Theorem rhmf1o 14132

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 14120	. . . . 5 |- ( F e. ( R RingHom S ) -> S e. Ring )
2		rhmrcl1 14119	. . . . 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 14126	. . . . . . 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 13812	. . . . . . 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 2230	. . . . . . 7 |- ( F e. ( R RingHom S ) -> F = F )
15		eqid 2229	. . . . . . . . 9 |- (mulGrp ` R ) = (mulGrp ` R )
16	15, 8	mgpbasg 13889	. . . . . . . 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 2229	. . . . . . . . 9 |- (mulGrp ` S ) = (mulGrp ` S )
19	18, 9	mgpbasg 13889	. . . . . . . 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 5566	. . . . . 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 14123	. . . . . . 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 2229	. . . . . . . 8 |- ( Base ` (mulGrp ` R ) ) = ( Base ` (mulGrp ` R ) )
26		eqid 2229	. . . . . . . 8 |- ( Base ` (mulGrp ` S ) ) = ( Base ` (mulGrp ` S ) )
27	25, 26	mhmf1o 13503	. . . . . . 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 14122	. . 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 14127	. . . . 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 5474	. . 3 |- ( ( F e. ( R RingHom S ) /\ `' F e. ( S RingHom R ) ) -> F Fn B )
37	9, 8	rhmf 14127	. . . . 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 5474	. . 3 |- ( ( F e. ( R RingHom S ) /\ `' F e. ( S RingHom R ) ) -> `' F Fn C )
40		dff1o4 5580	. . 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 598	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 1395   e. wcel 2200   `'ccnv 4718   Fn wfn 5313   -->wf 5314   -1-1-onto->wf1o 5317   ` cfv 5318  (class class class)co 6001   Basecbs 13032   MndHom cmhm 13490   GrpHom cghm 13777  mulGrpcmgp 13883   Ringcrg 13959   RingHom crh 14114

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-addcom 8099  ax-addass 8101  ax-i2m1 8104  ax-0lt1 8105  ax-0id 8107  ax-rnegex 8108  ax-pre-ltirr 8111  ax-pre-ltadd 8115

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-1st 6286  df-2nd 6287  df-map 6797  df-pnf 8183  df-mnf 8184  df-ltxr 8186  df-inn 9111  df-2 9169  df-3 9170  df-ndx 13035  df-slot 13036  df-base 13038  df-sets 13039  df-plusg 13123  df-mulr 13124  df-0g 13291  df-mgm 13389  df-sgrp 13435  df-mnd 13450  df-mhm 13492  df-grp 13536  df-ghm 13778  df-mgp 13884  df-ur 13923  df-ring 13961  df-rhm 14116

This theorem is referenced by:  isrim  14133