Theorem odlem1 17954

Description: The group element order is either zero or a nonzero multiplier that annihilates the element. (Contributed by Mario Carneiro, 14-Jan-2015.) (Revised by Stefan O'Rear, 5-Sep-2015.) (Revised by AV, 5-Oct-2020.)

Hypotheses

Ref	Expression
odval.1	|- X = ( Base ` G )
odval.2	|- .x. = (.g ` G )
odval.3	|- .0. = ( 0g ` G )
odval.4	|- O = ( od ` G )
odval.i	|- I = { y e. NN | ( y .x. A ) = .0. }

Assertion

Ref	Expression
odlem1	|- ( A e. X -> ( ( ( O ` A ) = 0 /\ I = (/) ) \/ ( O ` A ) e. I ) )

Distinct variable groups:   y, A   y, G   y, .x.   y, .0.

Allowed substitution hints:   I( y)   O( y)   X( y)

Proof of Theorem odlem1

Step	Hyp	Ref	Expression
1		odval.1	. . 3 |- X = ( Base ` G )
2		odval.2	. . 3 |- .x. = (.g ` G )
3		odval.3	. . 3 |- .0. = ( 0g ` G )
4		odval.4	. . 3 |- O = ( od ` G )
5		odval.i	. . 3 |- I = { y e. NN | ( y .x. A ) = .0. }
6	1, 2, 3, 4, 5	odval 17953	. 2 |- ( A e. X -> ( O ` A ) = if ( I = (/) , 0 , inf ( I , RR , < ) ) )
7		eqeq2 2633	. . . 4 |- ( 0 = if ( I = (/) , 0 , inf ( I , RR , < ) ) -> ( ( O ` A ) = 0 <-> ( O ` A ) = if ( I = (/) , 0 , inf ( I , RR , < ) ) ) )
8	7	imbi1d 331	. . 3 |- ( 0 = if ( I = (/) , 0 , inf ( I , RR , < ) ) -> ( ( ( O ` A ) = 0 -> ( ( ( O ` A ) = 0 /\ I = (/) ) \/ ( O ` A ) e. I ) ) <-> ( ( O ` A ) = if ( I = (/) , 0 , inf ( I , RR , < ) ) -> ( ( ( O ` A ) = 0 /\ I = (/) ) \/ ( O ` A ) e. I ) ) ) )
9		eqeq2 2633	. . . 4 |- (inf ( I , RR , < ) = if ( I = (/) , 0 , inf ( I , RR , < ) ) -> ( ( O ` A ) = inf ( I , RR , < ) <-> ( O ` A ) = if ( I = (/) , 0 , inf ( I , RR , < ) ) ) )
10	9	imbi1d 331	. . 3 |- (inf ( I , RR , < ) = if ( I = (/) , 0 , inf ( I , RR , < ) ) -> ( ( ( O ` A ) = inf ( I , RR , < ) -> ( ( ( O ` A ) = 0 /\ I = (/) ) \/ ( O ` A ) e. I ) ) <-> ( ( O ` A ) = if ( I = (/) , 0 , inf ( I , RR , < ) ) -> ( ( ( O ` A ) = 0 /\ I = (/) ) \/ ( O ` A ) e. I ) ) ) )
11		orc 400	. . . . 5 |- ( ( ( O ` A ) = 0 /\ I = (/) ) -> ( ( ( O ` A ) = 0 /\ I = (/) ) \/ ( O ` A ) e. I ) )
12	11	expcom 451	. . . 4 |- ( I = (/) -> ( ( O ` A ) = 0 -> ( ( ( O ` A ) = 0 /\ I = (/) ) \/ ( O ` A ) e. I ) ) )
13	12	adantl 482	. . 3 |- ( ( A e. X /\ I = (/) ) -> ( ( O ` A ) = 0 -> ( ( ( O ` A ) = 0 /\ I = (/) ) \/ ( O ` A ) e. I ) ) )
14		ssrab2 3687	. . . . . . 7 |- { y e. NN | ( y .x. A ) = .0. } C_ NN
15		nnuz 11723	. . . . . . . 8 |- NN = ( ZZ>= ` 1 )
16	15	eqcomi 2631	. . . . . . 7 |- ( ZZ>= ` 1 ) = NN
17	14, 5, 16	3sstr4i 3644	. . . . . 6 |- I C_ ( ZZ>= ` 1 )
18		df-ne 2795	. . . . . . . 8 |- ( I =/= (/) <-> -. I = (/) )
19	18	biimpri 218	. . . . . . 7 |- ( -. I = (/) -> I =/= (/) )
20	19	adantl 482	. . . . . 6 |- ( ( A e. X /\ -. I = (/) ) -> I =/= (/) )
21		infssuzcl 11772	. . . . . 6 |- ( ( I C_ ( ZZ>= ` 1 ) /\ I =/= (/) ) -> inf ( I , RR , < ) e. I )
22	17, 20, 21	sylancr 695	. . . . 5 |- ( ( A e. X /\ -. I = (/) ) -> inf ( I , RR , < ) e. I )
23		eleq1a 2696	. . . . 5 |- (inf ( I , RR , < ) e. I -> ( ( O ` A ) = inf ( I , RR , < ) -> ( O ` A ) e. I ) )
24	22, 23	syl 17	. . . 4 |- ( ( A e. X /\ -. I = (/) ) -> ( ( O ` A ) = inf ( I , RR , < ) -> ( O ` A ) e. I ) )
25		olc 399	. . . 4 |- ( ( O ` A ) e. I -> ( ( ( O ` A ) = 0 /\ I = (/) ) \/ ( O ` A ) e. I ) )
26	24, 25	syl6 35	. . 3 |- ( ( A e. X /\ -. I = (/) ) -> ( ( O ` A ) = inf ( I , RR , < ) -> ( ( ( O ` A ) = 0 /\ I = (/) ) \/ ( O ` A ) e. I ) ) )
27	8, 10, 13, 26	ifbothda 4123	. 2 |- ( A e. X -> ( ( O ` A ) = if ( I = (/) , 0 , inf ( I , RR , < ) ) -> ( ( ( O ` A ) = 0 /\ I = (/) ) \/ ( O ` A ) e. I ) ) )
28	6, 27	mpd 15	1 |- ( A e. X -> ( ( ( O ` A ) = 0 /\ I = (/) ) \/ ( O ` A ) e. I ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   {crab 2916   C_ wss 3574   (/)c0 3915   ifcif 4086   ` cfv 5888  (class class class)co 6650  infcinf 8347   RRcr 9935   0cc0 9936   1c1 9937   < clt 10074   NNcn 11020   ZZ>=cuz 11687   Basecbs 15857   0gc0g 16100  ._gcmg 17540   odcod 17944

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-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

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  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-uni 4437  df-iun 4522  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-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-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-od 17948

This theorem is referenced by:  odcl  17955  odid  17957