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Theorem mplind 21478

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 2736	. . . . . 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 21383	. . . . 5 ⊢ (𝜑 → (𝐼 mPwSer 𝑅) ∈ AssAlg)
5		inss2 4189	. . . . . 6 ⊢ (𝐻 ∩ 𝐵) ⊆ 𝐵
6		mplind.sy	. . . . . . . 8 ⊢ 𝑌 = (𝐼 mPoly 𝑅)
7		mplind.sb	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝑌)
8		crngring 19976	. . . . . . . . 9 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
9	3, 8	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ Ring)
10	1, 6, 7, 2, 9	mplsubrg 21411	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ (SubRing‘(𝐼 mPwSer 𝑅)))
11		eqid 2736	. . . . . . . 8 ⊢ (Base‘(𝐼 mPwSer 𝑅)) = (Base‘(𝐼 mPwSer 𝑅))
12	11	subrgss 20223	. . . . . . 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 3955	. . . . 5 ⊢ (𝜑 → (𝐻 ∩ 𝐵) ⊆ (Base‘(𝐼 mPwSer 𝑅)))
15		mplind.sv	. . . . . . . . 9 ⊢ 𝑉 = (𝐼 mVar 𝑅)
16	6, 15, 7, 2, 9	mvrf2 21468	. . . . . . . 8 ⊢ (𝜑 → 𝑉:𝐼⟶𝐵)
17	16	ffnd 6669	. . . . . . 7 ⊢ (𝜑 → 𝑉 Fn 𝐼)
18		mplind.v	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑉‘𝑥) ∈ 𝐻)
19	18	ralrimiva 3143	. . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ 𝐼 (𝑉‘𝑥) ∈ 𝐻)
20		fnfvrnss 7068	. . . . . . 7 ⊢ ((𝑉 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝑉‘𝑥) ∈ 𝐻) → ran 𝑉 ⊆ 𝐻)
21	17, 19, 20	syl2anc 584	. . . . . 6 ⊢ (𝜑 → ran 𝑉 ⊆ 𝐻)
22	16	frnd 6676	. . . . . 6 ⊢ (𝜑 → ran 𝑉 ⊆ 𝐵)
23	21, 22	ssind 4192	. . . . 5 ⊢ (𝜑 → ran 𝑉 ⊆ (𝐻 ∩ 𝐵))
24		eqid 2736	. . . . . 6 ⊢ (AlgSpan‘(𝐼 mPwSer 𝑅)) = (AlgSpan‘(𝐼 mPwSer 𝑅))
25	24, 11	aspss 21280	. . . . 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 1371	. . . 4 ⊢ (𝜑 → ((AlgSpan‘(𝐼 mPwSer 𝑅))‘ran 𝑉) ⊆ ((AlgSpan‘(𝐼 mPwSer 𝑅))‘(𝐻 ∩ 𝐵)))
27	6, 1, 15, 24, 2, 3	mplbas2 21443	. . . . 5 ⊢ (𝜑 → ((AlgSpan‘(𝐼 mPwSer 𝑅))‘ran 𝑉) = (Base‘𝑌))
28	27, 7	eqtr4di 2794	. . . 4 ⊢ (𝜑 → ((AlgSpan‘(𝐼 mPwSer 𝑅))‘ran 𝑉) = 𝐵)
29	5	a1i 11	. . . . . . . 8 ⊢ (𝜑 → (𝐻 ∩ 𝐵) ⊆ 𝐵)
30	6	mplassa 21427	. . . . . . . . . . . . . 14 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑅 ∈ CRing) → 𝑌 ∈ AssAlg)
31	2, 3, 30	syl2anc 584	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑌 ∈ AssAlg)
32		mplind.sc	. . . . . . . . . . . . . 14 ⊢ 𝐶 = (algSc‘𝑌)
33		eqid 2736	. . . . . . . . . . . . . 14 ⊢ (Scalar‘𝑌) = (Scalar‘𝑌)
34	32, 33	asclrhm 21293	. . . . . . . . . . . . 13 ⊢ (𝑌 ∈ AssAlg → 𝐶 ∈ ((Scalar‘𝑌) RingHom 𝑌))
35	31, 34	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐶 ∈ ((Scalar‘𝑌) RingHom 𝑌))
36		eqid 2736	. . . . . . . . . . . . 13 ⊢ (1_r‘(Scalar‘𝑌)) = (1_r‘(Scalar‘𝑌))
37		eqid 2736	. . . . . . . . . . . . 13 ⊢ (1_r‘𝑌) = (1_r‘𝑌)
38	36, 37	rhm1 20162	. . . . . . . . . . . 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 6842	. . . . . . . . . . . . 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 3143	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑥 ∈ 𝐾 (𝐶‘𝑥) ∈ 𝐻)
44	6, 2, 3	mplsca 21417	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑅 = (Scalar‘𝑌))
45	44, 9	eqeltrrd 2839	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (Scalar‘𝑌) ∈ Ring)
46		eqid 2736	. . . . . . . . . . . . . . 15 ⊢ (Base‘(Scalar‘𝑌)) = (Base‘(Scalar‘𝑌))
47	46, 36	ringidcl 19989	. . . . . . . . . . . . . 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 6846	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (Base‘𝑅) = (Base‘(Scalar‘𝑌)))
51	49, 50	eqtrid 2788	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐾 = (Base‘(Scalar‘𝑌)))
52	48, 51	eleqtrrd 2841	. . . . . . . . . . . 12 ⊢ (𝜑 → (1_r‘(Scalar‘𝑌)) ∈ 𝐾)
53	41, 43, 52	rspcdva 3582	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐶‘(1_r‘(Scalar‘𝑌))) ∈ 𝐻)
54	39, 53	eqeltrrd 2839	. . . . . . . . . 10 ⊢ (𝜑 → (1_r‘𝑌) ∈ 𝐻)
55		assaring 21267	. . . . . . . . . . . 12 ⊢ (𝑌 ∈ AssAlg → 𝑌 ∈ Ring)
56	31, 55	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑌 ∈ Ring)
57	7, 37	ringidcl 19989	. . . . . . . . . . 11 ⊢ (𝑌 ∈ Ring → (1_r‘𝑌) ∈ 𝐵)
58	56, 57	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (1_r‘𝑌) ∈ 𝐵)
59	54, 58	elind 4154	. . . . . . . . 9 ⊢ (𝜑 → (1_r‘𝑌) ∈ (𝐻 ∩ 𝐵))
60	59	ne0d 4295	. . . . . . . 8 ⊢ (𝜑 → (𝐻 ∩ 𝐵) ≠ ∅)
61		elinel1 4155	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ (𝐻 ∩ 𝐵) → 𝑧 ∈ 𝐻)
62		elinel1 4155	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∈ (𝐻 ∩ 𝐵) → 𝑤 ∈ 𝐻)
63	61, 62	anim12i 613	. . . . . . . . . . . . . 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 7546	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐻 ∧ 𝑤 ∈ 𝐻)) → (𝑧 + 𝑤) ∈ 𝐻)
66	63, 65	sylan2 593	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑧 ∈ (𝐻 ∩ 𝐵) ∧ 𝑤 ∈ (𝐻 ∩ 𝐵))) → (𝑧 + 𝑤) ∈ 𝐻)
67		assalmod 21266	. . . . . . . . . . . . . . . . 17 ⊢ (𝑌 ∈ AssAlg → 𝑌 ∈ LMod)
68	31, 67	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑌 ∈ LMod)
69		lmodgrp 20329	. . . . . . . . . . . . . . . 16 ⊢ (𝑌 ∈ LMod → 𝑌 ∈ Grp)
70	68, 69	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑌 ∈ Grp)
71	70	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑧 ∈ (𝐻 ∩ 𝐵) ∧ 𝑤 ∈ (𝐻 ∩ 𝐵))) → 𝑌 ∈ Grp)
72		simprl 769	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑧 ∈ (𝐻 ∩ 𝐵) ∧ 𝑤 ∈ (𝐻 ∩ 𝐵))) → 𝑧 ∈ (𝐻 ∩ 𝐵))
73	72	elin2d 4159	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑧 ∈ (𝐻 ∩ 𝐵) ∧ 𝑤 ∈ (𝐻 ∩ 𝐵))) → 𝑧 ∈ 𝐵)
74		simprr 771	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑧 ∈ (𝐻 ∩ 𝐵) ∧ 𝑤 ∈ (𝐻 ∩ 𝐵))) → 𝑤 ∈ (𝐻 ∩ 𝐵))
75	74	elin2d 4159	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑧 ∈ (𝐻 ∩ 𝐵) ∧ 𝑤 ∈ (𝐻 ∩ 𝐵))) → 𝑤 ∈ 𝐵)
76		mplind.sp	. . . . . . . . . . . . . . 15 ⊢ + = (+_g‘𝑌)
77	7, 76	grpcl 18756	. . . . . . . . . . . . . 14 ⊢ ((𝑌 ∈ Grp ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) → (𝑧 + 𝑤) ∈ 𝐵)
78	71, 73, 75, 77	syl3anc 1371	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑧 ∈ (𝐻 ∩ 𝐵) ∧ 𝑤 ∈ (𝐻 ∩ 𝐵))) → (𝑧 + 𝑤) ∈ 𝐵)
79	66, 78	elind 4154	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑧 ∈ (𝐻 ∩ 𝐵) ∧ 𝑤 ∈ (𝐻 ∩ 𝐵))) → (𝑧 + 𝑤) ∈ (𝐻 ∩ 𝐵))
80	79	anassrs 468	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝐻 ∩ 𝐵)) ∧ 𝑤 ∈ (𝐻 ∩ 𝐵)) → (𝑧 + 𝑤) ∈ (𝐻 ∩ 𝐵))
81	80	ralrimiva 3143	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐻 ∩ 𝐵)) → ∀𝑤 ∈ (𝐻 ∩ 𝐵)(𝑧 + 𝑤) ∈ (𝐻 ∩ 𝐵))
82		mplind.st	. . . . . . . . . . . . 13 ⊢ · = (._r‘𝑌)
83		eqid 2736	. . . . . . . . . . . . 13 ⊢ (inv_g‘𝑌) = (inv_g‘𝑌)
84	56	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐻 ∩ 𝐵)) → 𝑌 ∈ Ring)
85		simpr 485	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐻 ∩ 𝐵)) → 𝑧 ∈ (𝐻 ∩ 𝐵))
86	85	elin2d 4159	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐻 ∩ 𝐵)) → 𝑧 ∈ 𝐵)
87	7, 82, 37, 83, 84, 86	ringnegl 20018	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐻 ∩ 𝐵)) → (((inv_g‘𝑌)‘(1_r‘𝑌)) · 𝑧) = ((inv_g‘𝑌)‘𝑧))
88		simpl 483	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐻 ∩ 𝐵)) → 𝜑)
89		rhmghm 20157	. . . . . . . . . . . . . . . . . 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 2736	. . . . . . . . . . . . . . . . . 18 ⊢ (inv_g‘(Scalar‘𝑌)) = (inv_g‘(Scalar‘𝑌))
92	46, 91, 83	ghminv 19015	. . . . . . . . . . . . . . . . 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 584	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐶‘((inv_g‘(Scalar‘𝑌))‘(1_r‘(Scalar‘𝑌)))) = ((inv_g‘𝑌)‘(𝐶‘(1_r‘(Scalar‘𝑌)))))
94	39	fveq2d 6846	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((inv_g‘𝑌)‘(𝐶‘(1_r‘(Scalar‘𝑌)))) = ((inv_g‘𝑌)‘(1_r‘𝑌)))
95	93, 94	eqtrd 2776	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐶‘((inv_g‘(Scalar‘𝑌))‘(1_r‘(Scalar‘𝑌)))) = ((inv_g‘𝑌)‘(1_r‘𝑌)))
96		fveq2 6842	. . . . . . . . . . . . . . . . 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 19969	. . . . . . . . . . . . . . . . . . 19 ⊢ ((Scalar‘𝑌) ∈ Ring → (Scalar‘𝑌) ∈ Grp)
99	45, 98	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (Scalar‘𝑌) ∈ Grp)
100	46, 91	grpinvcl 18798	. . . . . . . . . . . . . . . . . 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 584	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((inv_g‘(Scalar‘𝑌))‘(1_r‘(Scalar‘𝑌))) ∈ (Base‘(Scalar‘𝑌)))
102	101, 51	eleqtrrd 2841	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((inv_g‘(Scalar‘𝑌))‘(1_r‘(Scalar‘𝑌))) ∈ 𝐾)
103	97, 43, 102	rspcdva 3582	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐶‘((inv_g‘(Scalar‘𝑌))‘(1_r‘(Scalar‘𝑌)))) ∈ 𝐻)
104	95, 103	eqeltrrd 2839	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((inv_g‘𝑌)‘(1_r‘𝑌)) ∈ 𝐻)
105	104	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐻 ∩ 𝐵)) → ((inv_g‘𝑌)‘(1_r‘𝑌)) ∈ 𝐻)
106	85	elin1d 4158	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐻 ∩ 𝐵)) → 𝑧 ∈ 𝐻)
107		mplind.t	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻)) → (𝑥 · 𝑦) ∈ 𝐻)
108	107	caovclg 7546	. . . . . . . . . . . . 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 835	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐻 ∩ 𝐵)) → (((inv_g‘𝑌)‘(1_r‘𝑌)) · 𝑧) ∈ 𝐻)
110	87, 109	eqeltrrd 2839	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐻 ∩ 𝐵)) → ((inv_g‘𝑌)‘𝑧) ∈ 𝐻)
111	70	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐻 ∩ 𝐵)) → 𝑌 ∈ Grp)
112	7, 83	grpinvcl 18798	. . . . . . . . . . . 12 ⊢ ((𝑌 ∈ Grp ∧ 𝑧 ∈ 𝐵) → ((inv_g‘𝑌)‘𝑧) ∈ 𝐵)
113	111, 86, 112	syl2anc 584	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐻 ∩ 𝐵)) → ((inv_g‘𝑌)‘𝑧) ∈ 𝐵)
114	110, 113	elind 4154	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐻 ∩ 𝐵)) → ((inv_g‘𝑌)‘𝑧) ∈ (𝐻 ∩ 𝐵))
115	81, 114	jca 512	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐻 ∩ 𝐵)) → (∀𝑤 ∈ (𝐻 ∩ 𝐵)(𝑧 + 𝑤) ∈ (𝐻 ∩ 𝐵) ∧ ((inv_g‘𝑌)‘𝑧) ∈ (𝐻 ∩ 𝐵)))
116	115	ralrimiva 3143	. . . . . . . 8 ⊢ (𝜑 → ∀𝑧 ∈ (𝐻 ∩ 𝐵)(∀𝑤 ∈ (𝐻 ∩ 𝐵)(𝑧 + 𝑤) ∈ (𝐻 ∩ 𝐵) ∧ ((inv_g‘𝑌)‘𝑧) ∈ (𝐻 ∩ 𝐵)))
117	7, 76, 83	issubg2 18943	. . . . . . . . 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 1342	. . . . . . 7 ⊢ (𝜑 → (𝐻 ∩ 𝐵) ∈ (SubGrp‘𝑌))
120		elinel1 4155	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝐻 ∩ 𝐵) → 𝑥 ∈ 𝐻)
121		elinel1 4155	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (𝐻 ∩ 𝐵) → 𝑦 ∈ 𝐻)
122	120, 121	anim12i 613	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (𝐻 ∩ 𝐵) ∧ 𝑦 ∈ (𝐻 ∩ 𝐵)) → (𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻))
123	122, 107	sylan2 593	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐻 ∩ 𝐵) ∧ 𝑦 ∈ (𝐻 ∩ 𝐵))) → (𝑥 · 𝑦) ∈ 𝐻)
124	56	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐻 ∩ 𝐵) ∧ 𝑦 ∈ (𝐻 ∩ 𝐵))) → 𝑌 ∈ Ring)
125		simprl 769	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐻 ∩ 𝐵) ∧ 𝑦 ∈ (𝐻 ∩ 𝐵))) → 𝑥 ∈ (𝐻 ∩ 𝐵))
126	125	elin2d 4159	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐻 ∩ 𝐵) ∧ 𝑦 ∈ (𝐻 ∩ 𝐵))) → 𝑥 ∈ 𝐵)
127		simprr 771	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐻 ∩ 𝐵) ∧ 𝑦 ∈ (𝐻 ∩ 𝐵))) → 𝑦 ∈ (𝐻 ∩ 𝐵))
128	127	elin2d 4159	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐻 ∩ 𝐵) ∧ 𝑦 ∈ (𝐻 ∩ 𝐵))) → 𝑦 ∈ 𝐵)
129	7, 82	ringcl 19981	. . . . . . . . . 10 ⊢ ((𝑌 ∈ Ring ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 · 𝑦) ∈ 𝐵)
130	124, 126, 128, 129	syl3anc 1371	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐻 ∩ 𝐵) ∧ 𝑦 ∈ (𝐻 ∩ 𝐵))) → (𝑥 · 𝑦) ∈ 𝐵)
131	123, 130	elind 4154	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐻 ∩ 𝐵) ∧ 𝑦 ∈ (𝐻 ∩ 𝐵))) → (𝑥 · 𝑦) ∈ (𝐻 ∩ 𝐵))
132	131	ralrimivva 3197	. . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ (𝐻 ∩ 𝐵)∀𝑦 ∈ (𝐻 ∩ 𝐵)(𝑥 · 𝑦) ∈ (𝐻 ∩ 𝐵))
133	7, 37, 82	issubrg2 20242	. . . . . . . 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 1342	. . . . . 6 ⊢ (𝜑 → (𝐻 ∩ 𝐵) ∈ (SubRing‘𝑌))
136	6, 1, 7	mplval2 21402	. . . . . . . 8 ⊢ 𝑌 = ((𝐼 mPwSer 𝑅) ↾_s 𝐵)
137	136	subsubrg 20248	. . . . . . 7 ⊢ (𝐵 ∈ (SubRing‘(𝐼 mPwSer 𝑅)) → ((𝐻 ∩ 𝐵) ∈ (SubRing‘𝑌) ↔ ((𝐻 ∩ 𝐵) ∈ (SubRing‘(𝐼 mPwSer 𝑅)) ∧ (𝐻 ∩ 𝐵) ⊆ 𝐵)))
138	137	simprbda 499	. . . . . 6 ⊢ ((𝐵 ∈ (SubRing‘(𝐼 mPwSer 𝑅)) ∧ (𝐻 ∩ 𝐵) ∈ (SubRing‘𝑌)) → (𝐻 ∩ 𝐵) ∈ (SubRing‘(𝐼 mPwSer 𝑅)))
139	10, 135, 138	syl2anc 584	. . . . 5 ⊢ (𝜑 → (𝐻 ∩ 𝐵) ∈ (SubRing‘(𝐼 mPwSer 𝑅)))
140		assalmod 21266	. . . . . . 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 21409	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ (LSubSp‘(𝐼 mPwSer 𝑅)))
143	31	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑧 ∈ (Base‘(Scalar‘𝑌)) ∧ 𝑤 ∈ (𝐻 ∩ 𝐵))) → 𝑌 ∈ AssAlg)
144		simprl 769	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑧 ∈ (Base‘(Scalar‘𝑌)) ∧ 𝑤 ∈ (𝐻 ∩ 𝐵))) → 𝑧 ∈ (Base‘(Scalar‘𝑌)))
145		simprr 771	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑧 ∈ (Base‘(Scalar‘𝑌)) ∧ 𝑤 ∈ (𝐻 ∩ 𝐵))) → 𝑤 ∈ (𝐻 ∩ 𝐵))
146	145	elin2d 4159	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑧 ∈ (Base‘(Scalar‘𝑌)) ∧ 𝑤 ∈ (𝐻 ∩ 𝐵))) → 𝑤 ∈ 𝐵)
147		eqid 2736	. . . . . . . . . . . 12 ⊢ ( ·_𝑠 ‘𝑌) = ( ·_𝑠 ‘𝑌)
148	32, 33, 46, 7, 82, 147	asclmul1 21289	. . . . . . . . . . 11 ⊢ ((𝑌 ∈ AssAlg ∧ 𝑧 ∈ (Base‘(Scalar‘𝑌)) ∧ 𝑤 ∈ 𝐵) → ((𝐶‘𝑧) · 𝑤) = (𝑧( ·_𝑠 ‘𝑌)𝑤))
149	143, 144, 146, 148	syl3anc 1371	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ (Base‘(Scalar‘𝑌)) ∧ 𝑤 ∈ (𝐻 ∩ 𝐵))) → ((𝐶‘𝑧) · 𝑤) = (𝑧( ·_𝑠 ‘𝑌)𝑤))
150		fveq2 6842	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑧 → (𝐶‘𝑥) = (𝐶‘𝑧))
151	150	eleq1d 2822	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑧 → ((𝐶‘𝑥) ∈ 𝐻 ↔ (𝐶‘𝑧) ∈ 𝐻))
152	43	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑧 ∈ (Base‘(Scalar‘𝑌)) ∧ 𝑤 ∈ (𝐻 ∩ 𝐵))) → ∀𝑥 ∈ 𝐾 (𝐶‘𝑥) ∈ 𝐻)
153	51	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑧 ∈ (Base‘(Scalar‘𝑌)) ∧ 𝑤 ∈ (𝐻 ∩ 𝐵))) → 𝐾 = (Base‘(Scalar‘𝑌)))
154	144, 153	eleqtrrd 2841	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑧 ∈ (Base‘(Scalar‘𝑌)) ∧ 𝑤 ∈ (𝐻 ∩ 𝐵))) → 𝑧 ∈ 𝐾)
155	151, 152, 154	rspcdva 3582	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑧 ∈ (Base‘(Scalar‘𝑌)) ∧ 𝑤 ∈ (𝐻 ∩ 𝐵))) → (𝐶‘𝑧) ∈ 𝐻)
156	145	elin1d 4158	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑧 ∈ (Base‘(Scalar‘𝑌)) ∧ 𝑤 ∈ (𝐻 ∩ 𝐵))) → 𝑤 ∈ 𝐻)
157	155, 156	jca 512	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑧 ∈ (Base‘(Scalar‘𝑌)) ∧ 𝑤 ∈ (𝐻 ∩ 𝐵))) → ((𝐶‘𝑧) ∈ 𝐻 ∧ 𝑤 ∈ 𝐻))
158	107	caovclg 7546	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝐶‘𝑧) ∈ 𝐻 ∧ 𝑤 ∈ 𝐻)) → ((𝐶‘𝑧) · 𝑤) ∈ 𝐻)
159	157, 158	syldan 591	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ (Base‘(Scalar‘𝑌)) ∧ 𝑤 ∈ (𝐻 ∩ 𝐵))) → ((𝐶‘𝑧) · 𝑤) ∈ 𝐻)
160	149, 159	eqeltrrd 2839	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ (Base‘(Scalar‘𝑌)) ∧ 𝑤 ∈ (𝐻 ∩ 𝐵))) → (𝑧( ·_𝑠 ‘𝑌)𝑤) ∈ 𝐻)
161	68	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ (Base‘(Scalar‘𝑌)) ∧ 𝑤 ∈ (𝐻 ∩ 𝐵))) → 𝑌 ∈ LMod)
162	7, 33, 147, 46	lmodvscl 20339	. . . . . . . . . 10 ⊢ ((𝑌 ∈ LMod ∧ 𝑧 ∈ (Base‘(Scalar‘𝑌)) ∧ 𝑤 ∈ 𝐵) → (𝑧( ·_𝑠 ‘𝑌)𝑤) ∈ 𝐵)
163	161, 144, 146, 162	syl3anc 1371	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ (Base‘(Scalar‘𝑌)) ∧ 𝑤 ∈ (𝐻 ∩ 𝐵))) → (𝑧( ·_𝑠 ‘𝑌)𝑤) ∈ 𝐵)
164	160, 163	elind 4154	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ (Base‘(Scalar‘𝑌)) ∧ 𝑤 ∈ (𝐻 ∩ 𝐵))) → (𝑧( ·_𝑠 ‘𝑌)𝑤) ∈ (𝐻 ∩ 𝐵))
165	164	ralrimivva 3197	. . . . . . 7 ⊢ (𝜑 → ∀𝑧 ∈ (Base‘(Scalar‘𝑌))∀𝑤 ∈ (𝐻 ∩ 𝐵)(𝑧( ·_𝑠 ‘𝑌)𝑤) ∈ (𝐻 ∩ 𝐵))
166		eqid 2736	. . . . . . . . 9 ⊢ (LSubSp‘𝑌) = (LSubSp‘𝑌)
167	33, 46, 7, 147, 166	islss4 20423	. . . . . . . 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 711	. . . . . 6 ⊢ (𝜑 → (𝐻 ∩ 𝐵) ∈ (LSubSp‘𝑌))
170		eqid 2736	. . . . . . . 8 ⊢ (LSubSp‘(𝐼 mPwSer 𝑅)) = (LSubSp‘(𝐼 mPwSer 𝑅))
171	136, 170, 166	lsslss 20422	. . . . . . 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 499	. . . . . 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 836	. . . . 5 ⊢ (𝜑 → (𝐻 ∩ 𝐵) ∈ (LSubSp‘(𝐼 mPwSer 𝑅)))
174	24, 11, 170	aspid 21278	. . . . 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 1371	. . . 4 ⊢ (𝜑 → ((AlgSpan‘(𝐼 mPwSer 𝑅))‘(𝐻 ∩ 𝐵)) = (𝐻 ∩ 𝐵))
176	26, 28, 175	3sstr3d 3990	. . 3 ⊢ (𝜑 → 𝐵 ⊆ (𝐻 ∩ 𝐵))
177		mplind.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
178	176, 177	sseldd 3945	. 2 ⊢ (𝜑 → 𝑋 ∈ (𝐻 ∩ 𝐵))
179	178	elin1d 4158	1 ⊢ (𝜑 → 𝑋 ∈ 𝐻)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2943 ∀wral 3064 ∩ cin 3909 ⊆ wss 3910 ∅c0 4282 ran crn 5634 Fn wfn 6491 ‘cfv 6496 (class class class)co 7357 Basecbs 17083 +_gcplusg 17133 ._rcmulr 17134 Scalarcsca 17136 ·_𝑠 cvsca 17137 Grpcgrp 18748 inv_gcminusg 18749 SubGrpcsubg 18922 GrpHom cghm 19005 1_rcur 19913 Ringcrg 19964 CRingccrg 19965 RingHom crh 20143 SubRingcsubrg 20218 LModclmod 20322 LSubSpclss 20392 AssAlgcasa 21256 AlgSpancasp 21257 algSccascl 21258 mPwSer cmps 21306 mVar cmvr 21307 mPoly cmpl 21308

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-iin 4957 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-se 5589 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-of 7617 df-ofr 7618 df-om 7803 df-1st 7921 df-2nd 7922 df-supp 8093 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-er 8648 df-map 8767 df-pm 8768 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9306 df-sup 9378 df-oi 9446 df-card 9875 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-5 12219 df-6 12220 df-7 12221 df-8 12222 df-9 12223 df-n0 12414 df-z 12500 df-dec 12619 df-uz 12764 df-fz 13425 df-fzo 13568 df-seq 13907 df-hash 14231 df-struct 17019 df-sets 17036 df-slot 17054 df-ndx 17066 df-base 17084 df-ress 17113 df-plusg 17146 df-mulr 17147 df-sca 17149 df-vsca 17150 df-ip 17151 df-tset 17152 df-ple 17153 df-ds 17155 df-hom 17157 df-cco 17158 df-0g 17323 df-gsum 17324 df-prds 17329 df-pws 17331 df-mre 17466 df-mrc 17467 df-acs 17469 df-mgm 18497 df-sgrp 18546 df-mnd 18557 df-mhm 18601 df-submnd 18602 df-grp 18751 df-minusg 18752 df-sbg 18753 df-mulg 18873 df-subg 18925 df-ghm 19006 df-cntz 19097 df-cmn 19564 df-abl 19565 df-mgp 19897 df-ur 19914 df-srg 19918 df-ring 19966 df-cring 19967 df-rnghom 20146 df-subrg 20220 df-lmod 20324 df-lss 20393 df-assa 21259 df-asp 21260 df-ascl 21261 df-psr 21311 df-mvr 21312 df-mpl 21313

This theorem is referenced by: mpfind 21517

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