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Theorem mplind 22409

Description: Prove a property of polynomials by "structural" induction, under a simplified model of structure which loses the sum of products structure. The commutativity condition is stronger than strictly needed. (Contributed by Stefan O'Rear, 11-Mar-2015.)

**Hypotheses**
Ref	Expression
mplind.sk	⊢ 𝐾 = (Base‘𝑅)
mplind.sv	⊢ 𝑉 = (𝐼 mVar 𝑅)
mplind.sy	⊢ 𝑌 = (𝐼 mPoly 𝑅)
mplind.sp	⊢ + = (+_g‘𝑌)
mplind.st	⊢ · = (._r‘𝑌)
mplind.sc	⊢ 𝐶 = (algSc‘𝑌)
mplind.sb	⊢ 𝐵 = (Base‘𝑌)
mplind.p	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻)) → (𝑥 + 𝑦) ∈ 𝐻)
mplind.t	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻)) → (𝑥 · 𝑦) ∈ 𝐻)
mplind.s	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → (𝐶‘𝑥) ∈ 𝐻)
mplind.v	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑉‘𝑥) ∈ 𝐻)
mplind.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
mplind.i	⊢ (𝜑 → 𝐼 ∈ 𝑊)
mplind.r	⊢ (𝜑 → 𝑅 ∈ CRing)

**Assertion**
Ref	Expression
mplind	⊢ (𝜑 → 𝑋 ∈ 𝐻)

Distinct variable groups: 𝑥,𝑦, + 𝑥,𝐵,𝑦 𝑥,𝐶,𝑦 𝑥,𝐼 𝜑,𝑥,𝑦 𝑥,𝐻,𝑦 𝑥,𝐾 𝑥, · ,𝑦 𝑥,𝑉 𝑥,𝑌,𝑦 Allowed substitution hints: 𝑅(𝑥, 𝑦) 𝐼(𝑦) 𝐾(𝑦) 𝑉(𝑦) 𝑊(𝑥, 𝑦) 𝑋(𝑥, 𝑦)

Proof of Theorem mplind Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2763	. . . . . 6 ⊢ (𝐼 mPwSer 𝑅) = (𝐼 mPwSer 𝑅)
2		mplind.i	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ 𝑊)
3		mplind.r	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ CRing)
4	1, 2, 3	psrassa 22310	. . . . 5 ⊢ (𝜑 → (𝐼 mPwSer 𝑅) ∈ AssAlg)
5		inss2 4137	. . . . . 6 ⊢ (𝐻 ∩ 𝐵) ⊆ 𝐵
6		mplind.sy	. . . . . . . 8 ⊢ 𝑌 = (𝐼 mPoly 𝑅)
7		mplind.sb	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝑌)
8		crngring 20502	. . . . . . . . 9 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
9	3, 8	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ Ring)
10	1, 6, 7, 2, 9	mplsubrg 22342	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ (SubRing‘(𝐼 mPwSer 𝑅)))
11		eqid 2763	. . . . . . . 8 ⊢ (Base‘(𝐼 mPwSer 𝑅)) = (Base‘(𝐼 mPwSer 𝑅))
12	11	subrgss 20854	. . . . . . 7 ⊢ (𝐵 ∈ (SubRing‘(𝐼 mPwSer 𝑅)) → 𝐵 ⊆ (Base‘(𝐼 mPwSer 𝑅)))
13	10, 12	syl 18	. . . . . 6 ⊢ (𝜑 → 𝐵 ⊆ (Base‘(𝐼 mPwSer 𝑅)))
14	5, 13	sstrid 3924	. . . . 5 ⊢ (𝜑 → (𝐻 ∩ 𝐵) ⊆ (Base‘(𝐼 mPwSer 𝑅)))
15		mplind.sv	. . . . . . . . 9 ⊢ 𝑉 = (𝐼 mVar 𝑅)
16	6, 15, 7, 2, 9	mvrf2 22330	. . . . . . . 8 ⊢ (𝜑 → 𝑉:𝐼⟶𝐵)
17	16	ffnd 6732	. . . . . . 7 ⊢ (𝜑 → 𝑉 Fn 𝐼)
18		mplind.v	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑉‘𝑥) ∈ 𝐻)
19	18	ralrimiva 3158	. . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ 𝐼 (𝑉‘𝑥) ∈ 𝐻)
20		fnfvrnss 7145	. . . . . . 7 ⊢ ((𝑉 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝑉‘𝑥) ∈ 𝐻) → ran 𝑉 ⊆ 𝐻)
21	17, 19, 20	syl2anc 596	. . . . . 6 ⊢ (𝜑 → ran 𝑉 ⊆ 𝐻)
22	16	frnd 6740	. . . . . 6 ⊢ (𝜑 → ran 𝑉 ⊆ 𝐵)
23	21, 22	ssind 4140	. . . . 5 ⊢ (𝜑 → ran 𝑉 ⊆ (𝐻 ∩ 𝐵))
24		eqid 2763	. . . . . 6 ⊢ (AlgSpan‘(𝐼 mPwSer 𝑅)) = (AlgSpan‘(𝐼 mPwSer 𝑅))
25	24, 11	aspss 22214	. . . . 5 ⊢ (((𝐼 mPwSer 𝑅) ∈ AssAlg ∧ (𝐻 ∩ 𝐵) ⊆ (Base‘(𝐼 mPwSer 𝑅)) ∧ ran 𝑉 ⊆ (𝐻 ∩ 𝐵)) → ((AlgSpan‘(𝐼 mPwSer 𝑅))‘ran 𝑉) ⊆ ((AlgSpan‘(𝐼 mPwSer 𝑅))‘(𝐻 ∩ 𝐵)))
26	4, 14, 23, 25	syl3anc 1399	. . . 4 ⊢ (𝜑 → ((AlgSpan‘(𝐼 mPwSer 𝑅))‘ran 𝑉) ⊆ ((AlgSpan‘(𝐼 mPwSer 𝑅))‘(𝐻 ∩ 𝐵)))
27	6, 1, 15, 24, 2, 3	mplbas2 22381	. . . . 5 ⊢ (𝜑 → ((AlgSpan‘(𝐼 mPwSer 𝑅))‘ran 𝑉) = (Base‘𝑌))
28	27, 7	eqtr4di 2816	. . . 4 ⊢ (𝜑 → ((AlgSpan‘(𝐼 mPwSer 𝑅))‘ran 𝑉) = 𝐵)
29	5	a1i 11	. . . . . . . 8 ⊢ (𝜑 → (𝐻 ∩ 𝐵) ⊆ 𝐵)
30	6	mplassa 22359	. . . . . . . . . . . . . 14 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑅 ∈ CRing) → 𝑌 ∈ AssAlg)
31	2, 3, 30	syl2anc 596	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑌 ∈ AssAlg)
32		mplind.sc	. . . . . . . . . . . . . 14 ⊢ 𝐶 = (algSc‘𝑌)
33		eqid 2763	. . . . . . . . . . . . . 14 ⊢ (Scalar‘𝑌) = (Scalar‘𝑌)
34	32, 33	asclrhm 22228	. . . . . . . . . . . . 13 ⊢ (𝑌 ∈ AssAlg → 𝐶 ∈ ((Scalar‘𝑌) RingHom 𝑌))
35	31, 34	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐶 ∈ ((Scalar‘𝑌) RingHom 𝑌))
36		eqid 2763	. . . . . . . . . . . . 13 ⊢ (1_r‘(Scalar‘𝑌)) = (1_r‘(Scalar‘𝑌))
37		eqid 2763	. . . . . . . . . . . . 13 ⊢ (1_r‘𝑌) = (1_r‘𝑌)
38	36, 37	rhm1 20754	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ ((Scalar‘𝑌) RingHom 𝑌) → (𝐶‘(1_r‘(Scalar‘𝑌))) = (1_r‘𝑌))
39	35, 38	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐶‘(1_r‘(Scalar‘𝑌))) = (1_r‘𝑌))
40		fveq2 6908	. . . . . . . . . . . . 13 ⊢ (𝑥 = (1_r‘(Scalar‘𝑌)) → (𝐶‘𝑥) = (𝐶‘(1_r‘(Scalar‘𝑌))))
41	40	eleq1d 2849	. . . . . . . . . . . 12 ⊢ (𝑥 = (1_r‘(Scalar‘𝑌)) → ((𝐶‘𝑥) ∈ 𝐻 ↔ (𝐶‘(1_r‘(Scalar‘𝑌))) ∈ 𝐻))
42		mplind.s	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → (𝐶‘𝑥) ∈ 𝐻)
43	42	ralrimiva 3158	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑥 ∈ 𝐾 (𝐶‘𝑥) ∈ 𝐻)
44	6, 2, 3	mplsca 22350	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑅 = (Scalar‘𝑌))
45	44, 9	eqeltrrd 2865	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (Scalar‘𝑌) ∈ Ring)
46		eqid 2763	. . . . . . . . . . . . . . 15 ⊢ (Base‘(Scalar‘𝑌)) = (Base‘(Scalar‘𝑌))
47	46, 36	ringidcl 20524	. . . . . . . . . . . . . 14 ⊢ ((Scalar‘𝑌) ∈ Ring → (1_r‘(Scalar‘𝑌)) ∈ (Base‘(Scalar‘𝑌)))
48	45, 47	syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → (1_r‘(Scalar‘𝑌)) ∈ (Base‘(Scalar‘𝑌)))
49		mplind.sk	. . . . . . . . . . . . . 14 ⊢ 𝐾 = (Base‘𝑅)
50	44	fveq2d 6912	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (Base‘𝑅) = (Base‘(Scalar‘𝑌)))
51	49, 50	eqtrid 2810	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐾 = (Base‘(Scalar‘𝑌)))
52	48, 51	eleqtrrd 2867	. . . . . . . . . . . 12 ⊢ (𝜑 → (1_r‘(Scalar‘𝑌)) ∈ 𝐾)
53	41, 43, 52	rspcdva 3581	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐶‘(1_r‘(Scalar‘𝑌))) ∈ 𝐻)
54	39, 53	eqeltrrd 2865	. . . . . . . . . 10 ⊢ (𝜑 → (1_r‘𝑌) ∈ 𝐻)
55		assaring 22199	. . . . . . . . . . . 12 ⊢ (𝑌 ∈ AssAlg → 𝑌 ∈ Ring)
56	31, 55	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑌 ∈ Ring)
57	7, 37	ringidcl 20524	. . . . . . . . . . 11 ⊢ (𝑌 ∈ Ring → (1_r‘𝑌) ∈ 𝐵)
58	56, 57	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → (1_r‘𝑌) ∈ 𝐵)
59	54, 58	elind 4100	. . . . . . . . 9 ⊢ (𝜑 → (1_r‘𝑌) ∈ (𝐻 ∩ 𝐵))
60	59	ne0d 4291	. . . . . . . 8 ⊢ (𝜑 → (𝐻 ∩ 𝐵) ≠ ∅)
61		elinel1 4101	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ (𝐻 ∩ 𝐵) → 𝑧 ∈ 𝐻)
62		elinel1 4101	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∈ (𝐻 ∩ 𝐵) → 𝑤 ∈ 𝐻)
63	61, 62	anim12i 625	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ (𝐻 ∩ 𝐵) ∧ 𝑤 ∈ (𝐻 ∩ 𝐵)) → (𝑧 ∈ 𝐻 ∧ 𝑤 ∈ 𝐻))
64		mplind.p	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻)) → (𝑥 + 𝑦) ∈ 𝐻)
65	64	caovclg 7637	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐻 ∧ 𝑤 ∈ 𝐻)) → (𝑧 + 𝑤) ∈ 𝐻)
66	63, 65	sylan2 605	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑧 ∈ (𝐻 ∩ 𝐵) ∧ 𝑤 ∈ (𝐻 ∩ 𝐵))) → (𝑧 + 𝑤) ∈ 𝐻)
67		assalmod 22198	. . . . . . . . . . . . . . . . 17 ⊢ (𝑌 ∈ AssAlg → 𝑌 ∈ LMod)
68	31, 67	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑌 ∈ LMod)
69		lmodgrp 21172	. . . . . . . . . . . . . . . 16 ⊢ (𝑌 ∈ LMod → 𝑌 ∈ Grp)
70	68, 69	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑌 ∈ Grp)
71	70	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑧 ∈ (𝐻 ∩ 𝐵) ∧ 𝑤 ∈ (𝐻 ∩ 𝐵))) → 𝑌 ∈ Grp)
72		simprl 783	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑧 ∈ (𝐻 ∩ 𝐵) ∧ 𝑤 ∈ (𝐻 ∩ 𝐵))) → 𝑧 ∈ (𝐻 ∩ 𝐵))
73	72	elin2d 4105	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑧 ∈ (𝐻 ∩ 𝐵) ∧ 𝑤 ∈ (𝐻 ∩ 𝐵))) → 𝑧 ∈ 𝐵)
74		simprr 785	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑧 ∈ (𝐻 ∩ 𝐵) ∧ 𝑤 ∈ (𝐻 ∩ 𝐵))) → 𝑤 ∈ (𝐻 ∩ 𝐵))
75	74	elin2d 4105	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑧 ∈ (𝐻 ∩ 𝐵) ∧ 𝑤 ∈ (𝐻 ∩ 𝐵))) → 𝑤 ∈ 𝐵)
76		mplind.sp	. . . . . . . . . . . . . . 15 ⊢ + = (+_g‘𝑌)
77	7, 76	grpcl 19181	. . . . . . . . . . . . . 14 ⊢ ((𝑌 ∈ Grp ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) → (𝑧 + 𝑤) ∈ 𝐵)
78	71, 73, 75, 77	syl3anc 1399	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑧 ∈ (𝐻 ∩ 𝐵) ∧ 𝑤 ∈ (𝐻 ∩ 𝐵))) → (𝑧 + 𝑤) ∈ 𝐵)
79	66, 78	elind 4100	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑧 ∈ (𝐻 ∩ 𝐵) ∧ 𝑤 ∈ (𝐻 ∩ 𝐵))) → (𝑧 + 𝑤) ∈ (𝐻 ∩ 𝐵))
80	79	anassrs 473	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝐻 ∩ 𝐵)) ∧ 𝑤 ∈ (𝐻 ∩ 𝐵)) → (𝑧 + 𝑤) ∈ (𝐻 ∩ 𝐵))
81	80	ralrimiva 3158	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐻 ∩ 𝐵)) → ∀𝑤 ∈ (𝐻 ∩ 𝐵)(𝑧 + 𝑤) ∈ (𝐻 ∩ 𝐵))
82		mplind.st	. . . . . . . . . . . . 13 ⊢ · = (._r‘𝑌)
83		eqid 2763	. . . . . . . . . . . . 13 ⊢ (inv_g‘𝑌) = (inv_g‘𝑌)
84	56	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐻 ∩ 𝐵)) → 𝑌 ∈ Ring)
85		simpr 490	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐻 ∩ 𝐵)) → 𝑧 ∈ (𝐻 ∩ 𝐵))
86	85	elin2d 4105	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐻 ∩ 𝐵)) → 𝑧 ∈ 𝐵)
87	7, 82, 37, 83, 84, 86	ringnegl 20563	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐻 ∩ 𝐵)) → (((inv_g‘𝑌)‘(1_r‘𝑌)) · 𝑧) = ((inv_g‘𝑌)‘𝑧))
88		simpl 488	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐻 ∩ 𝐵)) → 𝜑)
89		rhmghm 20744	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐶 ∈ ((Scalar‘𝑌) RingHom 𝑌) → 𝐶 ∈ ((Scalar‘𝑌) GrpHom 𝑌))
90	35, 89	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐶 ∈ ((Scalar‘𝑌) GrpHom 𝑌))
91		eqid 2763	. . . . . . . . . . . . . . . . . 18 ⊢ (inv_g‘(Scalar‘𝑌)) = (inv_g‘(Scalar‘𝑌))
92	46, 91, 83	ghminv 19466	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐶 ∈ ((Scalar‘𝑌) GrpHom 𝑌) ∧ (1_r‘(Scalar‘𝑌)) ∈ (Base‘(Scalar‘𝑌))) → (𝐶‘((inv_g‘(Scalar‘𝑌))‘(1_r‘(Scalar‘𝑌)))) = ((inv_g‘𝑌)‘(𝐶‘(1_r‘(Scalar‘𝑌)))))
93	90, 48, 92	syl2anc 596	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐶‘((inv_g‘(Scalar‘𝑌))‘(1_r‘(Scalar‘𝑌)))) = ((inv_g‘𝑌)‘(𝐶‘(1_r‘(Scalar‘𝑌)))))
94	39	fveq2d 6912	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((inv_g‘𝑌)‘(𝐶‘(1_r‘(Scalar‘𝑌)))) = ((inv_g‘𝑌)‘(1_r‘𝑌)))
95	93, 94	eqtrd 2798	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐶‘((inv_g‘(Scalar‘𝑌))‘(1_r‘(Scalar‘𝑌)))) = ((inv_g‘𝑌)‘(1_r‘𝑌)))
96		fveq2 6908	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = ((inv_g‘(Scalar‘𝑌))‘(1_r‘(Scalar‘𝑌))) → (𝐶‘𝑥) = (𝐶‘((inv_g‘(Scalar‘𝑌))‘(1_r‘(Scalar‘𝑌)))))
97	96	eleq1d 2849	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = ((inv_g‘(Scalar‘𝑌))‘(1_r‘(Scalar‘𝑌))) → ((𝐶‘𝑥) ∈ 𝐻 ↔ (𝐶‘((inv_g‘(Scalar‘𝑌))‘(1_r‘(Scalar‘𝑌)))) ∈ 𝐻))
98		ringgrp 20494	. . . . . . . . . . . . . . . . . . 19 ⊢ ((Scalar‘𝑌) ∈ Ring → (Scalar‘𝑌) ∈ Grp)
99	45, 98	syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (Scalar‘𝑌) ∈ Grp)
100	46, 91	grpinvcl 19227	. . . . . . . . . . . . . . . . . 18 ⊢ (((Scalar‘𝑌) ∈ Grp ∧ (1_r‘(Scalar‘𝑌)) ∈ (Base‘(Scalar‘𝑌))) → ((inv_g‘(Scalar‘𝑌))‘(1_r‘(Scalar‘𝑌))) ∈ (Base‘(Scalar‘𝑌)))
101	99, 48, 100	syl2anc 596	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((inv_g‘(Scalar‘𝑌))‘(1_r‘(Scalar‘𝑌))) ∈ (Base‘(Scalar‘𝑌)))
102	101, 51	eleqtrrd 2867	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((inv_g‘(Scalar‘𝑌))‘(1_r‘(Scalar‘𝑌))) ∈ 𝐾)
103	97, 43, 102	rspcdva 3581	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐶‘((inv_g‘(Scalar‘𝑌))‘(1_r‘(Scalar‘𝑌)))) ∈ 𝐻)
104	95, 103	eqeltrrd 2865	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((inv_g‘𝑌)‘(1_r‘𝑌)) ∈ 𝐻)
105	104	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐻 ∩ 𝐵)) → ((inv_g‘𝑌)‘(1_r‘𝑌)) ∈ 𝐻)
106	85	elin1d 4104	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐻 ∩ 𝐵)) → 𝑧 ∈ 𝐻)
107		mplind.t	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻)) → (𝑥 · 𝑦) ∈ 𝐻)
108	107	caovclg 7637	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (((inv_g‘𝑌)‘(1_r‘𝑌)) ∈ 𝐻 ∧ 𝑧 ∈ 𝐻)) → (((inv_g‘𝑌)‘(1_r‘𝑌)) · 𝑧) ∈ 𝐻)
109	88, 105, 106, 108	syl12anc 850	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐻 ∩ 𝐵)) → (((inv_g‘𝑌)‘(1_r‘𝑌)) · 𝑧) ∈ 𝐻)
110	87, 109	eqeltrrd 2865	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐻 ∩ 𝐵)) → ((inv_g‘𝑌)‘𝑧) ∈ 𝐻)
111	70	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐻 ∩ 𝐵)) → 𝑌 ∈ Grp)
112	7, 83	grpinvcl 19227	. . . . . . . . . . . 12 ⊢ ((𝑌 ∈ Grp ∧ 𝑧 ∈ 𝐵) → ((inv_g‘𝑌)‘𝑧) ∈ 𝐵)
113	111, 86, 112	syl2anc 596	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐻 ∩ 𝐵)) → ((inv_g‘𝑌)‘𝑧) ∈ 𝐵)
114	110, 113	elind 4100	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐻 ∩ 𝐵)) → ((inv_g‘𝑌)‘𝑧) ∈ (𝐻 ∩ 𝐵))
115	81, 114	jca 521	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐻 ∩ 𝐵)) → (∀𝑤 ∈ (𝐻 ∩ 𝐵)(𝑧 + 𝑤) ∈ (𝐻 ∩ 𝐵) ∧ ((inv_g‘𝑌)‘𝑧) ∈ (𝐻 ∩ 𝐵)))
116	115	ralrimiva 3158	. . . . . . . 8 ⊢ (𝜑 → ∀𝑧 ∈ (𝐻 ∩ 𝐵)(∀𝑤 ∈ (𝐻 ∩ 𝐵)(𝑧 + 𝑤) ∈ (𝐻 ∩ 𝐵) ∧ ((inv_g‘𝑌)‘𝑧) ∈ (𝐻 ∩ 𝐵)))
117	7, 76, 83	issubg2 19381	. . . . . . . . 9 ⊢ (𝑌 ∈ Grp → ((𝐻 ∩ 𝐵) ∈ (SubGrp‘𝑌) ↔ ((𝐻 ∩ 𝐵) ⊆ 𝐵 ∧ (𝐻 ∩ 𝐵) ≠ ∅ ∧ ∀𝑧 ∈ (𝐻 ∩ 𝐵)(∀𝑤 ∈ (𝐻 ∩ 𝐵)(𝑧 + 𝑤) ∈ (𝐻 ∩ 𝐵) ∧ ((inv_g‘𝑌)‘𝑧) ∈ (𝐻 ∩ 𝐵)))))
118	70, 117	syl 18	. . . . . . . 8 ⊢ (𝜑 → ((𝐻 ∩ 𝐵) ∈ (SubGrp‘𝑌) ↔ ((𝐻 ∩ 𝐵) ⊆ 𝐵 ∧ (𝐻 ∩ 𝐵) ≠ ∅ ∧ ∀𝑧 ∈ (𝐻 ∩ 𝐵)(∀𝑤 ∈ (𝐻 ∩ 𝐵)(𝑧 + 𝑤) ∈ (𝐻 ∩ 𝐵) ∧ ((inv_g‘𝑌)‘𝑧) ∈ (𝐻 ∩ 𝐵)))))
119	29, 60, 116, 118	mpbir3and 1361	. . . . . . 7 ⊢ (𝜑 → (𝐻 ∩ 𝐵) ∈ (SubGrp‘𝑌))
120		elinel1 4101	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝐻 ∩ 𝐵) → 𝑥 ∈ 𝐻)
121		elinel1 4101	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (𝐻 ∩ 𝐵) → 𝑦 ∈ 𝐻)
122	120, 121	anim12i 625	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (𝐻 ∩ 𝐵) ∧ 𝑦 ∈ (𝐻 ∩ 𝐵)) → (𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻))
123	122, 107	sylan2 605	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐻 ∩ 𝐵) ∧ 𝑦 ∈ (𝐻 ∩ 𝐵))) → (𝑥 · 𝑦) ∈ 𝐻)
124	56	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐻 ∩ 𝐵) ∧ 𝑦 ∈ (𝐻 ∩ 𝐵))) → 𝑌 ∈ Ring)
125		simprl 783	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐻 ∩ 𝐵) ∧ 𝑦 ∈ (𝐻 ∩ 𝐵))) → 𝑥 ∈ (𝐻 ∩ 𝐵))
126	125	elin2d 4105	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐻 ∩ 𝐵) ∧ 𝑦 ∈ (𝐻 ∩ 𝐵))) → 𝑥 ∈ 𝐵)
127		simprr 785	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐻 ∩ 𝐵) ∧ 𝑦 ∈ (𝐻 ∩ 𝐵))) → 𝑦 ∈ (𝐻 ∩ 𝐵))
128	127	elin2d 4105	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐻 ∩ 𝐵) ∧ 𝑦 ∈ (𝐻 ∩ 𝐵))) → 𝑦 ∈ 𝐵)
129	7, 82	ringcl 20507	. . . . . . . . . 10 ⊢ ((𝑌 ∈ Ring ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 · 𝑦) ∈ 𝐵)
130	124, 126, 128, 129	syl3anc 1399	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐻 ∩ 𝐵) ∧ 𝑦 ∈ (𝐻 ∩ 𝐵))) → (𝑥 · 𝑦) ∈ 𝐵)
131	123, 130	elind 4100	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐻 ∩ 𝐵) ∧ 𝑦 ∈ (𝐻 ∩ 𝐵))) → (𝑥 · 𝑦) ∈ (𝐻 ∩ 𝐵))
132	131	ralrimivva 3209	. . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ (𝐻 ∩ 𝐵)∀𝑦 ∈ (𝐻 ∩ 𝐵)(𝑥 · 𝑦) ∈ (𝐻 ∩ 𝐵))
133	7, 37, 82	issubrg2 20874	. . . . . . . 8 ⊢ (𝑌 ∈ Ring → ((𝐻 ∩ 𝐵) ∈ (SubRing‘𝑌) ↔ ((𝐻 ∩ 𝐵) ∈ (SubGrp‘𝑌) ∧ (1_r‘𝑌) ∈ (𝐻 ∩ 𝐵) ∧ ∀𝑥 ∈ (𝐻 ∩ 𝐵)∀𝑦 ∈ (𝐻 ∩ 𝐵)(𝑥 · 𝑦) ∈ (𝐻 ∩ 𝐵))))
134	56, 133	syl 18	. . . . . . 7 ⊢ (𝜑 → ((𝐻 ∩ 𝐵) ∈ (SubRing‘𝑌) ↔ ((𝐻 ∩ 𝐵) ∈ (SubGrp‘𝑌) ∧ (1_r‘𝑌) ∈ (𝐻 ∩ 𝐵) ∧ ∀𝑥 ∈ (𝐻 ∩ 𝐵)∀𝑦 ∈ (𝐻 ∩ 𝐵)(𝑥 · 𝑦) ∈ (𝐻 ∩ 𝐵))))
135	119, 59, 132, 134	mpbir3and 1361	. . . . . 6 ⊢ (𝜑 → (𝐻 ∩ 𝐵) ∈ (SubRing‘𝑌))
136	6, 1, 7	mplval2 22333	. . . . . . . 8 ⊢ 𝑌 = ((𝐼 mPwSer 𝑅) ↾_s 𝐵)
137	136	subsubrg 20880	. . . . . . 7 ⊢ (𝐵 ∈ (SubRing‘(𝐼 mPwSer 𝑅)) → ((𝐻 ∩ 𝐵) ∈ (SubRing‘𝑌) ↔ ((𝐻 ∩ 𝐵) ∈ (SubRing‘(𝐼 mPwSer 𝑅)) ∧ (𝐻 ∩ 𝐵) ⊆ 𝐵)))
138	137	simprbda 504	. . . . . 6 ⊢ ((𝐵 ∈ (SubRing‘(𝐼 mPwSer 𝑅)) ∧ (𝐻 ∩ 𝐵) ∈ (SubRing‘𝑌)) → (𝐻 ∩ 𝐵) ∈ (SubRing‘(𝐼 mPwSer 𝑅)))
139	10, 135, 138	syl2anc 596	. . . . 5 ⊢ (𝜑 → (𝐻 ∩ 𝐵) ∈ (SubRing‘(𝐼 mPwSer 𝑅)))
140		assalmod 22198	. . . . . . 7 ⊢ ((𝐼 mPwSer 𝑅) ∈ AssAlg → (𝐼 mPwSer 𝑅) ∈ LMod)
141	4, 140	syl 18	. . . . . 6 ⊢ (𝜑 → (𝐼 mPwSer 𝑅) ∈ LMod)
142	1, 6, 7, 2, 9	mpllss 22340	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ (LSubSp‘(𝐼 mPwSer 𝑅)))
143	31	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑧 ∈ (Base‘(Scalar‘𝑌)) ∧ 𝑤 ∈ (𝐻 ∩ 𝐵))) → 𝑌 ∈ AssAlg)
144		simprl 783	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑧 ∈ (Base‘(Scalar‘𝑌)) ∧ 𝑤 ∈ (𝐻 ∩ 𝐵))) → 𝑧 ∈ (Base‘(Scalar‘𝑌)))
145		simprr 785	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑧 ∈ (Base‘(Scalar‘𝑌)) ∧ 𝑤 ∈ (𝐻 ∩ 𝐵))) → 𝑤 ∈ (𝐻 ∩ 𝐵))
146	145	elin2d 4105	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑧 ∈ (Base‘(Scalar‘𝑌)) ∧ 𝑤 ∈ (𝐻 ∩ 𝐵))) → 𝑤 ∈ 𝐵)
147		eqid 2763	. . . . . . . . . . . 12 ⊢ ( ·_𝑠 ‘𝑌) = ( ·_𝑠 ‘𝑌)
148	32, 33, 46, 7, 82, 147	asclmul1 22224	. . . . . . . . . . 11 ⊢ ((𝑌 ∈ AssAlg ∧ 𝑧 ∈ (Base‘(Scalar‘𝑌)) ∧ 𝑤 ∈ 𝐵) → ((𝐶‘𝑧) · 𝑤) = (𝑧( ·_𝑠 ‘𝑌)𝑤))
149	143, 144, 146, 148	syl3anc 1399	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ (Base‘(Scalar‘𝑌)) ∧ 𝑤 ∈ (𝐻 ∩ 𝐵))) → ((𝐶‘𝑧) · 𝑤) = (𝑧( ·_𝑠 ‘𝑌)𝑤))
150		fveq2 6908	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑧 → (𝐶‘𝑥) = (𝐶‘𝑧))
151	150	eleq1d 2849	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑧 → ((𝐶‘𝑥) ∈ 𝐻 ↔ (𝐶‘𝑧) ∈ 𝐻))
152	43	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑧 ∈ (Base‘(Scalar‘𝑌)) ∧ 𝑤 ∈ (𝐻 ∩ 𝐵))) → ∀𝑥 ∈ 𝐾 (𝐶‘𝑥) ∈ 𝐻)
153	51	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑧 ∈ (Base‘(Scalar‘𝑌)) ∧ 𝑤 ∈ (𝐻 ∩ 𝐵))) → 𝐾 = (Base‘(Scalar‘𝑌)))
154	144, 153	eleqtrrd 2867	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑧 ∈ (Base‘(Scalar‘𝑌)) ∧ 𝑤 ∈ (𝐻 ∩ 𝐵))) → 𝑧 ∈ 𝐾)
155	151, 152, 154	rspcdva 3581	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑧 ∈ (Base‘(Scalar‘𝑌)) ∧ 𝑤 ∈ (𝐻 ∩ 𝐵))) → (𝐶‘𝑧) ∈ 𝐻)
156	145	elin1d 4104	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑧 ∈ (Base‘(Scalar‘𝑌)) ∧ 𝑤 ∈ (𝐻 ∩ 𝐵))) → 𝑤 ∈ 𝐻)
157	155, 156	jca 521	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑧 ∈ (Base‘(Scalar‘𝑌)) ∧ 𝑤 ∈ (𝐻 ∩ 𝐵))) → ((𝐶‘𝑧) ∈ 𝐻 ∧ 𝑤 ∈ 𝐻))
158	107	caovclg 7637	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝐶‘𝑧) ∈ 𝐻 ∧ 𝑤 ∈ 𝐻)) → ((𝐶‘𝑧) · 𝑤) ∈ 𝐻)
159	157, 158	syldan 603	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ (Base‘(Scalar‘𝑌)) ∧ 𝑤 ∈ (𝐻 ∩ 𝐵))) → ((𝐶‘𝑧) · 𝑤) ∈ 𝐻)
160	149, 159	eqeltrrd 2865	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ (Base‘(Scalar‘𝑌)) ∧ 𝑤 ∈ (𝐻 ∩ 𝐵))) → (𝑧( ·_𝑠 ‘𝑌)𝑤) ∈ 𝐻)
161	68	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ (Base‘(Scalar‘𝑌)) ∧ 𝑤 ∈ (𝐻 ∩ 𝐵))) → 𝑌 ∈ LMod)
162	7, 33, 147, 46	lmodvscl 21183	. . . . . . . . . 10 ⊢ ((𝑌 ∈ LMod ∧ 𝑧 ∈ (Base‘(Scalar‘𝑌)) ∧ 𝑤 ∈ 𝐵) → (𝑧( ·_𝑠 ‘𝑌)𝑤) ∈ 𝐵)
163	161, 144, 146, 162	syl3anc 1399	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ (Base‘(Scalar‘𝑌)) ∧ 𝑤 ∈ (𝐻 ∩ 𝐵))) → (𝑧( ·_𝑠 ‘𝑌)𝑤) ∈ 𝐵)
164	160, 163	elind 4100	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ (Base‘(Scalar‘𝑌)) ∧ 𝑤 ∈ (𝐻 ∩ 𝐵))) → (𝑧( ·_𝑠 ‘𝑌)𝑤) ∈ (𝐻 ∩ 𝐵))
165	164	ralrimivva 3209	. . . . . . 7 ⊢ (𝜑 → ∀𝑧 ∈ (Base‘(Scalar‘𝑌))∀𝑤 ∈ (𝐻 ∩ 𝐵)(𝑧( ·_𝑠 ‘𝑌)𝑤) ∈ (𝐻 ∩ 𝐵))
166		eqid 2763	. . . . . . . . 9 ⊢ (LSubSp‘𝑌) = (LSubSp‘𝑌)
167	33, 46, 7, 147, 166	islss4 21267	. . . . . . . 8 ⊢ (𝑌 ∈ LMod → ((𝐻 ∩ 𝐵) ∈ (LSubSp‘𝑌) ↔ ((𝐻 ∩ 𝐵) ∈ (SubGrp‘𝑌) ∧ ∀𝑧 ∈ (Base‘(Scalar‘𝑌))∀𝑤 ∈ (𝐻 ∩ 𝐵)(𝑧( ·_𝑠 ‘𝑌)𝑤) ∈ (𝐻 ∩ 𝐵))))
168	68, 167	syl 18	. . . . . . 7 ⊢ (𝜑 → ((𝐻 ∩ 𝐵) ∈ (LSubSp‘𝑌) ↔ ((𝐻 ∩ 𝐵) ∈ (SubGrp‘𝑌) ∧ ∀𝑧 ∈ (Base‘(Scalar‘𝑌))∀𝑤 ∈ (𝐻 ∩ 𝐵)(𝑧( ·_𝑠 ‘𝑌)𝑤) ∈ (𝐻 ∩ 𝐵))))
169	119, 165, 168	mpbir2and 726	. . . . . 6 ⊢ (𝜑 → (𝐻 ∩ 𝐵) ∈ (LSubSp‘𝑌))
170		eqid 2763	. . . . . . . 8 ⊢ (LSubSp‘(𝐼 mPwSer 𝑅)) = (LSubSp‘(𝐼 mPwSer 𝑅))
171	136, 170, 166	lsslss 21266	. . . . . . 7 ⊢ (((𝐼 mPwSer 𝑅) ∈ LMod ∧ 𝐵 ∈ (LSubSp‘(𝐼 mPwSer 𝑅))) → ((𝐻 ∩ 𝐵) ∈ (LSubSp‘𝑌) ↔ ((𝐻 ∩ 𝐵) ∈ (LSubSp‘(𝐼 mPwSer 𝑅)) ∧ (𝐻 ∩ 𝐵) ⊆ 𝐵)))
172	171	simprbda 504	. . . . . 6 ⊢ ((((𝐼 mPwSer 𝑅) ∈ LMod ∧ 𝐵 ∈ (LSubSp‘(𝐼 mPwSer 𝑅))) ∧ (𝐻 ∩ 𝐵) ∈ (LSubSp‘𝑌)) → (𝐻 ∩ 𝐵) ∈ (LSubSp‘(𝐼 mPwSer 𝑅)))
173	141, 142, 169, 172	syl21anc 851	. . . . 5 ⊢ (𝜑 → (𝐻 ∩ 𝐵) ∈ (LSubSp‘(𝐼 mPwSer 𝑅)))
174	24, 11, 170	aspid 22212	. . . . 5 ⊢ (((𝐼 mPwSer 𝑅) ∈ AssAlg ∧ (𝐻 ∩ 𝐵) ∈ (SubRing‘(𝐼 mPwSer 𝑅)) ∧ (𝐻 ∩ 𝐵) ∈ (LSubSp‘(𝐼 mPwSer 𝑅))) → ((AlgSpan‘(𝐼 mPwSer 𝑅))‘(𝐻 ∩ 𝐵)) = (𝐻 ∩ 𝐵))
175	4, 139, 173, 174	syl3anc 1399	. . . 4 ⊢ (𝜑 → ((AlgSpan‘(𝐼 mPwSer 𝑅))‘(𝐻 ∩ 𝐵)) = (𝐻 ∩ 𝐵))
176	26, 28, 175	3sstr3d 3967	. . 3 ⊢ (𝜑 → 𝐵 ⊆ (𝐻 ∩ 𝐵))
177		mplind.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
178	176, 177	sseldd 3914	. 2 ⊢ (𝜑 → 𝑋 ∈ (𝐻 ∩ 𝐵))
179	178	elin1d 4104	1 ⊢ (𝜑 → 𝑋 ∈ 𝐻)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2147 ≠ wne 2959 ∀wral 3080 ⊆ wss 3900 ∩ cin 4092 ∅c0 4281 ran crn 5656 Fn wfn 6553 ‘cfv 6558 (class class class)co 7443 Basecbs 17416 +_gcplusg 17457 ._rcmulr 17458 Scalarcsca 17460 ·_𝑠 cvsca 17461 Grpcgrp 19173 inv_gcminusg 19174 SubGrpcsubg 19359 GrpHom cghm 19456 1_rcur 20437 Ringcrg 20489 CRingccrg 20490 RingHom crh 20729 SubRingcsubrg 20851 LModclmod 21165 LSubSpclss 21236 AssAlgcasa 22188 AlgSpancasp 22189 algSccascl 22190 mPwSer cmps 22242 mVar cmvr 22243 mPoly cmpl 22244

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 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5235 ax-sep 5252 ax-nul 5263 ax-pow 5331 ax-pr 5395 ax-un 7775 ax-cnex 11283 ax-resscn 11284 ax-1cn 11285 ax-icn 11286 ax-addcl 11287 ax-addrcl 11288 ax-mulcl 11289 ax-mulrcl 11290 ax-mulcom 11291 ax-addass 11292 ax-mulass 11293 ax-distr 11294 ax-i2m1 11295 ax-1ne0 11296 ax-1rid 11297 ax-rnegex 11298 ax-rrecex 11299 ax-cnre 11300 ax-pre-lttri 11301 ax-pre-lttrn 11302 ax-pre-ltadd 11303 ax-pre-mulgt0 11304

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 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4839 df-uni 4871 df-int 4911 df-iun 4956 df-iin 4957 df-br 5107 df-opab 5171 df-mpt 5190 df-tr 5216 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-se 5609 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6322 df-ord 6383 df-on 6384 df-lim 6385 df-suc 6386 df-iota 6512 df-fun 6560 df-fn 6561 df-f 6562 df-f1 6563 df-fo 6564 df-f1o 6565 df-fv 6566 df-isom 6567 df-riota 7400 df-ov 7446 df-oprab 7447 df-mpo 7448 df-of 7717 df-ofr 7718 df-om 7902 df-1st 8025 df-2nd 8026 df-supp 8205 df-frecs 8327 df-wrecs 8358 df-recs 8407 df-rdg 8446 df-1o 8504 df-2o 8505 df-er 8745 df-map 8877 df-pm 8878 df-ixp 8954 df-en 9002 df-dom 9003 df-sdom 9004 df-fin 9005 df-fsupp 9382 df-sup 9461 df-oi 9531 df-card 10047 df-pnf 11372 df-mnf 11373 df-xr 11374 df-ltxr 11375 df-le 11376 df-sub 11570 df-neg 11571 df-nn 12363 df-2 12432 df-3 12433 df-4 12434 df-5 12435 df-6 12436 df-7 12437 df-8 12438 df-9 12439 df-n0 12634 df-z 12721 df-dec 12842 df-uz 12993 df-fz 13667 df-fzo 13816 df-seq 14172 df-hash 14502 df-struct 17354 df-sets 17371 df-slot 17389 df-ndx 17401 df-base 17417 df-ress 17438 df-plusg 17470 df-mulr 17471 df-sca 17473 df-vsca 17474 df-ip 17475 df-tset 17476 df-ple 17477 df-ds 17479 df-hom 17481 df-cco 17482 df-0g 17641 df-gsum 17642 df-prds 17647 df-pws 17649 df-mre 17785 df-mrc 17786 df-acs 17788 df-mgm 18845 df-sgrp 18937 df-mnd 18953 df-mhm 19007 df-submnd 19008 df-grp 19176 df-minusg 19177 df-sbg 19178 df-mulg 19307 df-subg 19362 df-ghm 19457 df-cntz 19561 df-cmn 20026 df-abl 20027 df-mgp 20391 df-rng 20405 df-ur 20438 df-srg 20443 df-ring 20491 df-cring 20492 df-rhm 20732 df-subrng 20828 df-subrg 20852 df-lmod 21167 df-lss 21237 df-assa 22191 df-asp 22192 df-ascl 22193 df-psr 22247 df-mvr 22248 df-mpl 22249

This theorem is used by: mpfind 22454

W3C validator