Theorem rhmf1o 14413

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 14401	. . . . 5 |- ( F e. ( R RingHom S ) -> S e. Ring )
2		rhmrcl1 14400	. . . . 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 14407	. . . . . . 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 14028	. . . . . . 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 2235	. . . . . . 7 |- ( F e. ( R RingHom S ) -> F = F )
15		eqid 2234	. . . . . . . . 9 |- (mulGrp ` R ) = (mulGrp ` R )
16	15, 8	mgpbasg 14165	. . . . . . . 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 2234	. . . . . . . . 9 |- (mulGrp ` S ) = (mulGrp ` S )
19	18, 9	mgpbasg 14165	. . . . . . . 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 5613	. . . . . 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 14404	. . . . . . 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 2234	. . . . . . . 8 |- ( Base ` (mulGrp ` R ) ) = ( Base ` (mulGrp ` R ) )
26		eqid 2234	. . . . . . . 8 |- ( Base ` (mulGrp ` S ) ) = ( Base ` (mulGrp ` S ) )
27	25, 26	mhmf1o 13725	. . . . . . 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 14403	. . 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 14408	. . . . 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 5514	. . 3 |- ( ( F e. ( R RingHom S ) /\ `' F e. ( S RingHom R ) ) -> F Fn B )
37	9, 8	rhmf 14408	. . . . 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 5514	. . 3 |- ( ( F e. ( R RingHom S ) /\ `' F e. ( S RingHom R ) ) -> `' F Fn C )
40		dff1o4 5627	. . 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 1398   e. wcel 2205   `'ccnv 4753   Fn wfn 5352   -->wf 5353   -1-1-onto->wf1o 5356   ` cfv 5357  (class class class)co 6058   Basecbs 13296   MndHom cmhm 13712   GrpHom cghm 13993  mulGrpcmgp 14159   Ringcrg 14239   RingHom crh 14395

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-addass 8245  ax-i2m1 8248  ax-0lt1 8249  ax-0id 8251  ax-rnegex 8252  ax-pre-ltirr 8255  ax-pre-ltadd 8259

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-map 6897  df-pnf 8326  df-mnf 8327  df-ltxr 8329  df-inn 9255  df-2 9313  df-3 9314  df-ndx 13299  df-slot 13300  df-base 13302  df-sets 13303  df-plusg 13387  df-mulr 13388  df-0g 13555  df-mgm 13619  df-sgrp 13665  df-mnd 13678  df-mhm 13714  df-grp 13758  df-ghm 13994  df-mgp 14160  df-ur 14203  df-ring 14241  df-rhm 14397

This theorem is referenced by:  isrim  14414