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Theorem zerdivemp1 25209
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 6293 . . . . . . 7  |-  ( ( A H B )  =  Z  ->  (
a H ( A H B ) )  =  ( a H Z ) )
2 simpl1 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 ) ) )  ->  R  e.  RingOps )
3 simpr1 1002 . . . . . . . . . 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 3626 . . . . . . . . . . . 12  |-  ( A  e.  ( X  \  Z )  ->  A  e.  X )
543ad2ant3 1019 . . . . . . . . . . 11  |-  ( ( a  e.  X  /\  ( a H A )  =  U  /\  A  e.  ( X  \  Z ) )  ->  A  e.  X )
65adantl 466 . . . . . . . . . 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 1001 . . . . . . . . . 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 25162 . . . . . . . . . 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 1230 . . . . . . . . 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 2493 . . . . . . . . . . . . 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 434 . . . . . . . . . . . 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 6292 . . . . . . . . . . . . . . . 16  |-  ( ( a H A )  =  U  ->  (
( a H A ) H B )  =  ( U H B ) )
16 eqtr 2493 . . . . . . . . . . . . . . . . . . . 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 25177 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( R  e.  RingOps  /\  a  e.  X )  ->  (
a H Z )  =  Z )
19183adant3 1016 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( R  e.  RingOps  /\  a  e.  X  /\  B  e.  X )  ->  (
a H Z )  =  Z )
208rneqi 5229 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ran  G  =  ran  ( 1st `  R
)
2110, 20eqtri 2496 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  X  =  ran  ( 1st `  R
)
22 zerdivemp.5 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  U  =  (GId `  H )
239, 21, 22rngolidm 25199 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( R  e.  RingOps  /\  B  e.  X )  ->  ( U H B )  =  B )
24233adant2 1015 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( R  e.  RingOps  /\  a  e.  X  /\  B  e.  X )  ->  ( U H B )  =  B )
25 simp1 996 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( ( ( U H B )  =  ( a H Z )  /\  ( U H B )  =  B  /\  (
a H Z )  =  Z )  -> 
( U H B )  =  ( a H Z ) )
26 simp2 997 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( ( ( U H B )  =  ( a H Z )  /\  ( U H B )  =  B  /\  (
a H Z )  =  Z )  -> 
( U H B )  =  B )
27 simp3 998 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( ( ( U H B )  =  ( a H Z )  /\  ( U H B )  =  B  /\  (
a H Z )  =  Z )  -> 
( a H Z )  =  Z )
2825, 26, 273eqtr3d 2516 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1195 . . . . . . . . . . . . . . . . . . . . . . . . . 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 1195 . . . . . . . . . . . . . . . . . . . . . 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 434 . . . . . . . . . . . . . . . . . 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 2479 . . . . . . . . . . . . . . . . 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 1190 . . . . . . . . . . . . 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 1209 . . . . . . . . 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 1212 . . . . . . 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 1195 . . . 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 2949 . . 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 1190 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 369    /\ w3a 973    = wceq 1379    e. wcel 1767   E.wrex 2815    \ cdif 3473   ran crn 5000   ` cfv 5588  (class class class)co 6285   1stc1st 6783   2ndc2nd 6784  GIdcgi 24962   RingOpscrngo 25150
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-fo 5594  df-fv 5596  df-riota 6246  df-ov 6288  df-1st 6785  df-2nd 6786  df-grpo 24966  df-gid 24967  df-ablo 25057  df-ass 25088  df-exid 25090  df-mgm 25094  df-sgr 25106  df-mndo 25113  df-rngo 25151
This theorem is referenced by: (None)
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