Theorem rhmf1o 14045

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 14033	. . . . 5 |- ( F e. ( R RingHom S ) -> S e. Ring )
2		rhmrcl1 14032	. . . . 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 14039	. . . . . . 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 13726	. . . . . . 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 2208	. . . . . . 7 |- ( F e. ( R RingHom S ) -> F = F )
15		eqid 2207	. . . . . . . . 9 |- (mulGrp ` R ) = (mulGrp ` R )
16	15, 8	mgpbasg 13803	. . . . . . . 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 2207	. . . . . . . . 9 |- (mulGrp ` S ) = (mulGrp ` S )
19	18, 9	mgpbasg 13803	. . . . . . . 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 5538	. . . . . 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 14036	. . . . . . 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 2207	. . . . . . . 8 |- ( Base ` (mulGrp ` R ) ) = ( Base ` (mulGrp ` R ) )
26		eqid 2207	. . . . . . . 8 |- ( Base ` (mulGrp ` S ) ) = ( Base ` (mulGrp ` S ) )
27	25, 26	mhmf1o 13417	. . . . . . 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 14035	. . 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 14040	. . . . 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 5446	. . 3 |- ( ( F e. ( R RingHom S ) /\ `' F e. ( S RingHom R ) ) -> F Fn B )
37	9, 8	rhmf 14040	. . . . 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 5446	. . 3 |- ( ( F e. ( R RingHom S ) /\ `' F e. ( S RingHom R ) ) -> `' F Fn C )
40		dff1o4 5552	. . 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 596	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 1373   e. wcel 2178   `'ccnv 4692   Fn wfn 5285   -->wf 5286   -1-1-onto->wf1o 5289   ` cfv 5290  (class class class)co 5967   Basecbs 12947   MndHom cmhm 13404   GrpHom cghm 13691  mulGrpcmgp 13797   Ringcrg 13873   RingHom crh 14027

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-addcom 8060  ax-addass 8062  ax-i2m1 8065  ax-0lt1 8066  ax-0id 8068  ax-rnegex 8069  ax-pre-ltirr 8072  ax-pre-ltadd 8076

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rmo 2494  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-map 6760  df-pnf 8144  df-mnf 8145  df-ltxr 8147  df-inn 9072  df-2 9130  df-3 9131  df-ndx 12950  df-slot 12951  df-base 12953  df-sets 12954  df-plusg 13037  df-mulr 13038  df-0g 13205  df-mgm 13303  df-sgrp 13349  df-mnd 13364  df-mhm 13406  df-grp 13450  df-ghm 13692  df-mgp 13798  df-ur 13837  df-ring 13875  df-rhm 14029

This theorem is referenced by:  isrim  14046