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Theorem 2zrngnmrid 32900
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 4157 . . . 4  |-  ( a  e.  ( E  \  { 0 } )  <-> 
( a  e.  E  /\  a  =/=  0
) )
2 eqeq1 2461 . . . . . . . 8  |-  ( z  =  a  ->  (
z  =  ( 2  x.  x )  <->  a  =  ( 2  x.  x
) ) )
32rexbidv 2968 . . . . . . 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 3259 . . . . . 6  |-  ( a  e.  E  <->  ( a  e.  ZZ  /\  E. x  e.  ZZ  a  =  ( 2  x.  x ) ) )
6 zcn 10890 . . . . . . 7  |-  ( a  e.  ZZ  ->  a  e.  CC )
76adantr 465 . . . . . 6  |-  ( ( a  e.  ZZ  /\  E. x  e.  ZZ  a  =  ( 2  x.  x ) )  -> 
a  e.  CC )
85, 7sylbi 195 . . . . 5  |-  ( a  e.  E  ->  a  e.  CC )
98anim1i 568 . . . 4  |-  ( ( a  e.  E  /\  a  =/=  0 )  -> 
( a  e.  CC  /\  a  =/=  0 ) )
101, 9sylbi 195 . . 3  |-  ( a  e.  ( E  \  { 0 } )  ->  ( a  e.  CC  /\  a  =/=  0 ) )
11 eqeq1 2461 . . . . . . 7  |-  ( z  =  b  ->  (
z  =  ( 2  x.  x )  <->  b  =  ( 2  x.  x
) ) )
1211rexbidv 2968 . . . . . 6  |-  ( z  =  b  ->  ( E. x  e.  ZZ  z  =  ( 2  x.  x )  <->  E. x  e.  ZZ  b  =  ( 2  x.  x ) ) )
1312, 4elrab2 3259 . . . . 5  |-  ( b  e.  E  <->  ( b  e.  ZZ  /\  E. x  e.  ZZ  b  =  ( 2  x.  x ) ) )
14 zcn 10890 . . . . . 6  |-  ( b  e.  ZZ  ->  b  e.  CC )
1514adantr 465 . . . . 5  |-  ( ( b  e.  ZZ  /\  E. x  e.  ZZ  b  =  ( 2  x.  x ) )  -> 
b  e.  CC )
1613, 15sylbi 195 . . . 4  |-  ( b  e.  E  ->  b  e.  CC )
1716ancli 551 . . 3  |-  ( b  e.  E  ->  (
b  e.  E  /\  b  e.  CC )
)
1841neven 32882 . . . . . . 7  |-  1  e/  E
19 elnelne2 2805 . . . . . . 7  |-  ( ( b  e.  E  /\  1  e/  E )  -> 
b  =/=  1 )
2018, 19mpan2 671 . . . . . 6  |-  ( b  e.  E  ->  b  =/=  1 )
2120ad2antrl 727 . . . . 5  |-  ( ( ( a  e.  CC  /\  a  =/=  0 )  /\  ( b  e.  E  /\  b  e.  CC ) )  -> 
b  =/=  1 )
22 simpr 461 . . . . . . . 8  |-  ( ( b  e.  E  /\  b  e.  CC )  ->  b  e.  CC )
2322anim2i 569 . . . . . . 7  |-  ( ( ( a  e.  CC  /\  a  =/=  0 )  /\  ( b  e.  E  /\  b  e.  CC ) )  -> 
( ( a  e.  CC  /\  a  =/=  0 )  /\  b  e.  CC ) )
24 3anass 977 . . . . . . . 8  |-  ( ( b  e.  CC  /\  a  e.  CC  /\  a  =/=  0 )  <->  ( b  e.  CC  /\  ( a  e.  CC  /\  a  =/=  0 ) ) )
25 ancom 450 . . . . . . . 8  |-  ( ( b  e.  CC  /\  ( a  e.  CC  /\  a  =/=  0 ) )  <->  ( ( a  e.  CC  /\  a  =/=  0 )  /\  b  e.  CC ) )
2624, 25bitri 249 . . . . . . 7  |-  ( ( b  e.  CC  /\  a  e.  CC  /\  a  =/=  0 )  <->  ( (
a  e.  CC  /\  a  =/=  0 )  /\  b  e.  CC )
)
2723, 26sylibr 212 . . . . . 6  |-  ( ( ( a  e.  CC  /\  a  =/=  0 )  /\  ( b  e.  E  /\  b  e.  CC ) )  -> 
( b  e.  CC  /\  a  e.  CC  /\  a  =/=  0 ) )
28 divcan3 10252 . . . . . 6  |-  ( ( b  e.  CC  /\  a  e.  CC  /\  a  =/=  0 )  ->  (
( a  x.  b
)  /  a )  =  b )
2927, 28syl 16 . . . . 5  |-  ( ( ( a  e.  CC  /\  a  =/=  0 )  /\  ( b  e.  E  /\  b  e.  CC ) )  -> 
( ( a  x.  b )  /  a
)  =  b )
30 divid 10255 . . . . . 6  |-  ( ( a  e.  CC  /\  a  =/=  0 )  -> 
( a  /  a
)  =  1 )
3130adantr 465 . . . . 5  |-  ( ( ( a  e.  CC  /\  a  =/=  0 )  /\  ( b  e.  E  /\  b  e.  CC ) )  -> 
( a  /  a
)  =  1 )
3221, 29, 313netr4d 2762 . . . 4  |-  ( ( ( a  e.  CC  /\  a  =/=  0 )  /\  ( b  e.  E  /\  b  e.  CC ) )  -> 
( ( a  x.  b )  /  a
)  =/=  ( a  /  a ) )
33 simpl 457 . . . . . . . 8  |-  ( ( a  e.  CC  /\  a  =/=  0 )  -> 
a  e.  CC )
34 mulcl 9593 . . . . . . . 8  |-  ( ( a  e.  CC  /\  b  e.  CC )  ->  ( a  x.  b
)  e.  CC )
3533, 22, 34syl2an 477 . . . . . . 7  |-  ( ( ( a  e.  CC  /\  a  =/=  0 )  /\  ( b  e.  E  /\  b  e.  CC ) )  -> 
( a  x.  b
)  e.  CC )
3633adantr 465 . . . . . . 7  |-  ( ( ( a  e.  CC  /\  a  =/=  0 )  /\  ( b  e.  E  /\  b  e.  CC ) )  -> 
a  e.  CC )
37 simpl 457 . . . . . . 7  |-  ( ( ( a  e.  CC  /\  a  =/=  0 )  /\  ( b  e.  E  /\  b  e.  CC ) )  -> 
( a  e.  CC  /\  a  =/=  0 ) )
38 div11 10254 . . . . . . 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 1228 . . . . . 6  |-  ( ( ( a  e.  CC  /\  a  =/=  0 )  /\  ( b  e.  E  /\  b  e.  CC ) )  -> 
( ( ( a  x.  b )  / 
a )  =  ( a  /  a )  <-> 
( a  x.  b
)  =  a ) )
4039biimprd 223 . . . . 5  |-  ( ( ( a  e.  CC  /\  a  =/=  0 )  /\  ( b  e.  E  /\  b  e.  CC ) )  -> 
( ( a  x.  b )  =  a  ->  ( ( a  x.  b )  / 
a )  =  ( a  /  a ) ) )
4140necon3d 2681 . . . 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 477 . 2  |-  ( ( a  e.  ( E 
\  { 0 } )  /\  b  e.  E )  ->  (
a  x.  b )  =/=  a )
4443rgen2 2882 1  |-  A. a  e.  ( E  \  {
0 } ) A. b  e.  E  (
a  x.  b )  =/=  a
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819    =/= wne 2652    e/ wnel 2653   A.wral 2807   E.wrex 2808   {crab 2811    \ cdif 3468   {csn 4032   ` cfv 5594  (class class class)co 6296   CCcc 9507   0cc0 9509   1c1 9510    x. cmul 9514    / cdiv 10227   2c2 10606   ZZcz 10885   ↾s cress 14645  mulGrpcmgp 17268  ℂfldccnfld 18547
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-recs 7060  df-rdg 7094  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-n0 10817  df-z 10886
This theorem is referenced by:  2zrngnmlid2  32901
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