Theorem ghmcyg 18297

Description: The image of a cyclic group under a surjective group homomorphism is cyclic. (Contributed by Mario Carneiro, 21-Apr-2016.)

Hypotheses

Ref	Expression
cygctb.1	|- B = ( Base ` G )
ghmcyg.1	|- C = ( Base ` H )

Assertion

Ref	Expression
ghmcyg	|- ( ( F e. ( G GrpHom H ) /\ F : B -onto-> C ) -> ( G e. CycGrp -> H e. CycGrp ) )

Proof of Theorem ghmcyg

Dummy variables m n x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cygctb.1	. . . 4 |- B = ( Base ` G )
2		eqid 2622	. . . 4 |- (.g ` G ) = (.g ` G )
3	1, 2	iscyg 18281	. . 3 |- ( G e. CycGrp <-> ( G e. Grp /\ E. x e. B ran ( n e. ZZ |-> ( n (.g ` G ) x ) ) = B ) )
4	3	simprbi 480	. 2 |- ( G e. CycGrp -> E. x e. B ran ( n e. ZZ |-> ( n (.g ` G ) x ) ) = B )
5		ghmcyg.1	. . . 4 |- C = ( Base ` H )
6		eqid 2622	. . . 4 |- (.g ` H ) = (.g ` H )
7		ghmgrp2 17663	. . . . 5 |- ( F e. ( G GrpHom H ) -> H e. Grp )
8	7	ad2antrr 762	. . . 4 |- ( ( ( F e. ( G GrpHom H ) /\ F : B -onto-> C ) /\ ( x e. B /\ ran ( n e. ZZ |-> ( n (.g ` G ) x ) ) = B ) ) -> H e. Grp )
9		fof 6115	. . . . . 6 |- ( F : B -onto-> C -> F : B --> C )
10	9	ad2antlr 763	. . . . 5 |- ( ( ( F e. ( G GrpHom H ) /\ F : B -onto-> C ) /\ ( x e. B /\ ran ( n e. ZZ |-> ( n (.g ` G ) x ) ) = B ) ) -> F : B --> C )
11		simprl 794	. . . . 5 |- ( ( ( F e. ( G GrpHom H ) /\ F : B -onto-> C ) /\ ( x e. B /\ ran ( n e. ZZ |-> ( n (.g ` G ) x ) ) = B ) ) -> x e. B )
12	10, 11	ffvelrnd 6360	. . . 4 |- ( ( ( F e. ( G GrpHom H ) /\ F : B -onto-> C ) /\ ( x e. B /\ ran ( n e. ZZ |-> ( n (.g ` G ) x ) ) = B ) ) -> ( F ` x ) e. C )
13		simplr 792	. . . . . . . 8 |- ( ( ( F e. ( G GrpHom H ) /\ F : B -onto-> C ) /\ ( x e. B /\ ran ( n e. ZZ |-> ( n (.g ` G ) x ) ) = B ) ) -> F : B -onto-> C )
14		foeq2 6112	. . . . . . . . 9 |- ( ran ( n e. ZZ |-> ( n (.g ` G ) x ) ) = B -> ( F : ran ( n e. ZZ |-> ( n (.g ` G ) x ) ) -onto-> C <-> F : B -onto-> C ) )
15	14	ad2antll 765	. . . . . . . 8 |- ( ( ( F e. ( G GrpHom H ) /\ F : B -onto-> C ) /\ ( x e. B /\ ran ( n e. ZZ |-> ( n (.g ` G ) x ) ) = B ) ) -> ( F : ran ( n e. ZZ |-> ( n (.g ` G ) x ) ) -onto-> C <-> F : B -onto-> C ) )
16	13, 15	mpbird 247	. . . . . . 7 |- ( ( ( F e. ( G GrpHom H ) /\ F : B -onto-> C ) /\ ( x e. B /\ ran ( n e. ZZ |-> ( n (.g ` G ) x ) ) = B ) ) -> F : ran ( n e. ZZ |-> ( n (.g ` G ) x ) ) -onto-> C )
17		foelrn 6378	. . . . . . 7 |- ( ( F : ran ( n e. ZZ |-> ( n (.g ` G ) x ) ) -onto-> C /\ y e. C ) -> E. z e. ran ( n e. ZZ |-> ( n (.g ` G ) x ) ) y = ( F ` z ) )
18	16, 17	sylan 488	. . . . . 6 |- ( ( ( ( F e. ( G GrpHom H ) /\ F : B -onto-> C ) /\ ( x e. B /\ ran ( n e. ZZ |-> ( n (.g ` G ) x ) ) = B ) ) /\ y e. C ) -> E. z e. ran ( n e. ZZ |-> ( n (.g ` G ) x ) ) y = ( F ` z ) )
19		ovex 6678	. . . . . . . 8 |- ( m (.g ` G ) x ) e. _V
20	19	rgenw 2924	. . . . . . 7 |- A. m e. ZZ ( m (.g ` G ) x ) e. _V
21		oveq1 6657	. . . . . . . . 9 |- ( n = m -> ( n (.g ` G ) x ) = ( m (.g ` G ) x ) )
22	21	cbvmptv 4750	. . . . . . . 8 |- ( n e. ZZ |-> ( n (.g ` G ) x ) ) = ( m e. ZZ |-> ( m (.g ` G ) x ) )
23		fveq2 6191	. . . . . . . . 9 |- ( z = ( m (.g ` G ) x ) -> ( F ` z ) = ( F ` ( m (.g ` G ) x ) ) )
24	23	eqeq2d 2632	. . . . . . . 8 |- ( z = ( m (.g ` G ) x ) -> ( y = ( F ` z ) <-> y = ( F ` ( m (.g ` G ) x ) ) ) )
25	22, 24	rexrnmpt 6369	. . . . . . 7 |- ( A. m e. ZZ ( m (.g ` G ) x ) e. _V -> ( E. z e. ran ( n e. ZZ |-> ( n (.g ` G ) x ) ) y = ( F ` z ) <-> E. m e. ZZ y = ( F ` ( m (.g ` G ) x ) ) ) )
26	20, 25	ax-mp 5	. . . . . 6 |- ( E. z e. ran ( n e. ZZ |-> ( n (.g ` G ) x ) ) y = ( F ` z ) <-> E. m e. ZZ y = ( F ` ( m (.g ` G ) x ) ) )
27	18, 26	sylib 208	. . . . 5 |- ( ( ( ( F e. ( G GrpHom H ) /\ F : B -onto-> C ) /\ ( x e. B /\ ran ( n e. ZZ |-> ( n (.g ` G ) x ) ) = B ) ) /\ y e. C ) -> E. m e. ZZ y = ( F ` ( m (.g ` G ) x ) ) )
28		simp-4l 806	. . . . . . . 8 |- ( ( ( ( ( F e. ( G GrpHom H ) /\ F : B -onto-> C ) /\ ( x e. B /\ ran ( n e. ZZ |-> ( n (.g ` G ) x ) ) = B ) ) /\ y e. C ) /\ m e. ZZ ) -> F e. ( G GrpHom H ) )
29		simpr 477	. . . . . . . 8 |- ( ( ( ( ( F e. ( G GrpHom H ) /\ F : B -onto-> C ) /\ ( x e. B /\ ran ( n e. ZZ |-> ( n (.g ` G ) x ) ) = B ) ) /\ y e. C ) /\ m e. ZZ ) -> m e. ZZ )
30	11	ad2antrr 762	. . . . . . . 8 |- ( ( ( ( ( F e. ( G GrpHom H ) /\ F : B -onto-> C ) /\ ( x e. B /\ ran ( n e. ZZ |-> ( n (.g ` G ) x ) ) = B ) ) /\ y e. C ) /\ m e. ZZ ) -> x e. B )
31	1, 2, 6	ghmmulg 17672	. . . . . . . 8 |- ( ( F e. ( G GrpHom H ) /\ m e. ZZ /\ x e. B ) -> ( F ` ( m (.g ` G ) x ) ) = ( m (.g ` H ) ( F ` x ) ) )
32	28, 29, 30, 31	syl3anc 1326	. . . . . . 7 |- ( ( ( ( ( F e. ( G GrpHom H ) /\ F : B -onto-> C ) /\ ( x e. B /\ ran ( n e. ZZ |-> ( n (.g ` G ) x ) ) = B ) ) /\ y e. C ) /\ m e. ZZ ) -> ( F ` ( m (.g ` G ) x ) ) = ( m (.g ` H ) ( F ` x ) ) )
33	32	eqeq2d 2632	. . . . . 6 |- ( ( ( ( ( F e. ( G GrpHom H ) /\ F : B -onto-> C ) /\ ( x e. B /\ ran ( n e. ZZ |-> ( n (.g ` G ) x ) ) = B ) ) /\ y e. C ) /\ m e. ZZ ) -> ( y = ( F ` ( m (.g ` G ) x ) ) <-> y = ( m (.g ` H ) ( F ` x ) ) ) )
34	33	rexbidva 3049	. . . . 5 |- ( ( ( ( F e. ( G GrpHom H ) /\ F : B -onto-> C ) /\ ( x e. B /\ ran ( n e. ZZ |-> ( n (.g ` G ) x ) ) = B ) ) /\ y e. C ) -> ( E. m e. ZZ y = ( F ` ( m (.g ` G ) x ) ) <-> E. m e. ZZ y = ( m (.g ` H ) ( F ` x ) ) ) )
35	27, 34	mpbid 222	. . . 4 |- ( ( ( ( F e. ( G GrpHom H ) /\ F : B -onto-> C ) /\ ( x e. B /\ ran ( n e. ZZ |-> ( n (.g ` G ) x ) ) = B ) ) /\ y e. C ) -> E. m e. ZZ y = ( m (.g ` H ) ( F ` x ) ) )
36	5, 6, 8, 12, 35	iscygd 18289	. . 3 |- ( ( ( F e. ( G GrpHom H ) /\ F : B -onto-> C ) /\ ( x e. B /\ ran ( n e. ZZ |-> ( n (.g ` G ) x ) ) = B ) ) -> H e. CycGrp )
37	36	rexlimdvaa 3032	. 2 |- ( ( F e. ( G GrpHom H ) /\ F : B -onto-> C ) -> ( E. x e. B ran ( n e. ZZ |-> ( n (.g ` G ) x ) ) = B -> H e. CycGrp ) )
38	4, 37	syl5 34	1 |- ( ( F e. ( G GrpHom H ) /\ F : B -onto-> C ) -> ( G e. CycGrp -> H e. CycGrp ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   _Vcvv 3200   |-> cmpt 4729   ran crn 5115   -->wf 5884   -onto->wfo 5886   ` cfv 5888  (class class class)co 6650   ZZcz 11377   Basecbs 15857   Grpcgrp 17422  ._gcmg 17540   GrpHom cghm 17657  CycGrpccyg 18279

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

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-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  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-fz 12327  df-seq 12802  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mhm 17335  df-grp 17425  df-minusg 17426  df-mulg 17541  df-ghm 17658  df-cyg 18280

This theorem is referenced by:  giccyg  18301