Theorem lmimco 20183

Description: The composition of two isomorphisms of modules is an isomorphism of modules. (Contributed by AV, 10-Mar-2019.)

Assertion

Ref	Expression
lmimco	|- ( ( F e. ( S LMIso T ) /\ G e. ( R LMIso S ) ) -> ( F o. G ) e. ( R LMIso T ) )

Proof of Theorem lmimco

Step	Hyp	Ref	Expression
1		eqid 2622	. . 3 |- ( Base ` S ) = ( Base ` S )
2		eqid 2622	. . 3 |- ( Base ` T ) = ( Base ` T )
3	1, 2	islmim 19062	. 2 |- ( F e. ( S LMIso T ) <-> ( F e. ( S LMHom T ) /\ F : ( Base ` S ) -1-1-onto-> ( Base ` T ) ) )
4		eqid 2622	. . 3 |- ( Base ` R ) = ( Base ` R )
5	4, 1	islmim 19062	. 2 |- ( G e. ( R LMIso S ) <-> ( G e. ( R LMHom S ) /\ G : ( Base ` R ) -1-1-onto-> ( Base ` S ) ) )
6		lmhmco 19043	. . . 4 |- ( ( F e. ( S LMHom T ) /\ G e. ( R LMHom S ) ) -> ( F o. G ) e. ( R LMHom T ) )
7	6	ad2ant2r 783	. . 3 |- ( ( ( F e. ( S LMHom T ) /\ F : ( Base ` S ) -1-1-onto-> ( Base ` T ) ) /\ ( G e. ( R LMHom S ) /\ G : ( Base ` R ) -1-1-onto-> ( Base ` S ) ) ) -> ( F o. G ) e. ( R LMHom T ) )
8		f1oco 6159	. . . 4 |- ( ( F : ( Base ` S ) -1-1-onto-> ( Base ` T ) /\ G : ( Base ` R ) -1-1-onto-> ( Base ` S ) ) -> ( F o. G ) : ( Base ` R ) -1-1-onto-> ( Base ` T ) )
9	8	ad2ant2l 782	. . 3 |- ( ( ( F e. ( S LMHom T ) /\ F : ( Base ` S ) -1-1-onto-> ( Base ` T ) ) /\ ( G e. ( R LMHom S ) /\ G : ( Base ` R ) -1-1-onto-> ( Base ` S ) ) ) -> ( F o. G ) : ( Base ` R ) -1-1-onto-> ( Base ` T ) )
10	4, 2	islmim 19062	. . 3 |- ( ( F o. G ) e. ( R LMIso T ) <-> ( ( F o. G ) e. ( R LMHom T ) /\ ( F o. G ) : ( Base ` R ) -1-1-onto-> ( Base ` T ) ) )
11	7, 9, 10	sylanbrc 698	. 2 |- ( ( ( F e. ( S LMHom T ) /\ F : ( Base ` S ) -1-1-onto-> ( Base ` T ) ) /\ ( G e. ( R LMHom S ) /\ G : ( Base ` R ) -1-1-onto-> ( Base ` S ) ) ) -> ( F o. G ) e. ( R LMIso T ) )
12	3, 5, 11	syl2anb 496	1 |- ( ( F e. ( S LMIso T ) /\ G e. ( R LMIso S ) ) -> ( F o. G ) e. ( R LMIso T ) )

Syntax hints:   -> wi 4   /\ wa 384   e. wcel 1990   o. ccom 5118   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   Basecbs 15857   LMHom clmhm 19019   LMIso clmim 19020

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-rep 4771  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-map 7859  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mhm 17335  df-grp 17425  df-ghm 17658  df-lmod 18865  df-lmhm 19022  df-lmim 19023

This theorem is referenced by:  lmictra  20184