Theorem zrhpsgnmhm 18109

Description: Embedding of permutation signs into an arbitrary ring is a homomorphism. (Contributed by SO, 9-Jul-2018.)

Assertion

Ref	Expression
zrhpsgnmhm	|- ( ( R e. Ring /\ A e. Fin ) -> ( ( ZRHom ` R ) o. (pmSgn ` A ) ) e. ( ( SymGrp ` A ) MndHom (mulGrp ` R ) ) )

Proof of Theorem zrhpsgnmhm

Step	Hyp	Ref	Expression
1		eqid 2450	. . . 4 |- ( ZRHom ` R ) = ( ZRHom ` R )
2	1	zrhrhm 18038	. . 3 |- ( R e. Ring -> ( ZRHom ` R ) e. (ℤring RingHom R ) )
3		eqid 2450	. . . 4 |- (mulGrp ` ℤring ) = (mulGrp ` ℤring )
4		eqid 2450	. . . 4 |- (mulGrp ` R ) = (mulGrp ` R )
5	3, 4	rhmmhm 16904	. . 3 |- ( ( ZRHom ` R ) e. (ℤring RingHom R ) -> ( ZRHom ` R ) e. ( (mulGrp ` ℤring ) MndHom (mulGrp ` R ) ) )
6	2, 5	syl 16	. 2 |- ( R e. Ring -> ( ZRHom ` R ) e. ( (mulGrp ` ℤring ) MndHom (mulGrp ` R ) ) )
7		eqid 2450	. . . . 5 |- ( SymGrp ` A ) = ( SymGrp ` A )
8		eqid 2450	. . . . 5 |- (pmSgn ` A ) = (pmSgn ` A )
9		eqid 2450	. . . . 5 |- ( (mulGrp ` ℂfld ) ↾s { 1 , -u 1 } ) = ( (mulGrp ` ℂfld ) ↾s { 1 , -u 1 } )
10	7, 8, 9	psgnghm2 18106	. . . 4 |- ( A e. Fin -> (pmSgn ` A ) e. ( ( SymGrp ` A ) GrpHom ( (mulGrp ` ℂfld ) ↾s { 1 , -u 1 } ) ) )
11		ghmmhm 15845	. . . 4 |- ( (pmSgn ` A ) e. ( ( SymGrp ` A ) GrpHom ( (mulGrp ` ℂfld ) ↾s { 1 , -u 1 } ) ) -> (pmSgn ` A ) e. ( ( SymGrp ` A ) MndHom ( (mulGrp ` ℂfld ) ↾s { 1 , -u 1 } ) ) )
12	10, 11	syl 16	. . 3 |- ( A e. Fin -> (pmSgn ` A ) e. ( ( SymGrp ` A ) MndHom ( (mulGrp ` ℂfld ) ↾s { 1 , -u 1 } ) ) )
13		eqid 2450	. . . . . . . 8 |- ( (mulGrp ` ℂfld ) ↾s ( CC \ { 0 } ) ) = ( (mulGrp ` ℂfld ) ↾s ( CC \ { 0 } ) )
14	13	cnmsgnsubg 18102	. . . . . . 7 |- { 1 , -u 1 } e. (SubGrp ` ( (mulGrp ` ℂfld ) ↾s ( CC \ { 0 } ) ) )
15		subgsubm 15791	. . . . . . 7 |- ( { 1 , -u 1 } e. (SubGrp ` ( (mulGrp ` ℂfld ) ↾s ( CC \ { 0 } ) ) ) -> { 1 , -u 1 } e. (SubMnd ` ( (mulGrp ` ℂfld ) ↾s ( CC \ { 0 } ) ) ) )
16	14, 15	ax-mp 5	. . . . . 6 |- { 1 , -u 1 } e. (SubMnd ` ( (mulGrp ` ℂfld ) ↾s ( CC \ { 0 } ) ) )
17		cnrng 17933	. . . . . . 7 |- ℂfld e. Ring
18		cnfldbas 17917	. . . . . . . . 9 |- CC = ( Base ` ℂfld )
19		cnfld0 17935	. . . . . . . . 9 |- 0 = ( 0g ` ℂfld )
20		cndrng 17940	. . . . . . . . 9 |- ℂfld e. DivRing
21	18, 19, 20	drngui 16930	. . . . . . . 8 |- ( CC \ { 0 } ) = (Unit ` ℂfld )
22		eqid 2450	. . . . . . . 8 |- (mulGrp ` ℂfld ) = (mulGrp ` ℂfld )
23	21, 22	unitsubm 16854	. . . . . . 7 |- (ℂfld e. Ring -> ( CC \ { 0 } ) e. (SubMnd ` (mulGrp ` ℂfld ) ) )
24	13	subsubm 15573	. . . . . . 7 |- ( ( CC \ { 0 } ) e. (SubMnd ` (mulGrp ` ℂfld ) ) -> ( { 1 , -u 1 } e. (SubMnd ` ( (mulGrp ` ℂfld ) ↾s ( CC \ { 0 } ) ) ) <-> ( { 1 , -u 1 } e. (SubMnd ` (mulGrp ` ℂfld ) ) /\ { 1 , -u 1 } C_ ( CC \ { 0 } ) ) ) )
25	17, 23, 24	mp2b 10	. . . . . 6 |- ( { 1 , -u 1 } e. (SubMnd ` ( (mulGrp ` ℂfld ) ↾s ( CC \ { 0 } ) ) ) <-> ( { 1 , -u 1 } e. (SubMnd ` (mulGrp ` ℂfld ) ) /\ { 1 , -u 1 } C_ ( CC \ { 0 } ) ) )
26	16, 25	mpbi 208	. . . . 5 |- ( { 1 , -u 1 } e. (SubMnd ` (mulGrp ` ℂfld ) ) /\ { 1 , -u 1 } C_ ( CC \ { 0 } ) )
27	26	simpli 458	. . . 4 |- { 1 , -u 1 } e. (SubMnd ` (mulGrp ` ℂfld ) )
28		1z 10763	. . . . 5 |- 1 e. ZZ
29		neg1z 10768	. . . . 5 |- -u 1 e. ZZ
30		prssi 4113	. . . . 5 |- ( ( 1 e. ZZ /\ -u 1 e. ZZ ) -> { 1 , -u 1 } C_ ZZ )
31	28, 29, 30	mp2an 672	. . . 4 |- { 1 , -u 1 } C_ ZZ
32		zsubrg 17961	. . . . 5 |- ZZ e. (SubRing ` ℂfld )
33	22	subrgsubm 16970	. . . . 5 |- ( ZZ e. (SubRing ` ℂfld ) -> ZZ e. (SubMnd ` (mulGrp ` ℂfld ) ) )
34		zringmpg 18011	. . . . . . 7 |- ( (mulGrp ` ℂfld ) ↾s ZZ ) = (mulGrp ` ℤring )
35	34	eqcomi 2462	. . . . . 6 |- (mulGrp ` ℤring ) = ( (mulGrp ` ℂfld ) ↾s ZZ )
36	35	subsubm 15573	. . . . 5 |- ( ZZ e. (SubMnd ` (mulGrp ` ℂfld ) ) -> ( { 1 , -u 1 } e. (SubMnd ` (mulGrp ` ℤring ) ) <-> ( { 1 , -u 1 } e. (SubMnd ` (mulGrp ` ℂfld ) ) /\ { 1 , -u 1 } C_ ZZ ) ) )
37	32, 33, 36	mp2b 10	. . . 4 |- ( { 1 , -u 1 } e. (SubMnd ` (mulGrp ` ℤring ) ) <-> ( { 1 , -u 1 } e. (SubMnd ` (mulGrp ` ℂfld ) ) /\ { 1 , -u 1 } C_ ZZ ) )
38	27, 31, 37	mpbir2an 911	. . 3 |- { 1 , -u 1 } e. (SubMnd ` (mulGrp ` ℤring ) )
39		zex 10742	. . . . . 6 |- ZZ e. _V
40		ressabs 14324	. . . . . 6 |- ( ( ZZ e. _V /\ { 1 , -u 1 } C_ ZZ ) -> ( ( (mulGrp ` ℂfld ) ↾s ZZ ) ↾s { 1 , -u 1 } ) = ( (mulGrp ` ℂfld ) ↾s { 1 , -u 1 } ) )
41	39, 31, 40	mp2an 672	. . . . 5 |- ( ( (mulGrp ` ℂfld ) ↾s ZZ ) ↾s { 1 , -u 1 } ) = ( (mulGrp ` ℂfld ) ↾s { 1 , -u 1 } )
42	34	oveq1i 6186	. . . . 5 |- ( ( (mulGrp ` ℂfld ) ↾s ZZ ) ↾s { 1 , -u 1 } ) = ( (mulGrp ` ℤring ) ↾s { 1 , -u 1 } )
43	41, 42	eqtr3i 2480	. . . 4 |- ( (mulGrp ` ℂfld ) ↾s { 1 , -u 1 } ) = ( (mulGrp ` ℤring ) ↾s { 1 , -u 1 } )
44	43	resmhm2 15576	. . 3 |- ( ( (pmSgn ` A ) e. ( ( SymGrp ` A ) MndHom ( (mulGrp ` ℂfld ) ↾s { 1 , -u 1 } ) ) /\ { 1 , -u 1 } e. (SubMnd ` (mulGrp ` ℤring ) ) ) -> (pmSgn ` A ) e. ( ( SymGrp ` A ) MndHom (mulGrp ` ℤring ) ) )
45	12, 38, 44	sylancl 662	. 2 |- ( A e. Fin -> (pmSgn ` A ) e. ( ( SymGrp ` A ) MndHom (mulGrp ` ℤring ) ) )
46		mhmco 15578	. 2 |- ( ( ( ZRHom ` R ) e. ( (mulGrp ` ℤring ) MndHom (mulGrp ` R ) ) /\ (pmSgn ` A ) e. ( ( SymGrp ` A ) MndHom (mulGrp ` ℤring ) ) ) -> ( ( ZRHom ` R ) o. (pmSgn ` A ) ) e. ( ( SymGrp ` A ) MndHom (mulGrp ` R ) ) )
47	6, 45, 46	syl2an 477	1 |- ( ( R e. Ring /\ A e. Fin ) -> ( ( ZRHom ` R ) o. (pmSgn ` A ) ) e. ( ( SymGrp ` A ) MndHom (mulGrp ` R ) ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1370   e. wcel 1757   _Vcvv 3054   \ cdif 3409   C_ wss 3412   {csn 3961   {cpr 3963   o. ccom 4928   ` cfv 5502  (class class class)co 6176   Fincfn 7396   CCcc 9367   0cc0 9369   1c1 9370   -ucneg 9683   ZZcz 10733   ↾_s cress 14263   MndHom cmhm 15550  SubMndcsubmnd 15551  SubGrpcsubg 15763   GrpHom cghm 15832   SymGrpcsymg 15970  pmSgncpsgn 16083  mulGrpcmgp 16682   Ringcrg 16737   RingHom crh 16896  SubRingcsubrg 16953  ℂ_fldccnfld 17913  ℤ_ringzring 17978   ZRHomczrh 18026

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1709  ax-7 1729  ax-8 1759  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429  ax-rep 4487  ax-sep 4497  ax-nul 4505  ax-pow 4554  ax-pr 4615  ax-un 6458  ax-inf2 7934  ax-cnex 9425  ax-resscn 9426  ax-1cn 9427  ax-icn 9428  ax-addcl 9429  ax-addrcl 9430  ax-mulcl 9431  ax-mulrcl 9432  ax-mulcom 9433  ax-addass 9434  ax-mulass 9435  ax-distr 9436  ax-i2m1 9437  ax-1ne0 9438  ax-1rid 9439  ax-rnegex 9440  ax-rrecex 9441  ax-cnre 9442  ax-pre-lttri 9443  ax-pre-lttrn 9444  ax-pre-ltadd 9445  ax-pre-mulgt0 9446  ax-addf 9448  ax-mulf 9449

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-xor 1352  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1702  df-eu 2263  df-mo 2264  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-ne 2643  df-nel 2644  df-ral 2797  df-rex 2798  df-reu 2799  df-rmo 2800  df-rab 2801  df-v 3056  df-sbc 3271  df-csb 3373  df-dif 3415  df-un 3417  df-in 3419  df-ss 3426  df-pss 3428  df-nul 3722  df-if 3876  df-pw 3946  df-sn 3962  df-pr 3964  df-tp 3966  df-op 3968  df-ot 3970  df-uni 4176  df-int 4213  df-iun 4257  df-iin 4258  df-br 4377  df-opab 4435  df-mpt 4436  df-tr 4470  df-eprel 4716  df-id 4720  df-po 4725  df-so 4726  df-fr 4763  df-se 4764  df-we 4765  df-ord 4806  df-on 4807  df-lim 4808  df-suc 4809  df-xp 4930  df-rel 4931  df-cnv 4932  df-co 4933  df-dm 4934  df-rn 4935  df-res 4936  df-ima 4937  df-iota 5465  df-fun 5504  df-fn 5505  df-f 5506  df-f1 5507  df-fo 5508  df-f1o 5509  df-fv 5510  df-isom 5511  df-riota 6137  df-ov 6179  df-oprab 6180  df-mpt2 6181  df-om 6563  df-1st 6663  df-2nd 6664  df-tpos 6831  df-recs 6918  df-rdg 6952  df-1o 7006  df-2o 7007  df-oadd 7010  df-er 7187  df-map 7302  df-en 7397  df-dom 7398  df-sdom 7399  df-fin 7400  df-card 8196  df-pnf 9507  df-mnf 9508  df-xr 9509  df-ltxr 9510  df-le 9511  df-sub 9684  df-neg 9685  df-div 10081  df-nn 10410  df-2 10467  df-3 10468  df-4 10469  df-5 10470  df-6 10471  df-7 10472  df-8 10473  df-9 10474  df-10 10475  df-n0 10667  df-z 10734  df-dec 10843  df-uz 10949  df-rp 11079  df-fz 11525  df-fzo 11636  df-seq 11894  df-exp 11953  df-hash 12191  df-word 12317  df-concat 12319  df-s1 12320  df-substr 12321  df-splice 12322  df-reverse 12323  df-s2 12563  df-struct 14264  df-ndx 14265  df-slot 14266  df-base 14267  df-sets 14268  df-ress 14269  df-plusg 14339  df-mulr 14340  df-starv 14341  df-tset 14345  df-ple 14346  df-ds 14348  df-unif 14349  df-0g 14468  df-gsum 14469  df-mre 14612  df-mrc 14613  df-acs 14615  df-mnd 15503  df-mhm 15552  df-submnd 15553  df-grp 15633  df-minusg 15634  df-mulg 15636  df-subg 15766  df-ghm 15833  df-gim 15875  df-oppg 15949  df-symg 15971  df-pmtr 16036  df-psgn 16085  df-cmn 16369  df-abl 16370  df-mgp 16683  df-ur 16695  df-rng 16739  df-cring 16740  df-oppr 16807  df-dvdsr 16825  df-unit 16826  df-invr 16856  df-dvr 16867  df-rnghom 16898  df-drng 16926  df-subrg 16955  df-cnfld 17914  df-zring 17979  df-zrh 18030

This theorem is referenced by:  madetsumid  18443  mdetleib2  18496  mdetf  18503  mdetdiaglem  18506  mdetrlin  18510  mdetrsca  18511  mdetralt  18516  mdetunilem7  18526  mdetunilem8  18527