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Theorem mplind 22026

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 2737	. . . . . 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 21929	. . . . 5 ⊢ (𝜑 → (𝐼 mPwSer 𝑅) ∈ AssAlg)
5		inss2 4179	. . . . . 6 ⊢ (𝐻 ∩ 𝐵) ⊆ 𝐵
6		mplind.sy	. . . . . . . 8 ⊢ 𝑌 = (𝐼 mPoly 𝑅)
7		mplind.sb	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝑌)
8		crngring 20184	. . . . . . . . 9 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
9	3, 8	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ Ring)
10	1, 6, 7, 2, 9	mplsubrg 21961	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ (SubRing‘(𝐼 mPwSer 𝑅)))
11		eqid 2737	. . . . . . . 8 ⊢ (Base‘(𝐼 mPwSer 𝑅)) = (Base‘(𝐼 mPwSer 𝑅))
12	11	subrgss 20507	. . . . . . 7 ⊢ (𝐵 ∈ (SubRing‘(𝐼 mPwSer 𝑅)) → 𝐵 ⊆ (Base‘(𝐼 mPwSer 𝑅)))
13	10, 12	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐵 ⊆ (Base‘(𝐼 mPwSer 𝑅)))
14	5, 13	sstrid 3934	. . . . 5 ⊢ (𝜑 → (𝐻 ∩ 𝐵) ⊆ (Base‘(𝐼 mPwSer 𝑅)))
15		mplind.sv	. . . . . . . . 9 ⊢ 𝑉 = (𝐼 mVar 𝑅)
16	6, 15, 7, 2, 9	mvrf2 21949	. . . . . . . 8 ⊢ (𝜑 → 𝑉:𝐼⟶𝐵)
17	16	ffnd 6661	. . . . . . 7 ⊢ (𝜑 → 𝑉 Fn 𝐼)
18		mplind.v	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑉‘𝑥) ∈ 𝐻)
19	18	ralrimiva 3130	. . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ 𝐼 (𝑉‘𝑥) ∈ 𝐻)
20		fnfvrnss 7065	. . . . . . 7 ⊢ ((𝑉 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝑉‘𝑥) ∈ 𝐻) → ran 𝑉 ⊆ 𝐻)
21	17, 19, 20	syl2anc 585	. . . . . 6 ⊢ (𝜑 → ran 𝑉 ⊆ 𝐻)
22	16	frnd 6668	. . . . . 6 ⊢ (𝜑 → ran 𝑉 ⊆ 𝐵)
23	21, 22	ssind 4182	. . . . 5 ⊢ (𝜑 → ran 𝑉 ⊆ (𝐻 ∩ 𝐵))
24		eqid 2737	. . . . . 6 ⊢ (AlgSpan‘(𝐼 mPwSer 𝑅)) = (AlgSpan‘(𝐼 mPwSer 𝑅))
25	24, 11	aspss 21833	. . . . 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 1374	. . . 4 ⊢ (𝜑 → ((AlgSpan‘(𝐼 mPwSer 𝑅))‘ran 𝑉) ⊆ ((AlgSpan‘(𝐼 mPwSer 𝑅))‘(𝐻 ∩ 𝐵)))
27	6, 1, 15, 24, 2, 3	mplbas2 21998	. . . . 5 ⊢ (𝜑 → ((AlgSpan‘(𝐼 mPwSer 𝑅))‘ran 𝑉) = (Base‘𝑌))
28	27, 7	eqtr4di 2790	. . . 4 ⊢ (𝜑 → ((AlgSpan‘(𝐼 mPwSer 𝑅))‘ran 𝑉) = 𝐵)
29	5	a1i 11	. . . . . . . 8 ⊢ (𝜑 → (𝐻 ∩ 𝐵) ⊆ 𝐵)
30	6	mplassa 21978	. . . . . . . . . . . . . 14 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑅 ∈ CRing) → 𝑌 ∈ AssAlg)
31	2, 3, 30	syl2anc 585	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑌 ∈ AssAlg)
32		mplind.sc	. . . . . . . . . . . . . 14 ⊢ 𝐶 = (algSc‘𝑌)
33		eqid 2737	. . . . . . . . . . . . . 14 ⊢ (Scalar‘𝑌) = (Scalar‘𝑌)
34	32, 33	asclrhm 21847	. . . . . . . . . . . . 13 ⊢ (𝑌 ∈ AssAlg → 𝐶 ∈ ((Scalar‘𝑌) RingHom 𝑌))
35	31, 34	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐶 ∈ ((Scalar‘𝑌) RingHom 𝑌))
36		eqid 2737	. . . . . . . . . . . . 13 ⊢ (1_r‘(Scalar‘𝑌)) = (1_r‘(Scalar‘𝑌))
37		eqid 2737	. . . . . . . . . . . . 13 ⊢ (1_r‘𝑌) = (1_r‘𝑌)
38	36, 37	rhm1 20426	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ ((Scalar‘𝑌) RingHom 𝑌) → (𝐶‘(1_r‘(Scalar‘𝑌))) = (1_r‘𝑌))
39	35, 38	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐶‘(1_r‘(Scalar‘𝑌))) = (1_r‘𝑌))
40		fveq2 6832	. . . . . . . . . . . . 13 ⊢ (𝑥 = (1_r‘(Scalar‘𝑌)) → (𝐶‘𝑥) = (𝐶‘(1_r‘(Scalar‘𝑌))))
41	40	eleq1d 2822	. . . . . . . . . . . 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 3130	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑥 ∈ 𝐾 (𝐶‘𝑥) ∈ 𝐻)
44	6, 2, 3	mplsca 21969	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑅 = (Scalar‘𝑌))
45	44, 9	eqeltrrd 2838	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (Scalar‘𝑌) ∈ Ring)
46		eqid 2737	. . . . . . . . . . . . . . 15 ⊢ (Base‘(Scalar‘𝑌)) = (Base‘(Scalar‘𝑌))
47	46, 36	ringidcl 20204	. . . . . . . . . . . . . 14 ⊢ ((Scalar‘𝑌) ∈ Ring → (1_r‘(Scalar‘𝑌)) ∈ (Base‘(Scalar‘𝑌)))
48	45, 47	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (1_r‘(Scalar‘𝑌)) ∈ (Base‘(Scalar‘𝑌)))
49		mplind.sk	. . . . . . . . . . . . . 14 ⊢ 𝐾 = (Base‘𝑅)
50	44	fveq2d 6836	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (Base‘𝑅) = (Base‘(Scalar‘𝑌)))
51	49, 50	eqtrid 2784	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐾 = (Base‘(Scalar‘𝑌)))
52	48, 51	eleqtrrd 2840	. . . . . . . . . . . 12 ⊢ (𝜑 → (1_r‘(Scalar‘𝑌)) ∈ 𝐾)
53	41, 43, 52	rspcdva 3566	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐶‘(1_r‘(Scalar‘𝑌))) ∈ 𝐻)
54	39, 53	eqeltrrd 2838	. . . . . . . . . 10 ⊢ (𝜑 → (1_r‘𝑌) ∈ 𝐻)
55		assaring 21818	. . . . . . . . . . . 12 ⊢ (𝑌 ∈ AssAlg → 𝑌 ∈ Ring)
56	31, 55	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑌 ∈ Ring)
57	7, 37	ringidcl 20204	. . . . . . . . . . 11 ⊢ (𝑌 ∈ Ring → (1_r‘𝑌) ∈ 𝐵)
58	56, 57	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (1_r‘𝑌) ∈ 𝐵)
59	54, 58	elind 4141	. . . . . . . . 9 ⊢ (𝜑 → (1_r‘𝑌) ∈ (𝐻 ∩ 𝐵))
60	59	ne0d 4283	. . . . . . . 8 ⊢ (𝜑 → (𝐻 ∩ 𝐵) ≠ ∅)
61		elinel1 4142	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ (𝐻 ∩ 𝐵) → 𝑧 ∈ 𝐻)
62		elinel1 4142	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∈ (𝐻 ∩ 𝐵) → 𝑤 ∈ 𝐻)
63	61, 62	anim12i 614	. . . . . . . . . . . . . 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 7550	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐻 ∧ 𝑤 ∈ 𝐻)) → (𝑧 + 𝑤) ∈ 𝐻)
66	63, 65	sylan2 594	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑧 ∈ (𝐻 ∩ 𝐵) ∧ 𝑤 ∈ (𝐻 ∩ 𝐵))) → (𝑧 + 𝑤) ∈ 𝐻)
67		assalmod 21817	. . . . . . . . . . . . . . . . 17 ⊢ (𝑌 ∈ AssAlg → 𝑌 ∈ LMod)
68	31, 67	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑌 ∈ LMod)
69		lmodgrp 20820	. . . . . . . . . . . . . . . 16 ⊢ (𝑌 ∈ LMod → 𝑌 ∈ Grp)
70	68, 69	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑌 ∈ Grp)
71	70	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑧 ∈ (𝐻 ∩ 𝐵) ∧ 𝑤 ∈ (𝐻 ∩ 𝐵))) → 𝑌 ∈ Grp)
72		simprl 771	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑧 ∈ (𝐻 ∩ 𝐵) ∧ 𝑤 ∈ (𝐻 ∩ 𝐵))) → 𝑧 ∈ (𝐻 ∩ 𝐵))
73	72	elin2d 4146	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑧 ∈ (𝐻 ∩ 𝐵) ∧ 𝑤 ∈ (𝐻 ∩ 𝐵))) → 𝑧 ∈ 𝐵)
74		simprr 773	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑧 ∈ (𝐻 ∩ 𝐵) ∧ 𝑤 ∈ (𝐻 ∩ 𝐵))) → 𝑤 ∈ (𝐻 ∩ 𝐵))
75	74	elin2d 4146	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑧 ∈ (𝐻 ∩ 𝐵) ∧ 𝑤 ∈ (𝐻 ∩ 𝐵))) → 𝑤 ∈ 𝐵)
76		mplind.sp	. . . . . . . . . . . . . . 15 ⊢ + = (+_g‘𝑌)
77	7, 76	grpcl 18875	. . . . . . . . . . . . . 14 ⊢ ((𝑌 ∈ Grp ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) → (𝑧 + 𝑤) ∈ 𝐵)
78	71, 73, 75, 77	syl3anc 1374	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑧 ∈ (𝐻 ∩ 𝐵) ∧ 𝑤 ∈ (𝐻 ∩ 𝐵))) → (𝑧 + 𝑤) ∈ 𝐵)
79	66, 78	elind 4141	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑧 ∈ (𝐻 ∩ 𝐵) ∧ 𝑤 ∈ (𝐻 ∩ 𝐵))) → (𝑧 + 𝑤) ∈ (𝐻 ∩ 𝐵))
80	79	anassrs 467	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝐻 ∩ 𝐵)) ∧ 𝑤 ∈ (𝐻 ∩ 𝐵)) → (𝑧 + 𝑤) ∈ (𝐻 ∩ 𝐵))
81	80	ralrimiva 3130	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐻 ∩ 𝐵)) → ∀𝑤 ∈ (𝐻 ∩ 𝐵)(𝑧 + 𝑤) ∈ (𝐻 ∩ 𝐵))
82		mplind.st	. . . . . . . . . . . . 13 ⊢ · = (._r‘𝑌)
83		eqid 2737	. . . . . . . . . . . . 13 ⊢ (inv_g‘𝑌) = (inv_g‘𝑌)
84	56	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐻 ∩ 𝐵)) → 𝑌 ∈ Ring)
85		simpr 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐻 ∩ 𝐵)) → 𝑧 ∈ (𝐻 ∩ 𝐵))
86	85	elin2d 4146	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐻 ∩ 𝐵)) → 𝑧 ∈ 𝐵)
87	7, 82, 37, 83, 84, 86	ringnegl 20241	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐻 ∩ 𝐵)) → (((inv_g‘𝑌)‘(1_r‘𝑌)) · 𝑧) = ((inv_g‘𝑌)‘𝑧))
88		simpl 482	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐻 ∩ 𝐵)) → 𝜑)
89		rhmghm 20421	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐶 ∈ ((Scalar‘𝑌) RingHom 𝑌) → 𝐶 ∈ ((Scalar‘𝑌) GrpHom 𝑌))
90	35, 89	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐶 ∈ ((Scalar‘𝑌) GrpHom 𝑌))
91		eqid 2737	. . . . . . . . . . . . . . . . . 18 ⊢ (inv_g‘(Scalar‘𝑌)) = (inv_g‘(Scalar‘𝑌))
92	46, 91, 83	ghminv 19156	. . . . . . . . . . . . . . . . 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 585	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐶‘((inv_g‘(Scalar‘𝑌))‘(1_r‘(Scalar‘𝑌)))) = ((inv_g‘𝑌)‘(𝐶‘(1_r‘(Scalar‘𝑌)))))
94	39	fveq2d 6836	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((inv_g‘𝑌)‘(𝐶‘(1_r‘(Scalar‘𝑌)))) = ((inv_g‘𝑌)‘(1_r‘𝑌)))
95	93, 94	eqtrd 2772	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐶‘((inv_g‘(Scalar‘𝑌))‘(1_r‘(Scalar‘𝑌)))) = ((inv_g‘𝑌)‘(1_r‘𝑌)))
96		fveq2 6832	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = ((inv_g‘(Scalar‘𝑌))‘(1_r‘(Scalar‘𝑌))) → (𝐶‘𝑥) = (𝐶‘((inv_g‘(Scalar‘𝑌))‘(1_r‘(Scalar‘𝑌)))))
97	96	eleq1d 2822	. . . . . . . . . . . . . . . 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 20177	. . . . . . . . . . . . . . . . . . 19 ⊢ ((Scalar‘𝑌) ∈ Ring → (Scalar‘𝑌) ∈ Grp)
99	45, 98	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (Scalar‘𝑌) ∈ Grp)
100	46, 91	grpinvcl 18921	. . . . . . . . . . . . . . . . . 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 585	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((inv_g‘(Scalar‘𝑌))‘(1_r‘(Scalar‘𝑌))) ∈ (Base‘(Scalar‘𝑌)))
102	101, 51	eleqtrrd 2840	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((inv_g‘(Scalar‘𝑌))‘(1_r‘(Scalar‘𝑌))) ∈ 𝐾)
103	97, 43, 102	rspcdva 3566	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐶‘((inv_g‘(Scalar‘𝑌))‘(1_r‘(Scalar‘𝑌)))) ∈ 𝐻)
104	95, 103	eqeltrrd 2838	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((inv_g‘𝑌)‘(1_r‘𝑌)) ∈ 𝐻)
105	104	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐻 ∩ 𝐵)) → ((inv_g‘𝑌)‘(1_r‘𝑌)) ∈ 𝐻)
106	85	elin1d 4145	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐻 ∩ 𝐵)) → 𝑧 ∈ 𝐻)
107		mplind.t	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻)) → (𝑥 · 𝑦) ∈ 𝐻)
108	107	caovclg 7550	. . . . . . . . . . . . 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 837	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐻 ∩ 𝐵)) → (((inv_g‘𝑌)‘(1_r‘𝑌)) · 𝑧) ∈ 𝐻)
110	87, 109	eqeltrrd 2838	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐻 ∩ 𝐵)) → ((inv_g‘𝑌)‘𝑧) ∈ 𝐻)
111	70	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐻 ∩ 𝐵)) → 𝑌 ∈ Grp)
112	7, 83	grpinvcl 18921	. . . . . . . . . . . 12 ⊢ ((𝑌 ∈ Grp ∧ 𝑧 ∈ 𝐵) → ((inv_g‘𝑌)‘𝑧) ∈ 𝐵)
113	111, 86, 112	syl2anc 585	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐻 ∩ 𝐵)) → ((inv_g‘𝑌)‘𝑧) ∈ 𝐵)
114	110, 113	elind 4141	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐻 ∩ 𝐵)) → ((inv_g‘𝑌)‘𝑧) ∈ (𝐻 ∩ 𝐵))
115	81, 114	jca 511	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐻 ∩ 𝐵)) → (∀𝑤 ∈ (𝐻 ∩ 𝐵)(𝑧 + 𝑤) ∈ (𝐻 ∩ 𝐵) ∧ ((inv_g‘𝑌)‘𝑧) ∈ (𝐻 ∩ 𝐵)))
116	115	ralrimiva 3130	. . . . . . . 8 ⊢ (𝜑 → ∀𝑧 ∈ (𝐻 ∩ 𝐵)(∀𝑤 ∈ (𝐻 ∩ 𝐵)(𝑧 + 𝑤) ∈ (𝐻 ∩ 𝐵) ∧ ((inv_g‘𝑌)‘𝑧) ∈ (𝐻 ∩ 𝐵)))
117	7, 76, 83	issubg2 19075	. . . . . . . . 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 17	. . . . . . . 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 1344	. . . . . . 7 ⊢ (𝜑 → (𝐻 ∩ 𝐵) ∈ (SubGrp‘𝑌))
120		elinel1 4142	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝐻 ∩ 𝐵) → 𝑥 ∈ 𝐻)
121		elinel1 4142	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (𝐻 ∩ 𝐵) → 𝑦 ∈ 𝐻)
122	120, 121	anim12i 614	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (𝐻 ∩ 𝐵) ∧ 𝑦 ∈ (𝐻 ∩ 𝐵)) → (𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻))
123	122, 107	sylan2 594	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐻 ∩ 𝐵) ∧ 𝑦 ∈ (𝐻 ∩ 𝐵))) → (𝑥 · 𝑦) ∈ 𝐻)
124	56	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐻 ∩ 𝐵) ∧ 𝑦 ∈ (𝐻 ∩ 𝐵))) → 𝑌 ∈ Ring)
125		simprl 771	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐻 ∩ 𝐵) ∧ 𝑦 ∈ (𝐻 ∩ 𝐵))) → 𝑥 ∈ (𝐻 ∩ 𝐵))
126	125	elin2d 4146	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐻 ∩ 𝐵) ∧ 𝑦 ∈ (𝐻 ∩ 𝐵))) → 𝑥 ∈ 𝐵)
127		simprr 773	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐻 ∩ 𝐵) ∧ 𝑦 ∈ (𝐻 ∩ 𝐵))) → 𝑦 ∈ (𝐻 ∩ 𝐵))
128	127	elin2d 4146	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐻 ∩ 𝐵) ∧ 𝑦 ∈ (𝐻 ∩ 𝐵))) → 𝑦 ∈ 𝐵)
129	7, 82	ringcl 20189	. . . . . . . . . 10 ⊢ ((𝑌 ∈ Ring ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 · 𝑦) ∈ 𝐵)
130	124, 126, 128, 129	syl3anc 1374	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐻 ∩ 𝐵) ∧ 𝑦 ∈ (𝐻 ∩ 𝐵))) → (𝑥 · 𝑦) ∈ 𝐵)
131	123, 130	elind 4141	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐻 ∩ 𝐵) ∧ 𝑦 ∈ (𝐻 ∩ 𝐵))) → (𝑥 · 𝑦) ∈ (𝐻 ∩ 𝐵))
132	131	ralrimivva 3181	. . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ (𝐻 ∩ 𝐵)∀𝑦 ∈ (𝐻 ∩ 𝐵)(𝑥 · 𝑦) ∈ (𝐻 ∩ 𝐵))
133	7, 37, 82	issubrg2 20527	. . . . . . . 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 17	. . . . . . 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 1344	. . . . . 6 ⊢ (𝜑 → (𝐻 ∩ 𝐵) ∈ (SubRing‘𝑌))
136	6, 1, 7	mplval2 21952	. . . . . . . 8 ⊢ 𝑌 = ((𝐼 mPwSer 𝑅) ↾_s 𝐵)
137	136	subsubrg 20533	. . . . . . 7 ⊢ (𝐵 ∈ (SubRing‘(𝐼 mPwSer 𝑅)) → ((𝐻 ∩ 𝐵) ∈ (SubRing‘𝑌) ↔ ((𝐻 ∩ 𝐵) ∈ (SubRing‘(𝐼 mPwSer 𝑅)) ∧ (𝐻 ∩ 𝐵) ⊆ 𝐵)))
138	137	simprbda 498	. . . . . 6 ⊢ ((𝐵 ∈ (SubRing‘(𝐼 mPwSer 𝑅)) ∧ (𝐻 ∩ 𝐵) ∈ (SubRing‘𝑌)) → (𝐻 ∩ 𝐵) ∈ (SubRing‘(𝐼 mPwSer 𝑅)))
139	10, 135, 138	syl2anc 585	. . . . 5 ⊢ (𝜑 → (𝐻 ∩ 𝐵) ∈ (SubRing‘(𝐼 mPwSer 𝑅)))
140		assalmod 21817	. . . . . . 7 ⊢ ((𝐼 mPwSer 𝑅) ∈ AssAlg → (𝐼 mPwSer 𝑅) ∈ LMod)
141	4, 140	syl 17	. . . . . 6 ⊢ (𝜑 → (𝐼 mPwSer 𝑅) ∈ LMod)
142	1, 6, 7, 2, 9	mpllss 21959	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ (LSubSp‘(𝐼 mPwSer 𝑅)))
143	31	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑧 ∈ (Base‘(Scalar‘𝑌)) ∧ 𝑤 ∈ (𝐻 ∩ 𝐵))) → 𝑌 ∈ AssAlg)
144		simprl 771	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑧 ∈ (Base‘(Scalar‘𝑌)) ∧ 𝑤 ∈ (𝐻 ∩ 𝐵))) → 𝑧 ∈ (Base‘(Scalar‘𝑌)))
145		simprr 773	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑧 ∈ (Base‘(Scalar‘𝑌)) ∧ 𝑤 ∈ (𝐻 ∩ 𝐵))) → 𝑤 ∈ (𝐻 ∩ 𝐵))
146	145	elin2d 4146	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑧 ∈ (Base‘(Scalar‘𝑌)) ∧ 𝑤 ∈ (𝐻 ∩ 𝐵))) → 𝑤 ∈ 𝐵)
147		eqid 2737	. . . . . . . . . . . 12 ⊢ ( ·_𝑠 ‘𝑌) = ( ·_𝑠 ‘𝑌)
148	32, 33, 46, 7, 82, 147	asclmul1 21843	. . . . . . . . . . 11 ⊢ ((𝑌 ∈ AssAlg ∧ 𝑧 ∈ (Base‘(Scalar‘𝑌)) ∧ 𝑤 ∈ 𝐵) → ((𝐶‘𝑧) · 𝑤) = (𝑧( ·_𝑠 ‘𝑌)𝑤))
149	143, 144, 146, 148	syl3anc 1374	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ (Base‘(Scalar‘𝑌)) ∧ 𝑤 ∈ (𝐻 ∩ 𝐵))) → ((𝐶‘𝑧) · 𝑤) = (𝑧( ·_𝑠 ‘𝑌)𝑤))
150		fveq2 6832	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑧 → (𝐶‘𝑥) = (𝐶‘𝑧))
151	150	eleq1d 2822	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑧 → ((𝐶‘𝑥) ∈ 𝐻 ↔ (𝐶‘𝑧) ∈ 𝐻))
152	43	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑧 ∈ (Base‘(Scalar‘𝑌)) ∧ 𝑤 ∈ (𝐻 ∩ 𝐵))) → ∀𝑥 ∈ 𝐾 (𝐶‘𝑥) ∈ 𝐻)
153	51	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑧 ∈ (Base‘(Scalar‘𝑌)) ∧ 𝑤 ∈ (𝐻 ∩ 𝐵))) → 𝐾 = (Base‘(Scalar‘𝑌)))
154	144, 153	eleqtrrd 2840	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑧 ∈ (Base‘(Scalar‘𝑌)) ∧ 𝑤 ∈ (𝐻 ∩ 𝐵))) → 𝑧 ∈ 𝐾)
155	151, 152, 154	rspcdva 3566	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑧 ∈ (Base‘(Scalar‘𝑌)) ∧ 𝑤 ∈ (𝐻 ∩ 𝐵))) → (𝐶‘𝑧) ∈ 𝐻)
156	145	elin1d 4145	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑧 ∈ (Base‘(Scalar‘𝑌)) ∧ 𝑤 ∈ (𝐻 ∩ 𝐵))) → 𝑤 ∈ 𝐻)
157	155, 156	jca 511	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑧 ∈ (Base‘(Scalar‘𝑌)) ∧ 𝑤 ∈ (𝐻 ∩ 𝐵))) → ((𝐶‘𝑧) ∈ 𝐻 ∧ 𝑤 ∈ 𝐻))
158	107	caovclg 7550	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝐶‘𝑧) ∈ 𝐻 ∧ 𝑤 ∈ 𝐻)) → ((𝐶‘𝑧) · 𝑤) ∈ 𝐻)
159	157, 158	syldan 592	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ (Base‘(Scalar‘𝑌)) ∧ 𝑤 ∈ (𝐻 ∩ 𝐵))) → ((𝐶‘𝑧) · 𝑤) ∈ 𝐻)
160	149, 159	eqeltrrd 2838	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ (Base‘(Scalar‘𝑌)) ∧ 𝑤 ∈ (𝐻 ∩ 𝐵))) → (𝑧( ·_𝑠 ‘𝑌)𝑤) ∈ 𝐻)
161	68	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ (Base‘(Scalar‘𝑌)) ∧ 𝑤 ∈ (𝐻 ∩ 𝐵))) → 𝑌 ∈ LMod)
162	7, 33, 147, 46	lmodvscl 20831	. . . . . . . . . 10 ⊢ ((𝑌 ∈ LMod ∧ 𝑧 ∈ (Base‘(Scalar‘𝑌)) ∧ 𝑤 ∈ 𝐵) → (𝑧( ·_𝑠 ‘𝑌)𝑤) ∈ 𝐵)
163	161, 144, 146, 162	syl3anc 1374	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ (Base‘(Scalar‘𝑌)) ∧ 𝑤 ∈ (𝐻 ∩ 𝐵))) → (𝑧( ·_𝑠 ‘𝑌)𝑤) ∈ 𝐵)
164	160, 163	elind 4141	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ (Base‘(Scalar‘𝑌)) ∧ 𝑤 ∈ (𝐻 ∩ 𝐵))) → (𝑧( ·_𝑠 ‘𝑌)𝑤) ∈ (𝐻 ∩ 𝐵))
165	164	ralrimivva 3181	. . . . . . 7 ⊢ (𝜑 → ∀𝑧 ∈ (Base‘(Scalar‘𝑌))∀𝑤 ∈ (𝐻 ∩ 𝐵)(𝑧( ·_𝑠 ‘𝑌)𝑤) ∈ (𝐻 ∩ 𝐵))
166		eqid 2737	. . . . . . . . 9 ⊢ (LSubSp‘𝑌) = (LSubSp‘𝑌)
167	33, 46, 7, 147, 166	islss4 20915	. . . . . . . 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 17	. . . . . . 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 714	. . . . . 6 ⊢ (𝜑 → (𝐻 ∩ 𝐵) ∈ (LSubSp‘𝑌))
170		eqid 2737	. . . . . . . 8 ⊢ (LSubSp‘(𝐼 mPwSer 𝑅)) = (LSubSp‘(𝐼 mPwSer 𝑅))
171	136, 170, 166	lsslss 20914	. . . . . . 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 498	. . . . . 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 838	. . . . 5 ⊢ (𝜑 → (𝐻 ∩ 𝐵) ∈ (LSubSp‘(𝐼 mPwSer 𝑅)))
174	24, 11, 170	aspid 21831	. . . . 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 1374	. . . 4 ⊢ (𝜑 → ((AlgSpan‘(𝐼 mPwSer 𝑅))‘(𝐻 ∩ 𝐵)) = (𝐻 ∩ 𝐵))
176	26, 28, 175	3sstr3d 3977	. . 3 ⊢ (𝜑 → 𝐵 ⊆ (𝐻 ∩ 𝐵))
177		mplind.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
178	176, 177	sseldd 3923	. 2 ⊢ (𝜑 → 𝑋 ∈ (𝐻 ∩ 𝐵))
179	178	elin1d 4145	1 ⊢ (𝜑 → 𝑋 ∈ 𝐻)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 ∀wral 3052 ∩ cin 3889 ⊆ wss 3890 ∅c0 4274 ran crn 5623 Fn wfn 6485 ‘cfv 6490 (class class class)co 7358 Basecbs 17137 +_gcplusg 17178 ._rcmulr 17179 Scalarcsca 17181 ·_𝑠 cvsca 17182 Grpcgrp 18867 inv_gcminusg 18868 SubGrpcsubg 19054 GrpHom cghm 19145 1_rcur 20120 Ringcrg 20172 CRingccrg 20173 RingHom crh 20407 SubRingcsubrg 20504 LModclmod 20813 LSubSpclss 20884 AssAlgcasa 21807 AlgSpancasp 21808 algSccascl 21809 mPwSer cmps 21861 mVar cmvr 21862 mPoly cmpl 21863

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 5212 ax-sep 5231 ax-nul 5241 ax-pow 5300 ax-pr 5368 ax-un 7680 ax-cnex 11083 ax-resscn 11084 ax-1cn 11085 ax-icn 11086 ax-addcl 11087 ax-addrcl 11088 ax-mulcl 11089 ax-mulrcl 11090 ax-mulcom 11091 ax-addass 11092 ax-mulass 11093 ax-distr 11094 ax-i2m1 11095 ax-1ne0 11096 ax-1rid 11097 ax-rnegex 11098 ax-rrecex 11099 ax-cnre 11100 ax-pre-lttri 11101 ax-pre-lttrn 11102 ax-pre-ltadd 11103 ax-pre-mulgt0 11104

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-rmo 3343 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-tp 4573 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-iin 4937 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5517 df-eprel 5522 df-po 5530 df-so 5531 df-fr 5575 df-se 5576 df-we 5577 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-pred 6257 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-isom 6499 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7622 df-ofr 7623 df-om 7809 df-1st 7933 df-2nd 7934 df-supp 8102 df-frecs 8222 df-wrecs 8253 df-recs 8302 df-rdg 8340 df-1o 8396 df-2o 8397 df-er 8634 df-map 8766 df-pm 8767 df-ixp 8837 df-en 8885 df-dom 8886 df-sdom 8887 df-fin 8888 df-fsupp 9266 df-sup 9346 df-oi 9416 df-card 9852 df-pnf 11169 df-mnf 11170 df-xr 11171 df-ltxr 11172 df-le 11173 df-sub 11367 df-neg 11368 df-nn 12147 df-2 12209 df-3 12210 df-4 12211 df-5 12212 df-6 12213 df-7 12214 df-8 12215 df-9 12216 df-n0 12403 df-z 12490 df-dec 12609 df-uz 12753 df-fz 13425 df-fzo 13572 df-seq 13926 df-hash 14255 df-struct 17075 df-sets 17092 df-slot 17110 df-ndx 17122 df-base 17138 df-ress 17159 df-plusg 17191 df-mulr 17192 df-sca 17194 df-vsca 17195 df-ip 17196 df-tset 17197 df-ple 17198 df-ds 17200 df-hom 17202 df-cco 17203 df-0g 17362 df-gsum 17363 df-prds 17368 df-pws 17370 df-mre 17506 df-mrc 17507 df-acs 17509 df-mgm 18566 df-sgrp 18645 df-mnd 18661 df-mhm 18709 df-submnd 18710 df-grp 18870 df-minusg 18871 df-sbg 18872 df-mulg 19002 df-subg 19057 df-ghm 19146 df-cntz 19250 df-cmn 19715 df-abl 19716 df-mgp 20080 df-rng 20092 df-ur 20121 df-srg 20126 df-ring 20174 df-cring 20175 df-rhm 20410 df-subrng 20481 df-subrg 20505 df-lmod 20815 df-lss 20885 df-assa 21810 df-asp 21811 df-ascl 21812 df-psr 21866 df-mvr 21867 df-mpl 21868

This theorem is referenced by: mpfind 22071

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