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Theorem 2zrngmmgm 32896
Description: R is a (multiplicative) magma. (Contributed by AV, 11-Feb-2020.)
Hypotheses
Ref Expression
2zrng.e  |-  E  =  { z  e.  ZZ  |  E. x  e.  ZZ  z  =  ( 2  x.  x ) }
2zrngbas.r  |-  R  =  (flds  E )
2zrngmmgm.1  |-  M  =  (mulGrp `  R )
Assertion
Ref Expression
2zrngmmgm  |-  M  e. Mgm
Distinct variable groups:    x, z, R    x, E, z
Allowed substitution hints:    M( x, z)

Proof of Theorem 2zrngmmgm
Dummy variables  a 
b  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqeq1 2461 . . . . . 6  |-  ( z  =  a  ->  (
z  =  ( 2  x.  x )  <->  a  =  ( 2  x.  x
) ) )
21rexbidv 2968 . . . . 5  |-  ( z  =  a  ->  ( E. x  e.  ZZ  z  =  ( 2  x.  x )  <->  E. x  e.  ZZ  a  =  ( 2  x.  x ) ) )
3 2zrng.e . . . . 5  |-  E  =  { z  e.  ZZ  |  E. x  e.  ZZ  z  =  ( 2  x.  x ) }
42, 3elrab2 3259 . . . 4  |-  ( a  e.  E  <->  ( a  e.  ZZ  /\  E. x  e.  ZZ  a  =  ( 2  x.  x ) ) )
5 eqeq1 2461 . . . . . 6  |-  ( z  =  b  ->  (
z  =  ( 2  x.  x )  <->  b  =  ( 2  x.  x
) ) )
65rexbidv 2968 . . . . 5  |-  ( z  =  b  ->  ( E. x  e.  ZZ  z  =  ( 2  x.  x )  <->  E. x  e.  ZZ  b  =  ( 2  x.  x ) ) )
76, 3elrab2 3259 . . . 4  |-  ( b  e.  E  <->  ( b  e.  ZZ  /\  E. x  e.  ZZ  b  =  ( 2  x.  x ) ) )
8 zmulcl 10933 . . . . . 6  |-  ( ( a  e.  ZZ  /\  b  e.  ZZ )  ->  ( a  x.  b
)  e.  ZZ )
98ad2ant2r 746 . . . . 5  |-  ( ( ( a  e.  ZZ  /\ 
E. x  e.  ZZ  a  =  ( 2  x.  x ) )  /\  ( b  e.  ZZ  /\  E. x  e.  ZZ  b  =  ( 2  x.  x ) ) )  ->  (
a  x.  b )  e.  ZZ )
10 nfv 1708 . . . . . . . . 9  |-  F/ x  a  e.  ZZ
11 nfv 1708 . . . . . . . . . . 11  |-  F/ x  b  e.  ZZ
12 nfre1 2918 . . . . . . . . . . 11  |-  F/ x E. x  e.  ZZ  b  =  ( 2  x.  x )
1311, 12nfan 1929 . . . . . . . . . 10  |-  F/ x
( b  e.  ZZ  /\ 
E. x  e.  ZZ  b  =  ( 2  x.  x ) )
14 nfv 1708 . . . . . . . . . 10  |-  F/ x E. y  e.  ZZ  ( a  x.  b
)  =  ( 2  x.  y )
1513, 14nfim 1921 . . . . . . . . 9  |-  F/ x
( ( b  e.  ZZ  /\  E. x  e.  ZZ  b  =  ( 2  x.  x ) )  ->  E. y  e.  ZZ  ( a  x.  b )  =  ( 2  x.  y ) )
1610, 15nfim 1921 . . . . . . . 8  |-  F/ x
( a  e.  ZZ  ->  ( ( b  e.  ZZ  /\  E. x  e.  ZZ  b  =  ( 2  x.  x ) )  ->  E. y  e.  ZZ  ( a  x.  b )  =  ( 2  x.  y ) ) )
17 simpll 753 . . . . . . . . . . 11  |-  ( ( ( x  e.  ZZ  /\  a  =  ( 2  x.  x ) )  /\  a  e.  ZZ )  ->  x  e.  ZZ )
18 simpl 457 . . . . . . . . . . 11  |-  ( ( b  e.  ZZ  /\  E. x  e.  ZZ  b  =  ( 2  x.  x ) )  -> 
b  e.  ZZ )
19 zmulcl 10933 . . . . . . . . . . 11  |-  ( ( x  e.  ZZ  /\  b  e.  ZZ )  ->  ( x  x.  b
)  e.  ZZ )
2017, 18, 19syl2an 477 . . . . . . . . . 10  |-  ( ( ( ( x  e.  ZZ  /\  a  =  ( 2  x.  x
) )  /\  a  e.  ZZ )  /\  (
b  e.  ZZ  /\  E. x  e.  ZZ  b  =  ( 2  x.  x ) ) )  ->  ( x  x.  b )  e.  ZZ )
21 oveq2 6304 . . . . . . . . . . . 12  |-  ( y  =  ( x  x.  b )  ->  (
2  x.  y )  =  ( 2  x.  ( x  x.  b
) ) )
2221eqeq2d 2471 . . . . . . . . . . 11  |-  ( y  =  ( x  x.  b )  ->  (
( a  x.  b
)  =  ( 2  x.  y )  <->  ( a  x.  b )  =  ( 2  x.  ( x  x.  b ) ) ) )
2322adantl 466 . . . . . . . . . 10  |-  ( ( ( ( ( x  e.  ZZ  /\  a  =  ( 2  x.  x ) )  /\  a  e.  ZZ )  /\  ( b  e.  ZZ  /\ 
E. x  e.  ZZ  b  =  ( 2  x.  x ) ) )  /\  y  =  ( x  x.  b
) )  ->  (
( a  x.  b
)  =  ( 2  x.  y )  <->  ( a  x.  b )  =  ( 2  x.  ( x  x.  b ) ) ) )
24 oveq1 6303 . . . . . . . . . . . 12  |-  ( a  =  ( 2  x.  x )  ->  (
a  x.  b )  =  ( ( 2  x.  x )  x.  b ) )
2524ad3antlr 730 . . . . . . . . . . 11  |-  ( ( ( ( x  e.  ZZ  /\  a  =  ( 2  x.  x
) )  /\  a  e.  ZZ )  /\  (
b  e.  ZZ  /\  E. x  e.  ZZ  b  =  ( 2  x.  x ) ) )  ->  ( a  x.  b )  =  ( ( 2  x.  x
)  x.  b ) )
26 2cnd 10629 . . . . . . . . . . . 12  |-  ( ( ( ( x  e.  ZZ  /\  a  =  ( 2  x.  x
) )  /\  a  e.  ZZ )  /\  (
b  e.  ZZ  /\  E. x  e.  ZZ  b  =  ( 2  x.  x ) ) )  ->  2  e.  CC )
27 zcn 10890 . . . . . . . . . . . . 13  |-  ( x  e.  ZZ  ->  x  e.  CC )
2827ad3antrrr 729 . . . . . . . . . . . 12  |-  ( ( ( ( x  e.  ZZ  /\  a  =  ( 2  x.  x
) )  /\  a  e.  ZZ )  /\  (
b  e.  ZZ  /\  E. x  e.  ZZ  b  =  ( 2  x.  x ) ) )  ->  x  e.  CC )
29 zcn 10890 . . . . . . . . . . . . . 14  |-  ( b  e.  ZZ  ->  b  e.  CC )
3029adantr 465 . . . . . . . . . . . . 13  |-  ( ( b  e.  ZZ  /\  E. x  e.  ZZ  b  =  ( 2  x.  x ) )  -> 
b  e.  CC )
3130adantl 466 . . . . . . . . . . . 12  |-  ( ( ( ( x  e.  ZZ  /\  a  =  ( 2  x.  x
) )  /\  a  e.  ZZ )  /\  (
b  e.  ZZ  /\  E. x  e.  ZZ  b  =  ( 2  x.  x ) ) )  ->  b  e.  CC )
3226, 28, 31mulassd 9636 . . . . . . . . . . 11  |-  ( ( ( ( x  e.  ZZ  /\  a  =  ( 2  x.  x
) )  /\  a  e.  ZZ )  /\  (
b  e.  ZZ  /\  E. x  e.  ZZ  b  =  ( 2  x.  x ) ) )  ->  ( ( 2  x.  x )  x.  b )  =  ( 2  x.  ( x  x.  b ) ) )
3325, 32eqtrd 2498 . . . . . . . . . 10  |-  ( ( ( ( x  e.  ZZ  /\  a  =  ( 2  x.  x
) )  /\  a  e.  ZZ )  /\  (
b  e.  ZZ  /\  E. x  e.  ZZ  b  =  ( 2  x.  x ) ) )  ->  ( a  x.  b )  =  ( 2  x.  ( x  x.  b ) ) )
3420, 23, 33rspcedvd 3215 . . . . . . . . 9  |-  ( ( ( ( x  e.  ZZ  /\  a  =  ( 2  x.  x
) )  /\  a  e.  ZZ )  /\  (
b  e.  ZZ  /\  E. x  e.  ZZ  b  =  ( 2  x.  x ) ) )  ->  E. y  e.  ZZ  ( a  x.  b
)  =  ( 2  x.  y ) )
3534exp41 610 . . . . . . . 8  |-  ( x  e.  ZZ  ->  (
a  =  ( 2  x.  x )  -> 
( a  e.  ZZ  ->  ( ( b  e.  ZZ  /\  E. x  e.  ZZ  b  =  ( 2  x.  x ) )  ->  E. y  e.  ZZ  ( a  x.  b )  =  ( 2  x.  y ) ) ) ) )
3616, 35rexlimi 2939 . . . . . . 7  |-  ( E. x  e.  ZZ  a  =  ( 2  x.  x )  ->  (
a  e.  ZZ  ->  ( ( b  e.  ZZ  /\ 
E. x  e.  ZZ  b  =  ( 2  x.  x ) )  ->  E. y  e.  ZZ  ( a  x.  b
)  =  ( 2  x.  y ) ) ) )
3736impcom 430 . . . . . 6  |-  ( ( a  e.  ZZ  /\  E. x  e.  ZZ  a  =  ( 2  x.  x ) )  -> 
( ( b  e.  ZZ  /\  E. x  e.  ZZ  b  =  ( 2  x.  x ) )  ->  E. y  e.  ZZ  ( a  x.  b )  =  ( 2  x.  y ) ) )
3837imp 429 . . . . 5  |-  ( ( ( a  e.  ZZ  /\ 
E. x  e.  ZZ  a  =  ( 2  x.  x ) )  /\  ( b  e.  ZZ  /\  E. x  e.  ZZ  b  =  ( 2  x.  x ) ) )  ->  E. y  e.  ZZ  ( a  x.  b )  =  ( 2  x.  y ) )
39 eqeq1 2461 . . . . . . . 8  |-  ( z  =  ( a  x.  b )  ->  (
z  =  ( 2  x.  x )  <->  ( a  x.  b )  =  ( 2  x.  x ) ) )
4039rexbidv 2968 . . . . . . 7  |-  ( z  =  ( a  x.  b )  ->  ( E. x  e.  ZZ  z  =  ( 2  x.  x )  <->  E. x  e.  ZZ  ( a  x.  b )  =  ( 2  x.  x ) ) )
4140, 3elrab2 3259 . . . . . 6  |-  ( ( a  x.  b )  e.  E  <->  ( (
a  x.  b )  e.  ZZ  /\  E. x  e.  ZZ  (
a  x.  b )  =  ( 2  x.  x ) ) )
42 oveq2 6304 . . . . . . . . 9  |-  ( x  =  y  ->  (
2  x.  x )  =  ( 2  x.  y ) )
4342eqeq2d 2471 . . . . . . . 8  |-  ( x  =  y  ->  (
( a  x.  b
)  =  ( 2  x.  x )  <->  ( a  x.  b )  =  ( 2  x.  y ) ) )
4443cbvrexv 3085 . . . . . . 7  |-  ( E. x  e.  ZZ  (
a  x.  b )  =  ( 2  x.  x )  <->  E. y  e.  ZZ  ( a  x.  b )  =  ( 2  x.  y ) )
4544anbi2i 694 . . . . . 6  |-  ( ( ( a  x.  b
)  e.  ZZ  /\  E. x  e.  ZZ  (
a  x.  b )  =  ( 2  x.  x ) )  <->  ( (
a  x.  b )  e.  ZZ  /\  E. y  e.  ZZ  (
a  x.  b )  =  ( 2  x.  y ) ) )
4641, 45bitri 249 . . . . 5  |-  ( ( a  x.  b )  e.  E  <->  ( (
a  x.  b )  e.  ZZ  /\  E. y  e.  ZZ  (
a  x.  b )  =  ( 2  x.  y ) ) )
479, 38, 46sylanbrc 664 . . . 4  |-  ( ( ( a  e.  ZZ  /\ 
E. x  e.  ZZ  a  =  ( 2  x.  x ) )  /\  ( b  e.  ZZ  /\  E. x  e.  ZZ  b  =  ( 2  x.  x ) ) )  ->  (
a  x.  b )  e.  E )
484, 7, 47syl2anb 479 . . 3  |-  ( ( a  e.  E  /\  b  e.  E )  ->  ( a  x.  b
)  e.  E )
4948rgen2a 2884 . 2  |-  A. a  e.  E  A. b  e.  E  ( a  x.  b )  e.  E
5030even 32881 . . 3  |-  0  e.  E
51 2zrngmmgm.1 . . . . 5  |-  M  =  (mulGrp `  R )
52 2zrngbas.r . . . . . 6  |-  R  =  (flds  E )
533, 522zrngbas 32886 . . . . 5  |-  E  =  ( Base `  R
)
5451, 53mgpbas 17274 . . . 4  |-  E  =  ( Base `  M
)
553, 522zrngmul 32895 . . . . 5  |-  x.  =  ( .r `  R )
5651, 55mgpplusg 17272 . . . 4  |-  x.  =  ( +g  `  M )
5754, 56ismgmn0 16001 . . 3  |-  ( 0  e.  E  ->  ( M  e. Mgm  <->  A. a  e.  E  A. b  e.  E  ( a  x.  b
)  e.  E ) )
5850, 57ax-mp 5 . 2  |-  ( M  e. Mgm 
<-> 
A. a  e.  E  A. b  e.  E  ( a  x.  b
)  e.  E )
5949, 58mpbir 209 1  |-  M  e. Mgm
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1395    e. wcel 1819   A.wral 2807   E.wrex 2808   {crab 2811   ` cfv 5594  (class class class)co 6296   CCcc 9507   0cc0 9509    x. cmul 9514   2c2 10606   ZZcz 10885   ↾s cress 14645  Mgmcmgm 15997  mulGrpcmgp 17268  ℂfldccnfld 18547
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-mulf 9589
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-5 10618  df-6 10619  df-7 10620  df-8 10621  df-9 10622  df-10 10623  df-n0 10817  df-z 10886  df-dec 11001  df-uz 11107  df-fz 11698  df-struct 14646  df-ndx 14647  df-slot 14648  df-base 14649  df-sets 14650  df-ress 14651  df-plusg 14725  df-mulr 14726  df-starv 14727  df-tset 14731  df-ple 14732  df-ds 14734  df-unif 14735  df-mgm 15999  df-mgp 17269  df-cnfld 18548
This theorem is referenced by:  2zrngmsgrp  32897
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