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Theorem cyggeninv 18285
Description: The inverse of a cyclic generator is a generator. (Contributed by Mario Carneiro, 21-Apr-2016.)
Hypotheses
Ref Expression
iscyg.1 |- B = ( Base ` G )
iscyg.2 |- .x. = (.g ` G )
iscyg3.e |- E = { x e. B | ran ( n e. ZZ |-> ( n .x. x ) ) = B }
cyggeninv.n |- N = ( invg ` G )
Assertion
Ref Expression
cyggeninv |- ( ( G e. Grp /\ X e. E ) -> ( N ` X ) e. E )
Distinct variable groups:   x, n, B   n, N, x   n, X, x   n, G, x   .x. , n, x
Allowed substitution hints:   E( x, n)
Proof of Theorem cyggeninv
Dummy variables m y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 iscyg.1 . . . . 5 |- B = ( Base ` G )
2 iscyg.2 . . . . 5 |- .x. = (.g ` G )
3 iscyg3.e . . . . 5 |- E = { x e. B | ran ( n e. ZZ |-> ( n .x. x ) ) = B }
41, 2, 3iscyggen2 18283 . . . 4 |- ( G e. Grp -> ( X e. E <-> ( X e. B /\ A. y e. B E. n e. ZZ y = ( n .x. X ) ) ) )
54simprbda 653 . . 3 |- ( ( G e. Grp /\ X e. E ) -> X e. B )
6 cyggeninv.n . . . 4 |- N = ( invg ` G )
71, 6grpinvcl 17467 . . 3 |- ( ( G e. Grp /\ X e. B ) -> ( N ` X ) e. B )
85, 7syldan 487 . 2 |- ( ( G e. Grp /\ X e. E ) -> ( N ` X ) e. B )
94simplbda 654 . . 3 |- ( ( G e. Grp /\ X e. E ) -> A. y e. B E. n e. ZZ y = ( n .x. X ) )
10 oveq1 6657 . . . . . . 7 |- ( n = m -> ( n .x. X ) = ( m .x. X ) )
1110eqeq2d 2632 . . . . . 6 |- ( n = m -> ( y = ( n .x. X ) <-> y = ( m .x. X ) ) )
1211cbvrexv 3172 . . . . 5 |- ( E. n e. ZZ y = ( n .x. X ) <-> E. m e. ZZ y = ( m .x. X ) )
13 znegcl 11412 . . . . . . . . 9 |- ( m e. ZZ -> -u m e. ZZ )
1413adantl 482 . . . . . . . 8 |- ( ( ( ( G e. Grp /\ X e. E ) /\ y e. B ) /\ m e. ZZ ) -> -u m e. ZZ )
15 simpr 477 . . . . . . . . . . . 12 |- ( ( ( ( G e. Grp /\ X e. E ) /\ y e. B ) /\ m e. ZZ ) -> m e. ZZ )
1615zcnd 11483 . . . . . . . . . . 11 |- ( ( ( ( G e. Grp /\ X e. E ) /\ y e. B ) /\ m e. ZZ ) -> m e. CC )
1716negnegd 10383 . . . . . . . . . 10 |- ( ( ( ( G e. Grp /\ X e. E ) /\ y e. B ) /\ m e. ZZ ) -> -u -u m = m )
1817oveq1d 6665 . . . . . . . . 9 |- ( ( ( ( G e. Grp /\ X e. E ) /\ y e. B ) /\ m e. ZZ ) -> ( -u -u m .x. X ) = ( m .x. X ) )
19 simplll 798 . . . . . . . . . 10 |- ( ( ( ( G e. Grp /\ X e. E ) /\ y e. B ) /\ m e. ZZ ) -> G e. Grp )
205ad2antrr 762 . . . . . . . . . 10 |- ( ( ( ( G e. Grp /\ X e. E ) /\ y e. B ) /\ m e. ZZ ) -> X e. B )
211, 2, 6mulgneg2 17575 . . . . . . . . . 10 |- ( ( G e. Grp /\ -u m e. ZZ /\ X e. B ) -> ( -u -u m .x. X ) = ( -u m .x. ( N ` X ) ) )
2219, 14, 20, 21syl3anc 1326 . . . . . . . . 9 |- ( ( ( ( G e. Grp /\ X e. E ) /\ y e. B ) /\ m e. ZZ ) -> ( -u -u m .x. X ) = ( -u m .x. ( N ` X ) ) )
2318, 22eqtr3d 2658 . . . . . . . 8 |- ( ( ( ( G e. Grp /\ X e. E ) /\ y e. B ) /\ m e. ZZ ) -> ( m .x. X ) = ( -u m .x. ( N ` X ) ) )
24 oveq1 6657 . . . . . . . . . 10 |- ( n = -u m -> ( n .x. ( N ` X ) ) = ( -u m .x. ( N ` X ) ) )
2524eqeq2d 2632 . . . . . . . . 9 |- ( n = -u m -> ( ( m .x. X ) = ( n .x. ( N ` X ) ) <-> ( m .x. X ) = ( -u m .x. ( N ` X ) ) ) )
2625rspcev 3309 . . . . . . . 8 |- ( ( -u m e. ZZ /\ ( m .x. X ) = ( -u m .x. ( N ` X ) ) ) -> E. n e. ZZ ( m .x. X ) = ( n .x. ( N ` X ) ) )
2714, 23, 26syl2anc 693 . . . . . . 7 |- ( ( ( ( G e. Grp /\ X e. E ) /\ y e. B ) /\ m e. ZZ ) -> E. n e. ZZ ( m .x. X ) = ( n .x. ( N ` X ) ) )
28 eqeq1 2626 . . . . . . . 8 |- ( y = ( m .x. X ) -> ( y = ( n .x. ( N ` X ) ) <-> ( m .x. X ) = ( n .x. ( N ` X ) ) ) )
2928rexbidv 3052 . . . . . . 7 |- ( y = ( m .x. X ) -> ( E. n e. ZZ y = ( n .x. ( N ` X ) ) <-> E. n e. ZZ ( m .x. X ) = ( n .x. ( N ` X ) ) ) )
3027, 29syl5ibrcom 237 . . . . . 6 |- ( ( ( ( G e. Grp /\ X e. E ) /\ y e. B ) /\ m e. ZZ ) -> ( y = ( m .x. X ) -> E. n e. ZZ y = ( n .x. ( N ` X ) ) ) )
3130rexlimdva 3031 . . . . 5 |- ( ( ( G e. Grp /\ X e. E ) /\ y e. B ) -> ( E. m e. ZZ y = ( m .x. X ) -> E. n e. ZZ y = ( n .x. ( N ` X ) ) ) )
3212, 31syl5bi 232 . . . 4 |- ( ( ( G e. Grp /\ X e. E ) /\ y e. B ) -> ( E. n e. ZZ y = ( n .x. X ) -> E. n e. ZZ y = ( n .x. ( N ` X ) ) ) )
3332ralimdva 2962 . . 3 |- ( ( G e. Grp /\ X e. E ) -> ( A. y e. B E. n e. ZZ y = ( n .x. X ) -> A. y e. B E. n e. ZZ y = ( n .x. ( N ` X ) ) ) )
349, 33mpd 15 . 2 |- ( ( G e. Grp /\ X e. E ) -> A. y e. B E. n e. ZZ y = ( n .x. ( N ` X ) ) )
351, 2, 3iscyggen2 18283 . . 3 |- ( G e. Grp -> ( ( N ` X ) e. E <-> ( ( N ` X ) e. B /\ A. y e. B E. n e. ZZ y = ( n .x. ( N ` X ) ) ) ) )
3635adantr 481 . 2 |- ( ( G e. Grp /\ X e. E ) -> ( ( N ` X ) e. E <-> ( ( N ` X ) e. B /\ A. y e. B E. n e. ZZ y = ( n .x. ( N ` X ) ) ) ) )
378, 34, 36mpbir2and 957 1 |- ( ( G e. Grp /\ X e. E ) -> ( N ` X ) e. E )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   {crab 2916   |-> cmpt 4729   ran crn 5115   ` cfv 5888  (class class class)co 6650   -ucneg 10267   ZZcz 11377   Basecbs 15857   Grpcgrp 17422   invgcminusg 17423  .gcmg 17540
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-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-grp 17425  df-minusg 17426  df-mulg 17541
This theorem is referenced by: (None)
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