Theorem 2zrngmmgm 41946

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 = (ℂfld ↾s 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.

Step	Hyp	Ref	Expression
1		eqeq1 2626	. . . . . 6 |- ( z = a -> ( z = ( 2 x. x ) <-> a = ( 2 x. x ) ) )
2	1	rexbidv 3052	. . . . 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 ) }
4	2, 3	elrab2 3366	. . . 4 |- ( a e. E <-> ( a e. ZZ /\ E. x e. ZZ a = ( 2 x. x ) ) )
5		eqeq1 2626	. . . . . 6 |- ( z = b -> ( z = ( 2 x. x ) <-> b = ( 2 x. x ) ) )
6	5	rexbidv 3052	. . . . 5 |- ( z = b -> ( E. x e. ZZ z = ( 2 x. x ) <-> E. x e. ZZ b = ( 2 x. x ) ) )
7	6, 3	elrab2 3366	. . . 4 |- ( b e. E <-> ( b e. ZZ /\ E. x e. ZZ b = ( 2 x. x ) ) )
8		zmulcl 11426	. . . . . 6 |- ( ( a e. ZZ /\ b e. ZZ ) -> ( a x. b ) e. ZZ )
9	8	ad2ant2r 783	. . . . 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 1843	. . . . . . . . 9 |- F/ x a e. ZZ
11		nfv 1843	. . . . . . . . . . 11 |- F/ x b e. ZZ
12		nfre1 3005	. . . . . . . . . . 11 |- F/ x E. x e. ZZ b = ( 2 x. x )
13	11, 12	nfan 1828	. . . . . . . . . 10 |- F/ x ( b e. ZZ /\ E. x e. ZZ b = ( 2 x. x ) )
14		nfv 1843	. . . . . . . . . 10 |- F/ x E. y e. ZZ ( a x. b ) = ( 2 x. y )
15	13, 14	nfim 1825	. . . . . . . . 9 |- F/ x ( ( b e. ZZ /\ E. x e. ZZ b = ( 2 x. x ) ) -> E. y e. ZZ ( a x. b ) = ( 2 x. y ) )
16	10, 15	nfim 1825	. . . . . . . 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 790	. . . . . . . . . . 11 |- ( ( ( x e. ZZ /\ a = ( 2 x. x ) ) /\ a e. ZZ ) -> x e. ZZ )
18		simpl 473	. . . . . . . . . . 11 |- ( ( b e. ZZ /\ E. x e. ZZ b = ( 2 x. x ) ) -> b e. ZZ )
19		zmulcl 11426	. . . . . . . . . . 11 |- ( ( x e. ZZ /\ b e. ZZ ) -> ( x x. b ) e. ZZ )
20	17, 18, 19	syl2an 494	. . . . . . . . . 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 6658	. . . . . . . . . . . 12 |- ( y = ( x x. b ) -> ( 2 x. y ) = ( 2 x. ( x x. b ) ) )
22	21	eqeq2d 2632	. . . . . . . . . . 11 |- ( y = ( x x. b ) -> ( ( a x. b ) = ( 2 x. y ) <-> ( a x. b ) = ( 2 x. ( x x. b ) ) ) )
23	22	adantl 482	. . . . . . . . . 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 6657	. . . . . . . . . . . 12 |- ( a = ( 2 x. x ) -> ( a x. b ) = ( ( 2 x. x ) x. b ) )
25	24	ad3antlr 767	. . . . . . . . . . 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 11093	. . . . . . . . . . . 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 11382	. . . . . . . . . . . . 13 |- ( x e. ZZ -> x e. CC )
28	27	ad3antrrr 766	. . . . . . . . . . . 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 11382	. . . . . . . . . . . . . 14 |- ( b e. ZZ -> b e. CC )
30	29	adantr 481	. . . . . . . . . . . . 13 |- ( ( b e. ZZ /\ E. x e. ZZ b = ( 2 x. x ) ) -> b e. CC )
31	30	adantl 482	. . . . . . . . . . . 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 )
32	26, 28, 31	mulassd 10063	. . . . . . . . . . 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 ) ) )
33	25, 32	eqtrd 2656	. . . . . . . . . 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 ) ) )
34	20, 23, 33	rspcedvd 3317	. . . . . . . . 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 ) )
35	34	exp41 638	. . . . . . . 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 ) ) ) ) )
36	16, 35	rexlimi 3024	. . . . . . 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 ) ) ) )
37	36	impcom 446	. . . . . 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 ) ) )
38	37	imp 445	. . . . 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 2626	. . . . . . . 8 |- ( z = ( a x. b ) -> ( z = ( 2 x. x ) <-> ( a x. b ) = ( 2 x. x ) ) )
40	39	rexbidv 3052	. . . . . . 7 |- ( z = ( a x. b ) -> ( E. x e. ZZ z = ( 2 x. x ) <-> E. x e. ZZ ( a x. b ) = ( 2 x. x ) ) )
41	40, 3	elrab2 3366	. . . . . 6 |- ( ( a x. b ) e. E <-> ( ( a x. b ) e. ZZ /\ E. x e. ZZ ( a x. b ) = ( 2 x. x ) ) )
42		oveq2 6658	. . . . . . . . 9 |- ( x = y -> ( 2 x. x ) = ( 2 x. y ) )
43	42	eqeq2d 2632	. . . . . . . 8 |- ( x = y -> ( ( a x. b ) = ( 2 x. x ) <-> ( a x. b ) = ( 2 x. y ) ) )
44	43	cbvrexv 3172	. . . . . . 7 |- ( E. x e. ZZ ( a x. b ) = ( 2 x. x ) <-> E. y e. ZZ ( a x. b ) = ( 2 x. y ) )
45	44	anbi2i 730	. . . . . 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 ) ) )
46	41, 45	bitri 264	. . . . 5 |- ( ( a x. b ) e. E <-> ( ( a x. b ) e. ZZ /\ E. y e. ZZ ( a x. b ) = ( 2 x. y ) ) )
47	9, 38, 46	sylanbrc 698	. . . 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 )
48	4, 7, 47	syl2anb 496	. . 3 |- ( ( a e. E /\ b e. E ) -> ( a x. b ) e. E )
49	48	rgen2a 2977	. 2 |- A. a e. E A. b e. E ( a x. b ) e. E
50	3	0even 41931	. . 3 |- 0 e. E
51		2zrngmmgm.1	. . . . 5 |- M = (mulGrp ` R )
52		2zrngbas.r	. . . . . 6 |- R = (ℂfld ↾s E )
53	3, 52	2zrngbas 41936	. . . . 5 |- E = ( Base ` R )
54	51, 53	mgpbas 18495	. . . 4 |- E = ( Base ` M )
55	3, 52	2zrngmul 41945	. . . . 5 |- x. = ( .r ` R )
56	51, 55	mgpplusg 18493	. . . 4 |- x. = ( +g ` M )
57	54, 56	ismgmn0 17244	. . 3 |- ( 0 e. E -> ( M e. Mgm <-> A. a e. E A. b e. E ( a x. b ) e. E ) )
58	50, 57	ax-mp 5	. 2 |- ( M e. Mgm <-> A. a e. E A. b e. E ( a x. b ) e. E )
59	49, 58	mpbir 221	1 |- M e. Mgm

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   {crab 2916   ` cfv 5888  (class class class)co 6650   CCcc 9934   0cc0 9936   x. cmul 9941   2c2 11070   ZZcz 11377   ↾_s cress 15858  Mgmcmgm 17240  mulGrpcmgp 18489  ℂ_fldccnfld 19746

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-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  ax-mulf 10016

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-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-int 4476  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-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  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-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-fz 12327  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-mgm 17242  df-mgp 18490  df-cnfld 19747

This theorem is referenced by:  2zrngmsgrp  41947