Theorem zrhpsgnmhm 18387

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 2467	. . . 4 |- ( ZRHom ` R ) = ( ZRHom ` R )
2	1	zrhrhm 18316	. . 3 |- ( R e. Ring -> ( ZRHom ` R ) e. (ℤring RingHom R ) )
3		eqid 2467	. . . 4 |- (mulGrp ` ℤring ) = (mulGrp ` ℤring )
4		eqid 2467	. . . 4 |- (mulGrp ` R ) = (mulGrp ` R )
5	3, 4	rhmmhm 17155	. . 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 2467	. . . . 5 |- ( SymGrp ` A ) = ( SymGrp ` A )
8		eqid 2467	. . . . 5 |- (pmSgn ` A ) = (pmSgn ` A )
9		eqid 2467	. . . . 5 |- ( (mulGrp ` ℂfld ) ↾s { 1 , -u 1 } ) = ( (mulGrp ` ℂfld ) ↾s { 1 , -u 1 } )
10	7, 8, 9	psgnghm2 18384	. . . 4 |- ( A e. Fin -> (pmSgn ` A ) e. ( ( SymGrp ` A ) GrpHom ( (mulGrp ` ℂfld ) ↾s { 1 , -u 1 } ) ) )
11		ghmmhm 16072	. . . 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 2467	. . . . . . . 8 |- ( (mulGrp ` ℂfld ) ↾s ( CC \ { 0 } ) ) = ( (mulGrp ` ℂfld ) ↾s ( CC \ { 0 } ) )
14	13	cnmsgnsubg 18380	. . . . . . 7 |- { 1 , -u 1 } e. (SubGrp ` ( (mulGrp ` ℂfld ) ↾s ( CC \ { 0 } ) ) )
15		subgsubm 16018	. . . . . . 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 18211	. . . . . . 7 |- ℂfld e. Ring
18		cnfldbas 18195	. . . . . . . . 9 |- CC = ( Base ` ℂfld )
19		cnfld0 18213	. . . . . . . . 9 |- 0 = ( 0g ` ℂfld )
20		cndrng 18218	. . . . . . . . 9 |- ℂfld e. DivRing
21	18, 19, 20	drngui 17185	. . . . . . . 8 |- ( CC \ { 0 } ) = (Unit ` ℂfld )
22		eqid 2467	. . . . . . . 8 |- (mulGrp ` ℂfld ) = (mulGrp ` ℂfld )
23	21, 22	unitsubm 17103	. . . . . . 7 |- (ℂfld e. Ring -> ( CC \ { 0 } ) e. (SubMnd ` (mulGrp ` ℂfld ) ) )
24	13	subsubm 15798	. . . . . . 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 10890	. . . . 5 |- 1 e. ZZ
29		neg1z 10895	. . . . 5 |- -u 1 e. ZZ
30		prssi 4183	. . . . 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 18239	. . . . 5 |- ZZ e. (SubRing ` ℂfld )
33	22	subrgsubm 17225	. . . . 5 |- ( ZZ e. (SubRing ` ℂfld ) -> ZZ e. (SubMnd ` (mulGrp ` ℂfld ) ) )
34		zringmpg 18289	. . . . . . 7 |- ( (mulGrp ` ℂfld ) ↾s ZZ ) = (mulGrp ` ℤring )
35	34	eqcomi 2480	. . . . . 6 |- (mulGrp ` ℤring ) = ( (mulGrp ` ℂfld ) ↾s ZZ )
36	35	subsubm 15798	. . . . 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 918	. . 3 |- { 1 , -u 1 } e. (SubMnd ` (mulGrp ` ℤring ) )
39		zex 10869	. . . . . 6 |- ZZ e. _V
40		ressabs 14549	. . . . . 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 6292	. . . . 5 |- ( ( (mulGrp ` ℂfld ) ↾s ZZ ) ↾s { 1 , -u 1 } ) = ( (mulGrp ` ℤring ) ↾s { 1 , -u 1 } )
43	41, 42	eqtr3i 2498	. . . 4 |- ( (mulGrp ` ℂfld ) ↾s { 1 , -u 1 } ) = ( (mulGrp ` ℤring ) ↾s { 1 , -u 1 } )
44	43	resmhm2 15801	. . 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 15803	. 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 1379   e. wcel 1767   _Vcvv 3113   \ cdif 3473   C_ wss 3476   {csn 4027   {cpr 4029   o. ccom 5003   ` cfv 5586  (class class class)co 6282   Fincfn 7513   CCcc 9486   0cc0 9488   1c1 9489   -ucneg 9802   ZZcz 10860   ↾_s cress 14487   MndHom cmhm 15775  SubMndcsubmnd 15776  SubGrpcsubg 15990   GrpHom cghm 16059   SymGrpcsymg 16197  pmSgncpsgn 16310  mulGrpcmgp 16931   Ringcrg 16986   RingHom crh 17145  SubRingcsubrg 17208  ℂ_fldccnfld 18191  ℤ_ringzring 18256   ZRHomczrh 18304

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-inf2 8054  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-addf 9567  ax-mulf 9568

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-xor 1361  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-ot 4036  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-tpos 6952  df-recs 7039  df-rdg 7073  df-1o 7127  df-2o 7128  df-oadd 7131  df-er 7308  df-map 7419  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-card 8316  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10973  df-uz 11079  df-rp 11217  df-fz 11669  df-fzo 11789  df-seq 12072  df-exp 12131  df-hash 12370  df-word 12504  df-concat 12506  df-s1 12507  df-substr 12508  df-splice 12509  df-reverse 12510  df-s2 12772  df-struct 14488  df-ndx 14489  df-slot 14490  df-base 14491  df-sets 14492  df-ress 14493  df-plusg 14564  df-mulr 14565  df-starv 14566  df-tset 14570  df-ple 14571  df-ds 14573  df-unif 14574  df-0g 14693  df-gsum 14694  df-mre 14837  df-mrc 14838  df-acs 14840  df-mnd 15728  df-mhm 15777  df-submnd 15778  df-grp 15858  df-minusg 15859  df-mulg 15861  df-subg 15993  df-ghm 16060  df-gim 16102  df-oppg 16176  df-symg 16198  df-pmtr 16263  df-psgn 16312  df-cmn 16596  df-abl 16597  df-mgp 16932  df-ur 16944  df-rng 16988  df-cring 16989  df-oppr 17056  df-dvdsr 17074  df-unit 17075  df-invr 17105  df-dvr 17116  df-rnghom 17148  df-drng 17181  df-subrg 17210  df-cnfld 18192  df-zring 18257  df-zrh 18308

This theorem is referenced by:  madetsumid  18730  mdetleib2  18857  mdetf  18864  mdetdiaglem  18867  mdetrlin  18871  mdetrsca  18872  mdetralt  18877  mdetunilem7  18887  mdetunilem8  18888