Theorem rngoisoco 33781

Description: The composition of two ring isomorphisms is a ring isomorphism. (Contributed by Jeff Madsen, 16-Jun-2011.)

Assertion

Ref	Expression
rngoisoco	|- ( ( ( R e. RingOps /\ S e. RingOps /\ T e. RingOps ) /\ ( F e. ( R RngIso S ) /\ G e. ( S RngIso T ) ) ) -> ( G o. F ) e. ( R RngIso T ) )

Proof of Theorem rngoisoco

Step	Hyp	Ref	Expression
1		rngoisohom 33779	. . . . . 6 |- ( ( R e. RingOps /\ S e. RingOps /\ F e. ( R RngIso S ) ) -> F e. ( R RngHom S ) )
2	1	3expa 1265	. . . . 5 |- ( ( ( R e. RingOps /\ S e. RingOps ) /\ F e. ( R RngIso S ) ) -> F e. ( R RngHom S ) )
3	2	3adantl3 1219	. . . 4 |- ( ( ( R e. RingOps /\ S e. RingOps /\ T e. RingOps ) /\ F e. ( R RngIso S ) ) -> F e. ( R RngHom S ) )
4		rngoisohom 33779	. . . . . 6 |- ( ( S e. RingOps /\ T e. RingOps /\ G e. ( S RngIso T ) ) -> G e. ( S RngHom T ) )
5	4	3expa 1265	. . . . 5 |- ( ( ( S e. RingOps /\ T e. RingOps ) /\ G e. ( S RngIso T ) ) -> G e. ( S RngHom T ) )
6	5	3adantl1 1217	. . . 4 |- ( ( ( R e. RingOps /\ S e. RingOps /\ T e. RingOps ) /\ G e. ( S RngIso T ) ) -> G e. ( S RngHom T ) )
7	3, 6	anim12da 33506	. . 3 |- ( ( ( R e. RingOps /\ S e. RingOps /\ T e. RingOps ) /\ ( F e. ( R RngIso S ) /\ G e. ( S RngIso T ) ) ) -> ( F e. ( R RngHom S ) /\ G e. ( S RngHom T ) ) )
8		rngohomco 33773	. . 3 |- ( ( ( R e. RingOps /\ S e. RingOps /\ T e. RingOps ) /\ ( F e. ( R RngHom S ) /\ G e. ( S RngHom T ) ) ) -> ( G o. F ) e. ( R RngHom T ) )
9	7, 8	syldan 487	. 2 |- ( ( ( R e. RingOps /\ S e. RingOps /\ T e. RingOps ) /\ ( F e. ( R RngIso S ) /\ G e. ( S RngIso T ) ) ) -> ( G o. F ) e. ( R RngHom T ) )
10		eqid 2622	. . . . . . 7 |- ( 1st ` S ) = ( 1st ` S )
11		eqid 2622	. . . . . . 7 |- ran ( 1st ` S ) = ran ( 1st ` S )
12		eqid 2622	. . . . . . 7 |- ( 1st ` T ) = ( 1st ` T )
13		eqid 2622	. . . . . . 7 |- ran ( 1st ` T ) = ran ( 1st ` T )
14	10, 11, 12, 13	rngoiso1o 33778	. . . . . 6 |- ( ( S e. RingOps /\ T e. RingOps /\ G e. ( S RngIso T ) ) -> G : ran ( 1st ` S ) -1-1-onto-> ran ( 1st ` T ) )
15	14	3expa 1265	. . . . 5 |- ( ( ( S e. RingOps /\ T e. RingOps ) /\ G e. ( S RngIso T ) ) -> G : ran ( 1st ` S ) -1-1-onto-> ran ( 1st ` T ) )
16	15	3adantl1 1217	. . . 4 |- ( ( ( R e. RingOps /\ S e. RingOps /\ T e. RingOps ) /\ G e. ( S RngIso T ) ) -> G : ran ( 1st ` S ) -1-1-onto-> ran ( 1st ` T ) )
17	16	adantrl 752	. . 3 |- ( ( ( R e. RingOps /\ S e. RingOps /\ T e. RingOps ) /\ ( F e. ( R RngIso S ) /\ G e. ( S RngIso T ) ) ) -> G : ran ( 1st ` S ) -1-1-onto-> ran ( 1st ` T ) )
18		eqid 2622	. . . . . . 7 |- ( 1st ` R ) = ( 1st ` R )
19		eqid 2622	. . . . . . 7 |- ran ( 1st ` R ) = ran ( 1st ` R )
20	18, 19, 10, 11	rngoiso1o 33778	. . . . . 6 |- ( ( R e. RingOps /\ S e. RingOps /\ F e. ( R RngIso S ) ) -> F : ran ( 1st ` R ) -1-1-onto-> ran ( 1st ` S ) )
21	20	3expa 1265	. . . . 5 |- ( ( ( R e. RingOps /\ S e. RingOps ) /\ F e. ( R RngIso S ) ) -> F : ran ( 1st ` R ) -1-1-onto-> ran ( 1st ` S ) )
22	21	3adantl3 1219	. . . 4 |- ( ( ( R e. RingOps /\ S e. RingOps /\ T e. RingOps ) /\ F e. ( R RngIso S ) ) -> F : ran ( 1st ` R ) -1-1-onto-> ran ( 1st ` S ) )
23	22	adantrr 753	. . 3 |- ( ( ( R e. RingOps /\ S e. RingOps /\ T e. RingOps ) /\ ( F e. ( R RngIso S ) /\ G e. ( S RngIso T ) ) ) -> F : ran ( 1st ` R ) -1-1-onto-> ran ( 1st ` S ) )
24		f1oco 6159	. . 3 |- ( ( G : ran ( 1st ` S ) -1-1-onto-> ran ( 1st ` T ) /\ F : ran ( 1st ` R ) -1-1-onto-> ran ( 1st ` S ) ) -> ( G o. F ) : ran ( 1st ` R ) -1-1-onto-> ran ( 1st ` T ) )
25	17, 23, 24	syl2anc 693	. 2 |- ( ( ( R e. RingOps /\ S e. RingOps /\ T e. RingOps ) /\ ( F e. ( R RngIso S ) /\ G e. ( S RngIso T ) ) ) -> ( G o. F ) : ran ( 1st ` R ) -1-1-onto-> ran ( 1st ` T ) )
26	18, 19, 12, 13	isrngoiso 33777	. . . 4 |- ( ( R e. RingOps /\ T e. RingOps ) -> ( ( G o. F ) e. ( R RngIso T ) <-> ( ( G o. F ) e. ( R RngHom T ) /\ ( G o. F ) : ran ( 1st ` R ) -1-1-onto-> ran ( 1st ` T ) ) ) )
27	26	3adant2 1080	. . 3 |- ( ( R e. RingOps /\ S e. RingOps /\ T e. RingOps ) -> ( ( G o. F ) e. ( R RngIso T ) <-> ( ( G o. F ) e. ( R RngHom T ) /\ ( G o. F ) : ran ( 1st ` R ) -1-1-onto-> ran ( 1st ` T ) ) ) )
28	27	adantr 481	. 2 |- ( ( ( R e. RingOps /\ S e. RingOps /\ T e. RingOps ) /\ ( F e. ( R RngIso S ) /\ G e. ( S RngIso T ) ) ) -> ( ( G o. F ) e. ( R RngIso T ) <-> ( ( G o. F ) e. ( R RngHom T ) /\ ( G o. F ) : ran ( 1st ` R ) -1-1-onto-> ran ( 1st ` T ) ) ) )
29	9, 25, 28	mpbir2and 957	1 |- ( ( ( R e. RingOps /\ S e. RingOps /\ T e. RingOps ) /\ ( F e. ( R RngIso S ) /\ G e. ( S RngIso T ) ) ) -> ( G o. F ) e. ( R RngIso T ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   e. wcel 1990   ran crn 5115   o. ccom 5118   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   1stc1st 7166   RingOpscrngo 33693   RngHom crnghom 33759   RngIso crngiso 33760

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-map 7859  df-grpo 27347  df-gid 27348  df-ablo 27399  df-ass 33642  df-exid 33644  df-mgmOLD 33648  df-sgrOLD 33660  df-mndo 33666  df-rngo 33694  df-rngohom 33762  df-rngoiso 33775

This theorem is referenced by:  riscer  33787