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Theorem lmictra 19047
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.
StepHypRef Expression
1 brlmic 17909 . 2  |-  ( R 
~=ph𝑚  S 
<->  ( R LMIso  S )  =/=  (/) )
2 brlmic 17909 . 2  |-  ( S 
~=ph𝑚  T 
<->  ( S LMIso  T )  =/=  (/) )
3 n0 3793 . . 3  |-  ( ( R LMIso  S )  =/=  (/) 
<->  E. g  g  e.  ( R LMIso  S ) )
4 n0 3793 . . 3  |-  ( ( S LMIso  T )  =/=  (/) 
<->  E. f  f  e.  ( S LMIso  T ) )
5 lmimco 19046 . . . . . . . . 9  |-  ( ( f  e.  ( S LMIso 
T )  /\  g  e.  ( R LMIso  S ) )  ->  ( f  o.  g )  e.  ( R LMIso  T ) )
6 brlmici 17910 . . . . . . . . 9  |-  ( ( f  o.  g )  e.  ( R LMIso  T
)  ->  R  ~=ph𝑚  T )
75, 6syl 16 . . . . . . . 8  |-  ( ( f  e.  ( S LMIso 
T )  /\  g  e.  ( R LMIso  S ) )  ->  R  ~=ph𝑚  T )
87ex 432 . . . . . . 7  |-  ( f  e.  ( S LMIso  T
)  ->  ( g  e.  ( R LMIso  S )  ->  R  ~=ph𝑚 
T ) )
98exlimiv 1727 . . . . . 6  |-  ( E. f  f  e.  ( S LMIso  T )  -> 
( g  e.  ( R LMIso  S )  ->  R  ~=ph𝑚 
T ) )
109com12 31 . . . . 5  |-  ( g  e.  ( R LMIso  S
)  ->  ( E. f  f  e.  ( S LMIso  T )  ->  R  ~=ph𝑚  T ) )
1110exlimiv 1727 . . . 4  |-  ( E. g  g  e.  ( R LMIso  S )  -> 
( E. f  f  e.  ( S LMIso  T
)  ->  R  ~=ph𝑚  T ) )
1211imp 427 . . 3  |-  ( ( E. g  g  e.  ( R LMIso  S )  /\  E. f  f  e.  ( S LMIso  T
) )  ->  R  ~=ph𝑚  T )
133, 4, 12syl2anb 477 . 2  |-  ( ( ( R LMIso  S )  =/=  (/)  /\  ( S LMIso 
T )  =/=  (/) )  ->  R  ~=ph𝑚 
T )
141, 2, 13syl2anb 477 1  |-  ( ( R  ~=ph𝑚 
S  /\  S  ~=ph𝑚  T )  ->  R  ~=ph𝑚 
T )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367   E.wex 1617    e. wcel 1823    =/= wne 2649   (/)c0 3783   class class class wbr 4439    o. ccom 4992  (class class class)co 6270   LMIso clmim 17861    ~=ph𝑚 clmic 17862
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-1st 6773  df-2nd 6774  df-1o 7122  df-map 7414  df-0g 14931  df-mgm 16071  df-sgrp 16110  df-mnd 16120  df-mhm 16165  df-grp 16256  df-ghm 16464  df-lmod 17709  df-lmhm 17863  df-lmim 17864  df-lmic 17865
This theorem is referenced by: (None)
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