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Theorem 2zrngnmrid 40003
Description: R has no multiplicative (right) identity. (Contributed by AV, 12-Feb-2020.)
Hypotheses
Ref Expression
2zrng.e  |-  E  =  { z  e.  ZZ  |  E. x  e.  ZZ  z  =  ( 2  x.  x ) }
2zrngbas.r  |-  R  =  (flds  E )
2zrngmmgm.1  |-  M  =  (mulGrp `  R )
Assertion
Ref Expression
2zrngnmrid  |-  A. a  e.  ( E  \  {
0 } ) A. b  e.  E  (
a  x.  b )  =/=  a
Distinct variable groups:    x, z    E, a, b    R, a, b, x, z    x, E, z    M, a, b
Allowed substitution hints:    M( x, z)

Proof of Theorem 2zrngnmrid
StepHypRef Expression
1 eldifsn 4097 . . . 4  |-  ( a  e.  ( E  \  { 0 } )  <-> 
( a  e.  E  /\  a  =/=  0
) )
2 eqeq1 2455 . . . . . . . 8  |-  ( z  =  a  ->  (
z  =  ( 2  x.  x )  <->  a  =  ( 2  x.  x
) ) )
32rexbidv 2901 . . . . . . 7  |-  ( z  =  a  ->  ( E. x  e.  ZZ  z  =  ( 2  x.  x )  <->  E. x  e.  ZZ  a  =  ( 2  x.  x ) ) )
4 2zrng.e . . . . . . 7  |-  E  =  { z  e.  ZZ  |  E. x  e.  ZZ  z  =  ( 2  x.  x ) }
53, 4elrab2 3198 . . . . . 6  |-  ( a  e.  E  <->  ( a  e.  ZZ  /\  E. x  e.  ZZ  a  =  ( 2  x.  x ) ) )
6 zcn 10942 . . . . . . 7  |-  ( a  e.  ZZ  ->  a  e.  CC )
76adantr 467 . . . . . 6  |-  ( ( a  e.  ZZ  /\  E. x  e.  ZZ  a  =  ( 2  x.  x ) )  -> 
a  e.  CC )
85, 7sylbi 199 . . . . 5  |-  ( a  e.  E  ->  a  e.  CC )
98anim1i 572 . . . 4  |-  ( ( a  e.  E  /\  a  =/=  0 )  -> 
( a  e.  CC  /\  a  =/=  0 ) )
101, 9sylbi 199 . . 3  |-  ( a  e.  ( E  \  { 0 } )  ->  ( a  e.  CC  /\  a  =/=  0 ) )
11 eqeq1 2455 . . . . . . 7  |-  ( z  =  b  ->  (
z  =  ( 2  x.  x )  <->  b  =  ( 2  x.  x
) ) )
1211rexbidv 2901 . . . . . 6  |-  ( z  =  b  ->  ( E. x  e.  ZZ  z  =  ( 2  x.  x )  <->  E. x  e.  ZZ  b  =  ( 2  x.  x ) ) )
1312, 4elrab2 3198 . . . . 5  |-  ( b  e.  E  <->  ( b  e.  ZZ  /\  E. x  e.  ZZ  b  =  ( 2  x.  x ) ) )
14 zcn 10942 . . . . . 6  |-  ( b  e.  ZZ  ->  b  e.  CC )
1514adantr 467 . . . . 5  |-  ( ( b  e.  ZZ  /\  E. x  e.  ZZ  b  =  ( 2  x.  x ) )  -> 
b  e.  CC )
1613, 15sylbi 199 . . . 4  |-  ( b  e.  E  ->  b  e.  CC )
1716ancli 554 . . 3  |-  ( b  e.  E  ->  (
b  e.  E  /\  b  e.  CC )
)
1841neven 39985 . . . . . . 7  |-  1  e/  E
19 elnelne2 2735 . . . . . . 7  |-  ( ( b  e.  E  /\  1  e/  E )  -> 
b  =/=  1 )
2018, 19mpan2 677 . . . . . 6  |-  ( b  e.  E  ->  b  =/=  1 )
2120ad2antrl 734 . . . . 5  |-  ( ( ( a  e.  CC  /\  a  =/=  0 )  /\  ( b  e.  E  /\  b  e.  CC ) )  -> 
b  =/=  1 )
22 simpr 463 . . . . . . . 8  |-  ( ( b  e.  E  /\  b  e.  CC )  ->  b  e.  CC )
2322anim2i 573 . . . . . . 7  |-  ( ( ( a  e.  CC  /\  a  =/=  0 )  /\  ( b  e.  E  /\  b  e.  CC ) )  -> 
( ( a  e.  CC  /\  a  =/=  0 )  /\  b  e.  CC ) )
24 3anass 989 . . . . . . . 8  |-  ( ( b  e.  CC  /\  a  e.  CC  /\  a  =/=  0 )  <->  ( b  e.  CC  /\  ( a  e.  CC  /\  a  =/=  0 ) ) )
25 ancom 452 . . . . . . . 8  |-  ( ( b  e.  CC  /\  ( a  e.  CC  /\  a  =/=  0 ) )  <->  ( ( a  e.  CC  /\  a  =/=  0 )  /\  b  e.  CC ) )
2624, 25bitri 253 . . . . . . 7  |-  ( ( b  e.  CC  /\  a  e.  CC  /\  a  =/=  0 )  <->  ( (
a  e.  CC  /\  a  =/=  0 )  /\  b  e.  CC )
)
2723, 26sylibr 216 . . . . . 6  |-  ( ( ( a  e.  CC  /\  a  =/=  0 )  /\  ( b  e.  E  /\  b  e.  CC ) )  -> 
( b  e.  CC  /\  a  e.  CC  /\  a  =/=  0 ) )
28 divcan3 10294 . . . . . 6  |-  ( ( b  e.  CC  /\  a  e.  CC  /\  a  =/=  0 )  ->  (
( a  x.  b
)  /  a )  =  b )
2927, 28syl 17 . . . . 5  |-  ( ( ( a  e.  CC  /\  a  =/=  0 )  /\  ( b  e.  E  /\  b  e.  CC ) )  -> 
( ( a  x.  b )  /  a
)  =  b )
30 divid 10297 . . . . . 6  |-  ( ( a  e.  CC  /\  a  =/=  0 )  -> 
( a  /  a
)  =  1 )
3130adantr 467 . . . . 5  |-  ( ( ( a  e.  CC  /\  a  =/=  0 )  /\  ( b  e.  E  /\  b  e.  CC ) )  -> 
( a  /  a
)  =  1 )
3221, 29, 313netr4d 2701 . . . 4  |-  ( ( ( a  e.  CC  /\  a  =/=  0 )  /\  ( b  e.  E  /\  b  e.  CC ) )  -> 
( ( a  x.  b )  /  a
)  =/=  ( a  /  a ) )
33 simpl 459 . . . . . . . 8  |-  ( ( a  e.  CC  /\  a  =/=  0 )  -> 
a  e.  CC )
34 mulcl 9623 . . . . . . . 8  |-  ( ( a  e.  CC  /\  b  e.  CC )  ->  ( a  x.  b
)  e.  CC )
3533, 22, 34syl2an 480 . . . . . . 7  |-  ( ( ( a  e.  CC  /\  a  =/=  0 )  /\  ( b  e.  E  /\  b  e.  CC ) )  -> 
( a  x.  b
)  e.  CC )
3633adantr 467 . . . . . . 7  |-  ( ( ( a  e.  CC  /\  a  =/=  0 )  /\  ( b  e.  E  /\  b  e.  CC ) )  -> 
a  e.  CC )
37 simpl 459 . . . . . . 7  |-  ( ( ( a  e.  CC  /\  a  =/=  0 )  /\  ( b  e.  E  /\  b  e.  CC ) )  -> 
( a  e.  CC  /\  a  =/=  0 ) )
38 div11 10296 . . . . . . 7  |-  ( ( ( a  x.  b
)  e.  CC  /\  a  e.  CC  /\  (
a  e.  CC  /\  a  =/=  0 ) )  ->  ( ( ( a  x.  b )  /  a )  =  ( a  /  a
)  <->  ( a  x.  b )  =  a ) )
3935, 36, 37, 38syl3anc 1268 . . . . . 6  |-  ( ( ( a  e.  CC  /\  a  =/=  0 )  /\  ( b  e.  E  /\  b  e.  CC ) )  -> 
( ( ( a  x.  b )  / 
a )  =  ( a  /  a )  <-> 
( a  x.  b
)  =  a ) )
4039biimprd 227 . . . . 5  |-  ( ( ( a  e.  CC  /\  a  =/=  0 )  /\  ( b  e.  E  /\  b  e.  CC ) )  -> 
( ( a  x.  b )  =  a  ->  ( ( a  x.  b )  / 
a )  =  ( a  /  a ) ) )
4140necon3d 2645 . . . 4  |-  ( ( ( a  e.  CC  /\  a  =/=  0 )  /\  ( b  e.  E  /\  b  e.  CC ) )  -> 
( ( ( a  x.  b )  / 
a )  =/=  (
a  /  a )  ->  ( a  x.  b )  =/=  a
) )
4232, 41mpd 15 . . 3  |-  ( ( ( a  e.  CC  /\  a  =/=  0 )  /\  ( b  e.  E  /\  b  e.  CC ) )  -> 
( a  x.  b
)  =/=  a )
4310, 17, 42syl2an 480 . 2  |-  ( ( a  e.  ( E 
\  { 0 } )  /\  b  e.  E )  ->  (
a  x.  b )  =/=  a )
4443rgen2 2813 1  |-  A. a  e.  ( E  \  {
0 } ) A. b  e.  E  (
a  x.  b )  =/=  a
Colors of variables: wff setvar class
Syntax hints:    <-> wb 188    /\ wa 371    /\ w3a 985    = wceq 1444    e. wcel 1887    =/= wne 2622    e/ wnel 2623   A.wral 2737   E.wrex 2738   {crab 2741    \ cdif 3401   {csn 3968   ` cfv 5582  (class class class)co 6290   CCcc 9537   0cc0 9539   1c1 9540    x. cmul 9544    / cdiv 10269   2c2 10659   ZZcz 10937   ↾s cress 15122  mulGrpcmgp 17723  ℂfldccnfld 18970
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-er 7363  df-en 7570  df-dom 7571  df-sdom 7572  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-n0 10870  df-z 10938
This theorem is referenced by:  2zrngnmlid2  40004
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