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Theorem cnfldmulg 18221
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 6289 . . . 4  |-  ( x  =  0  ->  (
x (.g ` fld ) B )  =  ( 0 (.g ` fld ) B ) )
2 oveq1 6289 . . . 4  |-  ( x  =  0  ->  (
x  x.  B )  =  ( 0  x.  B ) )
31, 2eqeq12d 2489 . . 3  |-  ( x  =  0  ->  (
( x (.g ` fld ) B )  =  ( x  x.  B
)  <->  ( 0 (.g ` fld ) B )  =  ( 0  x.  B ) ) )
4 oveq1 6289 . . . 4  |-  ( x  =  y  ->  (
x (.g ` fld ) B )  =  ( y (.g ` fld ) B ) )
5 oveq1 6289 . . . 4  |-  ( x  =  y  ->  (
x  x.  B )  =  ( y  x.  B ) )
64, 5eqeq12d 2489 . . 3  |-  ( x  =  y  ->  (
( x (.g ` fld ) B )  =  ( x  x.  B
)  <->  ( y (.g ` fld ) B )  =  ( y  x.  B ) ) )
7 oveq1 6289 . . . 4  |-  ( x  =  ( y  +  1 )  ->  (
x (.g ` fld ) B )  =  ( ( y  +  1 ) (.g ` fld ) B ) )
8 oveq1 6289 . . . 4  |-  ( x  =  ( y  +  1 )  ->  (
x  x.  B )  =  ( ( y  +  1 )  x.  B ) )
97, 8eqeq12d 2489 . . 3  |-  ( x  =  ( y  +  1 )  ->  (
( x (.g ` fld ) B )  =  ( x  x.  B
)  <->  ( ( y  +  1 ) (.g ` fld ) B )  =  ( ( y  +  1 )  x.  B ) ) )
10 oveq1 6289 . . . 4  |-  ( x  =  -u y  ->  (
x (.g ` fld ) B )  =  ( -u y (.g ` fld ) B ) )
11 oveq1 6289 . . . 4  |-  ( x  =  -u y  ->  (
x  x.  B )  =  ( -u y  x.  B ) )
1210, 11eqeq12d 2489 . . 3  |-  ( x  =  -u y  ->  (
( x (.g ` fld ) B )  =  ( x  x.  B
)  <->  ( -u y
(.g ` fld ) B )  =  ( -u y  x.  B ) ) )
13 oveq1 6289 . . . 4  |-  ( x  =  A  ->  (
x (.g ` fld ) B )  =  ( A (.g ` fld ) B ) )
14 oveq1 6289 . . . 4  |-  ( x  =  A  ->  (
x  x.  B )  =  ( A  x.  B ) )
1513, 14eqeq12d 2489 . . 3  |-  ( x  =  A  ->  (
( x (.g ` fld ) B )  =  ( x  x.  B
)  <->  ( A (.g ` fld ) B )  =  ( A  x.  B ) ) )
16 cnfldbas 18195 . . . . 5  |-  CC  =  ( Base ` fld )
17 cnfld0 18213 . . . . 5  |-  0  =  ( 0g ` fld )
18 eqid 2467 . . . . 5  |-  (.g ` fld )  =  (.g ` fld )
1916, 17, 18mulg0 15947 . . . 4  |-  ( B  e.  CC  ->  (
0 (.g ` fld ) B )  =  0 )
20 mul02 9753 . . . 4  |-  ( B  e.  CC  ->  (
0  x.  B )  =  0 )
2119, 20eqtr4d 2511 . . 3  |-  ( B  e.  CC  ->  (
0 (.g ` fld ) B )  =  ( 0  x.  B
) )
22 oveq1 6289 . . . . 5  |-  ( ( y (.g ` fld ) B )  =  ( y  x.  B
)  ->  ( (
y (.g ` fld ) B )  +  B )  =  ( ( y  x.  B
)  +  B ) )
23 cnrng 18211 . . . . . . . 8  |-fld  e.  Ring
24 rngmnd 16995 . . . . . . . 8  |-  (fld  e.  Ring  ->fld  e.  Mnd )
2523, 24ax-mp 5 . . . . . . 7  |-fld  e.  Mnd
26 cnfldadd 18196 . . . . . . . 8  |-  +  =  ( +g  ` fld )
2716, 18, 26mulgnn0p1 15953 . . . . . . 7  |-  ( (fld  e. 
Mnd  /\  y  e.  NN0 
/\  B  e.  CC )  ->  ( ( y  +  1 ) (.g ` fld ) B )  =  ( ( y (.g ` fld ) B )  +  B ) )
2825, 27mp3an1 1311 . . . . . 6  |-  ( ( y  e.  NN0  /\  B  e.  CC )  ->  ( ( y  +  1 ) (.g ` fld ) B )  =  ( ( y (.g ` fld ) B )  +  B
) )
29 nn0cn 10801 . . . . . . . . 9  |-  ( y  e.  NN0  ->  y  e.  CC )
3029adantr 465 . . . . . . . 8  |-  ( ( y  e.  NN0  /\  B  e.  CC )  ->  y  e.  CC )
31 1cnd 9608 . . . . . . . 8  |-  ( ( y  e.  NN0  /\  B  e.  CC )  ->  1  e.  CC )
32 simpr 461 . . . . . . . 8  |-  ( ( y  e.  NN0  /\  B  e.  CC )  ->  B  e.  CC )
3330, 31, 32adddird 9617 . . . . . . 7  |-  ( ( y  e.  NN0  /\  B  e.  CC )  ->  ( ( y  +  1 )  x.  B
)  =  ( ( y  x.  B )  +  ( 1  x.  B ) ) )
34 mulid2 9590 . . . . . . . . 9  |-  ( B  e.  CC  ->  (
1  x.  B )  =  B )
3534adantl 466 . . . . . . . 8  |-  ( ( y  e.  NN0  /\  B  e.  CC )  ->  ( 1  x.  B
)  =  B )
3635oveq2d 6298 . . . . . . 7  |-  ( ( y  e.  NN0  /\  B  e.  CC )  ->  ( ( y  x.  B )  +  ( 1  x.  B ) )  =  ( ( y  x.  B )  +  B ) )
3733, 36eqtrd 2508 . . . . . 6  |-  ( ( y  e.  NN0  /\  B  e.  CC )  ->  ( ( y  +  1 )  x.  B
)  =  ( ( y  x.  B )  +  B ) )
3828, 37eqeq12d 2489 . . . . 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 ) ) )
3922, 38syl5ibr 221 . . . 4  |-  ( ( y  e.  NN0  /\  B  e.  CC )  ->  ( ( y (.g ` fld ) B )  =  ( y  x.  B )  ->  ( ( y  +  1 ) (.g ` fld ) B )  =  ( ( y  +  1 )  x.  B ) ) )
4039expcom 435 . . 3  |-  ( B  e.  CC  ->  (
y  e.  NN0  ->  ( ( y (.g ` fld ) B )  =  ( y  x.  B
)  ->  ( (
y  +  1 ) (.g ` fld ) B )  =  ( ( y  +  1 )  x.  B
) ) ) )
41 fveq2 5864 . . . . 5  |-  ( ( y (.g ` fld ) B )  =  ( y  x.  B
)  ->  ( ( invg ` fld ) `  ( y (.g ` fld ) B ) )  =  ( ( invg ` fld ) `  ( y  x.  B ) ) )
42 eqid 2467 . . . . . . 7  |-  ( invg ` fld )  =  ( invg ` fld )
4316, 18, 42mulgnegnn 15952 . . . . . 6  |-  ( ( y  e.  NN  /\  B  e.  CC )  ->  ( -u y (.g ` fld ) B )  =  ( ( invg ` fld ) `  ( y (.g ` fld ) B ) ) )
44 nncn 10540 . . . . . . . 8  |-  ( y  e.  NN  ->  y  e.  CC )
45 mulneg1 9989 . . . . . . . 8  |-  ( ( y  e.  CC  /\  B  e.  CC )  ->  ( -u y  x.  B )  =  -u ( y  x.  B
) )
4644, 45sylan 471 . . . . . . 7  |-  ( ( y  e.  NN  /\  B  e.  CC )  ->  ( -u y  x.  B )  =  -u ( y  x.  B
) )
47 mulcl 9572 . . . . . . . . 9  |-  ( ( y  e.  CC  /\  B  e.  CC )  ->  ( y  x.  B
)  e.  CC )
4844, 47sylan 471 . . . . . . . 8  |-  ( ( y  e.  NN  /\  B  e.  CC )  ->  ( y  x.  B
)  e.  CC )
49 cnfldneg 18215 . . . . . . . 8  |-  ( ( y  x.  B )  e.  CC  ->  (
( invg ` fld ) `  ( y  x.  B
) )  =  -u ( y  x.  B
) )
5048, 49syl 16 . . . . . . 7  |-  ( ( y  e.  NN  /\  B  e.  CC )  ->  ( ( invg ` fld ) `  ( y  x.  B ) )  = 
-u ( y  x.  B ) )
5146, 50eqtr4d 2511 . . . . . 6  |-  ( ( y  e.  NN  /\  B  e.  CC )  ->  ( -u y  x.  B )  =  ( ( invg ` fld ) `  ( y  x.  B
) ) )
5243, 51eqeq12d 2489 . . . . 5  |-  ( ( y  e.  NN  /\  B  e.  CC )  ->  ( ( -u y
(.g ` fld ) B )  =  ( -u y  x.  B )  <->  ( ( invg ` fld ) `  ( y (.g ` fld ) B ) )  =  ( ( invg ` fld ) `  ( y  x.  B ) ) ) )
5341, 52syl5ibr 221 . . . 4  |-  ( ( y  e.  NN  /\  B  e.  CC )  ->  ( ( y (.g ` fld ) B )  =  ( y  x.  B )  ->  ( -u y
(.g ` fld ) B )  =  ( -u y  x.  B ) ) )
5453expcom 435 . . 3  |-  ( B  e.  CC  ->  (
y  e.  NN  ->  ( ( y (.g ` fld ) B )  =  ( y  x.  B
)  ->  ( -u y
(.g ` fld ) B )  =  ( -u y  x.  B ) ) ) )
553, 6, 9, 12, 15, 21, 40, 54zindd 10958 . 2  |-  ( B  e.  CC  ->  ( A  e.  ZZ  ->  ( A (.g ` fld ) B )  =  ( A  x.  B
) ) )
5655impcom 430 1  |-  ( ( A  e.  ZZ  /\  B  e.  CC )  ->  ( A (.g ` fld ) B )  =  ( A  x.  B
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   ` cfv 5586  (class class class)co 6282   CCcc 9486   0cc0 9488   1c1 9489    + caddc 9491    x. cmul 9493   -ucneg 9802   NNcn 10532   NN0cn0 10791   ZZcz 10860   Mndcmnd 15722   invgcminusg 15724  .gcmg 15727   Ringcrg 16986  ℂfldccnfld 18191
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-inf2 8054  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-addf 9567  ax-mulf 9568
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  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 10973  df-uz 11079  df-fz 11669  df-seq 12072  df-struct 14488  df-ndx 14489  df-slot 14490  df-base 14491  df-sets 14492  df-plusg 14564  df-mulr 14565  df-starv 14566  df-tset 14570  df-ple 14571  df-ds 14573  df-unif 14574  df-0g 14693  df-mnd 15728  df-grp 15858  df-minusg 15859  df-mulg 15861  df-cmn 16596  df-mgp 16932  df-rng 16988  df-cring 16989  df-cnfld 18192
This theorem is referenced by:  zsssubrg  18244  zringmulg  18264  zrngmulg  18270  zringcyg  18280  zcyg  18285  mulgrhm2  18300  mulgrhm2OLD  18303  remulg  18410  amgmlem  23047  cnzh  27587  rezh  27588
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