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Theorem mplind 22112

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 2735	. . . . . 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 22011	. . . . 5 ⊢ (𝜑 → (𝐼 mPwSer 𝑅) ∈ AssAlg)
5		inss2 4246	. . . . . 6 ⊢ (𝐻 ∩ 𝐵) ⊆ 𝐵
6		mplind.sy	. . . . . . . 8 ⊢ 𝑌 = (𝐼 mPoly 𝑅)
7		mplind.sb	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝑌)
8		crngring 20263	. . . . . . . . 9 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
9	3, 8	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ Ring)
10	1, 6, 7, 2, 9	mplsubrg 22043	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ (SubRing‘(𝐼 mPwSer 𝑅)))
11		eqid 2735	. . . . . . . 8 ⊢ (Base‘(𝐼 mPwSer 𝑅)) = (Base‘(𝐼 mPwSer 𝑅))
12	11	subrgss 20589	. . . . . . 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 4007	. . . . 5 ⊢ (𝜑 → (𝐻 ∩ 𝐵) ⊆ (Base‘(𝐼 mPwSer 𝑅)))
15		mplind.sv	. . . . . . . . 9 ⊢ 𝑉 = (𝐼 mVar 𝑅)
16	6, 15, 7, 2, 9	mvrf2 22031	. . . . . . . 8 ⊢ (𝜑 → 𝑉:𝐼⟶𝐵)
17	16	ffnd 6738	. . . . . . 7 ⊢ (𝜑 → 𝑉 Fn 𝐼)
18		mplind.v	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑉‘𝑥) ∈ 𝐻)
19	18	ralrimiva 3144	. . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ 𝐼 (𝑉‘𝑥) ∈ 𝐻)
20		fnfvrnss 7141	. . . . . . 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 6745	. . . . . 6 ⊢ (𝜑 → ran 𝑉 ⊆ 𝐵)
23	21, 22	ssind 4249	. . . . 5 ⊢ (𝜑 → ran 𝑉 ⊆ (𝐻 ∩ 𝐵))
24		eqid 2735	. . . . . 6 ⊢ (AlgSpan‘(𝐼 mPwSer 𝑅)) = (AlgSpan‘(𝐼 mPwSer 𝑅))
25	24, 11	aspss 21915	. . . . 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 1370	. . . 4 ⊢ (𝜑 → ((AlgSpan‘(𝐼 mPwSer 𝑅))‘ran 𝑉) ⊆ ((AlgSpan‘(𝐼 mPwSer 𝑅))‘(𝐻 ∩ 𝐵)))
27	6, 1, 15, 24, 2, 3	mplbas2 22078	. . . . 5 ⊢ (𝜑 → ((AlgSpan‘(𝐼 mPwSer 𝑅))‘ran 𝑉) = (Base‘𝑌))
28	27, 7	eqtr4di 2793	. . . 4 ⊢ (𝜑 → ((AlgSpan‘(𝐼 mPwSer 𝑅))‘ran 𝑉) = 𝐵)
29	5	a1i 11	. . . . . . . 8 ⊢ (𝜑 → (𝐻 ∩ 𝐵) ⊆ 𝐵)
30	6	mplassa 22060	. . . . . . . . . . . . . 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 2735	. . . . . . . . . . . . . 14 ⊢ (Scalar‘𝑌) = (Scalar‘𝑌)
34	32, 33	asclrhm 21928	. . . . . . . . . . . . 13 ⊢ (𝑌 ∈ AssAlg → 𝐶 ∈ ((Scalar‘𝑌) RingHom 𝑌))
35	31, 34	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐶 ∈ ((Scalar‘𝑌) RingHom 𝑌))
36		eqid 2735	. . . . . . . . . . . . 13 ⊢ (1_r‘(Scalar‘𝑌)) = (1_r‘(Scalar‘𝑌))
37		eqid 2735	. . . . . . . . . . . . 13 ⊢ (1_r‘𝑌) = (1_r‘𝑌)
38	36, 37	rhm1 20506	. . . . . . . . . . . 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 6907	. . . . . . . . . . . . 13 ⊢ (𝑥 = (1_r‘(Scalar‘𝑌)) → (𝐶‘𝑥) = (𝐶‘(1_r‘(Scalar‘𝑌))))
41	40	eleq1d 2824	. . . . . . . . . . . 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 3144	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑥 ∈ 𝐾 (𝐶‘𝑥) ∈ 𝐻)
44	6, 2, 3	mplsca 22051	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑅 = (Scalar‘𝑌))
45	44, 9	eqeltrrd 2840	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (Scalar‘𝑌) ∈ Ring)
46		eqid 2735	. . . . . . . . . . . . . . 15 ⊢ (Base‘(Scalar‘𝑌)) = (Base‘(Scalar‘𝑌))
47	46, 36	ringidcl 20280	. . . . . . . . . . . . . 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 6911	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (Base‘𝑅) = (Base‘(Scalar‘𝑌)))
51	49, 50	eqtrid 2787	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐾 = (Base‘(Scalar‘𝑌)))
52	48, 51	eleqtrrd 2842	. . . . . . . . . . . 12 ⊢ (𝜑 → (1_r‘(Scalar‘𝑌)) ∈ 𝐾)
53	41, 43, 52	rspcdva 3623	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐶‘(1_r‘(Scalar‘𝑌))) ∈ 𝐻)
54	39, 53	eqeltrrd 2840	. . . . . . . . . 10 ⊢ (𝜑 → (1_r‘𝑌) ∈ 𝐻)
55		assaring 21899	. . . . . . . . . . . 12 ⊢ (𝑌 ∈ AssAlg → 𝑌 ∈ Ring)
56	31, 55	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑌 ∈ Ring)
57	7, 37	ringidcl 20280	. . . . . . . . . . 11 ⊢ (𝑌 ∈ Ring → (1_r‘𝑌) ∈ 𝐵)
58	56, 57	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (1_r‘𝑌) ∈ 𝐵)
59	54, 58	elind 4210	. . . . . . . . 9 ⊢ (𝜑 → (1_r‘𝑌) ∈ (𝐻 ∩ 𝐵))
60	59	ne0d 4348	. . . . . . . 8 ⊢ (𝜑 → (𝐻 ∩ 𝐵) ≠ ∅)
61		elinel1 4211	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ (𝐻 ∩ 𝐵) → 𝑧 ∈ 𝐻)
62		elinel1 4211	. . . . . . . . . . . . . . 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 7625	. . . . . . . . . . . . . 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 21898	. . . . . . . . . . . . . . . . 17 ⊢ (𝑌 ∈ AssAlg → 𝑌 ∈ LMod)
68	31, 67	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑌 ∈ LMod)
69		lmodgrp 20882	. . . . . . . . . . . . . . . 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 4215	. . . . . . . . . . . . . 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 4215	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑧 ∈ (𝐻 ∩ 𝐵) ∧ 𝑤 ∈ (𝐻 ∩ 𝐵))) → 𝑤 ∈ 𝐵)
76		mplind.sp	. . . . . . . . . . . . . . 15 ⊢ + = (+_g‘𝑌)
77	7, 76	grpcl 18972	. . . . . . . . . . . . . 14 ⊢ ((𝑌 ∈ Grp ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) → (𝑧 + 𝑤) ∈ 𝐵)
78	71, 73, 75, 77	syl3anc 1370	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑧 ∈ (𝐻 ∩ 𝐵) ∧ 𝑤 ∈ (𝐻 ∩ 𝐵))) → (𝑧 + 𝑤) ∈ 𝐵)
79	66, 78	elind 4210	. . . . . . . . . . . 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 3144	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐻 ∩ 𝐵)) → ∀𝑤 ∈ (𝐻 ∩ 𝐵)(𝑧 + 𝑤) ∈ (𝐻 ∩ 𝐵))
82		mplind.st	. . . . . . . . . . . . 13 ⊢ · = (._r‘𝑌)
83		eqid 2735	. . . . . . . . . . . . 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 4215	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐻 ∩ 𝐵)) → 𝑧 ∈ 𝐵)
87	7, 82, 37, 83, 84, 86	ringnegl 20316	. . . . . . . . . . . 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 20501	. . . . . . . . . . . . . . . . . 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 2735	. . . . . . . . . . . . . . . . . 18 ⊢ (inv_g‘(Scalar‘𝑌)) = (inv_g‘(Scalar‘𝑌))
92	46, 91, 83	ghminv 19254	. . . . . . . . . . . . . . . . 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 6911	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((inv_g‘𝑌)‘(𝐶‘(1_r‘(Scalar‘𝑌)))) = ((inv_g‘𝑌)‘(1_r‘𝑌)))
95	93, 94	eqtrd 2775	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐶‘((inv_g‘(Scalar‘𝑌))‘(1_r‘(Scalar‘𝑌)))) = ((inv_g‘𝑌)‘(1_r‘𝑌)))
96		fveq2 6907	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = ((inv_g‘(Scalar‘𝑌))‘(1_r‘(Scalar‘𝑌))) → (𝐶‘𝑥) = (𝐶‘((inv_g‘(Scalar‘𝑌))‘(1_r‘(Scalar‘𝑌)))))
97	96	eleq1d 2824	. . . . . . . . . . . . . . . 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 20256	. . . . . . . . . . . . . . . . . . 19 ⊢ ((Scalar‘𝑌) ∈ Ring → (Scalar‘𝑌) ∈ Grp)
99	45, 98	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (Scalar‘𝑌) ∈ Grp)
100	46, 91	grpinvcl 19018	. . . . . . . . . . . . . . . . . 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 2842	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((inv_g‘(Scalar‘𝑌))‘(1_r‘(Scalar‘𝑌))) ∈ 𝐾)
103	97, 43, 102	rspcdva 3623	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐶‘((inv_g‘(Scalar‘𝑌))‘(1_r‘(Scalar‘𝑌)))) ∈ 𝐻)
104	95, 103	eqeltrrd 2840	. . . . . . . . . . . . . 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 4214	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐻 ∩ 𝐵)) → 𝑧 ∈ 𝐻)
107		mplind.t	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻)) → (𝑥 · 𝑦) ∈ 𝐻)
108	107	caovclg 7625	. . . . . . . . . . . . 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 2840	. . . . . . . . . . 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 19018	. . . . . . . . . . . 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 4210	. . . . . . . . . 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 3144	. . . . . . . 8 ⊢ (𝜑 → ∀𝑧 ∈ (𝐻 ∩ 𝐵)(∀𝑤 ∈ (𝐻 ∩ 𝐵)(𝑧 + 𝑤) ∈ (𝐻 ∩ 𝐵) ∧ ((inv_g‘𝑌)‘𝑧) ∈ (𝐻 ∩ 𝐵)))
117	7, 76, 83	issubg2 19172	. . . . . . . . 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 1341	. . . . . . 7 ⊢ (𝜑 → (𝐻 ∩ 𝐵) ∈ (SubGrp‘𝑌))
120		elinel1 4211	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝐻 ∩ 𝐵) → 𝑥 ∈ 𝐻)
121		elinel1 4211	. . . . . . . . . . 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 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 4215	. . . . . . . . . 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 4215	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐻 ∩ 𝐵) ∧ 𝑦 ∈ (𝐻 ∩ 𝐵))) → 𝑦 ∈ 𝐵)
129	7, 82	ringcl 20268	. . . . . . . . . 10 ⊢ ((𝑌 ∈ Ring ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 · 𝑦) ∈ 𝐵)
130	124, 126, 128, 129	syl3anc 1370	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐻 ∩ 𝐵) ∧ 𝑦 ∈ (𝐻 ∩ 𝐵))) → (𝑥 · 𝑦) ∈ 𝐵)
131	123, 130	elind 4210	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐻 ∩ 𝐵) ∧ 𝑦 ∈ (𝐻 ∩ 𝐵))) → (𝑥 · 𝑦) ∈ (𝐻 ∩ 𝐵))
132	131	ralrimivva 3200	. . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ (𝐻 ∩ 𝐵)∀𝑦 ∈ (𝐻 ∩ 𝐵)(𝑥 · 𝑦) ∈ (𝐻 ∩ 𝐵))
133	7, 37, 82	issubrg2 20609	. . . . . . . 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 1341	. . . . . 6 ⊢ (𝜑 → (𝐻 ∩ 𝐵) ∈ (SubRing‘𝑌))
136	6, 1, 7	mplval2 22034	. . . . . . . 8 ⊢ 𝑌 = ((𝐼 mPwSer 𝑅) ↾_s 𝐵)
137	136	subsubrg 20615	. . . . . . 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 584	. . . . 5 ⊢ (𝜑 → (𝐻 ∩ 𝐵) ∈ (SubRing‘(𝐼 mPwSer 𝑅)))
140		assalmod 21898	. . . . . . 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 22041	. . . . . 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 4215	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑧 ∈ (Base‘(Scalar‘𝑌)) ∧ 𝑤 ∈ (𝐻 ∩ 𝐵))) → 𝑤 ∈ 𝐵)
147		eqid 2735	. . . . . . . . . . . 12 ⊢ ( ·_𝑠 ‘𝑌) = ( ·_𝑠 ‘𝑌)
148	32, 33, 46, 7, 82, 147	asclmul1 21924	. . . . . . . . . . 11 ⊢ ((𝑌 ∈ AssAlg ∧ 𝑧 ∈ (Base‘(Scalar‘𝑌)) ∧ 𝑤 ∈ 𝐵) → ((𝐶‘𝑧) · 𝑤) = (𝑧( ·_𝑠 ‘𝑌)𝑤))
149	143, 144, 146, 148	syl3anc 1370	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ (Base‘(Scalar‘𝑌)) ∧ 𝑤 ∈ (𝐻 ∩ 𝐵))) → ((𝐶‘𝑧) · 𝑤) = (𝑧( ·_𝑠 ‘𝑌)𝑤))
150		fveq2 6907	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑧 → (𝐶‘𝑥) = (𝐶‘𝑧))
151	150	eleq1d 2824	. . . . . . . . . . . . 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 2842	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑧 ∈ (Base‘(Scalar‘𝑌)) ∧ 𝑤 ∈ (𝐻 ∩ 𝐵))) → 𝑧 ∈ 𝐾)
155	151, 152, 154	rspcdva 3623	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑧 ∈ (Base‘(Scalar‘𝑌)) ∧ 𝑤 ∈ (𝐻 ∩ 𝐵))) → (𝐶‘𝑧) ∈ 𝐻)
156	145	elin1d 4214	. . . . . . . . . . . 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 7625	. . . . . . . . . . 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 2840	. . . . . . . . 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 20893	. . . . . . . . . 10 ⊢ ((𝑌 ∈ LMod ∧ 𝑧 ∈ (Base‘(Scalar‘𝑌)) ∧ 𝑤 ∈ 𝐵) → (𝑧( ·_𝑠 ‘𝑌)𝑤) ∈ 𝐵)
163	161, 144, 146, 162	syl3anc 1370	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ (Base‘(Scalar‘𝑌)) ∧ 𝑤 ∈ (𝐻 ∩ 𝐵))) → (𝑧( ·_𝑠 ‘𝑌)𝑤) ∈ 𝐵)
164	160, 163	elind 4210	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ (Base‘(Scalar‘𝑌)) ∧ 𝑤 ∈ (𝐻 ∩ 𝐵))) → (𝑧( ·_𝑠 ‘𝑌)𝑤) ∈ (𝐻 ∩ 𝐵))
165	164	ralrimivva 3200	. . . . . . 7 ⊢ (𝜑 → ∀𝑧 ∈ (Base‘(Scalar‘𝑌))∀𝑤 ∈ (𝐻 ∩ 𝐵)(𝑧( ·_𝑠 ‘𝑌)𝑤) ∈ (𝐻 ∩ 𝐵))
166		eqid 2735	. . . . . . . . 9 ⊢ (LSubSp‘𝑌) = (LSubSp‘𝑌)
167	33, 46, 7, 147, 166	islss4 20978	. . . . . . . 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 713	. . . . . 6 ⊢ (𝜑 → (𝐻 ∩ 𝐵) ∈ (LSubSp‘𝑌))
170		eqid 2735	. . . . . . . 8 ⊢ (LSubSp‘(𝐼 mPwSer 𝑅)) = (LSubSp‘(𝐼 mPwSer 𝑅))
171	136, 170, 166	lsslss 20977	. . . . . . 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 21913	. . . . 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 1370	. . . 4 ⊢ (𝜑 → ((AlgSpan‘(𝐼 mPwSer 𝑅))‘(𝐻 ∩ 𝐵)) = (𝐻 ∩ 𝐵))
176	26, 28, 175	3sstr3d 4042	. . 3 ⊢ (𝜑 → 𝐵 ⊆ (𝐻 ∩ 𝐵))
177		mplind.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
178	176, 177	sseldd 3996	. 2 ⊢ (𝜑 → 𝑋 ∈ (𝐻 ∩ 𝐵))
179	178	elin1d 4214	1 ⊢ (𝜑 → 𝑋 ∈ 𝐻)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ∀wral 3059 ∩ cin 3962 ⊆ wss 3963 ∅c0 4339 ran crn 5690 Fn wfn 6558 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 +_gcplusg 17298 ._rcmulr 17299 Scalarcsca 17301 ·_𝑠 cvsca 17302 Grpcgrp 18964 inv_gcminusg 18965 SubGrpcsubg 19151 GrpHom cghm 19243 1_rcur 20199 Ringcrg 20251 CRingccrg 20252 RingHom crh 20486 SubRingcsubrg 20586 LModclmod 20875 LSubSpclss 20947 AssAlgcasa 21888 AlgSpancasp 21889 algSccascl 21890 mPwSer cmps 21942 mVar cmvr 21943 mPoly cmpl 21944

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-ofr 7698 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-map 8867 df-pm 8868 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-sup 9480 df-oi 9548 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-fz 13545 df-fzo 13692 df-seq 14040 df-hash 14367 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-hom 17322 df-cco 17323 df-0g 17488 df-gsum 17489 df-prds 17494 df-pws 17496 df-mre 17631 df-mrc 17632 df-acs 17634 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-mhm 18809 df-submnd 18810 df-grp 18967 df-minusg 18968 df-sbg 18969 df-mulg 19099 df-subg 19154 df-ghm 19244 df-cntz 19348 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-srg 20205 df-ring 20253 df-cring 20254 df-rhm 20489 df-subrng 20563 df-subrg 20587 df-lmod 20877 df-lss 20948 df-assa 21891 df-asp 21892 df-ascl 21893 df-psr 21947 df-mvr 21948 df-mpl 21949

This theorem is referenced by: mpfind 22149

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