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Theorem cnfldmulg 16406
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 5865 . . . 4  |-  ( x  =  0  ->  (
x (.g ` fld ) B )  =  ( 0 (.g ` fld ) B ) )
2 oveq1 5865 . . . 4  |-  ( x  =  0  ->  (
x  x.  B )  =  ( 0  x.  B ) )
31, 2eqeq12d 2297 . . 3  |-  ( x  =  0  ->  (
( x (.g ` fld ) B )  =  ( x  x.  B
)  <->  ( 0 (.g ` fld ) B )  =  ( 0  x.  B ) ) )
4 oveq1 5865 . . . 4  |-  ( x  =  y  ->  (
x (.g ` fld ) B )  =  ( y (.g ` fld ) B ) )
5 oveq1 5865 . . . 4  |-  ( x  =  y  ->  (
x  x.  B )  =  ( y  x.  B ) )
64, 5eqeq12d 2297 . . 3  |-  ( x  =  y  ->  (
( x (.g ` fld ) B )  =  ( x  x.  B
)  <->  ( y (.g ` fld ) B )  =  ( y  x.  B ) ) )
7 oveq1 5865 . . . 4  |-  ( x  =  ( y  +  1 )  ->  (
x (.g ` fld ) B )  =  ( ( y  +  1 ) (.g ` fld ) B ) )
8 oveq1 5865 . . . 4  |-  ( x  =  ( y  +  1 )  ->  (
x  x.  B )  =  ( ( y  +  1 )  x.  B ) )
97, 8eqeq12d 2297 . . 3  |-  ( x  =  ( y  +  1 )  ->  (
( x (.g ` fld ) B )  =  ( x  x.  B
)  <->  ( ( y  +  1 ) (.g ` fld ) B )  =  ( ( y  +  1 )  x.  B ) ) )
10 oveq1 5865 . . . 4  |-  ( x  =  -u y  ->  (
x (.g ` fld ) B )  =  ( -u y (.g ` fld ) B ) )
11 oveq1 5865 . . . 4  |-  ( x  =  -u y  ->  (
x  x.  B )  =  ( -u y  x.  B ) )
1210, 11eqeq12d 2297 . . 3  |-  ( x  =  -u y  ->  (
( x (.g ` fld ) B )  =  ( x  x.  B
)  <->  ( -u y
(.g ` fld ) B )  =  ( -u y  x.  B ) ) )
13 oveq1 5865 . . . 4  |-  ( x  =  A  ->  (
x (.g ` fld ) B )  =  ( A (.g ` fld ) B ) )
14 oveq1 5865 . . . 4  |-  ( x  =  A  ->  (
x  x.  B )  =  ( A  x.  B ) )
1513, 14eqeq12d 2297 . . 3  |-  ( x  =  A  ->  (
( x (.g ` fld ) B )  =  ( x  x.  B
)  <->  ( A (.g ` fld ) B )  =  ( A  x.  B ) ) )
16 cnfldbas 16383 . . . . 5  |-  CC  =  ( Base ` fld )
17 cnfld0 16398 . . . . 5  |-  0  =  ( 0g ` fld )
18 eqid 2283 . . . . 5  |-  (.g ` fld )  =  (.g ` fld )
1916, 17, 18mulg0 14572 . . . 4  |-  ( B  e.  CC  ->  (
0 (.g ` fld ) B )  =  0 )
20 mul02 8990 . . . 4  |-  ( B  e.  CC  ->  (
0  x.  B )  =  0 )
2119, 20eqtr4d 2318 . . 3  |-  ( B  e.  CC  ->  (
0 (.g ` fld ) B )  =  ( 0  x.  B
) )
22 oveq1 5865 . . . . 5  |-  ( ( y (.g ` fld ) B )  =  ( y  x.  B
)  ->  ( (
y (.g ` fld ) B )  +  B )  =  ( ( y  x.  B
)  +  B ) )
23 cnrng 16396 . . . . . . . 8  |-fld  e.  Ring
24 rngmnd 15350 . . . . . . . 8  |-  (fld  e.  Ring  ->fld  e.  Mnd )
2523, 24ax-mp 8 . . . . . . 7  |-fld  e.  Mnd
26 cnfldadd 16384 . . . . . . . 8  |-  +  =  ( +g  ` fld )
2716, 18, 26mulgnn0p1 14578 . . . . . . 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 9975 . . . . . . . . 9  |-  ( y  e.  NN0  ->  y  e.  CC )
3029adantr 451 . . . . . . . 8  |-  ( ( y  e.  NN0  /\  B  e.  CC )  ->  y  e.  CC )
31 ax-1cn 8795 . . . . . . . . 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 8860 . . . . . . 7  |-  ( ( y  e.  NN0  /\  B  e.  CC )  ->  ( ( y  +  1 )  x.  B
)  =  ( ( y  x.  B )  +  ( 1  x.  B ) ) )
35 mulid2 8836 . . . . . . . . 9  |-  ( B  e.  CC  ->  (
1  x.  B )  =  B )
3635adantl 452 . . . . . . . 8  |-  ( ( y  e.  NN0  /\  B  e.  CC )  ->  ( 1  x.  B
)  =  B )
3736oveq2d 5874 . . . . . . 7  |-  ( ( y  e.  NN0  /\  B  e.  CC )  ->  ( ( y  x.  B )  +  ( 1  x.  B ) )  =  ( ( y  x.  B )  +  B ) )
3834, 37eqtrd 2315 . . . . . 6  |-  ( ( y  e.  NN0  /\  B  e.  CC )  ->  ( ( y  +  1 )  x.  B
)  =  ( ( y  x.  B )  +  B ) )
3928, 38eqeq12d 2297 . . . . 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 5525 . . . . 5  |-  ( ( y (.g ` fld ) B )  =  ( y  x.  B
)  ->  ( ( inv g ` fld ) `  ( y (.g ` fld ) B ) )  =  ( ( inv g ` fld ) `  ( y  x.  B ) ) )
43 eqid 2283 . . . . . . 7  |-  ( inv g ` fld )  =  ( inv g ` fld )
4416, 18, 43mulgnegnn 14577 . . . . . 6  |-  ( ( y  e.  NN  /\  B  e.  CC )  ->  ( -u y (.g ` fld ) B )  =  ( ( inv g ` fld ) `  ( y (.g ` fld ) B ) ) )
45 nncn 9754 . . . . . . . 8  |-  ( y  e.  NN  ->  y  e.  CC )
46 mulneg1 9216 . . . . . . . 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 8821 . . . . . . . . 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 16400 . . . . . . . 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 2318 . . . . . 6  |-  ( ( y  e.  NN  /\  B  e.  CC )  ->  ( -u y  x.  B )  =  ( ( inv g ` fld ) `  ( y  x.  B
) ) )
5344, 52eqeq12d 2297 . . . . 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 10113 . 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 1623    e. wcel 1684   ` cfv 5255  (class class class)co 5858   CCcc 8735   0cc0 8737   1c1 8738    + caddc 8740    x. cmul 8742   -ucneg 9038   NNcn 9746   NN0cn0 9965   ZZcz 10024   Mndcmnd 14361   inv gcminusg 14363  .gcmg 14366   Ringcrg 15337  ℂfldccnfld 16377
This theorem is referenced by:  zsssubrg  16430  zcyg  16445  mulgrhm2  16461  amgmlem  20284
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-addf 8816  ax-mulf 8817
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-fz 10783  df-seq 11047  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-plusg 13221  df-mulr 13222  df-starv 13223  df-tset 13227  df-ple 13228  df-ds 13230  df-0g 13404  df-mnd 14367  df-grp 14489  df-minusg 14490  df-mulg 14492  df-cmn 15091  df-mgp 15326  df-rng 15340  df-cring 15341  df-cnfld 16378
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