cayleyhamiltonALT - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > cayleyhamiltonALT		Structured version Visualization version Unicode version

Theorem cayleyhamiltonALT 20696

Description: Alternate proof of cayleyhamilton 20695, the Cayley-Hamilton theorem. This proof does not use cayleyhamilton0 20694 directly, but has the same structure as the proof of cayleyhamilton0 20694. In contrast to the proof of cayleyhamilton0 20694, only the definitions required to formulate the theorem itself are used, causing the definitions used in the lemmas being expanded, which makes the proof longer and more difficult to read. (Contributed by AV, 25-Nov-2019.) (New usage is discouraged.) (Proof modification is discouraged.)

**Hypotheses**
Ref	Expression
cayleyhamilton.a	Mat
cayleyhamilton.b
cayleyhamilton.0
cayleyhamilton.c	CharPlyMat
cayleyhamilton.k	coe₁
cayleyhamilton.m
cayleyhamilton.e	._gmulGrp

**Assertion**
Ref	Expression
cayleyhamiltonALT	$/\$ $/\$ _g

Distinct variable groups:

Allowed substitution hints:

(

)

(

)

Proof of Theorem cayleyhamiltonALT Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cayleyhamilton.a	. . . 4 Mat
2		cayleyhamilton.b	. . . 4
3		eqid 2622	. . . 4 Poly₁ Poly₁
4		eqid 2622	. . . 4 Mat Poly₁ Mat Poly₁
5		eqid 2622	. . . 4 Mat Poly₁ Mat Poly₁
6		eqid 2622	. . . 4 Mat Poly₁ Mat Poly₁
7		eqid 2622	. . . 4 Mat Poly₁ Mat Poly₁
8		eqid 2622	. . . 4 matToPolyMat matToPolyMat
9		cayleyhamilton.c	. . . 4 CharPlyMat
10		eqid 2622	. . . 4
11		eqeq1 2626	. . . . . 6
12		eqeq1 2626	. . . . . . 7
13		breq2 4657	. . . . . . . 8
14		oveq1 6657	. . . . . . . . . . 11
15	14	fveq2d 6195	. . . . . . . . . 10
16	15	fveq2d 6195	. . . . . . . . 9 matToPolyMat matToPolyMat
17		fveq2 6191	. . . . . . . . . . 11
18	17	fveq2d 6195	. . . . . . . . . 10 matToPolyMat matToPolyMat
19	18	oveq2d 6666	. . . . . . . . 9 matToPolyMat Mat Poly₁ matToPolyMat matToPolyMat Mat Poly₁ matToPolyMat
20	16, 19	oveq12d 6668	. . . . . . . 8 matToPolyMat Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat matToPolyMat Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat
21	13, 20	ifbieq2d 4111	. . . . . . 7 Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat
22	12, 21	ifbieq2d 4111	. . . . . 6 matToPolyMat Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat matToPolyMat Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat
23	11, 22	ifbieq2d 4111	. . . . 5 Mat Poly₁ Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat matToPolyMat Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat Mat Poly₁ Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat matToPolyMat Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat
24	23	cbvmptv 4750	. . . 4 Mat Poly₁ Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat matToPolyMat Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat Mat Poly₁ Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat matToPolyMat Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat
25		eqid 2622	. . . 4 Mat Poly₁ Mat Poly₁
26		eqid 2622	. . . 4
27		cayleyhamilton.m	. . . 4
28		eqid 2622	. . . 4 cPolyMatToMat cPolyMatToMat
29		cayleyhamilton.e	. . . 4 ._gmulGrp
30		eqid 2622	. . . 4 ._gmulGrp Mat Poly₁ ._gmulGrp Mat Poly₁
31	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 24, 25, 26, 27, 28, 29, 30	cayhamlem4 20693	. . 3 $/\$ $/\$ _g coe₁ cPolyMatToMat Mat Poly₁ _g ._gmulGrp Mat Poly₁ matToPolyMat Mat Poly₁ Mat Poly₁ Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat matToPolyMat Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat
32		eqid 2622	. . . . . . . . 9 ConstPolyMat ConstPolyMat
33	28, 32	cpm2mfval 20554	. . . . . . . 8 $/\$ cPolyMatToMat ConstPolyMat coe₁
34	33	eqcomd 2628	. . . . . . 7 $/\$ ConstPolyMat coe₁ cPolyMatToMat
35	34	3adant3 1081	. . . . . 6 $/\$ $/\$ ConstPolyMat coe₁ cPolyMatToMat
36	35	fveq1d 6193	. . . . 5 $/\$ $/\$ ConstPolyMat coe₁ Mat Poly₁ _g ._gmulGrp Mat Poly₁ matToPolyMat Mat Poly₁ Mat Poly₁ Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat matToPolyMat Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat cPolyMatToMat Mat Poly₁ _g ._gmulGrp Mat Poly₁ matToPolyMat Mat Poly₁ Mat Poly₁ Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat matToPolyMat Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat
37	36	eqeq2d 2632	. . . 4 $/\$ $/\$ _g coe₁ ConstPolyMat coe₁ Mat Poly₁ _g ._gmulGrp Mat Poly₁ matToPolyMat Mat Poly₁ Mat Poly₁ Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat matToPolyMat Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat _g coe₁ cPolyMatToMat Mat Poly₁ _g ._gmulGrp Mat Poly₁ matToPolyMat Mat Poly₁ Mat Poly₁ Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat matToPolyMat Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat
38	37	2rexbidv 3057	. . 3 $/\$ $/\$ _g coe₁ ConstPolyMat coe₁ Mat Poly₁ _g ._gmulGrp Mat Poly₁ matToPolyMat Mat Poly₁ Mat Poly₁ Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat matToPolyMat Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat _g coe₁ cPolyMatToMat Mat Poly₁ _g ._gmulGrp Mat Poly₁ matToPolyMat Mat Poly₁ Mat Poly₁ Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat matToPolyMat Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat
39	31, 38	mpbird 247	. 2 $/\$ $/\$ _g coe₁ ConstPolyMat coe₁ Mat Poly₁ _g ._gmulGrp Mat Poly₁ matToPolyMat Mat Poly₁ Mat Poly₁ Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat matToPolyMat Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat
40		cayleyhamilton.k	. . . . . . . . . . . . 13 coe₁
41	40	eqcomi 2631	. . . . . . . . . . . 12 coe₁
42	41	a1i 11	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ coe₁
43	42	fveq1d 6193	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ coe₁
44	43	oveq1d 6665	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ coe₁
45	44	mpteq2dva 4744	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ coe₁
46	45	oveq2d 6666	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ _g coe₁ _g
47	46	eqeq1d 2624	. . . . . 6 $/\$ $/\$ $/\$ $/\$ _g coe₁ ConstPolyMat coe₁ Mat Poly₁ _g ._gmulGrp Mat Poly₁ matToPolyMat Mat Poly₁ Mat Poly₁ Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat matToPolyMat Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat _g ConstPolyMat coe₁ Mat Poly₁ _g ._gmulGrp Mat Poly₁ matToPolyMat Mat Poly₁ Mat Poly₁ Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat matToPolyMat Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat
48	47	biimpa 501	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ _g coe₁ ConstPolyMat coe₁ Mat Poly₁ _g ._gmulGrp Mat Poly₁ matToPolyMat Mat Poly₁ Mat Poly₁ Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat matToPolyMat Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat _g ConstPolyMat coe₁ Mat Poly₁ _g ._gmulGrp Mat Poly₁ matToPolyMat Mat Poly₁ Mat Poly₁ Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat matToPolyMat Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat
49		oveq1 6657	. . . . . . . . . . . 12 ._gmulGrp Mat Poly₁ matToPolyMat ._gmulGrp Mat Poly₁ matToPolyMat
50		fveq2 6191	. . . . . . . . . . . 12 Mat Poly₁ Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat matToPolyMat Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat Mat Poly₁ Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat matToPolyMat Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat
51	49, 50	oveq12d 6668	. . . . . . . . . . 11 ._gmulGrp Mat Poly₁ matToPolyMat Mat Poly₁ Mat Poly₁ Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat matToPolyMat Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat ._gmulGrp Mat Poly₁ matToPolyMat Mat Poly₁ Mat Poly₁ Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat matToPolyMat Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat
52	51	cbvmptv 4750	. . . . . . . . . 10 ._gmulGrp Mat Poly₁ matToPolyMat Mat Poly₁ Mat Poly₁ Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat matToPolyMat Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat ._gmulGrp Mat Poly₁ matToPolyMat Mat Poly₁ Mat Poly₁ Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat matToPolyMat Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat
53	52	oveq2i 6661	. . . . . . . . 9 Mat Poly₁ _g ._gmulGrp Mat Poly₁ matToPolyMat Mat Poly₁ Mat Poly₁ Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat matToPolyMat Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat Mat Poly₁ _g ._gmulGrp Mat Poly₁ matToPolyMat Mat Poly₁ Mat Poly₁ Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat matToPolyMat Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat
54	53	a1i 11	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ Mat Poly₁ _g ._gmulGrp Mat Poly₁ matToPolyMat Mat Poly₁ Mat Poly₁ Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat matToPolyMat Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat Mat Poly₁ _g ._gmulGrp Mat Poly₁ matToPolyMat Mat Poly₁ Mat Poly₁ Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat matToPolyMat Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat
55	1, 2, 3, 4, 5, 6, 7, 8, 24, 30	cayhamlem1 20671	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ Mat Poly₁ _g ._gmulGrp Mat Poly₁ matToPolyMat Mat Poly₁ Mat Poly₁ Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat matToPolyMat Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat Mat Poly₁
56	54, 55	eqtrd 2656	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ Mat Poly₁ _g ._gmulGrp Mat Poly₁ matToPolyMat Mat Poly₁ Mat Poly₁ Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat matToPolyMat Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat Mat Poly₁
57		fveq2 6191	. . . . . . . 8 Mat Poly₁ _g ._gmulGrp Mat Poly₁ matToPolyMat Mat Poly₁ Mat Poly₁ Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat matToPolyMat Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat Mat Poly₁ ConstPolyMat coe₁ Mat Poly₁ _g ._gmulGrp Mat Poly₁ matToPolyMat Mat Poly₁ Mat Poly₁ Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat matToPolyMat Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat ConstPolyMat coe₁ Mat Poly₁
58		crngring 18558	. . . . . . . . . . . . 13
59	58	anim2i 593	. . . . . . . . . . . 12 $/\$ $/\$
60	59	3adant3 1081	. . . . . . . . . . 11 $/\$ $/\$ $/\$
61	28, 32	cpm2mfval 20554	. . . . . . . . . . . . . 14 $/\$ cPolyMatToMat ConstPolyMat coe₁
62	61	eqcomd 2628	. . . . . . . . . . . . 13 $/\$ ConstPolyMat coe₁ cPolyMatToMat
63	62	fveq1d 6193	. . . . . . . . . . . 12 $/\$ ConstPolyMat coe₁ Mat Poly₁ cPolyMatToMat Mat Poly₁
64		eqid 2622	. . . . . . . . . . . . 13
65	1, 28, 3, 4, 64, 7	m2cpminv0 20566	. . . . . . . . . . . 12 $/\$ cPolyMatToMat Mat Poly₁
66	63, 65	eqtrd 2656	. . . . . . . . . . 11 $/\$ ConstPolyMat coe₁ Mat Poly₁
67	60, 66	syl 17	. . . . . . . . . 10 $/\$ $/\$ ConstPolyMat coe₁ Mat Poly₁
68		cayleyhamilton.0	. . . . . . . . . 10
69	67, 68	syl6eqr 2674	. . . . . . . . 9 $/\$ $/\$ ConstPolyMat coe₁ Mat Poly₁
70	69	adantr 481	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ ConstPolyMat coe₁ Mat Poly₁
71	57, 70	sylan9eqr 2678	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ Mat Poly₁ _g ._gmulGrp Mat Poly₁ matToPolyMat Mat Poly₁ Mat Poly₁ Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat matToPolyMat Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat Mat Poly₁ ConstPolyMat coe₁ Mat Poly₁ _g ._gmulGrp Mat Poly₁ matToPolyMat Mat Poly₁ Mat Poly₁ Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat matToPolyMat Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat
72	56, 71	mpdan 702	. . . . . 6 $/\$ $/\$ $/\$ $/\$ ConstPolyMat coe₁ Mat Poly₁ _g ._gmulGrp Mat Poly₁ matToPolyMat Mat Poly₁ Mat Poly₁ Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat matToPolyMat Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat
73	72	adantr 481	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ _g coe₁ ConstPolyMat coe₁ Mat Poly₁ _g ._gmulGrp Mat Poly₁ matToPolyMat Mat Poly₁ Mat Poly₁ Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat matToPolyMat Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat ConstPolyMat coe₁ Mat Poly₁ _g ._gmulGrp Mat Poly₁ matToPolyMat Mat Poly₁ Mat Poly₁ Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat matToPolyMat Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat
74	48, 73	eqtrd 2656	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ _g coe₁ ConstPolyMat coe₁ Mat Poly₁ _g ._gmulGrp Mat Poly₁ matToPolyMat Mat Poly₁ Mat Poly₁ Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat matToPolyMat Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat _g
75	74	ex 450	. . 3 $/\$ $/\$ $/\$ $/\$ _g coe₁ ConstPolyMat coe₁ Mat Poly₁ _g ._gmulGrp Mat Poly₁ matToPolyMat Mat Poly₁ Mat Poly₁ Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat matToPolyMat Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat _g
76	75	rexlimdvva 3038	. 2 $/\$ $/\$ _g coe₁ ConstPolyMat coe₁ Mat Poly₁ _g ._gmulGrp Mat Poly₁ matToPolyMat Mat Poly₁ Mat Poly₁ Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat matToPolyMat Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat Mat Poly₁ matToPolyMat _g
77	39, 76	mpd 15	1 $/\$ $/\$ _g

Colors of variables: wff setvar class

Syntax hints:

wi 4 $/\$ wa 384 $/\$ w3a 1037

wceq 1483

wcel 1990

wrex 2913

cif 4086 class class class wbr 4653

cmpt 4729

cfv 5888 (class class class)co 6650

cmpt2 6652

cmap 7857

cfn 7955

cc0 9936

c1 9937

caddc 9939

clt 10074

cmin 10266

cn 11020

cn0 11292

cfz 12326

cbs 15857

cmulr 15942

cvsca 15945

c0g 16100

_g cgsu 16101

csg 17424 ._gcmg 17540 mulGrpcmgp 18489

cur 18501

crg 18547

ccrg 18548 Poly₁cpl1 19547 coe₁cco1 19548 Mat cmat 20213 ConstPolyMat ccpmat 20508 matToPolyMat cmat2pmat 20509 cPolyMatToMat ccpmat2mat 20510 CharPlyMat cchpmat 20631

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-addf 10015 ax-mulf 10016

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-xor 1465 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-ot 4186 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-ofr 6898 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-tpos 7352 df-cur 7393 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-sup 8348 df-oi 8415 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-xnn0 11364 df-z 11378 df-dec 11494 df-uz 11688 df-rp 11833 df-fz 12327 df-fzo 12466 df-seq 12802 df-exp 12861 df-hash 13118 df-word 13299 df-lsw 13300 df-concat 13301 df-s1 13302 df-substr 13303 df-splice 13304 df-reverse 13305 df-s2 13593 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-hom 15966 df-cco 15967 df-0g 16102 df-gsum 16103 df-prds 16108 df-pws 16110 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-mhm 17335 df-submnd 17336 df-grp 17425 df-minusg 17426 df-sbg 17427 df-mulg 17541 df-subg 17591 df-ghm 17658 df-gim 17701 df-cntz 17750 df-oppg 17776 df-symg 17798 df-pmtr 17862 df-psgn 17911 df-evpm 17912 df-cmn 18195 df-abl 18196 df-mgp 18490 df-ur 18502 df-srg 18506 df-ring 18549 df-cring 18550 df-oppr 18623 df-dvdsr 18641 df-unit 18642 df-invr 18672 df-dvr 18683 df-rnghom 18715 df-drng 18749 df-subrg 18778 df-lmod 18865 df-lss 18933 df-sra 19172 df-rgmod 19173 df-assa 19312 df-ascl 19314 df-psr 19356 df-mvr 19357 df-mpl 19358 df-opsr 19360 df-psr1 19550 df-vr1 19551 df-ply1 19552 df-coe1 19553 df-cnfld 19747 df-zring 19819 df-zrh 19852 df-dsmm 20076 df-frlm 20091 df-mamu 20190 df-mat 20214 df-mdet 20391 df-madu 20440 df-cpmat 20511 df-mat2pmat 20512 df-cpmat2mat 20513 df-decpmat 20568 df-pm2mp 20598 df-chpmat 20632

This theorem is referenced by: (None)

W3C validator