Theorem rhmf1o 14181

Description: A ring homomorphism is bijective iff its converse is also a ring homomorphism. (Contributed by AV, 22-Oct-2019.)

Hypotheses

Ref	Expression
rhmf1o.b	|- B = ( Base ` R )
rhmf1o.c	|- C = ( Base ` S )

Assertion

Ref	Expression
rhmf1o	|- ( F e. ( R RingHom S ) -> ( F : B -1-1-onto-> C <-> `' F e. ( S RingHom R ) ) )

Proof of Theorem rhmf1o

Step	Hyp	Ref	Expression
1		rhmrcl2 14169	. . . . 5 |- ( F e. ( R RingHom S ) -> S e. Ring )
2		rhmrcl1 14168	. . . . 5 |- ( F e. ( R RingHom S ) -> R e. Ring )
3	1, 2	jca 306	. . . 4 |- ( F e. ( R RingHom S ) -> ( S e. Ring /\ R e. Ring ) )
4	3	adantr 276	. . 3 |- ( ( F e. ( R RingHom S ) /\ F : B -1-1-onto-> C ) -> ( S e. Ring /\ R e. Ring ) )
5		simpr 110	. . . . 5 |- ( ( F e. ( R RingHom S ) /\ F : B -1-1-onto-> C ) -> F : B -1-1-onto-> C )
6		rhmghm 14175	. . . . . . 7 |- ( F e. ( R RingHom S ) -> F e. ( R GrpHom S ) )
7	6	adantr 276	. . . . . 6 |- ( ( F e. ( R RingHom S ) /\ F : B -1-1-onto-> C ) -> F e. ( R GrpHom S ) )
8		rhmf1o.b	. . . . . . . 8 |- B = ( Base ` R )
9		rhmf1o.c	. . . . . . . 8 |- C = ( Base ` S )
10	8, 9	ghmf1o 13861	. . . . . . 7 |- ( F e. ( R GrpHom S ) -> ( F : B -1-1-onto-> C <-> `' F e. ( S GrpHom R ) ) )
11	10	bicomd 141	. . . . . 6 |- ( F e. ( R GrpHom S ) -> ( `' F e. ( S GrpHom R ) <-> F : B -1-1-onto-> C ) )
12	7, 11	syl 14	. . . . 5 |- ( ( F e. ( R RingHom S ) /\ F : B -1-1-onto-> C ) -> ( `' F e. ( S GrpHom R ) <-> F : B -1-1-onto-> C ) )
13	5, 12	mpbird 167	. . . 4 |- ( ( F e. ( R RingHom S ) /\ F : B -1-1-onto-> C ) -> `' F e. ( S GrpHom R ) )
14		eqidd 2232	. . . . . . 7 |- ( F e. ( R RingHom S ) -> F = F )
15		eqid 2231	. . . . . . . . 9 |- (mulGrp ` R ) = (mulGrp ` R )
16	15, 8	mgpbasg 13938	. . . . . . . 8 |- ( R e. Ring -> B = ( Base ` (mulGrp ` R ) ) )
17	2, 16	syl 14	. . . . . . 7 |- ( F e. ( R RingHom S ) -> B = ( Base ` (mulGrp ` R ) ) )
18		eqid 2231	. . . . . . . . 9 |- (mulGrp ` S ) = (mulGrp ` S )
19	18, 9	mgpbasg 13938	. . . . . . . 8 |- ( S e. Ring -> C = ( Base ` (mulGrp ` S ) ) )
20	1, 19	syl 14	. . . . . . 7 |- ( F e. ( R RingHom S ) -> C = ( Base ` (mulGrp ` S ) ) )
21	14, 17, 20	f1oeq123d 5577	. . . . . 6 |- ( F e. ( R RingHom S ) -> ( F : B -1-1-onto-> C <-> F : ( Base ` (mulGrp ` R ) ) -1-1-onto-> ( Base ` (mulGrp ` S ) ) ) )
22	21	biimpa 296	. . . . 5 |- ( ( F e. ( R RingHom S ) /\ F : B -1-1-onto-> C ) -> F : ( Base ` (mulGrp ` R ) ) -1-1-onto-> ( Base ` (mulGrp ` S ) ) )
23	15, 18	rhmmhm 14172	. . . . . . 7 |- ( F e. ( R RingHom S ) -> F e. ( (mulGrp ` R ) MndHom (mulGrp ` S ) ) )
24	23	adantr 276	. . . . . 6 |- ( ( F e. ( R RingHom S ) /\ F : B -1-1-onto-> C ) -> F e. ( (mulGrp ` R ) MndHom (mulGrp ` S ) ) )
25		eqid 2231	. . . . . . . 8 |- ( Base ` (mulGrp ` R ) ) = ( Base ` (mulGrp ` R ) )
26		eqid 2231	. . . . . . . 8 |- ( Base ` (mulGrp ` S ) ) = ( Base ` (mulGrp ` S ) )
27	25, 26	mhmf1o 13552	. . . . . . 7 |- ( F e. ( (mulGrp ` R ) MndHom (mulGrp ` S ) ) -> ( F : ( Base ` (mulGrp ` R ) ) -1-1-onto-> ( Base ` (mulGrp ` S ) ) <-> `' F e. ( (mulGrp ` S ) MndHom (mulGrp ` R ) ) ) )
28	27	bicomd 141	. . . . . 6 |- ( F e. ( (mulGrp ` R ) MndHom (mulGrp ` S ) ) -> ( `' F e. ( (mulGrp ` S ) MndHom (mulGrp ` R ) ) <-> F : ( Base ` (mulGrp ` R ) ) -1-1-onto-> ( Base ` (mulGrp ` S ) ) ) )
29	24, 28	syl 14	. . . . 5 |- ( ( F e. ( R RingHom S ) /\ F : B -1-1-onto-> C ) -> ( `' F e. ( (mulGrp ` S ) MndHom (mulGrp ` R ) ) <-> F : ( Base ` (mulGrp ` R ) ) -1-1-onto-> ( Base ` (mulGrp ` S ) ) ) )
30	22, 29	mpbird 167	. . . 4 |- ( ( F e. ( R RingHom S ) /\ F : B -1-1-onto-> C ) -> `' F e. ( (mulGrp ` S ) MndHom (mulGrp ` R ) ) )
31	13, 30	jca 306	. . 3 |- ( ( F e. ( R RingHom S ) /\ F : B -1-1-onto-> C ) -> ( `' F e. ( S GrpHom R ) /\ `' F e. ( (mulGrp ` S ) MndHom (mulGrp ` R ) ) ) )
32	18, 15	isrhm 14171	. . 3 |- ( `' F e. ( S RingHom R ) <-> ( ( S e. Ring /\ R e. Ring ) /\ ( `' F e. ( S GrpHom R ) /\ `' F e. ( (mulGrp ` S ) MndHom (mulGrp ` R ) ) ) ) )
33	4, 31, 32	sylanbrc 417	. 2 |- ( ( F e. ( R RingHom S ) /\ F : B -1-1-onto-> C ) -> `' F e. ( S RingHom R ) )
34	8, 9	rhmf 14176	. . . . 5 |- ( F e. ( R RingHom S ) -> F : B --> C )
35	34	adantr 276	. . . 4 |- ( ( F e. ( R RingHom S ) /\ `' F e. ( S RingHom R ) ) -> F : B --> C )
36	35	ffnd 5483	. . 3 |- ( ( F e. ( R RingHom S ) /\ `' F e. ( S RingHom R ) ) -> F Fn B )
37	9, 8	rhmf 14176	. . . . 5 |- ( `' F e. ( S RingHom R ) -> `' F : C --> B )
38	37	adantl 277	. . . 4 |- ( ( F e. ( R RingHom S ) /\ `' F e. ( S RingHom R ) ) -> `' F : C --> B )
39	38	ffnd 5483	. . 3 |- ( ( F e. ( R RingHom S ) /\ `' F e. ( S RingHom R ) ) -> `' F Fn C )
40		dff1o4 5591	. . 3 |- ( F : B -1-1-onto-> C <-> ( F Fn B /\ `' F Fn C ) )
41	36, 39, 40	sylanbrc 417	. 2 |- ( ( F e. ( R RingHom S ) /\ `' F e. ( S RingHom R ) ) -> F : B -1-1-onto-> C )
42	33, 41	impbida 600	1 |- ( F e. ( R RingHom S ) -> ( F : B -1-1-onto-> C <-> `' F e. ( S RingHom R ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1397   e. wcel 2202   `'ccnv 4724   Fn wfn 5321   -->wf 5322   -1-1-onto->wf1o 5325   ` cfv 5326  (class class class)co 6017   Basecbs 13081   MndHom cmhm 13539   GrpHom cghm 13826  mulGrpcmgp 13932   Ringcrg 14008   RingHom crh 14163

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-addcom 8131  ax-addass 8133  ax-i2m1 8136  ax-0lt1 8137  ax-0id 8139  ax-rnegex 8140  ax-pre-ltirr 8143  ax-pre-ltadd 8147

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-map 6818  df-pnf 8215  df-mnf 8216  df-ltxr 8218  df-inn 9143  df-2 9201  df-3 9202  df-ndx 13084  df-slot 13085  df-base 13087  df-sets 13088  df-plusg 13172  df-mulr 13173  df-0g 13340  df-mgm 13438  df-sgrp 13484  df-mnd 13499  df-mhm 13541  df-grp 13585  df-ghm 13827  df-mgp 13933  df-ur 13972  df-ring 14010  df-rhm 14165

This theorem is referenced by:  isrim  14182