Theorem lmictra 20184

Description: Module isomorphism is transitive. (Contributed by AV, 10-Mar-2019.)

Assertion

Ref	Expression
lmictra	|- ( ( R ~=ph𝑚 S /\ S ~=ph𝑚 T ) -> R ~=ph𝑚 T )

Proof of Theorem lmictra

Dummy variables f g are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		brlmic 19068	. 2 |- ( R ~=ph𝑚 S <-> ( R LMIso S ) =/= (/) )
2		brlmic 19068	. 2 |- ( S ~=ph𝑚 T <-> ( S LMIso T ) =/= (/) )
3		n0 3931	. . 3 |- ( ( R LMIso S ) =/= (/) <-> E. g g e. ( R LMIso S ) )
4		n0 3931	. . 3 |- ( ( S LMIso T ) =/= (/) <-> E. f f e. ( S LMIso T ) )
5		lmimco 20183	. . . . . . . . 9 |- ( ( f e. ( S LMIso T ) /\ g e. ( R LMIso S ) ) -> ( f o. g ) e. ( R LMIso T ) )
6		brlmici 19069	. . . . . . . . 9 |- ( ( f o. g ) e. ( R LMIso T ) -> R ~=ph𝑚 T )
7	5, 6	syl 17	. . . . . . . 8 |- ( ( f e. ( S LMIso T ) /\ g e. ( R LMIso S ) ) -> R ~=ph𝑚 T )
8	7	ex 450	. . . . . . 7 |- ( f e. ( S LMIso T ) -> ( g e. ( R LMIso S ) -> R ~=ph𝑚 T ) )
9	8	exlimiv 1858	. . . . . 6 |- ( E. f f e. ( S LMIso T ) -> ( g e. ( R LMIso S ) -> R ~=ph𝑚 T ) )
10	9	com12 32	. . . . 5 |- ( g e. ( R LMIso S ) -> ( E. f f e. ( S LMIso T ) -> R ~=ph𝑚 T ) )
11	10	exlimiv 1858	. . . 4 |- ( E. g g e. ( R LMIso S ) -> ( E. f f e. ( S LMIso T ) -> R ~=ph𝑚 T ) )
12	11	imp 445	. . 3 |- ( ( E. g g e. ( R LMIso S ) /\ E. f f e. ( S LMIso T ) ) -> R ~=ph𝑚 T )
13	3, 4, 12	syl2anb 496	. 2 |- ( ( ( R LMIso S ) =/= (/) /\ ( S LMIso T ) =/= (/) ) -> R ~=ph𝑚 T )
14	1, 2, 13	syl2anb 496	1 |- ( ( R ~=ph𝑚 S /\ S ~=ph𝑚 T ) -> R ~=ph𝑚 T )

Syntax hints:   -> wi 4   /\ wa 384   E.wex 1704   e. wcel 1990   =/= wne 2794   (/)c0 3915   class class class wbr 4653   o. ccom 5118  (class class class)co 6650   LMIso clmim 19020   ~=ph_𝑚 clmic 19021

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-suc 5729  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-1o 7560  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  df-lmic 19024

This theorem is referenced by: (None)