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Theorem cnfldmulg 17966
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 6200 . . . 4  |-  ( x  =  0  ->  (
x (.g ` fld ) B )  =  ( 0 (.g ` fld ) B ) )
2 oveq1 6200 . . . 4  |-  ( x  =  0  ->  (
x  x.  B )  =  ( 0  x.  B ) )
31, 2eqeq12d 2473 . . 3  |-  ( x  =  0  ->  (
( x (.g ` fld ) B )  =  ( x  x.  B
)  <->  ( 0 (.g ` fld ) B )  =  ( 0  x.  B ) ) )
4 oveq1 6200 . . . 4  |-  ( x  =  y  ->  (
x (.g ` fld ) B )  =  ( y (.g ` fld ) B ) )
5 oveq1 6200 . . . 4  |-  ( x  =  y  ->  (
x  x.  B )  =  ( y  x.  B ) )
64, 5eqeq12d 2473 . . 3  |-  ( x  =  y  ->  (
( x (.g ` fld ) B )  =  ( x  x.  B
)  <->  ( y (.g ` fld ) B )  =  ( y  x.  B ) ) )
7 oveq1 6200 . . . 4  |-  ( x  =  ( y  +  1 )  ->  (
x (.g ` fld ) B )  =  ( ( y  +  1 ) (.g ` fld ) B ) )
8 oveq1 6200 . . . 4  |-  ( x  =  ( y  +  1 )  ->  (
x  x.  B )  =  ( ( y  +  1 )  x.  B ) )
97, 8eqeq12d 2473 . . 3  |-  ( x  =  ( y  +  1 )  ->  (
( x (.g ` fld ) B )  =  ( x  x.  B
)  <->  ( ( y  +  1 ) (.g ` fld ) B )  =  ( ( y  +  1 )  x.  B ) ) )
10 oveq1 6200 . . . 4  |-  ( x  =  -u y  ->  (
x (.g ` fld ) B )  =  ( -u y (.g ` fld ) B ) )
11 oveq1 6200 . . . 4  |-  ( x  =  -u y  ->  (
x  x.  B )  =  ( -u y  x.  B ) )
1210, 11eqeq12d 2473 . . 3  |-  ( x  =  -u y  ->  (
( x (.g ` fld ) B )  =  ( x  x.  B
)  <->  ( -u y
(.g ` fld ) B )  =  ( -u y  x.  B ) ) )
13 oveq1 6200 . . . 4  |-  ( x  =  A  ->  (
x (.g ` fld ) B )  =  ( A (.g ` fld ) B ) )
14 oveq1 6200 . . . 4  |-  ( x  =  A  ->  (
x  x.  B )  =  ( A  x.  B ) )
1513, 14eqeq12d 2473 . . 3  |-  ( x  =  A  ->  (
( x (.g ` fld ) B )  =  ( x  x.  B
)  <->  ( A (.g ` fld ) B )  =  ( A  x.  B ) ) )
16 cnfldbas 17940 . . . . 5  |-  CC  =  ( Base ` fld )
17 cnfld0 17958 . . . . 5  |-  0  =  ( 0g ` fld )
18 eqid 2451 . . . . 5  |-  (.g ` fld )  =  (.g ` fld )
1916, 17, 18mulg0 15743 . . . 4  |-  ( B  e.  CC  ->  (
0 (.g ` fld ) B )  =  0 )
20 mul02 9651 . . . 4  |-  ( B  e.  CC  ->  (
0  x.  B )  =  0 )
2119, 20eqtr4d 2495 . . 3  |-  ( B  e.  CC  ->  (
0 (.g ` fld ) B )  =  ( 0  x.  B
) )
22 oveq1 6200 . . . . 5  |-  ( ( y (.g ` fld ) B )  =  ( y  x.  B
)  ->  ( (
y (.g ` fld ) B )  +  B )  =  ( ( y  x.  B
)  +  B ) )
23 cnrng 17956 . . . . . . . 8  |-fld  e.  Ring
24 rngmnd 16769 . . . . . . . 8  |-  (fld  e.  Ring  ->fld  e.  Mnd )
2523, 24ax-mp 5 . . . . . . 7  |-fld  e.  Mnd
26 cnfldadd 17941 . . . . . . . 8  |-  +  =  ( +g  ` fld )
2716, 18, 26mulgnn0p1 15749 . . . . . . 7  |-  ( (fld  e. 
Mnd  /\  y  e.  NN0 
/\  B  e.  CC )  ->  ( ( y  +  1 ) (.g ` fld ) B )  =  ( ( y (.g ` fld ) B )  +  B ) )
2825, 27mp3an1 1302 . . . . . 6  |-  ( ( y  e.  NN0  /\  B  e.  CC )  ->  ( ( y  +  1 ) (.g ` fld ) B )  =  ( ( y (.g ` fld ) B )  +  B
) )
29 nn0cn 10693 . . . . . . . . 9  |-  ( y  e.  NN0  ->  y  e.  CC )
3029adantr 465 . . . . . . . 8  |-  ( ( y  e.  NN0  /\  B  e.  CC )  ->  y  e.  CC )
31 1cnd 9506 . . . . . . . 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 9515 . . . . . . 7  |-  ( ( y  e.  NN0  /\  B  e.  CC )  ->  ( ( y  +  1 )  x.  B
)  =  ( ( y  x.  B )  +  ( 1  x.  B ) ) )
34 mulid2 9488 . . . . . . . . 9  |-  ( B  e.  CC  ->  (
1  x.  B )  =  B )
3534adantl 466 . . . . . . . 8  |-  ( ( y  e.  NN0  /\  B  e.  CC )  ->  ( 1  x.  B
)  =  B )
3635oveq2d 6209 . . . . . . 7  |-  ( ( y  e.  NN0  /\  B  e.  CC )  ->  ( ( y  x.  B )  +  ( 1  x.  B ) )  =  ( ( y  x.  B )  +  B ) )
3733, 36eqtrd 2492 . . . . . 6  |-  ( ( y  e.  NN0  /\  B  e.  CC )  ->  ( ( y  +  1 )  x.  B
)  =  ( ( y  x.  B )  +  B ) )
3828, 37eqeq12d 2473 . . . . 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 5792 . . . . 5  |-  ( ( y (.g ` fld ) B )  =  ( y  x.  B
)  ->  ( ( invg ` fld ) `  ( y (.g ` fld ) B ) )  =  ( ( invg ` fld ) `  ( y  x.  B ) ) )
42 eqid 2451 . . . . . . 7  |-  ( invg ` fld )  =  ( invg ` fld )
4316, 18, 42mulgnegnn 15748 . . . . . 6  |-  ( ( y  e.  NN  /\  B  e.  CC )  ->  ( -u y (.g ` fld ) B )  =  ( ( invg ` fld ) `  ( y (.g ` fld ) B ) ) )
44 nncn 10434 . . . . . . . 8  |-  ( y  e.  NN  ->  y  e.  CC )
45 mulneg1 9885 . . . . . . . 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 9470 . . . . . . . . 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 17960 . . . . . . . 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 2495 . . . . . 6  |-  ( ( y  e.  NN  /\  B  e.  CC )  ->  ( -u y  x.  B )  =  ( ( invg ` fld ) `  ( y  x.  B
) ) )
5243, 51eqeq12d 2473 . . . . 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 10847 . 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 1370    e. wcel 1758   ` cfv 5519  (class class class)co 6193   CCcc 9384   0cc0 9386   1c1 9387    + caddc 9389    x. cmul 9391   -ucneg 9700   NNcn 10426   NN0cn0 10683   ZZcz 10750   Mndcmnd 15520   invgcminusg 15522  .gcmg 15525   Ringcrg 16760  ℂfldccnfld 17936
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-inf2 7951  ax-cnex 9442  ax-resscn 9443  ax-1cn 9444  ax-icn 9445  ax-addcl 9446  ax-addrcl 9447  ax-mulcl 9448  ax-mulrcl 9449  ax-mulcom 9450  ax-addass 9451  ax-mulass 9452  ax-distr 9453  ax-i2m1 9454  ax-1ne0 9455  ax-1rid 9456  ax-rnegex 9457  ax-rrecex 9458  ax-cnre 9459  ax-pre-lttri 9460  ax-pre-lttrn 9461  ax-pre-ltadd 9462  ax-pre-mulgt0 9463  ax-addf 9465  ax-mulf 9466
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-int 4230  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-om 6580  df-1st 6680  df-2nd 6681  df-recs 6935  df-rdg 6969  df-1o 7023  df-oadd 7027  df-er 7204  df-en 7414  df-dom 7415  df-sdom 7416  df-fin 7417  df-pnf 9524  df-mnf 9525  df-xr 9526  df-ltxr 9527  df-le 9528  df-sub 9701  df-neg 9702  df-nn 10427  df-2 10484  df-3 10485  df-4 10486  df-5 10487  df-6 10488  df-7 10489  df-8 10490  df-9 10491  df-10 10492  df-n0 10684  df-z 10751  df-dec 10860  df-uz 10966  df-fz 11548  df-seq 11917  df-struct 14287  df-ndx 14288  df-slot 14289  df-base 14290  df-sets 14291  df-plusg 14362  df-mulr 14363  df-starv 14364  df-tset 14368  df-ple 14369  df-ds 14371  df-unif 14372  df-0g 14491  df-mnd 15526  df-grp 15656  df-minusg 15657  df-mulg 15659  df-cmn 16392  df-mgp 16706  df-rng 16762  df-cring 16763  df-cnfld 17937
This theorem is referenced by:  zsssubrg  17989  zringmulg  18009  zrngmulg  18015  zringcyg  18025  zcyg  18030  mulgrhm2  18045  mulgrhm2OLD  18048  remulg  18155  amgmlem  22509  cnzh  26537  rezh  26538
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