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Theorem gcdaddmlem 12983
Description: Lemma for gcdaddm 12984. (Contributed by Paul Chapman, 31-Mar-2011.)
Hypotheses
Ref Expression
gcdaddmlem.1  |-  K  e.  ZZ
gcdaddmlem.2  |-  M  e.  ZZ
gcdaddmlem.3  |-  N  e.  ZZ
Assertion
Ref Expression
gcdaddmlem  |-  ( M  gcd  N )  =  ( M  gcd  (
( K  x.  M
)  +  N ) )

Proof of Theorem gcdaddmlem
StepHypRef Expression
1 gcdaddmlem.2 . . . . . . 7  |-  M  e.  ZZ
2 gcdaddmlem.3 . . . . . . 7  |-  N  e.  ZZ
3 gcddvds 12970 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( M  gcd  N )  ||  M  /\  ( M  gcd  N ) 
||  N ) )
41, 2, 3mp2an 654 . . . . . 6  |-  ( ( M  gcd  N ) 
||  M  /\  ( M  gcd  N )  ||  N )
54simpli 445 . . . . 5  |-  ( M  gcd  N )  ||  M
6 gcdcl 12972 . . . . . . . . . 10  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  gcd  N
)  e.  NN0 )
71, 2, 6mp2an 654 . . . . . . . . 9  |-  ( M  gcd  N )  e. 
NN0
87nn0zi 10262 . . . . . . . 8  |-  ( M  gcd  N )  e.  ZZ
9 gcdaddmlem.1 . . . . . . . . 9  |-  K  e.  ZZ
10 1z 10267 . . . . . . . . 9  |-  1  e.  ZZ
11 dvds2ln 12835 . . . . . . . . 9  |-  ( ( ( K  e.  ZZ  /\  1  e.  ZZ )  /\  ( ( M  gcd  N )  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ ) )  -> 
( ( ( M  gcd  N )  ||  M  /\  ( M  gcd  N )  ||  N )  ->  ( M  gcd  N )  ||  ( ( K  x.  M )  +  ( 1  x.  N ) ) ) )
129, 10, 11mpanl12 664 . . . . . . . 8  |-  ( ( ( M  gcd  N
)  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( ( M  gcd  N )  ||  M  /\  ( M  gcd  N ) 
||  N )  -> 
( M  gcd  N
)  ||  ( ( K  x.  M )  +  ( 1  x.  N ) ) ) )
138, 1, 2, 12mp3an 1279 . . . . . . 7  |-  ( ( ( M  gcd  N
)  ||  M  /\  ( M  gcd  N ) 
||  N )  -> 
( M  gcd  N
)  ||  ( ( K  x.  M )  +  ( 1  x.  N ) ) )
144, 13ax-mp 8 . . . . . 6  |-  ( M  gcd  N )  ||  ( ( K  x.  M )  +  ( 1  x.  N ) )
15 zcn 10243 . . . . . . . . 9  |-  ( N  e.  ZZ  ->  N  e.  CC )
162, 15ax-mp 8 . . . . . . . 8  |-  N  e.  CC
1716mulid2i 9049 . . . . . . 7  |-  ( 1  x.  N )  =  N
1817oveq2i 6051 . . . . . 6  |-  ( ( K  x.  M )  +  ( 1  x.  N ) )  =  ( ( K  x.  M )  +  N
)
1914, 18breqtri 4195 . . . . 5  |-  ( M  gcd  N )  ||  ( ( K  x.  M )  +  N
)
20 zmulcl 10280 . . . . . . . 8  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ )  ->  ( K  x.  M
)  e.  ZZ )
219, 1, 20mp2an 654 . . . . . . 7  |-  ( K  x.  M )  e.  ZZ
22 zaddcl 10273 . . . . . . 7  |-  ( ( ( K  x.  M
)  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( K  x.  M )  +  N
)  e.  ZZ )
2321, 2, 22mp2an 654 . . . . . 6  |-  ( ( K  x.  M )  +  N )  e.  ZZ
24 dvdslegcd 12971 . . . . . . 7  |-  ( ( ( ( M  gcd  N )  e.  ZZ  /\  M  e.  ZZ  /\  (
( K  x.  M
)  +  N )  e.  ZZ )  /\  -.  ( M  =  0  /\  ( ( K  x.  M )  +  N )  =  0 ) )  ->  (
( ( M  gcd  N )  ||  M  /\  ( M  gcd  N ) 
||  ( ( K  x.  M )  +  N ) )  -> 
( M  gcd  N
)  <_  ( M  gcd  ( ( K  x.  M )  +  N
) ) ) )
2524ex 424 . . . . . 6  |-  ( ( ( M  gcd  N
)  e.  ZZ  /\  M  e.  ZZ  /\  (
( K  x.  M
)  +  N )  e.  ZZ )  -> 
( -.  ( M  =  0  /\  (
( K  x.  M
)  +  N )  =  0 )  -> 
( ( ( M  gcd  N )  ||  M  /\  ( M  gcd  N )  ||  ( ( K  x.  M )  +  N ) )  ->  ( M  gcd  N )  <_  ( M  gcd  ( ( K  x.  M )  +  N
) ) ) ) )
268, 1, 23, 25mp3an 1279 . . . . 5  |-  ( -.  ( M  =  0  /\  ( ( K  x.  M )  +  N )  =  0 )  ->  ( (
( M  gcd  N
)  ||  M  /\  ( M  gcd  N ) 
||  ( ( K  x.  M )  +  N ) )  -> 
( M  gcd  N
)  <_  ( M  gcd  ( ( K  x.  M )  +  N
) ) ) )
275, 19, 26mp2ani 660 . . . 4  |-  ( -.  ( M  =  0  /\  ( ( K  x.  M )  +  N )  =  0 )  ->  ( M  gcd  N )  <_  ( M  gcd  ( ( K  x.  M )  +  N ) ) )
28 gcddvds 12970 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  ( ( K  x.  M )  +  N
)  e.  ZZ )  ->  ( ( M  gcd  ( ( K  x.  M )  +  N ) )  ||  M  /\  ( M  gcd  ( ( K  x.  M )  +  N
) )  ||  (
( K  x.  M
)  +  N ) ) )
291, 23, 28mp2an 654 . . . . . 6  |-  ( ( M  gcd  ( ( K  x.  M )  +  N ) ) 
||  M  /\  ( M  gcd  ( ( K  x.  M )  +  N ) )  ||  ( ( K  x.  M )  +  N
) )
3029simpli 445 . . . . 5  |-  ( M  gcd  ( ( K  x.  M )  +  N ) )  ||  M
31 gcdcl 12972 . . . . . . . . . 10  |-  ( ( M  e.  ZZ  /\  ( ( K  x.  M )  +  N
)  e.  ZZ )  ->  ( M  gcd  ( ( K  x.  M )  +  N
) )  e.  NN0 )
321, 23, 31mp2an 654 . . . . . . . . 9  |-  ( M  gcd  ( ( K  x.  M )  +  N ) )  e. 
NN0
3332nn0zi 10262 . . . . . . . 8  |-  ( M  gcd  ( ( K  x.  M )  +  N ) )  e.  ZZ
34 znegcl 10269 . . . . . . . . . 10  |-  ( K  e.  ZZ  ->  -u K  e.  ZZ )
359, 34ax-mp 8 . . . . . . . . 9  |-  -u K  e.  ZZ
36 dvds2ln 12835 . . . . . . . . 9  |-  ( ( ( -u K  e.  ZZ  /\  1  e.  ZZ )  /\  (
( M  gcd  (
( K  x.  M
)  +  N ) )  e.  ZZ  /\  M  e.  ZZ  /\  (
( K  x.  M
)  +  N )  e.  ZZ ) )  ->  ( ( ( M  gcd  ( ( K  x.  M )  +  N ) ) 
||  M  /\  ( M  gcd  ( ( K  x.  M )  +  N ) )  ||  ( ( K  x.  M )  +  N
) )  ->  ( M  gcd  ( ( K  x.  M )  +  N ) )  ||  ( ( -u K  x.  M )  +  ( 1  x.  ( ( K  x.  M )  +  N ) ) ) ) )
3735, 10, 36mpanl12 664 . . . . . . . 8  |-  ( ( ( M  gcd  (
( K  x.  M
)  +  N ) )  e.  ZZ  /\  M  e.  ZZ  /\  (
( K  x.  M
)  +  N )  e.  ZZ )  -> 
( ( ( M  gcd  ( ( K  x.  M )  +  N ) )  ||  M  /\  ( M  gcd  ( ( K  x.  M )  +  N
) )  ||  (
( K  x.  M
)  +  N ) )  ->  ( M  gcd  ( ( K  x.  M )  +  N
) )  ||  (
( -u K  x.  M
)  +  ( 1  x.  ( ( K  x.  M )  +  N ) ) ) ) )
3833, 1, 23, 37mp3an 1279 . . . . . . 7  |-  ( ( ( M  gcd  (
( K  x.  M
)  +  N ) )  ||  M  /\  ( M  gcd  ( ( K  x.  M )  +  N ) ) 
||  ( ( K  x.  M )  +  N ) )  -> 
( M  gcd  (
( K  x.  M
)  +  N ) )  ||  ( (
-u K  x.  M
)  +  ( 1  x.  ( ( K  x.  M )  +  N ) ) ) )
3929, 38ax-mp 8 . . . . . 6  |-  ( M  gcd  ( ( K  x.  M )  +  N ) )  ||  ( ( -u K  x.  M )  +  ( 1  x.  ( ( K  x.  M )  +  N ) ) )
40 zcn 10243 . . . . . . . . . 10  |-  ( K  e.  ZZ  ->  K  e.  CC )
419, 40ax-mp 8 . . . . . . . . 9  |-  K  e.  CC
42 zcn 10243 . . . . . . . . . 10  |-  ( M  e.  ZZ  ->  M  e.  CC )
431, 42ax-mp 8 . . . . . . . . 9  |-  M  e.  CC
4441, 43mulneg1i 9435 . . . . . . . 8  |-  ( -u K  x.  M )  =  -u ( K  x.  M )
45 zcn 10243 . . . . . . . . . 10  |-  ( ( ( K  x.  M
)  +  N )  e.  ZZ  ->  (
( K  x.  M
)  +  N )  e.  CC )
4623, 45ax-mp 8 . . . . . . . . 9  |-  ( ( K  x.  M )  +  N )  e.  CC
4746mulid2i 9049 . . . . . . . 8  |-  ( 1  x.  ( ( K  x.  M )  +  N ) )  =  ( ( K  x.  M )  +  N
)
4844, 47oveq12i 6052 . . . . . . 7  |-  ( (
-u K  x.  M
)  +  ( 1  x.  ( ( K  x.  M )  +  N ) ) )  =  ( -u ( K  x.  M )  +  ( ( K  x.  M )  +  N ) )
4941, 43mulcli 9051 . . . . . . . . . 10  |-  ( K  x.  M )  e.  CC
5049negcli 9324 . . . . . . . . . 10  |-  -u ( K  x.  M )  e.  CC
5149negidi 9325 . . . . . . . . . 10  |-  ( ( K  x.  M )  +  -u ( K  x.  M ) )  =  0
5249, 50, 51addcomli 9214 . . . . . . . . 9  |-  ( -u ( K  x.  M
)  +  ( K  x.  M ) )  =  0
5352oveq1i 6050 . . . . . . . 8  |-  ( (
-u ( K  x.  M )  +  ( K  x.  M ) )  +  N )  =  ( 0  +  N )
5450, 49, 16addassi 9054 . . . . . . . 8  |-  ( (
-u ( K  x.  M )  +  ( K  x.  M ) )  +  N )  =  ( -u ( K  x.  M )  +  ( ( K  x.  M )  +  N ) )
5516addid2i 9210 . . . . . . . 8  |-  ( 0  +  N )  =  N
5653, 54, 553eqtr3i 2432 . . . . . . 7  |-  ( -u ( K  x.  M
)  +  ( ( K  x.  M )  +  N ) )  =  N
5748, 56eqtri 2424 . . . . . 6  |-  ( (
-u K  x.  M
)  +  ( 1  x.  ( ( K  x.  M )  +  N ) ) )  =  N
5839, 57breqtri 4195 . . . . 5  |-  ( M  gcd  ( ( K  x.  M )  +  N ) )  ||  N
59 dvdslegcd 12971 . . . . . . 7  |-  ( ( ( ( M  gcd  ( ( K  x.  M )  +  N
) )  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  /\  N  =  0 ) )  -> 
( ( ( M  gcd  ( ( K  x.  M )  +  N ) )  ||  M  /\  ( M  gcd  ( ( K  x.  M )  +  N
) )  ||  N
)  ->  ( M  gcd  ( ( K  x.  M )  +  N
) )  <_  ( M  gcd  N ) ) )
6059ex 424 . . . . . 6  |-  ( ( ( M  gcd  (
( K  x.  M
)  +  N ) )  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( -.  ( M  =  0  /\  N  =  0 )  ->  ( (
( M  gcd  (
( K  x.  M
)  +  N ) )  ||  M  /\  ( M  gcd  ( ( K  x.  M )  +  N ) ) 
||  N )  -> 
( M  gcd  (
( K  x.  M
)  +  N ) )  <_  ( M  gcd  N ) ) ) )
6133, 1, 2, 60mp3an 1279 . . . . 5  |-  ( -.  ( M  =  0  /\  N  =  0 )  ->  ( (
( M  gcd  (
( K  x.  M
)  +  N ) )  ||  M  /\  ( M  gcd  ( ( K  x.  M )  +  N ) ) 
||  N )  -> 
( M  gcd  (
( K  x.  M
)  +  N ) )  <_  ( M  gcd  N ) ) )
6230, 58, 61mp2ani 660 . . . 4  |-  ( -.  ( M  =  0  /\  N  =  0 )  ->  ( M  gcd  ( ( K  x.  M )  +  N
) )  <_  ( M  gcd  N ) )
6327, 62anim12i 550 . . 3  |-  ( ( -.  ( M  =  0  /\  ( ( K  x.  M )  +  N )  =  0 )  /\  -.  ( M  =  0  /\  N  =  0
) )  ->  (
( M  gcd  N
)  <_  ( M  gcd  ( ( K  x.  M )  +  N
) )  /\  ( M  gcd  ( ( K  x.  M )  +  N ) )  <_ 
( M  gcd  N
) ) )
648zrei 10244 . . . 4  |-  ( M  gcd  N )  e.  RR
6533zrei 10244 . . . 4  |-  ( M  gcd  ( ( K  x.  M )  +  N ) )  e.  RR
6664, 65letri3i 9145 . . 3  |-  ( ( M  gcd  N )  =  ( M  gcd  ( ( K  x.  M )  +  N
) )  <->  ( ( M  gcd  N )  <_ 
( M  gcd  (
( K  x.  M
)  +  N ) )  /\  ( M  gcd  ( ( K  x.  M )  +  N ) )  <_ 
( M  gcd  N
) ) )
6763, 66sylibr 204 . 2  |-  ( ( -.  ( M  =  0  /\  ( ( K  x.  M )  +  N )  =  0 )  /\  -.  ( M  =  0  /\  N  =  0
) )  ->  ( M  gcd  N )  =  ( M  gcd  (
( K  x.  M
)  +  N ) ) )
68 pm4.57 484 . . 3  |-  ( -.  ( -.  ( M  =  0  /\  (
( K  x.  M
)  +  N )  =  0 )  /\  -.  ( M  =  0  /\  N  =  0 ) )  <->  ( ( M  =  0  /\  ( ( K  x.  M )  +  N
)  =  0 )  \/  ( M  =  0  /\  N  =  0 ) ) )
69 oveq2 6048 . . . . . . . . . 10  |-  ( M  =  0  ->  ( K  x.  M )  =  ( K  x.  0 ) )
7041mul01i 9212 . . . . . . . . . 10  |-  ( K  x.  0 )  =  0
7169, 70syl6eq 2452 . . . . . . . . 9  |-  ( M  =  0  ->  ( K  x.  M )  =  0 )
7271oveq1d 6055 . . . . . . . 8  |-  ( M  =  0  ->  (
( K  x.  M
)  +  N )  =  ( 0  +  N ) )
7372, 55syl6eq 2452 . . . . . . 7  |-  ( M  =  0  ->  (
( K  x.  M
)  +  N )  =  N )
7473eqeq1d 2412 . . . . . 6  |-  ( M  =  0  ->  (
( ( K  x.  M )  +  N
)  =  0  <->  N  =  0 ) )
7574pm5.32i 619 . . . . 5  |-  ( ( M  =  0  /\  ( ( K  x.  M )  +  N
)  =  0 )  <-> 
( M  =  0  /\  N  =  0 ) )
76 oveq12 6049 . . . . . 6  |-  ( ( M  =  0  /\  N  =  0 )  ->  ( M  gcd  N )  =  ( 0  gcd  0 ) )
77 oveq12 6049 . . . . . . 7  |-  ( ( M  =  0  /\  ( ( K  x.  M )  +  N
)  =  0 )  ->  ( M  gcd  ( ( K  x.  M )  +  N
) )  =  ( 0  gcd  0 ) )
7875, 77sylbir 205 . . . . . 6  |-  ( ( M  =  0  /\  N  =  0 )  ->  ( M  gcd  ( ( K  x.  M )  +  N
) )  =  ( 0  gcd  0 ) )
7976, 78eqtr4d 2439 . . . . 5  |-  ( ( M  =  0  /\  N  =  0 )  ->  ( M  gcd  N )  =  ( M  gcd  ( ( K  x.  M )  +  N ) ) )
8075, 79sylbi 188 . . . 4  |-  ( ( M  =  0  /\  ( ( K  x.  M )  +  N
)  =  0 )  ->  ( M  gcd  N )  =  ( M  gcd  ( ( K  x.  M )  +  N ) ) )
8180, 79jaoi 369 . . 3  |-  ( ( ( M  =  0  /\  ( ( K  x.  M )  +  N )  =  0 )  \/  ( M  =  0  /\  N  =  0 ) )  ->  ( M  gcd  N )  =  ( M  gcd  ( ( K  x.  M )  +  N ) ) )
8268, 81sylbi 188 . 2  |-  ( -.  ( -.  ( M  =  0  /\  (
( K  x.  M
)  +  N )  =  0 )  /\  -.  ( M  =  0  /\  N  =  0 ) )  ->  ( M  gcd  N )  =  ( M  gcd  (
( K  x.  M
)  +  N ) ) )
8367, 82pm2.61i 158 1  |-  ( M  gcd  N )  =  ( M  gcd  (
( K  x.  M
)  +  N ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 358    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   class class class wbr 4172  (class class class)co 6040   CCcc 8944   0cc0 8946   1c1 8947    + caddc 8949    x. cmul 8951    <_ cle 9077   -ucneg 9248   NN0cn0 10177   ZZcz 10238    || cdivides 12807    gcd cgcd 12961
This theorem is referenced by:  gcdaddm  12984
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-sup 7404  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-n0 10178  df-z 10239  df-uz 10445  df-rp 10569  df-seq 11279  df-exp 11338  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-dvds 12808  df-gcd 12962
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