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Theorem 2zrngALT 33027
Description: The ring of integers restricted to the even integers is a (non-unital) ring, the "ring of even integers". Alternate version of 2zrng 33014, based on a restriction of the field of the complex numbers. The proof is based on the facts that the ring of even integers is an additive abelian group (see 2zrngaabl 33023) and a multiplicative semigroup (see 2zrngmsgrp 33026). (Contributed by AV, 11-Feb-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
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
2zrngALT  |-  R  e. Rng
Distinct variable groups:    x, z, R    x, E, z
Allowed substitution hints:    M( x, z)

Proof of Theorem 2zrngALT
Dummy variables  a 
b  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 2zrng.e . . 3  |-  E  =  { z  e.  ZZ  |  E. x  e.  ZZ  z  =  ( 2  x.  x ) }
2 2zrngbas.r . . 3  |-  R  =  (flds  E )
31, 22zrngaabl 33023 . 2  |-  R  e. 
Abel
4 2zrngmmgm.1 . . 3  |-  M  =  (mulGrp `  R )
51, 2, 42zrngmsgrp 33026 . 2  |-  M  e. SGrp
6 elrabi 3251 . . . . . 6  |-  ( a  e.  { z  e.  ZZ  |  E. x  e.  ZZ  z  =  ( 2  x.  x ) }  ->  a  e.  ZZ )
76zcnd 10966 . . . . 5  |-  ( a  e.  { z  e.  ZZ  |  E. x  e.  ZZ  z  =  ( 2  x.  x ) }  ->  a  e.  CC )
87, 1eleq2s 2562 . . . 4  |-  ( a  e.  E  ->  a  e.  CC )
9 elrabi 3251 . . . . . 6  |-  ( b  e.  { z  e.  ZZ  |  E. x  e.  ZZ  z  =  ( 2  x.  x ) }  ->  b  e.  ZZ )
109zcnd 10966 . . . . 5  |-  ( b  e.  { z  e.  ZZ  |  E. x  e.  ZZ  z  =  ( 2  x.  x ) }  ->  b  e.  CC )
1110, 1eleq2s 2562 . . . 4  |-  ( b  e.  E  ->  b  e.  CC )
12 elrabi 3251 . . . . . 6  |-  ( y  e.  { z  e.  ZZ  |  E. x  e.  ZZ  z  =  ( 2  x.  x ) }  ->  y  e.  ZZ )
1312zcnd 10966 . . . . 5  |-  ( y  e.  { z  e.  ZZ  |  E. x  e.  ZZ  z  =  ( 2  x.  x ) }  ->  y  e.  CC )
1413, 1eleq2s 2562 . . . 4  |-  ( y  e.  E  ->  y  e.  CC )
15 adddi 9570 . . . . 5  |-  ( ( a  e.  CC  /\  b  e.  CC  /\  y  e.  CC )  ->  (
a  x.  ( b  +  y ) )  =  ( ( a  x.  b )  +  ( a  x.  y
) ) )
16 adddir 9576 . . . . 5  |-  ( ( a  e.  CC  /\  b  e.  CC  /\  y  e.  CC )  ->  (
( a  +  b )  x.  y )  =  ( ( a  x.  y )  +  ( b  x.  y
) ) )
1715, 16jca 530 . . . 4  |-  ( ( a  e.  CC  /\  b  e.  CC  /\  y  e.  CC )  ->  (
( a  x.  (
b  +  y ) )  =  ( ( a  x.  b )  +  ( a  x.  y ) )  /\  ( ( a  +  b )  x.  y
)  =  ( ( a  x.  y )  +  ( b  x.  y ) ) ) )
188, 11, 14, 17syl3an 1268 . . 3  |-  ( ( a  e.  E  /\  b  e.  E  /\  y  e.  E )  ->  ( ( a  x.  ( b  +  y ) )  =  ( ( a  x.  b
)  +  ( a  x.  y ) )  /\  ( ( a  +  b )  x.  y )  =  ( ( a  x.  y
)  +  ( b  x.  y ) ) ) )
1918rgen3 2880 . 2  |-  A. a  e.  E  A. b  e.  E  A. y  e.  E  ( (
a  x.  ( b  +  y ) )  =  ( ( a  x.  b )  +  ( a  x.  y
) )  /\  (
( a  +  b )  x.  y )  =  ( ( a  x.  y )  +  ( b  x.  y
) ) )
201, 22zrngbas 33015 . . 3  |-  E  =  ( Base `  R
)
211, 22zrngadd 33016 . . 3  |-  +  =  ( +g  `  R )
221, 22zrngmul 33024 . . 3  |-  x.  =  ( .r `  R )
2320, 4, 21, 22isrng 32955 . 2  |-  ( R  e. Rng 
<->  ( R  e.  Abel  /\  M  e. SGrp  /\  A. a  e.  E  A. b  e.  E  A. y  e.  E  (
( a  x.  (
b  +  y ) )  =  ( ( a  x.  b )  +  ( a  x.  y ) )  /\  ( ( a  +  b )  x.  y
)  =  ( ( a  x.  y )  +  ( b  x.  y ) ) ) ) )
243, 5, 19, 23mpbir3an 1176 1  |-  R  e. Rng
Colors of variables: wff setvar class
Syntax hints:    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823   A.wral 2804   E.wrex 2805   {crab 2808   ` cfv 5570  (class class class)co 6270   CCcc 9479    + caddc 9484    x. cmul 9486   2c2 10581   ZZcz 10860   ↾s cress 14720  SGrpcsgrp 16112   Abelcabl 17001  mulGrpcmgp 17339  ℂfldccnfld 18618  Rngcrng 32953
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-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-addf 9560  ax-mulf 9561
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  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-nel 2652  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-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  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-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10977  df-uz 11083  df-fz 11676  df-struct 14721  df-ndx 14722  df-slot 14723  df-base 14724  df-sets 14725  df-ress 14726  df-plusg 14800  df-mulr 14801  df-starv 14802  df-tset 14806  df-ple 14807  df-ds 14809  df-unif 14810  df-0g 14934  df-mgm 16074  df-sgrp 16113  df-mnd 16123  df-grp 16259  df-cmn 17002  df-abl 17003  df-mgp 17340  df-ring 17398  df-cring 17399  df-cnfld 18619  df-rng0 32954
This theorem is referenced by: (None)
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