Theorem resssra 33743

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 2737	. . . . . . . 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 3944	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ⊆ 𝐴)
6	5, 1	sseqtrdi 3974	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ⊆ (Base‘𝑅))
7	2, 6	srabase 21129	. . . . . . 7 ⊢ (𝜑 → (Base‘𝑅) = (Base‘((subringAlg ‘𝑅)‘𝐶)))
8	1, 7	eqtrid 2783	. . . . . 6 ⊢ (𝜑 → 𝐴 = (Base‘((subringAlg ‘𝑅)‘𝐶)))
9	8	oveq2d 7374	. . . . 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 3951	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ⊆ 𝐵) → 𝐴 = 𝐵)
14	13	oveq2d 7374	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ⊆ 𝐵) → (((subringAlg ‘𝑅)‘𝐶) ↾s 𝐴) = (((subringAlg ‘𝑅)‘𝐶) ↾s 𝐵))
15		fvex 6847	. . . . 5 ⊢ ((subringAlg ‘𝑅)‘𝐶) ∈ V
16		eqid 2736	. . . . . 6 ⊢ (Base‘((subringAlg ‘𝑅)‘𝐶)) = (Base‘((subringAlg ‘𝑅)‘𝐶))
17	16	ressid 17171	. . . . 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 2779	. . 3 ⊢ ((𝜑 ∧ 𝐴 ⊆ 𝐵) → (((subringAlg ‘𝑅)‘𝐶) ↾s 𝐵) = ((subringAlg ‘𝑅)‘𝐶))
20	1	oveq2i 7369	. . . . . . . 8 ⊢ (𝑅 ↾s 𝐴) = (𝑅 ↾s (Base‘𝑅))
21		resssra.r	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ 𝑉)
22	21	elexd 3464	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 ∈ V)
23		eqid 2736	. . . . . . . . . 10 ⊢ (Base‘𝑅) = (Base‘𝑅)
24	23	ressid 17171	. . . . . . . . 9 ⊢ (𝑅 ∈ V → (𝑅 ↾s (Base‘𝑅)) = 𝑅)
25	22, 24	syl 17	. . . . . . . 8 ⊢ (𝜑 → (𝑅 ↾s (Base‘𝑅)) = 𝑅)
26	20, 25	eqtrid 2783	. . . . . . 7 ⊢ (𝜑 → (𝑅 ↾s 𝐴) = 𝑅)
27	26	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ⊆ 𝐵) → (𝑅 ↾s 𝐴) = 𝑅)
28	13	oveq2d 7374	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ⊆ 𝐵) → (𝑅 ↾s 𝐴) = (𝑅 ↾s 𝐵))
29		resssra.s	. . . . . . 7 ⊢ 𝑆 = (𝑅 ↾s 𝐵)
30	28, 29	eqtr4di 2789	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ⊆ 𝐵) → (𝑅 ↾s 𝐴) = 𝑆)
31	27, 30	eqtr3d 2773	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ⊆ 𝐵) → 𝑅 = 𝑆)
32	31	fveq2d 6838	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ⊆ 𝐵) → (subringAlg ‘𝑅) = (subringAlg ‘𝑆))
33	32	fveq1d 6836	. . 3 ⊢ ((𝜑 ∧ 𝐴 ⊆ 𝐵) → ((subringAlg ‘𝑅)‘𝐶) = ((subringAlg ‘𝑆)‘𝐶))
34	19, 33	eqtr2d 2772	. 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 6848	. . . . . . . . . . . . . 14 ⊢ 𝐴 ∈ V
38	37	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴 ∈ V)
39	38, 4	ssexd 5269	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 ∈ V)
40	39	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 𝐴 ⊆ 𝐵) → 𝐵 ∈ V)
41	29, 1	ressval2 17162	. . . . . . . . . . 11 ⊢ ((¬ 𝐴 ⊆ 𝐵 ∧ 𝑅 ∈ V ∧ 𝐵 ∈ V) → 𝑆 = (𝑅 sSet ⟨(Base‘ndx), (𝐵 ∩ 𝐴)⟩))
42	35, 36, 40, 41	syl3anc 1373	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝐴 ⊆ 𝐵) → 𝑆 = (𝑅 sSet ⟨(Base‘ndx), (𝐵 ∩ 𝐴)⟩))
43		dfss2 3919	. . . . . . . . . . . . . 14 ⊢ (𝐵 ⊆ 𝐴 ↔ (𝐵 ∩ 𝐴) = 𝐵)
44	4, 43	sylib 218	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐵 ∩ 𝐴) = 𝐵)
45	44	opeq2d 4836	. . . . . . . . . . . 12 ⊢ (𝜑 → ⟨(Base‘ndx), (𝐵 ∩ 𝐴)⟩ = ⟨(Base‘ndx), 𝐵⟩)
46	45	oveq2d 7374	. . . . . . . . . . 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 2771	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐴 ⊆ 𝐵) → 𝑆 = (𝑅 sSet ⟨(Base‘ndx), 𝐵⟩))
49	29	oveq1i 7368	. . . . . . . . . . . 12 ⊢ (𝑆 ↾s 𝐶) = ((𝑅 ↾s 𝐵) ↾s 𝐶)
50		ressabs 17175	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ V ∧ 𝐶 ⊆ 𝐵) → ((𝑅 ↾s 𝐵) ↾s 𝐶) = (𝑅 ↾s 𝐶))
51	39, 3, 50	syl2anc 584	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑅 ↾s 𝐵) ↾s 𝐶) = (𝑅 ↾s 𝐶))
52	49, 51	eqtrid 2783	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑆 ↾s 𝐶) = (𝑅 ↾s 𝐶))
53	52	opeq2d 4836	. . . . . . . . . 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 7376	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐴 ⊆ 𝐵) → (𝑆 sSet ⟨(Scalar‘ndx), (𝑆 ↾s 𝐶)⟩) = ((𝑅 sSet ⟨(Base‘ndx), 𝐵⟩) sSet ⟨(Scalar‘ndx), (𝑅 ↾s 𝐶)⟩))
56		scandxnbasendx 17236	. . . . . . . . . . 11 ⊢ (Scalar‘ndx) ≠ (Base‘ndx)
57	56	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → (Scalar‘ndx) ≠ (Base‘ndx))
58		ovexd 7393	. . . . . . . . . 10 ⊢ (𝜑 → (𝑅 ↾s 𝐶) ∈ V)
59		fvex 6847	. . . . . . . . . . 11 ⊢ (Scalar‘ndx) ∈ V
60		fvex 6847	. . . . . . . . . . 11 ⊢ (Base‘ndx) ∈ V
61	59, 60	setscom 17107	. . . . . . . . . 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 838	. . . . . . . . 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 2774	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝐴 ⊆ 𝐵) → (𝑆 sSet ⟨(Scalar‘ndx), (𝑆 ↾s 𝐶)⟩) = ((𝑅 sSet ⟨(Scalar‘ndx), (𝑅 ↾s 𝐶)⟩) sSet ⟨(Base‘ndx), 𝐵⟩))
65		eqid 2736	. . . . . . . . . . . 12 ⊢ (.r‘𝑅) = (.r‘𝑅)
66	29, 65	ressmulr 17227	. . . . . . . . . . 11 ⊢ (𝐵 ∈ V → (.r‘𝑅) = (.r‘𝑆))
67	39, 66	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (.r‘𝑅) = (.r‘𝑆))
68	67	eqcomd 2742	. . . . . . . . 9 ⊢ (𝜑 → (.r‘𝑆) = (.r‘𝑅))
69	68	opeq2d 4836	. . . . . . . 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 7376	. . . . . 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 7393	. . . . . . . 8 ⊢ (𝜑 → (𝑅 sSet ⟨(Scalar‘ndx), (𝑅 ↾s 𝐶)⟩) ∈ V)
73		vscandxnbasendx 17241	. . . . . . . . 9 ⊢ ( ·𝑠 ‘ndx) ≠ (Base‘ndx)
74	73	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ( ·𝑠 ‘ndx) ≠ (Base‘ndx))
75		fvexd 6849	. . . . . . . 8 ⊢ (𝜑 → (.r‘𝑅) ∈ V)
76		fvex 6847	. . . . . . . . 9 ⊢ ( ·𝑠 ‘ndx) ∈ V
77	76, 60	setscom 17107	. . . . . . . 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 838	. . . . . . 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 2774	. . . . 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 4836	. . . . . 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 7376	. . . 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 7393	. . . . . 6 ⊢ (𝜑 → ((𝑅 sSet ⟨(Scalar‘ndx), (𝑅 ↾s 𝐶)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r‘𝑅)⟩) ∈ V)
85		ipndxnbasendx 17252	. . . . . . 7 ⊢ (·𝑖‘ndx) ≠ (Base‘ndx)
86	85	a1i 11	. . . . . 6 ⊢ (𝜑 → (·𝑖‘ndx) ≠ (Base‘ndx))
87		fvex 6847	. . . . . . 7 ⊢ (·𝑖‘ndx) ∈ V
88	87, 60	setscom 17107	. . . . . 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 838	. . . . 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 2774	. . 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 7392	. . . 4 ⊢ 𝑆 ∈ V
93	29, 1	ressbas2 17165	. . . . . . 7 ⊢ (𝐵 ⊆ 𝐴 → 𝐵 = (Base‘𝑆))
94	4, 93	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐵 = (Base‘𝑆))
95	3, 94	sseqtrd 3970	. . . . 5 ⊢ (𝜑 → 𝐶 ⊆ (Base‘𝑆))
96	95	adantr 480	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐴 ⊆ 𝐵) → 𝐶 ⊆ (Base‘𝑆))
97		sraval 21127	. . . 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 587	. . 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 3965	. . . . . 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 6849	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐴 ⊆ 𝐵) → ((subringAlg ‘𝑅)‘𝐶) ∈ V)
103		eqid 2736	. . . . . 6 ⊢ (((subringAlg ‘𝑅)‘𝐶) ↾s 𝐵) = (((subringAlg ‘𝑅)‘𝐶) ↾s 𝐵)
104	103, 16	ressval2 17162	. . . . 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 1373	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐴 ⊆ 𝐵) → (((subringAlg ‘𝑅)‘𝐶) ↾s 𝐵) = (((subringAlg ‘𝑅)‘𝐶) sSet ⟨(Base‘ndx), (𝐵 ∩ (Base‘((subringAlg ‘𝑅)‘𝐶)))⟩))
106	8	ineq2d 4172	. . . . . . . 8 ⊢ (𝜑 → (𝐵 ∩ 𝐴) = (𝐵 ∩ (Base‘((subringAlg ‘𝑅)‘𝐶))))
107	106, 44	eqtr3d 2773	. . . . . . 7 ⊢ (𝜑 → (𝐵 ∩ (Base‘((subringAlg ‘𝑅)‘𝐶))) = 𝐵)
108	107	opeq2d 4836	. . . . . 6 ⊢ (𝜑 → ⟨(Base‘ndx), (𝐵 ∩ (Base‘((subringAlg ‘𝑅)‘𝐶)))⟩ = ⟨(Base‘ndx), 𝐵⟩)
109	108	oveq2d 7374	. . . . 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 21127	. . . . . . 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 584	. . . . . 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 7373	. . . . 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 2775	. . 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 2781	. 2 ⊢ ((𝜑 ∧ ¬ 𝐴 ⊆ 𝐵) → ((subringAlg ‘𝑆)‘𝐶) = (((subringAlg ‘𝑅)‘𝐶) ↾s 𝐵))
117	34, 116	pm2.61dan 812	1 ⊢ (𝜑 → ((subringAlg ‘𝑆)‘𝐶) = (((subringAlg ‘𝑅)‘𝐶) ↾s 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113   ≠ wne 2932  Vcvv 3440   ∩ cin 3900   ⊆ wss 3901  ⟨cop 4586  ‘cfv 6492  (class class class)co 7358   sSet csts 17090  ndxcnx 17120  Basecbs 17136   ↾_s cress 17157  ._rcmulr 17178  Scalarcsca 17180   ·_𝑠 cvsca 17181  ·_𝑖cip 17182  subringAlg csra 21123

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 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-cnex 11082  ax-resscn 11083  ax-1cn 11084  ax-icn 11085  ax-addcl 11086  ax-addrcl 11087  ax-mulcl 11088  ax-mulrcl 11089  ax-mulcom 11090  ax-addass 11091  ax-mulass 11092  ax-distr 11093  ax-i2m1 11094  ax-1ne0 11095  ax-1rid 11096  ax-rnegex 11097  ax-rrecex 11098  ax-cnre 11099  ax-pre-lttri 11100  ax-pre-lttrn 11101  ax-pre-ltadd 11102  ax-pre-mulgt0 11103

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 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 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-er 8635  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11168  df-mnf 11169  df-xr 11170  df-ltxr 11171  df-le 11172  df-sub 11366  df-neg 11367  df-nn 12146  df-2 12208  df-3 12209  df-4 12210  df-5 12211  df-6 12212  df-7 12213  df-8 12214  df-sets 17091  df-slot 17109  df-ndx 17121  df-base 17137  df-ress 17158  df-mulr 17191  df-sca 17193  df-vsca 17194  df-ip 17195  df-sra 21125

This theorem is referenced by:  lsssra  33744  fldextrspunlem1  33832  algextdeglem2  33875