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Theorem cnfldmulg 16735
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 6090 . . . 4  |-  ( x  =  0  ->  (
x (.g ` fld ) B )  =  ( 0 (.g ` fld ) B ) )
2 oveq1 6090 . . . 4  |-  ( x  =  0  ->  (
x  x.  B )  =  ( 0  x.  B ) )
31, 2eqeq12d 2452 . . 3  |-  ( x  =  0  ->  (
( x (.g ` fld ) B )  =  ( x  x.  B
)  <->  ( 0 (.g ` fld ) B )  =  ( 0  x.  B ) ) )
4 oveq1 6090 . . . 4  |-  ( x  =  y  ->  (
x (.g ` fld ) B )  =  ( y (.g ` fld ) B ) )
5 oveq1 6090 . . . 4  |-  ( x  =  y  ->  (
x  x.  B )  =  ( y  x.  B ) )
64, 5eqeq12d 2452 . . 3  |-  ( x  =  y  ->  (
( x (.g ` fld ) B )  =  ( x  x.  B
)  <->  ( y (.g ` fld ) B )  =  ( y  x.  B ) ) )
7 oveq1 6090 . . . 4  |-  ( x  =  ( y  +  1 )  ->  (
x (.g ` fld ) B )  =  ( ( y  +  1 ) (.g ` fld ) B ) )
8 oveq1 6090 . . . 4  |-  ( x  =  ( y  +  1 )  ->  (
x  x.  B )  =  ( ( y  +  1 )  x.  B ) )
97, 8eqeq12d 2452 . . 3  |-  ( x  =  ( y  +  1 )  ->  (
( x (.g ` fld ) B )  =  ( x  x.  B
)  <->  ( ( y  +  1 ) (.g ` fld ) B )  =  ( ( y  +  1 )  x.  B ) ) )
10 oveq1 6090 . . . 4  |-  ( x  =  -u y  ->  (
x (.g ` fld ) B )  =  ( -u y (.g ` fld ) B ) )
11 oveq1 6090 . . . 4  |-  ( x  =  -u y  ->  (
x  x.  B )  =  ( -u y  x.  B ) )
1210, 11eqeq12d 2452 . . 3  |-  ( x  =  -u y  ->  (
( x (.g ` fld ) B )  =  ( x  x.  B
)  <->  ( -u y
(.g ` fld ) B )  =  ( -u y  x.  B ) ) )
13 oveq1 6090 . . . 4  |-  ( x  =  A  ->  (
x (.g ` fld ) B )  =  ( A (.g ` fld ) B ) )
14 oveq1 6090 . . . 4  |-  ( x  =  A  ->  (
x  x.  B )  =  ( A  x.  B ) )
1513, 14eqeq12d 2452 . . 3  |-  ( x  =  A  ->  (
( x (.g ` fld ) B )  =  ( x  x.  B
)  <->  ( A (.g ` fld ) B )  =  ( A  x.  B ) ) )
16 cnfldbas 16709 . . . . 5  |-  CC  =  ( Base ` fld )
17 cnfld0 16727 . . . . 5  |-  0  =  ( 0g ` fld )
18 eqid 2438 . . . . 5  |-  (.g ` fld )  =  (.g ` fld )
1916, 17, 18mulg0 14897 . . . 4  |-  ( B  e.  CC  ->  (
0 (.g ` fld ) B )  =  0 )
20 mul02 9246 . . . 4  |-  ( B  e.  CC  ->  (
0  x.  B )  =  0 )
2119, 20eqtr4d 2473 . . 3  |-  ( B  e.  CC  ->  (
0 (.g ` fld ) B )  =  ( 0  x.  B
) )
22 oveq1 6090 . . . . 5  |-  ( ( y (.g ` fld ) B )  =  ( y  x.  B
)  ->  ( (
y (.g ` fld ) B )  +  B )  =  ( ( y  x.  B
)  +  B ) )
23 cnrng 16725 . . . . . . . 8  |-fld  e.  Ring
24 rngmnd 15675 . . . . . . . 8  |-  (fld  e.  Ring  ->fld  e.  Mnd )
2523, 24ax-mp 8 . . . . . . 7  |-fld  e.  Mnd
26 cnfldadd 16710 . . . . . . . 8  |-  +  =  ( +g  ` fld )
2716, 18, 26mulgnn0p1 14903 . . . . . . 7  |-  ( (fld  e. 
Mnd  /\  y  e.  NN0 
/\  B  e.  CC )  ->  ( ( y  +  1 ) (.g ` fld ) B )  =  ( ( y (.g ` fld ) B )  +  B ) )
2825, 27mp3an1 1267 . . . . . 6  |-  ( ( y  e.  NN0  /\  B  e.  CC )  ->  ( ( y  +  1 ) (.g ` fld ) B )  =  ( ( y (.g ` fld ) B )  +  B
) )
29 nn0cn 10233 . . . . . . . . 9  |-  ( y  e.  NN0  ->  y  e.  CC )
3029adantr 453 . . . . . . . 8  |-  ( ( y  e.  NN0  /\  B  e.  CC )  ->  y  e.  CC )
31 ax-1cn 9050 . . . . . . . . 9  |-  1  e.  CC
3231a1i 11 . . . . . . . 8  |-  ( ( y  e.  NN0  /\  B  e.  CC )  ->  1  e.  CC )
33 simpr 449 . . . . . . . 8  |-  ( ( y  e.  NN0  /\  B  e.  CC )  ->  B  e.  CC )
3430, 32, 33adddird 9115 . . . . . . 7  |-  ( ( y  e.  NN0  /\  B  e.  CC )  ->  ( ( y  +  1 )  x.  B
)  =  ( ( y  x.  B )  +  ( 1  x.  B ) ) )
35 mulid2 9091 . . . . . . . . 9  |-  ( B  e.  CC  ->  (
1  x.  B )  =  B )
3635adantl 454 . . . . . . . 8  |-  ( ( y  e.  NN0  /\  B  e.  CC )  ->  ( 1  x.  B
)  =  B )
3736oveq2d 6099 . . . . . . 7  |-  ( ( y  e.  NN0  /\  B  e.  CC )  ->  ( ( y  x.  B )  +  ( 1  x.  B ) )  =  ( ( y  x.  B )  +  B ) )
3834, 37eqtrd 2470 . . . . . 6  |-  ( ( y  e.  NN0  /\  B  e.  CC )  ->  ( ( y  +  1 )  x.  B
)  =  ( ( y  x.  B )  +  B ) )
3928, 38eqeq12d 2452 . . . . 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 214 . . . 4  |-  ( ( y  e.  NN0  /\  B  e.  CC )  ->  ( ( y (.g ` fld ) B )  =  ( y  x.  B )  ->  ( ( y  +  1 ) (.g ` fld ) B )  =  ( ( y  +  1 )  x.  B ) ) )
4140expcom 426 . . 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 5730 . . . . 5  |-  ( ( y (.g ` fld ) B )  =  ( y  x.  B
)  ->  ( ( inv g ` fld ) `  ( y (.g ` fld ) B ) )  =  ( ( inv g ` fld ) `  ( y  x.  B ) ) )
43 eqid 2438 . . . . . . 7  |-  ( inv g ` fld )  =  ( inv g ` fld )
4416, 18, 43mulgnegnn 14902 . . . . . 6  |-  ( ( y  e.  NN  /\  B  e.  CC )  ->  ( -u y (.g ` fld ) B )  =  ( ( inv g ` fld ) `  ( y (.g ` fld ) B ) ) )
45 nncn 10010 . . . . . . . 8  |-  ( y  e.  NN  ->  y  e.  CC )
46 mulneg1 9472 . . . . . . . 8  |-  ( ( y  e.  CC  /\  B  e.  CC )  ->  ( -u y  x.  B )  =  -u ( y  x.  B
) )
4745, 46sylan 459 . . . . . . 7  |-  ( ( y  e.  NN  /\  B  e.  CC )  ->  ( -u y  x.  B )  =  -u ( y  x.  B
) )
48 mulcl 9076 . . . . . . . . 9  |-  ( ( y  e.  CC  /\  B  e.  CC )  ->  ( y  x.  B
)  e.  CC )
4945, 48sylan 459 . . . . . . . 8  |-  ( ( y  e.  NN  /\  B  e.  CC )  ->  ( y  x.  B
)  e.  CC )
50 cnfldneg 16729 . . . . . . . 8  |-  ( ( y  x.  B )  e.  CC  ->  (
( inv g ` fld ) `  ( y  x.  B
) )  =  -u ( y  x.  B
) )
5149, 50syl 16 . . . . . . 7  |-  ( ( y  e.  NN  /\  B  e.  CC )  ->  ( ( inv g ` fld ) `  ( y  x.  B ) )  = 
-u ( y  x.  B ) )
5247, 51eqtr4d 2473 . . . . . 6  |-  ( ( y  e.  NN  /\  B  e.  CC )  ->  ( -u y  x.  B )  =  ( ( inv g ` fld ) `  ( y  x.  B
) ) )
5344, 52eqeq12d 2452 . . . . 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 214 . . . 4  |-  ( ( y  e.  NN  /\  B  e.  CC )  ->  ( ( y (.g ` fld ) B )  =  ( y  x.  B )  ->  ( -u y
(.g ` fld ) B )  =  ( -u y  x.  B ) ) )
5554expcom 426 . . 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 10373 . 2  |-  ( B  e.  CC  ->  ( A  e.  ZZ  ->  ( A (.g ` fld ) B )  =  ( A  x.  B
) ) )
5756impcom 421 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 360    = wceq 1653    e. wcel 1726   ` cfv 5456  (class class class)co 6083   CCcc 8990   0cc0 8992   1c1 8993    + caddc 8995    x. cmul 8997   -ucneg 9294   NNcn 10002   NN0cn0 10223   ZZcz 10284   Mndcmnd 14686   inv gcminusg 14688  .gcmg 14691   Ringcrg 15662  ℂfldccnfld 16705
This theorem is referenced by:  zsssubrg  16759  zcyg  16774  mulgrhm2  16790  amgmlem  20830  zzsmulg  24267  remulg  24272  cnzh  24356  rezh  24357
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-inf2 7598  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069  ax-addf 9071  ax-mulf 9072
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-oadd 6730  df-er 6907  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-nn 10003  df-2 10060  df-3 10061  df-4 10062  df-5 10063  df-6 10064  df-7 10065  df-8 10066  df-9 10067  df-10 10068  df-n0 10224  df-z 10285  df-dec 10385  df-uz 10491  df-fz 11046  df-seq 11326  df-struct 13473  df-ndx 13474  df-slot 13475  df-base 13476  df-sets 13477  df-plusg 13544  df-mulr 13545  df-starv 13546  df-tset 13550  df-ple 13551  df-ds 13553  df-unif 13554  df-0g 13729  df-mnd 14692  df-grp 14814  df-minusg 14815  df-mulg 14817  df-cmn 15416  df-mgp 15651  df-rng 15665  df-cring 15666  df-cnfld 16706
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