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Theorem zerdivemp1 25574
Description: In a unitary ring a left invertible element is not a zero divisor. (Contributed by FL, 18-Apr-2010.)
Hypotheses
Ref Expression
zerdivemp.1  |-  G  =  ( 1st `  R
)
zerdivemp.2  |-  H  =  ( 2nd `  R
)
zerdivemp.3  |-  Z  =  (GId `  G )
zerdivemp.4  |-  X  =  ran  G
zerdivemp.5  |-  U  =  (GId `  H )
Assertion
Ref Expression
zerdivemp1  |-  ( ( R  e.  RingOps  /\  A  e.  ( X  \  Z
)  /\  E. a  e.  X  ( a H A )  =  U )  ->  ( B  e.  X  ->  ( ( A H B )  =  Z  ->  B  =  Z ) ) )
Distinct variable groups:    A, a    B, a    H, a    R, a    X, a    Z, a
Allowed substitution hints:    U( a)    G( a)

Proof of Theorem zerdivemp1
StepHypRef Expression
1 oveq2 6222 . . . . . . 7  |-  ( ( A H B )  =  Z  ->  (
a H ( A H B ) )  =  ( a H Z ) )
2 simpl1 997 . . . . . . . . . 10  |-  ( ( ( R  e.  RingOps  /\  ( a H ( A H B ) )  =  ( a H Z )  /\  B  e.  X )  /\  ( a  e.  X  /\  ( a H A )  =  U  /\  A  e.  ( X  \  Z ) ) )  ->  R  e.  RingOps )
3 simpr1 1000 . . . . . . . . . 10  |-  ( ( ( R  e.  RingOps  /\  ( a H ( A H B ) )  =  ( a H Z )  /\  B  e.  X )  /\  ( a  e.  X  /\  ( a H A )  =  U  /\  A  e.  ( X  \  Z ) ) )  ->  a  e.  X
)
4 eldifi 3553 . . . . . . . . . . . 12  |-  ( A  e.  ( X  \  Z )  ->  A  e.  X )
543ad2ant3 1017 . . . . . . . . . . 11  |-  ( ( a  e.  X  /\  ( a H A )  =  U  /\  A  e.  ( X  \  Z ) )  ->  A  e.  X )
65adantl 464 . . . . . . . . . 10  |-  ( ( ( R  e.  RingOps  /\  ( a H ( A H B ) )  =  ( a H Z )  /\  B  e.  X )  /\  ( a  e.  X  /\  ( a H A )  =  U  /\  A  e.  ( X  \  Z ) ) )  ->  A  e.  X
)
7 simpl3 999 . . . . . . . . . 10  |-  ( ( ( R  e.  RingOps  /\  ( a H ( A H B ) )  =  ( a H Z )  /\  B  e.  X )  /\  ( a  e.  X  /\  ( a H A )  =  U  /\  A  e.  ( X  \  Z ) ) )  ->  B  e.  X
)
8 zerdivemp.1 . . . . . . . . . . 11  |-  G  =  ( 1st `  R
)
9 zerdivemp.2 . . . . . . . . . . 11  |-  H  =  ( 2nd `  R
)
10 zerdivemp.4 . . . . . . . . . . 11  |-  X  =  ran  G
118, 9, 10rngoass 25527 . . . . . . . . . 10  |-  ( ( R  e.  RingOps  /\  (
a  e.  X  /\  A  e.  X  /\  B  e.  X )
)  ->  ( (
a H A ) H B )  =  ( a H ( A H B ) ) )
122, 3, 6, 7, 11syl13anc 1228 . . . . . . . . 9  |-  ( ( ( R  e.  RingOps  /\  ( a H ( A H B ) )  =  ( a H Z )  /\  B  e.  X )  /\  ( a  e.  X  /\  ( a H A )  =  U  /\  A  e.  ( X  \  Z ) ) )  ->  ( ( a H A ) H B )  =  ( a H ( A H B ) ) )
13 eqtr 2418 . . . . . . . . . . . . 13  |-  ( ( ( ( a H A ) H B )  =  ( a H ( A H B ) )  /\  ( a H ( A H B ) )  =  ( a H Z ) )  ->  ( ( a H A ) H B )  =  ( a H Z ) )
1413ex 432 . . . . . . . . . . . 12  |-  ( ( ( a H A ) H B )  =  ( a H ( A H B ) )  ->  (
( a H ( A H B ) )  =  ( a H Z )  -> 
( ( a H A ) H B )  =  ( a H Z ) ) )
15 oveq1 6221 . . . . . . . . . . . . . . . 16  |-  ( ( a H A )  =  U  ->  (
( a H A ) H B )  =  ( U H B ) )
16 eqtr 2418 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( U H B )  =  ( ( a H A ) H B )  /\  ( ( a H A ) H B )  =  ( a H Z ) )  ->  ( U H B )  =  ( a H Z ) )
17 zerdivemp.3 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  Z  =  (GId `  G )
1817, 10, 8, 9rngorz 25542 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( R  e.  RingOps  /\  a  e.  X )  ->  (
a H Z )  =  Z )
19183adant3 1014 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( R  e.  RingOps  /\  a  e.  X  /\  B  e.  X )  ->  (
a H Z )  =  Z )
208rneqi 5155 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ran  G  =  ran  ( 1st `  R
)
2110, 20eqtri 2421 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  X  =  ran  ( 1st `  R
)
22 zerdivemp.5 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  U  =  (GId `  H )
239, 21, 22rngolidm 25564 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( R  e.  RingOps  /\  B  e.  X )  ->  ( U H B )  =  B )
24233adant2 1013 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( R  e.  RingOps  /\  a  e.  X  /\  B  e.  X )  ->  ( U H B )  =  B )
25 simp1 994 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( ( ( U H B )  =  ( a H Z )  /\  ( U H B )  =  B  /\  (
a H Z )  =  Z )  -> 
( U H B )  =  ( a H Z ) )
26 simp2 995 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( ( ( U H B )  =  ( a H Z )  /\  ( U H B )  =  B  /\  (
a H Z )  =  Z )  -> 
( U H B )  =  B )
27 simp3 996 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( ( ( U H B )  =  ( a H Z )  /\  ( U H B )  =  B  /\  (
a H Z )  =  Z )  -> 
( a H Z )  =  Z )
2825, 26, 273eqtr3d 2441 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( ( ( U H B )  =  ( a H Z )  /\  ( U H B )  =  B  /\  (
a H Z )  =  Z )  ->  B  =  Z )
2928a1d 25 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( ( U H B )  =  ( a H Z )  /\  ( U H B )  =  B  /\  (
a H Z )  =  Z )  -> 
( A  e.  ( X  \  Z )  ->  B  =  Z ) )
30293exp 1193 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( U H B )  =  ( a H Z )  ->  (
( U H B )  =  B  -> 
( ( a H Z )  =  Z  ->  ( A  e.  ( X  \  Z
)  ->  B  =  Z ) ) ) )
3130com14 88 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( A  e.  ( X  \  Z )  ->  (
( U H B )  =  B  -> 
( ( a H Z )  =  Z  ->  ( ( U H B )  =  ( a H Z )  ->  B  =  Z ) ) ) )
3231com13 80 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( a H Z )  =  Z  ->  (
( U H B )  =  B  -> 
( A  e.  ( X  \  Z )  ->  ( ( U H B )  =  ( a H Z )  ->  B  =  Z ) ) ) )
3319, 24, 32sylc 60 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( R  e.  RingOps  /\  a  e.  X  /\  B  e.  X )  ->  ( A  e.  ( X  \  Z )  ->  (
( U H B )  =  ( a H Z )  ->  B  =  Z )
) )
34333exp 1193 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( R  e.  RingOps  ->  ( a  e.  X  ->  ( B  e.  X  ->  ( A  e.  ( X  \  Z )  ->  (
( U H B )  =  ( a H Z )  ->  B  =  Z )
) ) ) )
3534com15 93 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( U H B )  =  ( a H Z )  ->  (
a  e.  X  -> 
( B  e.  X  ->  ( A  e.  ( X  \  Z )  ->  ( R  e.  RingOps 
->  B  =  Z
) ) ) ) )
3635com24 87 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( U H B )  =  ( a H Z )  ->  ( A  e.  ( X  \  Z )  ->  ( B  e.  X  ->  ( a  e.  X  -> 
( R  e.  RingOps  ->  B  =  Z )
) ) ) )
3716, 36syl 16 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( U H B )  =  ( ( a H A ) H B )  /\  ( ( a H A ) H B )  =  ( a H Z ) )  ->  ( A  e.  ( X  \  Z
)  ->  ( B  e.  X  ->  ( a  e.  X  ->  ( R  e.  RingOps  ->  B  =  Z ) ) ) ) )
3837ex 432 . . . . . . . . . . . . . . . . . 18  |-  ( ( U H B )  =  ( ( a H A ) H B )  ->  (
( ( a H A ) H B )  =  ( a H Z )  -> 
( A  e.  ( X  \  Z )  ->  ( B  e.  X  ->  ( a  e.  X  ->  ( R  e.  RingOps  ->  B  =  Z ) ) ) ) ) )
3938eqcoms 2404 . . . . . . . . . . . . . . . . 17  |-  ( ( ( a H A ) H B )  =  ( U H B )  ->  (
( ( a H A ) H B )  =  ( a H Z )  -> 
( A  e.  ( X  \  Z )  ->  ( B  e.  X  ->  ( a  e.  X  ->  ( R  e.  RingOps  ->  B  =  Z ) ) ) ) ) )
4039com25 91 . . . . . . . . . . . . . . . 16  |-  ( ( ( a H A ) H B )  =  ( U H B )  ->  (
a  e.  X  -> 
( A  e.  ( X  \  Z )  ->  ( B  e.  X  ->  ( (
( a H A ) H B )  =  ( a H Z )  ->  ( R  e.  RingOps  ->  B  =  Z ) ) ) ) ) )
4115, 40syl 16 . . . . . . . . . . . . . . 15  |-  ( ( a H A )  =  U  ->  (
a  e.  X  -> 
( A  e.  ( X  \  Z )  ->  ( B  e.  X  ->  ( (
( a H A ) H B )  =  ( a H Z )  ->  ( R  e.  RingOps  ->  B  =  Z ) ) ) ) ) )
4241com12 31 . . . . . . . . . . . . . 14  |-  ( a  e.  X  ->  (
( a H A )  =  U  -> 
( A  e.  ( X  \  Z )  ->  ( B  e.  X  ->  ( (
( a H A ) H B )  =  ( a H Z )  ->  ( R  e.  RingOps  ->  B  =  Z ) ) ) ) ) )
43423imp 1188 . . . . . . . . . . . . 13  |-  ( ( a  e.  X  /\  ( a H A )  =  U  /\  A  e.  ( X  \  Z ) )  -> 
( B  e.  X  ->  ( ( ( a H A ) H B )  =  ( a H Z )  ->  ( R  e.  RingOps 
->  B  =  Z
) ) ) )
4443com13 80 . . . . . . . . . . . 12  |-  ( ( ( a H A ) H B )  =  ( a H Z )  ->  ( B  e.  X  ->  ( ( a  e.  X  /\  ( a H A )  =  U  /\  A  e.  ( X  \  Z ) )  -> 
( R  e.  RingOps  ->  B  =  Z )
) ) )
4514, 44syl6 33 . . . . . . . . . . 11  |-  ( ( ( a H A ) H B )  =  ( a H ( A H B ) )  ->  (
( a H ( A H B ) )  =  ( a H Z )  -> 
( B  e.  X  ->  ( ( a  e.  X  /\  ( a H A )  =  U  /\  A  e.  ( X  \  Z
) )  ->  ( R  e.  RingOps  ->  B  =  Z ) ) ) ) )
4645com15 93 . . . . . . . . . 10  |-  ( R  e.  RingOps  ->  ( ( a H ( A H B ) )  =  ( a H Z )  ->  ( B  e.  X  ->  ( ( a  e.  X  /\  ( a H A )  =  U  /\  A  e.  ( X  \  Z ) )  -> 
( ( ( a H A ) H B )  =  ( a H ( A H B ) )  ->  B  =  Z ) ) ) ) )
47463imp1 1207 . . . . . . . . 9  |-  ( ( ( R  e.  RingOps  /\  ( a H ( A H B ) )  =  ( a H Z )  /\  B  e.  X )  /\  ( a  e.  X  /\  ( a H A )  =  U  /\  A  e.  ( X  \  Z ) ) )  ->  ( ( ( a H A ) H B )  =  ( a H ( A H B ) )  ->  B  =  Z ) )
4812, 47mpd 15 . . . . . . . 8  |-  ( ( ( R  e.  RingOps  /\  ( a H ( A H B ) )  =  ( a H Z )  /\  B  e.  X )  /\  ( a  e.  X  /\  ( a H A )  =  U  /\  A  e.  ( X  \  Z ) ) )  ->  B  =  Z )
49483exp1 1210 . . . . . . 7  |-  ( R  e.  RingOps  ->  ( ( a H ( A H B ) )  =  ( a H Z )  ->  ( B  e.  X  ->  ( ( a  e.  X  /\  ( a H A )  =  U  /\  A  e.  ( X  \  Z ) )  ->  B  =  Z )
) ) )
501, 49syl5com 30 . . . . . 6  |-  ( ( A H B )  =  Z  ->  ( R  e.  RingOps  ->  ( B  e.  X  ->  ( ( a  e.  X  /\  ( a H A )  =  U  /\  A  e.  ( X  \  Z ) )  ->  B  =  Z )
) ) )
5150com14 88 . . . . 5  |-  ( ( a  e.  X  /\  ( a H A )  =  U  /\  A  e.  ( X  \  Z ) )  -> 
( R  e.  RingOps  -> 
( B  e.  X  ->  ( ( A H B )  =  Z  ->  B  =  Z ) ) ) )
52513exp 1193 . . . 4  |-  ( a  e.  X  ->  (
( a H A )  =  U  -> 
( A  e.  ( X  \  Z )  ->  ( R  e.  RingOps 
->  ( B  e.  X  ->  ( ( A H B )  =  Z  ->  B  =  Z ) ) ) ) ) )
5352rexlimiv 2878 . . 3  |-  ( E. a  e.  X  ( a H A )  =  U  ->  ( A  e.  ( X  \  Z )  ->  ( R  e.  RingOps  ->  ( B  e.  X  ->  ( ( A H B )  =  Z  ->  B  =  Z )
) ) ) )
5453com13 80 . 2  |-  ( R  e.  RingOps  ->  ( A  e.  ( X  \  Z
)  ->  ( E. a  e.  X  (
a H A )  =  U  ->  ( B  e.  X  ->  ( ( A H B )  =  Z  ->  B  =  Z )
) ) ) )
55543imp 1188 1  |-  ( ( R  e.  RingOps  /\  A  e.  ( X  \  Z
)  /\  E. a  e.  X  ( a H A )  =  U )  ->  ( B  e.  X  ->  ( ( A H B )  =  Z  ->  B  =  Z ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 971    = wceq 1399    e. wcel 1836   E.wrex 2743    \ cdif 3399   ran crn 4927   ` cfv 5509  (class class class)co 6214   1stc1st 6715   2ndc2nd 6716  GIdcgi 25327   RingOpscrngo 25515
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-8 1838  ax-9 1840  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2016  ax-ext 2370  ax-sep 4501  ax-nul 4509  ax-pow 4556  ax-pr 4614  ax-un 6509
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-eu 2232  df-mo 2233  df-clab 2378  df-cleq 2384  df-clel 2387  df-nfc 2542  df-ne 2589  df-ral 2747  df-rex 2748  df-reu 2749  df-rmo 2750  df-rab 2751  df-v 3049  df-sbc 3266  df-csb 3362  df-dif 3405  df-un 3407  df-in 3409  df-ss 3416  df-nul 3725  df-if 3871  df-sn 3958  df-pr 3960  df-op 3964  df-uni 4177  df-iun 4258  df-br 4381  df-opab 4439  df-mpt 4440  df-id 4722  df-xp 4932  df-rel 4933  df-cnv 4934  df-co 4935  df-dm 4936  df-rn 4937  df-iota 5473  df-fun 5511  df-fn 5512  df-f 5513  df-fo 5515  df-fv 5517  df-riota 6176  df-ov 6217  df-1st 6717  df-2nd 6718  df-grpo 25331  df-gid 25332  df-ablo 25422  df-ass 25453  df-exid 25455  df-mgmOLD 25459  df-sgrOLD 25471  df-mndo 25478  df-rngo 25516
This theorem is referenced by: (None)
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