Theorem rngoisoco 28926

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 28924	. . . . . 6 |- ( ( R e. RingOps /\ S e. RingOps /\ F e. ( R RngIso S ) ) -> F e. ( R RngHom S ) )
2	1	3expa 1188	. . . . 5 |- ( ( ( R e. RingOps /\ S e. RingOps ) /\ F e. ( R RngIso S ) ) -> F e. ( R RngHom S ) )
3	2	3adantl3 1146	. . . 4 |- ( ( ( R e. RingOps /\ S e. RingOps /\ T e. RingOps ) /\ F e. ( R RngIso S ) ) -> F e. ( R RngHom S ) )
4		rngoisohom 28924	. . . . . 6 |- ( ( S e. RingOps /\ T e. RingOps /\ G e. ( S RngIso T ) ) -> G e. ( S RngHom T ) )
5	4	3expa 1188	. . . . 5 |- ( ( ( S e. RingOps /\ T e. RingOps ) /\ G e. ( S RngIso T ) ) -> G e. ( S RngHom T ) )
6	5	3adantl1 1144	. . . 4 |- ( ( ( R e. RingOps /\ S e. RingOps /\ T e. RingOps ) /\ G e. ( S RngIso T ) ) -> G e. ( S RngHom T ) )
7	3, 6	anim12da 28742	. . 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 28918	. . 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 470	. 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 2451	. . . . . . 7 |- ( 1st ` S ) = ( 1st ` S )
11		eqid 2451	. . . . . . 7 |- ran ( 1st ` S ) = ran ( 1st ` S )
12		eqid 2451	. . . . . . 7 |- ( 1st ` T ) = ( 1st ` T )
13		eqid 2451	. . . . . . 7 |- ran ( 1st ` T ) = ran ( 1st ` T )
14	10, 11, 12, 13	rngoiso1o 28923	. . . . . 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 1188	. . . . 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 1144	. . . 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 715	. . 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 2451	. . . . . . 7 |- ( 1st ` R ) = ( 1st ` R )
19		eqid 2451	. . . . . . 7 |- ran ( 1st ` R ) = ran ( 1st ` R )
20	18, 19, 10, 11	rngoiso1o 28923	. . . . . 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 1188	. . . . 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 1146	. . . 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 716	. . 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 5761	. . 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 661	. 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 28922	. . . 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 1007	. . 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 465	. 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 913	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 184   /\ wa 369   /\ w3a 965   e. wcel 1758   ran crn 4939   o. ccom 4942   -1-1-onto->wf1o 5515   ` cfv 5516  (class class class)co 6190   1stc1st 6675   RingOpscrngo 23997   RngHom crnghom 28904   RngIso crngiso 28905

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-op 3982  df-uni 4190  df-iun 4271  df-br 4391  df-opab 4449  df-mpt 4450  df-id 4734  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-riota 6151  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-1st 6677  df-2nd 6678  df-map 7316  df-grpo 23813  df-gid 23814  df-ablo 23904  df-ass 23935  df-exid 23937  df-mgm 23941  df-sgr 23953  df-mndo 23960  df-rngo 23998  df-rngohom 28907  df-rngoiso 28920

This theorem is referenced by:  riscer  28932