‘ndx), (𝑅 ↾s 𝐶)⟩) ∈ V)
73		vscandxnbasendx 17278	. . . . . . . . 9 ⊢ ( ·𝑠 ‘ndx) ≠ (Base‘ndx)
74	73	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ( ·𝑠 ‘ndx) ≠ (Base‘ndx))
75		fvexd 6850	. . . . . . . 8 ⊢ (𝜑 → (.r‘𝑅) ∈ V)
76		fvex 6848	. . . . . . . . 9 ⊢ ( ·𝑠 ‘ndx) ∈ V
77	76, 60	setscom 17144	. . . . . . . 8 ⊢ ((((𝑅 sSet ⟨(Scalar‘ndx), (𝑅 ↾s 𝐶)⟩) ∈ V ∧ ( ·𝑠 ‘ndx) ≠ (Base‘ndx)) ∧ ((.r‘𝑅) ∈ V ∧ 𝐵 ∈ V)) → (((𝑅 sSet ⟨(Scalar‘ndx), (𝑅 ↾s 𝐶)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r‘𝑅)⟩) sSet ⟨(Base‘ndx), 𝐵⟩) = (((𝑅 sSet ⟨(Scalar‘ndx), (𝑅 ↾s 𝐶)⟩) sSet ⟨(Base‘ndx), 𝐵⟩) sSet ⟨( ·𝑠 ‘ndx), (.r‘𝑅)⟩))
78	72, 74, 75, 39, 77	syl22anc 839	. . . . . . 7 ⊢ (𝜑 → (((𝑅 sSet ⟨(Scalar‘ndx), (𝑅 ↾s 𝐶)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r‘𝑅)⟩) sSet ⟨(Base‘ndx), 𝐵⟩) = (((𝑅 sSet ⟨(Scalar‘ndx), (𝑅 ↾s 𝐶)⟩) sSet ⟨(Base‘ndx), 𝐵⟩) sSet ⟨( ·𝑠 ‘ndx), (.r‘𝑅)⟩))
79	78	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐴 ⊆ 𝐵) → (((𝑅 sSet ⟨(Scalar‘ndx), (𝑅 ↾s 𝐶)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r‘𝑅)⟩) sSet ⟨(Base‘ndx), 𝐵⟩) = (((𝑅 sSet ⟨(Scalar‘ndx), (𝑅 ↾s 𝐶)⟩) sSet ⟨(Base‘ndx), 𝐵⟩) sSet ⟨( ·𝑠 ‘ndx), (.r‘𝑅)⟩))
80	71, 79	eqtr4d 2775	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐴 ⊆ 𝐵) → ((𝑆 sSet ⟨(Scalar‘ndx), (𝑆 ↾s 𝐶)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r‘𝑆)⟩) = (((𝑅 sSet ⟨(Scalar‘ndx), (𝑅 ↾s 𝐶)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r‘𝑅)⟩) sSet ⟨(Base‘ndx), 𝐵⟩))
81	68	opeq2d 4824	. . . . . 6 ⊢ (𝜑 → ⟨(·𝑖‘ndx), (.r‘𝑆)⟩ = ⟨(·𝑖‘ndx), (.r‘𝑅)⟩)
82	81	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐴 ⊆ 𝐵) → ⟨(·𝑖‘ndx), (.r‘𝑆)⟩ = ⟨(·𝑖‘ndx), (.r‘𝑅)⟩)
83	80, 82	oveq12d 7379	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐴 ⊆ 𝐵) → (((𝑆 sSet ⟨(Scalar‘ndx), (𝑆 ↾s 𝐶)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r‘𝑆)⟩) sSet ⟨(·𝑖‘ndx), (.r‘𝑆)⟩) = ((((𝑅 sSet ⟨(Scalar‘ndx), (𝑅 ↾s 𝐶)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r‘𝑅)⟩) sSet ⟨(Base‘ndx), 𝐵⟩) sSet ⟨(·𝑖‘ndx), (.r‘𝑅)⟩))
84		ovexd 7396	. . . . . 6 ⊢ (𝜑 → ((𝑅 sSet ⟨(Scalar‘ndx), (𝑅 ↾s 𝐶)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r‘𝑅)⟩) ∈ V)
85		ipndxnbasendx 17289	. . . . . . 7 ⊢ (·𝑖‘ndx) ≠ (Base‘ndx)
86	85	a1i 11	. . . . . 6 ⊢ (𝜑 → (·𝑖‘ndx) ≠ (Base‘ndx))
87		fvex 6848	. . . . . . 7 ⊢ (·𝑖‘ndx) ∈ V
88	87, 60	setscom 17144	. . . . . 6 ⊢ (((((𝑅 sSet ⟨(Scalar‘ndx), (𝑅 ↾s 𝐶)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r‘𝑅)⟩) ∈ V ∧ (·𝑖‘ndx) ≠ (Base‘ndx)) ∧ ((.r‘𝑅) ∈ V ∧ 𝐵 ∈ V)) → ((((𝑅 sSet ⟨(Scalar‘ndx), (𝑅 ↾s 𝐶)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r‘𝑅)⟩) sSet ⟨(·𝑖‘ndx), (.r‘𝑅)⟩) sSet ⟨(Base‘ndx), 𝐵⟩) = ((((𝑅 sSet ⟨(Scalar‘ndx), (𝑅 ↾s 𝐶)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r‘𝑅)⟩) sSet ⟨(Base‘ndx), 𝐵⟩) sSet ⟨(·𝑖‘ndx), (.r‘𝑅)⟩))
89	84, 86, 75, 39, 88	syl22anc 839	. . . . 5 ⊢ (𝜑 → ((((𝑅 sSet ⟨(Scalar‘ndx), (𝑅 ↾s 𝐶)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r‘𝑅)⟩) sSet ⟨(·𝑖‘ndx), (.r‘𝑅)⟩) sSet ⟨(Base‘ndx), 𝐵⟩) = ((((𝑅 sSet ⟨(Scalar‘ndx), (𝑅 ↾s 𝐶)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r‘𝑅)⟩) sSet ⟨(Base‘ndx), 𝐵⟩) sSet ⟨(·𝑖‘ndx), (.r‘𝑅)⟩))
90	89	adantr 480	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐴 ⊆ 𝐵) → ((((𝑅 sSet ⟨(Scalar‘ndx), (𝑅 ↾s 𝐶)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r‘𝑅)⟩) sSet ⟨(·𝑖‘ndx), (.r‘𝑅)⟩) sSet ⟨(Base‘ndx), 𝐵⟩) = ((((𝑅 sSet ⟨(Scalar‘ndx), (𝑅 ↾s 𝐶)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r‘𝑅)⟩) sSet ⟨(Base‘ndx), 𝐵⟩) sSet ⟨(·𝑖‘ndx), (.r‘𝑅)⟩))
91	83, 90	eqtr4d 2775	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐴 ⊆ 𝐵) → (((𝑆 sSet ⟨(Scalar‘ndx), (𝑆 ↾s 𝐶)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r‘𝑆)⟩) sSet ⟨(·𝑖‘ndx), (.r‘𝑆)⟩) = ((((𝑅 sSet ⟨(Scalar‘ndx), (𝑅 ↾s 𝐶)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r‘𝑅)⟩) sSet ⟨(·𝑖‘ndx), (.r‘𝑅)⟩) sSet ⟨(Base‘ndx), 𝐵⟩))
92	29	ovexi 7395	. . . 4 ⊢ 𝑆 ∈ V
93	29, 1	ressbas2 17202	. . . . . . 7 ⊢ (𝐵 ⊆ 𝐴 → 𝐵 = (Base‘𝑆))
94	4, 93	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐵 = (Base‘𝑆))
95	3, 94	sseqtrd 3959	. . . . 5 ⊢ (𝜑 → 𝐶 ⊆ (Base‘𝑆))
96	95	adantr 480	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐴 ⊆ 𝐵) → 𝐶 ⊆ (Base‘𝑆))
97		sraval 21165	. . . 4 ⊢ ((𝑆 ∈ V ∧ 𝐶 ⊆ (Base‘𝑆)) → ((subringAlg ‘𝑆)‘𝐶) = (((𝑆 sSet ⟨(Scalar‘ndx), (𝑆 ↾s 𝐶)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r‘𝑆)⟩) sSet ⟨(·𝑖‘ndx), (.r‘𝑆)⟩))
98	92, 96, 97	sylancr 588	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐴 ⊆ 𝐵) → ((subringAlg ‘𝑆)‘𝐶) = (((𝑆 sSet ⟨(Scalar‘ndx), (𝑆 ↾s 𝐶)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r‘𝑆)⟩) sSet ⟨(·𝑖‘ndx), (.r‘𝑆)⟩))
99	8	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝐴 ⊆ 𝐵) → 𝐴 = (Base‘((subringAlg ‘𝑅)‘𝐶)))
100	99	sseq1d 3954	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐴 ⊆ 𝐵) → (𝐴 ⊆ 𝐵 ↔ (Base‘((subringAlg ‘𝑅)‘𝐶)) ⊆ 𝐵))
101	35, 100	mtbid 324	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐴 ⊆ 𝐵) → ¬ (Base‘((subringAlg ‘𝑅)‘𝐶)) ⊆ 𝐵)
102		fvexd 6850	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐴 ⊆ 𝐵) → ((subringAlg ‘𝑅)‘𝐶) ∈ V)
103		eqid 2737	. . . . . 6 ⊢ (((subringAlg ‘𝑅)‘𝐶) ↾s 𝐵) = (((subringAlg ‘𝑅)‘𝐶) ↾s 𝐵)
104	103, 16	ressval2 17199	. . . . 5 ⊢ ((¬ (Base‘((subringAlg ‘𝑅)‘𝐶)) ⊆ 𝐵 ∧ ((subringAlg ‘𝑅)‘𝐶) ∈ V ∧ 𝐵 ∈ V) → (((subringAlg ‘𝑅)‘𝐶) ↾s 𝐵) = (((subringAlg ‘𝑅)‘𝐶) sSet ⟨(Base‘ndx), (𝐵 ∩ (Base‘((subringAlg ‘𝑅)‘𝐶)))⟩))
105	101, 102, 40, 104	syl3anc 1374	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐴 ⊆ 𝐵) → (((subringAlg ‘𝑅)‘𝐶) ↾s 𝐵) = (((subringAlg ‘𝑅)‘𝐶) sSet ⟨(Base‘ndx), (𝐵 ∩ (Base‘((subringAlg ‘𝑅)‘𝐶)))⟩))
106	8	ineq2d 4161	. . . . . . . 8 ⊢ (𝜑 → (𝐵 ∩ 𝐴) = (𝐵 ∩ (Base‘((subringAlg ‘𝑅)‘𝐶))))
107	106, 44	eqtr3d 2774	. . . . . . 7 ⊢ (𝜑 → (𝐵 ∩ (Base‘((subringAlg ‘𝑅)‘𝐶))) = 𝐵)
108	107	opeq2d 4824	. . . . . 6 ⊢ (𝜑 → ⟨(Base‘ndx), (𝐵 ∩ (Base‘((subringAlg ‘𝑅)‘𝐶)))⟩ = ⟨(Base‘ndx), 𝐵⟩)
109	108	oveq2d 7377	. . . . 5 ⊢ (𝜑 → (((subringAlg ‘𝑅)‘𝐶) sSet ⟨(Base‘ndx), (𝐵 ∩ (Base‘((subringAlg ‘𝑅)‘𝐶)))⟩) = (((subringAlg ‘𝑅)‘𝐶) sSet ⟨(Base‘ndx), 𝐵⟩))
110	109	adantr 480	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐴 ⊆ 𝐵) → (((subringAlg ‘𝑅)‘𝐶) sSet ⟨(Base‘ndx), (𝐵 ∩ (Base‘((subringAlg ‘𝑅)‘𝐶)))⟩) = (((subringAlg ‘𝑅)‘𝐶) sSet ⟨(Base‘ndx), 𝐵⟩))
111		sraval 21165	. . . . . . 7 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐶 ⊆ (Base‘𝑅)) → ((subringAlg ‘𝑅)‘𝐶) = (((𝑅 sSet ⟨(Scalar‘ndx), (𝑅 ↾s 𝐶)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r‘𝑅)⟩) sSet ⟨(·𝑖‘ndx), (.r‘𝑅)⟩))
112	21, 6, 111	syl2anc 585	. . . . . 6 ⊢ (𝜑 → ((subringAlg ‘𝑅)‘𝐶) = (((𝑅 sSet ⟨(Scalar‘ndx), (𝑅 ↾s 𝐶)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r‘𝑅)⟩) sSet ⟨(·𝑖‘ndx), (.r‘𝑅)⟩))
113	112	oveq1d 7376	. . . . 5 ⊢ (𝜑 → (((subringAlg ‘𝑅)‘𝐶) sSet ⟨(Base‘ndx), 𝐵⟩) = ((((𝑅 sSet ⟨(Scalar‘ndx), (𝑅 ↾s 𝐶)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r‘𝑅)⟩) sSet ⟨(·𝑖‘ndx), (.r‘𝑅)⟩) sSet ⟨(Base‘ndx), 𝐵⟩))
114	113	adantr 480	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐴 ⊆ 𝐵) → (((subringAlg ‘𝑅)‘𝐶) sSet ⟨(Base‘ndx), 𝐵⟩) = ((((𝑅 sSet ⟨(Scalar‘ndx), (𝑅 ↾s 𝐶)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r‘𝑅)⟩) sSet ⟨(·𝑖‘ndx), (.r‘𝑅)⟩) sSet ⟨(Base‘ndx), 𝐵⟩))
115	105, 110, 114	3eqtrd 2776	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐴 ⊆ 𝐵) → (((subringAlg ‘𝑅)‘𝐶) ↾s 𝐵) = ((((𝑅 sSet ⟨(Scalar‘ndx), (𝑅 ↾s 𝐶)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r‘𝑅)⟩) sSet ⟨(·𝑖‘ndx), (.r‘𝑅)⟩) sSet ⟨(Base‘ndx), 𝐵⟩))
116	91, 98, 115	3eqtr4d 2782	. 2 ⊢ ((𝜑 ∧ ¬ 𝐴 ⊆ 𝐵) → ((subringAlg ‘𝑆)‘𝐶) = (((subringAlg ‘𝑅)‘𝐶) ↾s 𝐵))
117	34, 116	pm2.61dan 813	1 ⊢ (𝜑 → ((subringAlg ‘𝑆)‘𝐶) = (((subringAlg ‘𝑅)‘𝐶) ↾s 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  Vcvv 3430   ∩ cin 3889   ⊆ wss 3890  ⟨cop 4574  ‘cfv 6493  (class class class)co 7361   sSet csts 17127  ndxcnx 17157  Basecbs 17173   ↾_s cress 17194  ._rcmulr 17215  Scalarcsca 17217   ·_𝑠 cvsca 17218  ·_𝑖cip 17219  subringAlg csra 21161

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 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5303  ax-pr 5371  ax-un 7683  ax-cnex 11088  ax-resscn 11089  ax-1cn 11090  ax-icn 11091  ax-addcl 11092  ax-addrcl 11093  ax-mulcl 11094  ax-mulrcl 11095  ax-mulcom 11096  ax-addass 11097  ax-mulass 11098  ax-distr 11099  ax-i2m1 11100  ax-1ne0 11101  ax-1rid 11102  ax-rnegex 11103  ax-rrecex 11104  ax-cnre 11105  ax-pre-lttri 11106  ax-pre-lttrn 11107  ax-pre-ltadd 11108  ax-pre-mulgt0 11109

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 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7318  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7812  df-2nd 7937  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-er 8637  df-en 8888  df-dom 8889  df-sdom 8890  df-pnf 11175  df-mnf 11176  df-xr 11177  df-ltxr 11178  df-le 11179  df-sub 11373  df-neg 11374  df-nn 12169  df-2 12238  df-3 12239  df-4 12240  df-5 12241  df-6 12242  df-7 12243  df-8 12244  df-sets 17128  df-slot 17146  df-ndx 17158  df-base 17174  df-ress 17195  df-mulr 17228  df-sca 17230  df-vsca 17231  df-ip 17232  df-sra 21163

This theorem is referenced by:  lsssra  33750  fldextrspunlem1  33838  algextdeglem2  33881