Theorem rngisoco 16136

Description: The composition of two ring isomorphisms is a ring isomorphism.

Assertion

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

Proof of Theorem rngisoco

Step	Hyp	Ref	Expression
1		eqid 1884	. . . . 5 |- (1st` R) = (1st` R)
2		eqid 1884	. . . . 5 |- ran (1st` R) = ran (1st` R)
3		eqid 1884	. . . . 5 |- (1st` T) = (1st` T)
4		eqid 1884	. . . . 5 |- ran (1st` T) = ran (1st` T)
5	1, 2, 3, 4	isrngiso 16132	. . . 4 |- ((R e. Ring /\ T e. Ring) -> ((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))))
6	5	3adant2 895	. . 3 |- ((R e. Ring /\ S e. Ring /\ T e. Ring) -> ((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))))
7	6	adantr 425	. 2 |- (((R e. Ring /\ S e. Ring /\ T e. Ring) /\ (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))))
8		rngisohom 16134	. . . . . 6 |- ((R e. Ring /\ S e. Ring /\ F e. (R RngIso S)) -> F e. (R RngHom S))
9	8	3expa 1067	. . . . 5 |- (((R e. Ring /\ S e. Ring) /\ F e. (R RngIso S)) -> F e. (R RngHom S))
10	9	3adantl3 1034	. . . 4 |- (((R e. Ring /\ S e. Ring /\ T e. Ring) /\ F e. (R RngIso S)) -> F e. (R RngHom S))
11		rngisohom 16134	. . . . . 6 |- ((S e. Ring /\ T e. Ring /\ G e. (S RngIso T)) -> G e. (S RngHom T))
12	11	3expa 1067	. . . . 5 |- (((S e. Ring /\ T e. Ring) /\ G e. (S RngIso T)) -> G e. (S RngHom T))
13	12	3adantl1 1032	. . . 4 |- (((R e. Ring /\ S e. Ring /\ T e. Ring) /\ G e. (S RngIso T)) -> G e. (S RngHom T))
14	10, 13	anim12da 15647	. . 3 |- (((R e. Ring /\ S e. Ring /\ T e. Ring) /\ (F e. (R RngIso S) /\ G e. (S RngIso T))) -> (F e. (R RngHom S) /\ G e. (S RngHom T)))
15		rnghomco 16128	. . 3 |- (((R e. Ring /\ S e. Ring /\ T e. Ring) /\ (F e. (R RngHom S) /\ G e. (S RngHom T))) -> (G o. F) e. (R RngHom T))
16	14, 15	syldan 516	. 2 |- (((R e. Ring /\ S e. Ring /\ T e. Ring) /\ (F e. (R RngIso S) /\ G e. (S RngIso T))) -> (G o. F) e. (R RngHom T))
17		eqid 1884	. . . . . . 7 |- (1st` S) = (1st` S)
18		eqid 1884	. . . . . . 7 |- ran (1st` S) = ran (1st` S)
19	17, 18, 3, 4	rngiso1o 16133	. . . . . 6 |- ((S e. Ring /\ T e. Ring /\ G e. (S RngIso T)) -> G:ran (1st` S)-1-1-onto->ran (1st` T))
20	19	3expa 1067	. . . . 5 |- (((S e. Ring /\ T e. Ring) /\ G e. (S RngIso T)) -> G:ran (1st` S)-1-1-onto->ran (1st` T))
21	20	3adantl1 1032	. . . 4 |- (((R e. Ring /\ S e. Ring /\ T e. Ring) /\ G e. (S RngIso T)) -> G:ran (1st` S)-1-1-onto->ran (1st` T))
22	21	adantrl 430	. . 3 |- (((R e. Ring /\ S e. Ring /\ T e. Ring) /\ (F e. (R RngIso S) /\ G e. (S RngIso T))) -> G:ran (1st` S)-1-1-onto->ran (1st` T))
23	1, 2, 17, 18	rngiso1o 16133	. . . . . 6 |- ((R e. Ring /\ S e. Ring /\ F e. (R RngIso S)) -> F:ran (1st` R)-1-1-onto->ran (1st` S))
24	23	3expa 1067	. . . . 5 |- (((R e. Ring /\ S e. Ring) /\ F e. (R RngIso S)) -> F:ran (1st` R)-1-1-onto->ran (1st` S))
25	24	3adantl3 1034	. . . 4 |- (((R e. Ring /\ S e. Ring /\ T e. Ring) /\ F e. (R RngIso S)) -> F:ran (1st` R)-1-1-onto->ran (1st` S))
26	25	adantrr 431	. . 3 |- (((R e. Ring /\ S e. Ring /\ T e. Ring) /\ (F e. (R RngIso S) /\ G e. (S RngIso T))) -> F:ran (1st` R)-1-1-onto->ran (1st` S))
27		f1oco 4661	. . 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))
28	22, 26, 27	syl11anc 524	. 2 |- (((R e. Ring /\ S e. Ring /\ T e. Ring) /\ (F e. (R RngIso S) /\ G e. (S RngIso T))) -> (G o. F):ran (1st` R)-1-1-onto->ran (1st` T))
29	7, 16, 28	mpbir2and 802	1 |- (((R e. Ring /\ S e. Ring /\ T e. Ring) /\ (F e. (R RngIso S) /\ G e. (S RngIso T))) -> (G o. F) e. (R RngIso T))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   e. wcel 1300  ran crn 3987   o. ccom 3990  -1-1-onto->wf1o 3997  ` cfv 3998  (class class class)co 4884  1stc1st 5018  Ringcring 9463   RngHom crnghom 16114   RngIso crngiso 16115

This theorem is referenced by:  riscer 16142

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-1st 5020  df-2nd 5021  df-grp 9316  df-gid 9317  df-abl 9408  df-ring 9464  df-ass 10360  df-exid 10362  df-mgm 10366  df-sgr 10378  df-mnd 10385  df-rnghom 16117  df-rngiso 16130

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