Theorem rhmf1o 14475

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 14463	. . . . 5 |- ( F e. ( R RingHom S ) -> S e. Ring )
2		rhmrcl1 14462	. . . . 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 14469	. . . . . . 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 14078	. . . . . . 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 2239	. . . . . . 7 |- ( F e. ( R RingHom S ) -> F = F )
15		eqid 2238	. . . . . . . . 9 |- (mulGrp ` R ) = (mulGrp ` R )
16	15, 8	mgpbasg 14224	. . . . . . . 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 2238	. . . . . . . . 9 |- (mulGrp ` S ) = (mulGrp ` S )
19	18, 9	mgpbasg 14224	. . . . . . . 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 5633	. . . . . 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 14466	. . . . . . 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 2238	. . . . . . . 8 |- ( Base ` (mulGrp ` R ) ) = ( Base ` (mulGrp ` R ) )
26		eqid 2238	. . . . . . . 8 |- ( Base ` (mulGrp ` S ) ) = ( Base ` (mulGrp ` S ) )
27	25, 26	mhmf1o 13777	. . . . . . 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 14465	. . 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 421	. 2 |- ( ( F e. ( R RingHom S ) /\ F : B -1-1-onto-> C ) -> `' F e. ( S RingHom R ) )
34	8, 9	rhmf 14470	. . . . 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 5534	. . 3 |- ( ( F e. ( R RingHom S ) /\ `' F e. ( S RingHom R ) ) -> F Fn B )
37	9, 8	rhmf 14470	. . . . 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 5534	. . 3 |- ( ( F e. ( R RingHom S ) /\ `' F e. ( S RingHom R ) ) -> `' F Fn C )
40		dff1o4 5647	. . 3 |- ( F : B -1-1-onto-> C <-> ( F Fn B /\ `' F Fn C ) )
41	36, 39, 40	sylanbrc 421	. 2 |- ( ( F e. ( R RingHom S ) /\ `' F e. ( S RingHom R ) ) -> F : B -1-1-onto-> C )
42	33, 41	impbida 604	1 |- ( F e. ( R RingHom S ) -> ( F : B -1-1-onto-> C <-> `' F e. ( S RingHom R ) ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   `'ccnv 4773   Fn wfn 5372   -->wf 5373   -1-1-onto->wf1o 5376   ` cfv 5377  (class class class)co 6085   Basecbs 13352   MndHom cmhm 13764   GrpHom cghm 14043  mulGrpcmgp 14217   Ringcrg 14300   RingHom crh 14457

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-addass 8281  ax-i2m1 8284  ax-0lt1 8285  ax-0id 8287  ax-rnegex 8288  ax-pre-ltirr 8291  ax-pre-ltadd 8295

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-map 6924  df-pnf 8362  df-mnf 8363  df-ltxr 8365  df-inn 9305  df-2 9363  df-3 9364  df-ndx 13355  df-slot 13356  df-base 13358  df-sets 13359  df-plusg 13444  df-mulr 13445  df-0g 13612  df-mgm 13676  df-sgrp 13717  df-mnd 13730  df-mhm 13766  df-grp 13808  df-ghm 14044  df-mgp 14218  df-ur 14263  df-ring 14302  df-rhm 14459

This theorem is used by:  isrim  14476