Theorem lgsqrlem1 20580

Description: Lemma for lgsqr 20585. (Contributed by Mario Carneiro, 15-Jun-2015.)

Hypotheses

Ref	Expression
lgsqr.y	|- Y = (ℤ/nℤ ` P )
lgsqr.s	|- S = (Poly1 ` Y )
lgsqr.b	|- B = ( Base ` S )
lgsqr.d	|- D = ( deg1 ` Y )
lgsqr.o	|- O = (eval1 ` Y )
lgsqr.e	|- .^ = (.g ` (mulGrp ` S ) )
lgsqr.x	|- X = (var1 ` Y )
lgsqr.m	|- .- = ( -g ` S )
lgsqr.u	|- .1. = ( 1r ` S )
lgsqr.t	|- T = ( ( ( ( P - 1 ) / 2 ) .^ X ) .- .1. )
lgsqr.l	|- L = ( ZRHom ` Y )
lgsqr.1	|- ( ph -> P e. ( Prime \ { 2 } ) )
lgsqrlem1.3	|- ( ph -> A e. ZZ )
lgsqrlem1.4	|- ( ph -> ( ( A ^ ( ( P - 1 ) / 2 ) ) mod P ) = ( 1 mod P ) )

Assertion

Ref	Expression
lgsqrlem1	|- ( ph -> ( ( O ` T ) ` ( L ` A ) ) = ( 0g ` Y ) )

Proof of Theorem lgsqrlem1

Step	Hyp	Ref	Expression
1		lgsqr.t	. . . . 5 |- T = ( ( ( ( P - 1 ) / 2 ) .^ X ) .- .1. )
2	1	fveq2i 5528	. . . 4 |- ( O ` T ) = ( O ` ( ( ( ( P - 1 ) / 2 ) .^ X ) .- .1. ) )
3	2	fveq1i 5526	. . 3 |- ( ( O ` T ) ` ( L ` A ) ) = ( ( O ` ( ( ( ( P - 1 ) / 2 ) .^ X ) .- .1. ) ) ` ( L ` A ) )
4		lgsqr.o	. . . . 5 |- O = (eval1 ` Y )
5		lgsqr.s	. . . . 5 |- S = (Poly1 ` Y )
6		eqid 2283	. . . . 5 |- ( Base ` Y ) = ( Base ` Y )
7		lgsqr.b	. . . . 5 |- B = ( Base ` S )
8		lgsqr.1	. . . . . . . . 9 |- ( ph -> P e. ( Prime \ { 2 } ) )
9		eldifi 3298	. . . . . . . . 9 |- ( P e. ( Prime \ { 2 } ) -> P e. Prime )
10	8, 9	syl 15	. . . . . . . 8 |- ( ph -> P e. Prime )
11		lgsqr.y	. . . . . . . . 9 |- Y = (ℤ/nℤ ` P )
12	11	znfld 16514	. . . . . . . 8 |- ( P e. Prime -> Y e. Field )
13	10, 12	syl 15	. . . . . . 7 |- ( ph -> Y e. Field )
14		fldidom 16046	. . . . . . 7 |- ( Y e. Field -> Y e. IDomn )
15	13, 14	syl 15	. . . . . 6 |- ( ph -> Y e. IDomn )
16		isidom 16045	. . . . . . 7 |- ( Y e. IDomn <-> ( Y e. CRing /\ Y e. Domn ) )
17	16	simplbi 446	. . . . . 6 |- ( Y e. IDomn -> Y e. CRing )
18	15, 17	syl 15	. . . . 5 |- ( ph -> Y e. CRing )
19		crngrng 15351	. . . . . . . . 9 |- ( Y e. CRing -> Y e. Ring )
20	18, 19	syl 15	. . . . . . . 8 |- ( ph -> Y e. Ring )
21		eqid 2283	. . . . . . . . 9 |- (ℂfld ↾s ZZ ) = (ℂfld ↾s ZZ )
22		lgsqr.l	. . . . . . . . 9 |- L = ( ZRHom ` Y )
23	21, 22	zrhrhm 16466	. . . . . . . 8 |- ( Y e. Ring -> L e. ( (ℂfld ↾s ZZ ) RingHom Y ) )
24	20, 23	syl 15	. . . . . . 7 |- ( ph -> L e. ( (ℂfld ↾s ZZ ) RingHom Y ) )
25		zsubrg 16425	. . . . . . . . 9 |- ZZ e. (SubRing ` ℂfld )
26	21	subrgbas 15554	. . . . . . . . 9 |- ( ZZ e. (SubRing ` ℂfld ) -> ZZ = ( Base ` (ℂfld ↾s ZZ ) ) )
27	25, 26	ax-mp 8	. . . . . . . 8 |- ZZ = ( Base ` (ℂfld ↾s ZZ ) )
28	27, 6	rhmf 15504	. . . . . . 7 |- ( L e. ( (ℂfld ↾s ZZ ) RingHom Y ) -> L : ZZ --> ( Base ` Y ) )
29	24, 28	syl 15	. . . . . 6 |- ( ph -> L : ZZ --> ( Base ` Y ) )
30		lgsqrlem1.3	. . . . . 6 |- ( ph -> A e. ZZ )
31		ffvelrn 5663	. . . . . 6 |- ( ( L : ZZ --> ( Base ` Y ) /\ A e. ZZ ) -> ( L ` A ) e. ( Base ` Y ) )
32	29, 30, 31	syl2anc 642	. . . . 5 |- ( ph -> ( L ` A ) e. ( Base ` Y ) )
33		lgsqr.x	. . . . . . . 8 |- X = (var1 ` Y )
34	4, 33, 6, 5, 7, 18, 32	evl1vard 19416	. . . . . . 7 |- ( ph -> ( X e. B /\ ( ( O ` X ) ` ( L ` A ) ) = ( L ` A ) ) )
35		lgsqr.e	. . . . . . 7 |- .^ = (.g ` (mulGrp ` S ) )
36		eqid 2283	. . . . . . 7 |- (.g ` (mulGrp ` Y ) ) = (.g ` (mulGrp ` Y ) )
37		oddprm 12868	. . . . . . . . 9 |- ( P e. ( Prime \ { 2 } ) -> ( ( P - 1 ) / 2 ) e. NN )
38	8, 37	syl 15	. . . . . . . 8 |- ( ph -> ( ( P - 1 ) / 2 ) e. NN )
39	38	nnnn0d 10018	. . . . . . 7 |- ( ph -> ( ( P - 1 ) / 2 ) e. NN0 )
40	4, 5, 6, 7, 18, 32, 34, 35, 36, 39	evl1expd 19421	. . . . . 6 |- ( ph -> ( ( ( ( P - 1 ) / 2 ) .^ X ) e. B /\ ( ( O ` ( ( ( P - 1 ) / 2 ) .^ X ) ) ` ( L ` A ) ) = ( ( ( P - 1 ) / 2 ) (.g ` (mulGrp ` Y ) ) ( L ` A ) ) ) )
41		cnrng 16396	. . . . . . . . . . . . 13 |- ℂfld e. Ring
42		eqid 2283	. . . . . . . . . . . . . 14 |- (mulGrp ` ℂfld ) = (mulGrp ` ℂfld )
43	21, 42	mgpress 15336	. . . . . . . . . . . . 13 |- ( (ℂfld e. Ring /\ ZZ e. (SubRing ` ℂfld ) ) -> ( (mulGrp ` ℂfld ) ↾s ZZ ) = (mulGrp ` (ℂfld ↾s ZZ ) ) )
44	41, 25, 43	mp2an 653	. . . . . . . . . . . 12 |- ( (mulGrp ` ℂfld ) ↾s ZZ ) = (mulGrp ` (ℂfld ↾s ZZ ) )
45		eqid 2283	. . . . . . . . . . . 12 |- (mulGrp ` Y ) = (mulGrp ` Y )
46	44, 45	rhmmhm 15502	. . . . . . . . . . 11 |- ( L e. ( (ℂfld ↾s ZZ ) RingHom Y ) -> L e. ( ( (mulGrp ` ℂfld ) ↾s ZZ ) MndHom (mulGrp ` Y ) ) )
47	24, 46	syl 15	. . . . . . . . . 10 |- ( ph -> L e. ( ( (mulGrp ` ℂfld ) ↾s ZZ ) MndHom (mulGrp ` Y ) ) )
48	44, 27	mgpbas 15331	. . . . . . . . . . 11 |- ZZ = ( Base ` ( (mulGrp ` ℂfld ) ↾s ZZ ) )
49		eqid 2283	. . . . . . . . . . 11 |- (.g ` ( (mulGrp ` ℂfld ) ↾s ZZ ) ) = (.g ` ( (mulGrp ` ℂfld ) ↾s ZZ ) )
50	48, 49, 36	mhmmulg 14599	. . . . . . . . . 10 |- ( ( L e. ( ( (mulGrp ` ℂfld ) ↾s ZZ ) MndHom (mulGrp ` Y ) ) /\ ( ( P - 1 ) / 2 ) e. NN0 /\ A e. ZZ ) -> ( L ` ( ( ( P - 1 ) / 2 ) (.g ` ( (mulGrp ` ℂfld ) ↾s ZZ ) ) A ) ) = ( ( ( P - 1 ) / 2 ) (.g ` (mulGrp ` Y ) ) ( L ` A ) ) )
51	47, 39, 30, 50	syl3anc 1182	. . . . . . . . 9 |- ( ph -> ( L ` ( ( ( P - 1 ) / 2 ) (.g ` ( (mulGrp ` ℂfld ) ↾s ZZ ) ) A ) ) = ( ( ( P - 1 ) / 2 ) (.g ` (mulGrp ` Y ) ) ( L ` A ) ) )
52	42	subrgsubm 15558	. . . . . . . . . . . . . 14 |- ( ZZ e. (SubRing ` ℂfld ) -> ZZ e. (SubMnd ` (mulGrp ` ℂfld ) ) )
53	25, 52	mp1i 11	. . . . . . . . . . . . 13 |- ( ph -> ZZ e. (SubMnd ` (mulGrp ` ℂfld ) ) )
54		eqid 2283	. . . . . . . . . . . . . 14 |- (.g ` (mulGrp ` ℂfld ) ) = (.g ` (mulGrp ` ℂfld ) )
55		eqid 2283	. . . . . . . . . . . . . 14 |- ( (mulGrp ` ℂfld ) ↾s ZZ ) = ( (mulGrp ` ℂfld ) ↾s ZZ )
56	54, 55, 49	submmulg 14602	. . . . . . . . . . . . 13 |- ( ( ZZ e. (SubMnd ` (mulGrp ` ℂfld ) ) /\ ( ( P - 1 ) / 2 ) e. NN0 /\ A e. ZZ ) -> ( ( ( P - 1 ) / 2 ) (.g ` (mulGrp ` ℂfld ) ) A ) = ( ( ( P - 1 ) / 2 ) (.g ` ( (mulGrp ` ℂfld ) ↾s ZZ ) ) A ) )
57	53, 39, 30, 56	syl3anc 1182	. . . . . . . . . . . 12 |- ( ph -> ( ( ( P - 1 ) / 2 ) (.g ` (mulGrp ` ℂfld ) ) A ) = ( ( ( P - 1 ) / 2 ) (.g ` ( (mulGrp ` ℂfld ) ↾s ZZ ) ) A ) )
58	30	zcnd 10118	. . . . . . . . . . . . 13 |- ( ph -> A e. CC )
59		cnfldexp 16407	. . . . . . . . . . . . 13 |- ( ( A e. CC /\ ( ( P - 1 ) / 2 ) e. NN0 ) -> ( ( ( P - 1 ) / 2 ) (.g ` (mulGrp ` ℂfld ) ) A ) = ( A ^ ( ( P - 1 ) / 2 ) ) )
60	58, 39, 59	syl2anc 642	. . . . . . . . . . . 12 |- ( ph -> ( ( ( P - 1 ) / 2 ) (.g ` (mulGrp ` ℂfld ) ) A ) = ( A ^ ( ( P - 1 ) / 2 ) ) )
61	57, 60	eqtr3d 2317	. . . . . . . . . . 11 |- ( ph -> ( ( ( P - 1 ) / 2 ) (.g ` ( (mulGrp ` ℂfld ) ↾s ZZ ) ) A ) = ( A ^ ( ( P - 1 ) / 2 ) ) )
62	61	fveq2d 5529	. . . . . . . . . 10 |- ( ph -> ( L ` ( ( ( P - 1 ) / 2 ) (.g ` ( (mulGrp ` ℂfld ) ↾s ZZ ) ) A ) ) = ( L ` ( A ^ ( ( P - 1 ) / 2 ) ) ) )
63		lgsqrlem1.4	. . . . . . . . . . . 12 |- ( ph -> ( ( A ^ ( ( P - 1 ) / 2 ) ) mod P ) = ( 1 mod P ) )
64		prmnn 12761	. . . . . . . . . . . . . 14 |- ( P e. Prime -> P e. NN )
65	10, 64	syl 15	. . . . . . . . . . . . 13 |- ( ph -> P e. NN )
66		zexpcl 11118	. . . . . . . . . . . . . 14 |- ( ( A e. ZZ /\ ( ( P - 1 ) / 2 ) e. NN0 ) -> ( A ^ ( ( P - 1 ) / 2 ) ) e. ZZ )
67	30, 39, 66	syl2anc 642	. . . . . . . . . . . . 13 |- ( ph -> ( A ^ ( ( P - 1 ) / 2 ) ) e. ZZ )
68		1z 10053	. . . . . . . . . . . . . 14 |- 1 e. ZZ
69	68	a1i 10	. . . . . . . . . . . . 13 |- ( ph -> 1 e. ZZ )
70		moddvds 12538	. . . . . . . . . . . . 13 |- ( ( P e. NN /\ ( A ^ ( ( P - 1 ) / 2 ) ) e. ZZ /\ 1 e. ZZ ) -> ( ( ( A ^ ( ( P - 1 ) / 2 ) ) mod P ) = ( 1 mod P ) <-> P || ( ( A ^ ( ( P - 1 ) / 2 ) ) - 1 ) ) )
71	65, 67, 69, 70	syl3anc 1182	. . . . . . . . . . . 12 |- ( ph -> ( ( ( A ^ ( ( P - 1 ) / 2 ) ) mod P ) = ( 1 mod P ) <-> P || ( ( A ^ ( ( P - 1 ) / 2 ) ) - 1 ) ) )
72	63, 71	mpbid 201	. . . . . . . . . . 11 |- ( ph -> P || ( ( A ^ ( ( P - 1 ) / 2 ) ) - 1 ) )
73	65	nnnn0d 10018	. . . . . . . . . . . 12 |- ( ph -> P e. NN0 )
74	11, 22	zndvds 16503	. . . . . . . . . . . 12 |- ( ( P e. NN0 /\ ( A ^ ( ( P - 1 ) / 2 ) ) e. ZZ /\ 1 e. ZZ ) -> ( ( L ` ( A ^ ( ( P - 1 ) / 2 ) ) ) = ( L ` 1 ) <-> P || ( ( A ^ ( ( P - 1 ) / 2 ) ) - 1 ) ) )
75	73, 67, 69, 74	syl3anc 1182	. . . . . . . . . . 11 |- ( ph -> ( ( L ` ( A ^ ( ( P - 1 ) / 2 ) ) ) = ( L ` 1 ) <-> P || ( ( A ^ ( ( P - 1 ) / 2 ) ) - 1 ) ) )
76	72, 75	mpbird 223	. . . . . . . . . 10 |- ( ph -> ( L ` ( A ^ ( ( P - 1 ) / 2 ) ) ) = ( L ` 1 ) )
77		cnfld1 16399	. . . . . . . . . . . . . 14 |- 1 = ( 1r ` ℂfld )
78	21, 77	subrg1 15555	. . . . . . . . . . . . 13 |- ( ZZ e. (SubRing ` ℂfld ) -> 1 = ( 1r ` (ℂfld ↾s ZZ ) ) )
79	25, 78	ax-mp 8	. . . . . . . . . . . 12 |- 1 = ( 1r ` (ℂfld ↾s ZZ ) )
80		eqid 2283	. . . . . . . . . . . 12 |- ( 1r ` Y ) = ( 1r ` Y )
81	79, 80	rhm1 15508	. . . . . . . . . . 11 |- ( L e. ( (ℂfld ↾s ZZ ) RingHom Y ) -> ( L ` 1 ) = ( 1r ` Y ) )
82	24, 81	syl 15	. . . . . . . . . 10 |- ( ph -> ( L ` 1 ) = ( 1r ` Y ) )
83	62, 76, 82	3eqtrd 2319	. . . . . . . . 9 |- ( ph -> ( L ` ( ( ( P - 1 ) / 2 ) (.g ` ( (mulGrp ` ℂfld ) ↾s ZZ ) ) A ) ) = ( 1r ` Y ) )
84	51, 83	eqtr3d 2317	. . . . . . . 8 |- ( ph -> ( ( ( P - 1 ) / 2 ) (.g ` (mulGrp ` Y ) ) ( L ` A ) ) = ( 1r ` Y ) )
85	84	eqeq2d 2294	. . . . . . 7 |- ( ph -> ( ( ( O ` ( ( ( P - 1 ) / 2 ) .^ X ) ) ` ( L ` A ) ) = ( ( ( P - 1 ) / 2 ) (.g ` (mulGrp ` Y ) ) ( L ` A ) ) <-> ( ( O ` ( ( ( P - 1 ) / 2 ) .^ X ) ) ` ( L ` A ) ) = ( 1r ` Y ) ) )
86	85	anbi2d 684	. . . . . 6 |- ( ph -> ( ( ( ( ( P - 1 ) / 2 ) .^ X ) e. B /\ ( ( O ` ( ( ( P - 1 ) / 2 ) .^ X ) ) ` ( L ` A ) ) = ( ( ( P - 1 ) / 2 ) (.g ` (mulGrp ` Y ) ) ( L ` A ) ) ) <-> ( ( ( ( P - 1 ) / 2 ) .^ X ) e. B /\ ( ( O ` ( ( ( P - 1 ) / 2 ) .^ X ) ) ` ( L ` A ) ) = ( 1r ` Y ) ) ) )
87	40, 86	mpbid 201	. . . . 5 |- ( ph -> ( ( ( ( P - 1 ) / 2 ) .^ X ) e. B /\ ( ( O ` ( ( ( P - 1 ) / 2 ) .^ X ) ) ` ( L ` A ) ) = ( 1r ` Y ) ) )
88		eqid 2283	. . . . . . 7 |- (algSc ` S ) = (algSc ` S )
89	6, 80	rngidcl 15361	. . . . . . . 8 |- ( Y e. Ring -> ( 1r ` Y ) e. ( Base ` Y ) )
90	20, 89	syl 15	. . . . . . 7 |- ( ph -> ( 1r ` Y ) e. ( Base ` Y ) )
91	4, 5, 6, 88, 7, 18, 90, 32	evl1scad 19414	. . . . . 6 |- ( ph -> ( ( (algSc ` S ) ` ( 1r ` Y ) ) e. B /\ ( ( O ` ( (algSc ` S ) ` ( 1r ` Y ) ) ) ` ( L ` A ) ) = ( 1r ` Y ) ) )
92		lgsqr.u	. . . . . . . . . 10 |- .1. = ( 1r ` S )
93	5, 88, 80, 92	ply1scl1 16367	. . . . . . . . 9 |- ( Y e. Ring -> ( (algSc ` S ) ` ( 1r ` Y ) ) = .1. )
94	20, 93	syl 15	. . . . . . . 8 |- ( ph -> ( (algSc ` S ) ` ( 1r ` Y ) ) = .1. )
95	94	eleq1d 2349	. . . . . . 7 |- ( ph -> ( ( (algSc ` S ) ` ( 1r ` Y ) ) e. B <-> .1. e. B ) )
96	94	fveq2d 5529	. . . . . . . . 9 |- ( ph -> ( O ` ( (algSc ` S ) ` ( 1r ` Y ) ) ) = ( O ` .1. ) )
97	96	fveq1d 5527	. . . . . . . 8 |- ( ph -> ( ( O ` ( (algSc ` S ) ` ( 1r ` Y ) ) ) ` ( L ` A ) ) = ( ( O ` .1. ) ` ( L ` A ) ) )
98	97	eqeq1d 2291	. . . . . . 7 |- ( ph -> ( ( ( O ` ( (algSc ` S ) ` ( 1r ` Y ) ) ) ` ( L ` A ) ) = ( 1r ` Y ) <-> ( ( O ` .1. ) ` ( L ` A ) ) = ( 1r ` Y ) ) )
99	95, 98	anbi12d 691	. . . . . 6 |- ( ph -> ( ( ( (algSc ` S ) ` ( 1r ` Y ) ) e. B /\ ( ( O ` ( (algSc ` S ) ` ( 1r ` Y ) ) ) ` ( L ` A ) ) = ( 1r ` Y ) ) <-> ( .1. e. B /\ ( ( O ` .1. ) ` ( L ` A ) ) = ( 1r ` Y ) ) ) )
100	91, 99	mpbid 201	. . . . 5 |- ( ph -> ( .1. e. B /\ ( ( O ` .1. ) ` ( L ` A ) ) = ( 1r ` Y ) ) )
101		lgsqr.m	. . . . 5 |- .- = ( -g ` S )
102		eqid 2283	. . . . 5 |- ( -g ` Y ) = ( -g ` Y )
103	4, 5, 6, 7, 18, 32, 87, 100, 101, 102	evl1subd 19418	. . . 4 |- ( ph -> ( ( ( ( ( P - 1 ) / 2 ) .^ X ) .- .1. ) e. B /\ ( ( O ` ( ( ( ( P - 1 ) / 2 ) .^ X ) .- .1. ) ) ` ( L ` A ) ) = ( ( 1r ` Y ) ( -g ` Y ) ( 1r ` Y ) ) ) )
104	103	simprd 449	. . 3 |- ( ph -> ( ( O ` ( ( ( ( P - 1 ) / 2 ) .^ X ) .- .1. ) ) ` ( L ` A ) ) = ( ( 1r ` Y ) ( -g ` Y ) ( 1r ` Y ) ) )
105	3, 104	syl5eq 2327	. 2 |- ( ph -> ( ( O ` T ) ` ( L ` A ) ) = ( ( 1r ` Y ) ( -g ` Y ) ( 1r ` Y ) ) )
106		rnggrp 15346	. . . 4 |- ( Y e. Ring -> Y e. Grp )
107	20, 106	syl 15	. . 3 |- ( ph -> Y e. Grp )
108		eqid 2283	. . . 4 |- ( 0g ` Y ) = ( 0g ` Y )
109	6, 108, 102	grpsubid 14550	. . 3 |- ( ( Y e. Grp /\ ( 1r ` Y ) e. ( Base ` Y ) ) -> ( ( 1r ` Y ) ( -g ` Y ) ( 1r ` Y ) ) = ( 0g ` Y ) )
110	107, 90, 109	syl2anc 642	. 2 |- ( ph -> ( ( 1r ` Y ) ( -g ` Y ) ( 1r ` Y ) ) = ( 0g ` Y ) )
111	105, 110	eqtrd 2315	1 |- ( ph -> ( ( O ` T ) ` ( L ` A ) ) = ( 0g ` Y ) )

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358   = wceq 1623   e. wcel 1684   \ cdif 3149   {csn 3640   class class class wbr 4023   -->wf 5251   ` cfv 5255  (class class class)co 5858   CCcc 8735   1c1 8738   - cmin 9037   / cdiv 9423   NNcn 9746   2c2 9795   NN0cn0 9965   ZZcz 10024   mod cmo 10973   ^cexp 11104   || cdivides 12531   Primecprime 12758   Basecbs 13148   ↾_s cress 13149   0gc0g 13400   Grpcgrp 14362   -gcsg 14365  ._gcmg 14366   MndHom cmhm 14413  SubMndcsubmnd 14414  mulGrpcmgp 15325   Ringcrg 15337   CRingccrg 15338   1rcur 15339   RingHom crh 15494  Fieldcfield 15513  SubRingcsubrg 15541  Domncdomn 16021  IDomncidom 16022  algSccascl 16052  var₁cv1 16251  Poly₁cpl1 16252  eval₁ce1 16254  ℂ_fldccnfld 16377   ZRHomczrh 16451  ℤ/nℤczn 16454   deg₁ cdg1 19440

This theorem is referenced by:  lgsqrlem2  20581  lgsqrlem3  20582

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-ofr 6079  df-1st 6122  df-2nd 6123  df-tpos 6234  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-ec 6662  df-qs 6666  df-map 6774  df-pm 6775  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-rp 10355  df-fz 10783  df-fzo 10871  df-fl 10925  df-mod 10974  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-dvds 12532  df-gcd 12686  df-prm 12759  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-prds 13348  df-pws 13350  df-0g 13404  df-gsum 13405  df-imas 13411  df-divs 13412  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-mhm 14415  df-submnd 14416  df-grp 14489  df-minusg 14490  df-sbg 14491  df-mulg 14492  df-subg 14618  df-nsg 14619  df-eqg 14620  df-ghm 14681  df-cntz 14793  df-cmn 15091  df-abl 15092  df-mgp 15326  df-rng 15340  df-cring 15341  df-ur 15342  df-oppr 15405  df-dvdsr 15423  df-unit 15424  df-invr 15454  df-rnghom 15496  df-drng 15514  df-field 15515  df-subrg 15543  df-lmod 15629  df-lss 15690  df-lsp 15729  df-sra 15925  df-rgmod 15926  df-lidl 15927  df-rsp 15928  df-2idl 15984  df-nzr 16010  df-rlreg 16024  df-domn 16025  df-idom 16026  df-assa 16053  df-asp 16054  df-ascl 16055  df-psr 16098  df-mvr 16099  df-mpl 16100  df-evls 16101  df-evl 16102  df-opsr 16106  df-psr1 16257  df-vr1 16258  df-ply1 16259  df-evl1 16261  df-cnfld 16378  df-zrh 16455  df-zn 16458