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Theorem cnfldmulg 16422
Description: The group multiple function in the field of complex numbers. (Contributed by Mario Carneiro, 14-Jun-2015.)
Assertion
Ref Expression
cnfldmulg  |-  ( ( A  e.  ZZ  /\  B  e.  CC )  ->  ( A (.g ` fld ) B )  =  ( A  x.  B
) )

Proof of Theorem cnfldmulg
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq1 5881 . . . 4  |-  ( x  =  0  ->  (
x (.g ` fld ) B )  =  ( 0 (.g ` fld ) B ) )
2 oveq1 5881 . . . 4  |-  ( x  =  0  ->  (
x  x.  B )  =  ( 0  x.  B ) )
31, 2eqeq12d 2310 . . 3  |-  ( x  =  0  ->  (
( x (.g ` fld ) B )  =  ( x  x.  B
)  <->  ( 0 (.g ` fld ) B )  =  ( 0  x.  B ) ) )
4 oveq1 5881 . . . 4  |-  ( x  =  y  ->  (
x (.g ` fld ) B )  =  ( y (.g ` fld ) B ) )
5 oveq1 5881 . . . 4  |-  ( x  =  y  ->  (
x  x.  B )  =  ( y  x.  B ) )
64, 5eqeq12d 2310 . . 3  |-  ( x  =  y  ->  (
( x (.g ` fld ) B )  =  ( x  x.  B
)  <->  ( y (.g ` fld ) B )  =  ( y  x.  B ) ) )
7 oveq1 5881 . . . 4  |-  ( x  =  ( y  +  1 )  ->  (
x (.g ` fld ) B )  =  ( ( y  +  1 ) (.g ` fld ) B ) )
8 oveq1 5881 . . . 4  |-  ( x  =  ( y  +  1 )  ->  (
x  x.  B )  =  ( ( y  +  1 )  x.  B ) )
97, 8eqeq12d 2310 . . 3  |-  ( x  =  ( y  +  1 )  ->  (
( x (.g ` fld ) B )  =  ( x  x.  B
)  <->  ( ( y  +  1 ) (.g ` fld ) B )  =  ( ( y  +  1 )  x.  B ) ) )
10 oveq1 5881 . . . 4  |-  ( x  =  -u y  ->  (
x (.g ` fld ) B )  =  ( -u y (.g ` fld ) B ) )
11 oveq1 5881 . . . 4  |-  ( x  =  -u y  ->  (
x  x.  B )  =  ( -u y  x.  B ) )
1210, 11eqeq12d 2310 . . 3  |-  ( x  =  -u y  ->  (
( x (.g ` fld ) B )  =  ( x  x.  B
)  <->  ( -u y
(.g ` fld ) B )  =  ( -u y  x.  B ) ) )
13 oveq1 5881 . . . 4  |-  ( x  =  A  ->  (
x (.g ` fld ) B )  =  ( A (.g ` fld ) B ) )
14 oveq1 5881 . . . 4  |-  ( x  =  A  ->  (
x  x.  B )  =  ( A  x.  B ) )
1513, 14eqeq12d 2310 . . 3  |-  ( x  =  A  ->  (
( x (.g ` fld ) B )  =  ( x  x.  B
)  <->  ( A (.g ` fld ) B )  =  ( A  x.  B ) ) )
16 cnfldbas 16399 . . . . 5  |-  CC  =  ( Base ` fld )
17 cnfld0 16414 . . . . 5  |-  0  =  ( 0g ` fld )
18 eqid 2296 . . . . 5  |-  (.g ` fld )  =  (.g ` fld )
1916, 17, 18mulg0 14588 . . . 4  |-  ( B  e.  CC  ->  (
0 (.g ` fld ) B )  =  0 )
20 mul02 9006 . . . 4  |-  ( B  e.  CC  ->  (
0  x.  B )  =  0 )
2119, 20eqtr4d 2331 . . 3  |-  ( B  e.  CC  ->  (
0 (.g ` fld ) B )  =  ( 0  x.  B
) )
22 oveq1 5881 . . . . 5  |-  ( ( y (.g ` fld ) B )  =  ( y  x.  B
)  ->  ( (
y (.g ` fld ) B )  +  B )  =  ( ( y  x.  B
)  +  B ) )
23 cnrng 16412 . . . . . . . 8  |-fld  e.  Ring
24 rngmnd 15366 . . . . . . . 8  |-  (fld  e.  Ring  ->fld  e.  Mnd )
2523, 24ax-mp 8 . . . . . . 7  |-fld  e.  Mnd
26 cnfldadd 16400 . . . . . . . 8  |-  +  =  ( +g  ` fld )
2716, 18, 26mulgnn0p1 14594 . . . . . . 7  |-  ( (fld  e. 
Mnd  /\  y  e.  NN0 
/\  B  e.  CC )  ->  ( ( y  +  1 ) (.g ` fld ) B )  =  ( ( y (.g ` fld ) B )  +  B ) )
2825, 27mp3an1 1264 . . . . . 6  |-  ( ( y  e.  NN0  /\  B  e.  CC )  ->  ( ( y  +  1 ) (.g ` fld ) B )  =  ( ( y (.g ` fld ) B )  +  B
) )
29 nn0cn 9991 . . . . . . . . 9  |-  ( y  e.  NN0  ->  y  e.  CC )
3029adantr 451 . . . . . . . 8  |-  ( ( y  e.  NN0  /\  B  e.  CC )  ->  y  e.  CC )
31 ax-1cn 8811 . . . . . . . . 9  |-  1  e.  CC
3231a1i 10 . . . . . . . 8  |-  ( ( y  e.  NN0  /\  B  e.  CC )  ->  1  e.  CC )
33 simpr 447 . . . . . . . 8  |-  ( ( y  e.  NN0  /\  B  e.  CC )  ->  B  e.  CC )
3430, 32, 33adddird 8876 . . . . . . 7  |-  ( ( y  e.  NN0  /\  B  e.  CC )  ->  ( ( y  +  1 )  x.  B
)  =  ( ( y  x.  B )  +  ( 1  x.  B ) ) )
35 mulid2 8852 . . . . . . . . 9  |-  ( B  e.  CC  ->  (
1  x.  B )  =  B )
3635adantl 452 . . . . . . . 8  |-  ( ( y  e.  NN0  /\  B  e.  CC )  ->  ( 1  x.  B
)  =  B )
3736oveq2d 5890 . . . . . . 7  |-  ( ( y  e.  NN0  /\  B  e.  CC )  ->  ( ( y  x.  B )  +  ( 1  x.  B ) )  =  ( ( y  x.  B )  +  B ) )
3834, 37eqtrd 2328 . . . . . 6  |-  ( ( y  e.  NN0  /\  B  e.  CC )  ->  ( ( y  +  1 )  x.  B
)  =  ( ( y  x.  B )  +  B ) )
3928, 38eqeq12d 2310 . . . . 5  |-  ( ( y  e.  NN0  /\  B  e.  CC )  ->  ( ( ( y  +  1 ) (.g ` fld ) B )  =  ( ( y  +  1 )  x.  B )  <-> 
( ( y (.g ` fld ) B )  +  B
)  =  ( ( y  x.  B )  +  B ) ) )
4022, 39syl5ibr 212 . . . 4  |-  ( ( y  e.  NN0  /\  B  e.  CC )  ->  ( ( y (.g ` fld ) B )  =  ( y  x.  B )  ->  ( ( y  +  1 ) (.g ` fld ) B )  =  ( ( y  +  1 )  x.  B ) ) )
4140expcom 424 . . 3  |-  ( B  e.  CC  ->  (
y  e.  NN0  ->  ( ( y (.g ` fld ) B )  =  ( y  x.  B
)  ->  ( (
y  +  1 ) (.g ` fld ) B )  =  ( ( y  +  1 )  x.  B
) ) ) )
42 fveq2 5541 . . . . 5  |-  ( ( y (.g ` fld ) B )  =  ( y  x.  B
)  ->  ( ( inv g ` fld ) `  ( y (.g ` fld ) B ) )  =  ( ( inv g ` fld ) `  ( y  x.  B ) ) )
43 eqid 2296 . . . . . . 7  |-  ( inv g ` fld )  =  ( inv g ` fld )
4416, 18, 43mulgnegnn 14593 . . . . . 6  |-  ( ( y  e.  NN  /\  B  e.  CC )  ->  ( -u y (.g ` fld ) B )  =  ( ( inv g ` fld ) `  ( y (.g ` fld ) B ) ) )
45 nncn 9770 . . . . . . . 8  |-  ( y  e.  NN  ->  y  e.  CC )
46 mulneg1 9232 . . . . . . . 8  |-  ( ( y  e.  CC  /\  B  e.  CC )  ->  ( -u y  x.  B )  =  -u ( y  x.  B
) )
4745, 46sylan 457 . . . . . . 7  |-  ( ( y  e.  NN  /\  B  e.  CC )  ->  ( -u y  x.  B )  =  -u ( y  x.  B
) )
48 mulcl 8837 . . . . . . . . 9  |-  ( ( y  e.  CC  /\  B  e.  CC )  ->  ( y  x.  B
)  e.  CC )
4945, 48sylan 457 . . . . . . . 8  |-  ( ( y  e.  NN  /\  B  e.  CC )  ->  ( y  x.  B
)  e.  CC )
50 cnfldneg 16416 . . . . . . . 8  |-  ( ( y  x.  B )  e.  CC  ->  (
( inv g ` fld ) `  ( y  x.  B
) )  =  -u ( y  x.  B
) )
5149, 50syl 15 . . . . . . 7  |-  ( ( y  e.  NN  /\  B  e.  CC )  ->  ( ( inv g ` fld ) `  ( y  x.  B ) )  = 
-u ( y  x.  B ) )
5247, 51eqtr4d 2331 . . . . . 6  |-  ( ( y  e.  NN  /\  B  e.  CC )  ->  ( -u y  x.  B )  =  ( ( inv g ` fld ) `  ( y  x.  B
) ) )
5344, 52eqeq12d 2310 . . . . 5  |-  ( ( y  e.  NN  /\  B  e.  CC )  ->  ( ( -u y
(.g ` fld ) B )  =  ( -u y  x.  B )  <->  ( ( inv g ` fld ) `  ( y (.g ` fld ) B ) )  =  ( ( inv g ` fld ) `  ( y  x.  B ) ) ) )
5442, 53syl5ibr 212 . . . 4  |-  ( ( y  e.  NN  /\  B  e.  CC )  ->  ( ( y (.g ` fld ) B )  =  ( y  x.  B )  ->  ( -u y
(.g ` fld ) B )  =  ( -u y  x.  B ) ) )
5554expcom 424 . . 3  |-  ( B  e.  CC  ->  (
y  e.  NN  ->  ( ( y (.g ` fld ) B )  =  ( y  x.  B
)  ->  ( -u y
(.g ` fld ) B )  =  ( -u y  x.  B ) ) ) )
563, 6, 9, 12, 15, 21, 41, 55zindd 10129 . 2  |-  ( B  e.  CC  ->  ( A  e.  ZZ  ->  ( A (.g ` fld ) B )  =  ( A  x.  B
) ) )
5756impcom 419 1  |-  ( ( A  e.  ZZ  /\  B  e.  CC )  ->  ( A (.g ` fld ) B )  =  ( A  x.  B
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   ` cfv 5271  (class class class)co 5874   CCcc 8751   0cc0 8753   1c1 8754    + caddc 8756    x. cmul 8758   -ucneg 9054   NNcn 9762   NN0cn0 9981   ZZcz 10040   Mndcmnd 14377   inv gcminusg 14379  .gcmg 14382   Ringcrg 15353  ℂfldccnfld 16393
This theorem is referenced by:  zsssubrg  16446  zcyg  16461  mulgrhm2  16477  amgmlem  20300
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-addf 8832  ax-mulf 8833
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-fz 10799  df-seq 11063  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-plusg 13237  df-mulr 13238  df-starv 13239  df-tset 13243  df-ple 13244  df-ds 13246  df-0g 13420  df-mnd 14383  df-grp 14505  df-minusg 14506  df-mulg 14508  df-cmn 15107  df-mgp 15342  df-rng 15356  df-cring 15357  df-cnfld 16394
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