Theorem modgcd 12718

Description: The gcd remains unchanged if one operand is replaced with its remainder modulo the other. (Contributed by Paul Chapman, 31-Mar-2011.)

Assertion

Ref	Expression
modgcd	|- ( ( M e. ZZ /\ N e. NN ) -> ( ( M mod N ) gcd N ) = ( M gcd N ) )

Proof of Theorem modgcd

Step	Hyp	Ref	Expression
1		zq 9981	. . . . . . 7 |- ( M e. ZZ -> M e. QQ )
2	1	adantr 276	. . . . . 6 |- ( ( M e. ZZ /\ N e. NN ) -> M e. QQ )
3		nnq 9988	. . . . . . 7 |- ( N e. NN -> N e. QQ )
4	3	adantl 277	. . . . . 6 |- ( ( M e. ZZ /\ N e. NN ) -> N e. QQ )
5		nngt0 9284	. . . . . . 7 |- ( N e. NN -> 0 < N )
6	5	adantl 277	. . . . . 6 |- ( ( M e. ZZ /\ N e. NN ) -> 0 < N )
7		modqval 10715	. . . . . 6 |- ( ( M e. QQ /\ N e. QQ /\ 0 < N ) -> ( M mod N ) = ( M - ( N x. ( |_ ` ( M / N ) ) ) ) )
8	2, 4, 6, 7	syl3anc 1274	. . . . 5 |- ( ( M e. ZZ /\ N e. NN ) -> ( M mod N ) = ( M - ( N x. ( |_ ` ( M / N ) ) ) ) )
9		zcn 9604	. . . . . . 7 |- ( M e. ZZ -> M e. CC )
10	9	adantr 276	. . . . . 6 |- ( ( M e. ZZ /\ N e. NN ) -> M e. CC )
11		nncn 9267	. . . . . . 7 |- ( N e. NN -> N e. CC )
12	11	adantl 277	. . . . . 6 |- ( ( M e. ZZ /\ N e. NN ) -> N e. CC )
13		znq 9979	. . . . . . . 8 |- ( ( M e. ZZ /\ N e. NN ) -> ( M / N ) e. QQ )
14	13	flqcld 10666	. . . . . . 7 |- ( ( M e. ZZ /\ N e. NN ) -> ( |_ ` ( M / N ) ) e. ZZ )
15	14	zcnd 9724	. . . . . 6 |- ( ( M e. ZZ /\ N e. NN ) -> ( |_ ` ( M / N ) ) e. CC )
16		mulneg1 8688	. . . . . . . . . . 11 |- ( ( ( |_ ` ( M / N ) ) e. CC /\ N e. CC ) -> ( -u ( |_ ` ( M / N ) ) x. N ) = -u ( ( |_ ` ( M / N ) ) x. N ) )
17		mulcom 8274	. . . . . . . . . . . 12 |- ( ( ( |_ ` ( M / N ) ) e. CC /\ N e. CC ) -> ( ( |_ ` ( M / N ) ) x. N ) = ( N x. ( |_ ` ( M / N ) ) ) )
18	17	negeqd 8487	. . . . . . . . . . 11 |- ( ( ( |_ ` ( M / N ) ) e. CC /\ N e. CC ) -> -u ( ( |_ ` ( M / N ) ) x. N ) = -u ( N x. ( |_ ` ( M / N ) ) ) )
19	16, 18	eqtrd 2267	. . . . . . . . . 10 |- ( ( ( |_ ` ( M / N ) ) e. CC /\ N e. CC ) -> ( -u ( |_ ` ( M / N ) ) x. N ) = -u ( N x. ( |_ ` ( M / N ) ) ) )
20	19	ancoms 268	. . . . . . . . 9 |- ( ( N e. CC /\ ( |_ ` ( M / N ) ) e. CC ) -> ( -u ( |_ ` ( M / N ) ) x. N ) = -u ( N x. ( |_ ` ( M / N ) ) ) )
21	20	3adant1 1042	. . . . . . . 8 |- ( ( M e. CC /\ N e. CC /\ ( |_ ` ( M / N ) ) e. CC ) -> ( -u ( |_ ` ( M / N ) ) x. N ) = -u ( N x. ( |_ ` ( M / N ) ) ) )
22	21	oveq2d 6076	. . . . . . 7 |- ( ( M e. CC /\ N e. CC /\ ( |_ ` ( M / N ) ) e. CC ) -> ( M + ( -u ( |_ ` ( M / N ) ) x. N ) ) = ( M + -u ( N x. ( |_ ` ( M / N ) ) ) ) )
23		mulcl 8272	. . . . . . . . 9 |- ( ( N e. CC /\ ( |_ ` ( M / N ) ) e. CC ) -> ( N x. ( |_ ` ( M / N ) ) ) e. CC )
24		negsub 8540	. . . . . . . . 9 |- ( ( M e. CC /\ ( N x. ( |_ ` ( M / N ) ) ) e. CC ) -> ( M + -u ( N x. ( |_ ` ( M / N ) ) ) ) = ( M - ( N x. ( |_ ` ( M / N ) ) ) ) )
25	23, 24	sylan2 286	. . . . . . . 8 |- ( ( M e. CC /\ ( N e. CC /\ ( |_ ` ( M / N ) ) e. CC ) ) -> ( M + -u ( N x. ( |_ ` ( M / N ) ) ) ) = ( M - ( N x. ( |_ ` ( M / N ) ) ) ) )
26	25	3impb 1226	. . . . . . 7 |- ( ( M e. CC /\ N e. CC /\ ( |_ ` ( M / N ) ) e. CC ) -> ( M + -u ( N x. ( |_ ` ( M / N ) ) ) ) = ( M - ( N x. ( |_ ` ( M / N ) ) ) ) )
27	22, 26	eqtrd 2267	. . . . . 6 |- ( ( M e. CC /\ N e. CC /\ ( |_ ` ( M / N ) ) e. CC ) -> ( M + ( -u ( |_ ` ( M / N ) ) x. N ) ) = ( M - ( N x. ( |_ ` ( M / N ) ) ) ) )
28	10, 12, 15, 27	syl3anc 1274	. . . . 5 |- ( ( M e. ZZ /\ N e. NN ) -> ( M + ( -u ( |_ ` ( M / N ) ) x. N ) ) = ( M - ( N x. ( |_ ` ( M / N ) ) ) ) )
29	8, 28	eqtr4d 2270	. . . 4 |- ( ( M e. ZZ /\ N e. NN ) -> ( M mod N ) = ( M + ( -u ( |_ ` ( M / N ) ) x. N ) ) )
30	29	oveq2d 6076	. . 3 |- ( ( M e. ZZ /\ N e. NN ) -> ( N gcd ( M mod N ) ) = ( N gcd ( M + ( -u ( |_ ` ( M / N ) ) x. N ) ) ) )
31	14	znegcld 9725	. . . 4 |- ( ( M e. ZZ /\ N e. NN ) -> -u ( |_ ` ( M / N ) ) e. ZZ )
32		nnz 9618	. . . . 5 |- ( N e. NN -> N e. ZZ )
33	32	adantl 277	. . . 4 |- ( ( M e. ZZ /\ N e. NN ) -> N e. ZZ )
34		simpl 109	. . . 4 |- ( ( M e. ZZ /\ N e. NN ) -> M e. ZZ )
35		gcdaddm 12711	. . . 4 |- ( ( -u ( |_ ` ( M / N ) ) e. ZZ /\ N e. ZZ /\ M e. ZZ ) -> ( N gcd M ) = ( N gcd ( M + ( -u ( |_ ` ( M / N ) ) x. N ) ) ) )
36	31, 33, 34, 35	syl3anc 1274	. . 3 |- ( ( M e. ZZ /\ N e. NN ) -> ( N gcd M ) = ( N gcd ( M + ( -u ( |_ ` ( M / N ) ) x. N ) ) ) )
37	30, 36	eqtr4d 2270	. 2 |- ( ( M e. ZZ /\ N e. NN ) -> ( N gcd ( M mod N ) ) = ( N gcd M ) )
38		zmodcl 10735	. . . 4 |- ( ( M e. ZZ /\ N e. NN ) -> ( M mod N ) e. NN0 )
39	38	nn0zd 9721	. . 3 |- ( ( M e. ZZ /\ N e. NN ) -> ( M mod N ) e. ZZ )
40		gcdcom 12700	. . 3 |- ( ( N e. ZZ /\ ( M mod N ) e. ZZ ) -> ( N gcd ( M mod N ) ) = ( ( M mod N ) gcd N ) )
41	33, 39, 40	syl2anc 411	. 2 |- ( ( M e. ZZ /\ N e. NN ) -> ( N gcd ( M mod N ) ) = ( ( M mod N ) gcd N ) )
42		gcdcom 12700	. . 3 |- ( ( N e. ZZ /\ M e. ZZ ) -> ( N gcd M ) = ( M gcd N ) )
43	33, 34, 42	syl2anc 411	. 2 |- ( ( M e. ZZ /\ N e. NN ) -> ( N gcd M ) = ( M gcd N ) )
44	37, 41, 43	3eqtr3d 2275	1 |- ( ( M e. ZZ /\ N e. NN ) -> ( ( M mod N ) gcd N ) = ( M gcd N ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1005   = wceq 1398   e. wcel 2205   class class class wbr 4115   ` cfv 5359  (class class class)co 6060   CCcc 8143   0cc0 8145   + caddc 8148   x. cmul 8150   < clt 8326   - cmin 8463   -ucneg 8464   / cdiv 8968   NNcn 9259   ZZcz 9599   QQcq 9974   |_cfl 10657   mod cmo 10713   gcd cgcd 12680

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-coll 4231  ax-sep 4234  ax-nul 4242  ax-pow 4293  ax-pr 4328  ax-un 4560  ax-setind 4666  ax-iinf 4717  ax-cnex 8236  ax-resscn 8237  ax-1cn 8238  ax-1re 8239  ax-icn 8240  ax-addcl 8241  ax-addrcl 8242  ax-mulcl 8243  ax-mulrcl 8244  ax-addcom 8245  ax-mulcom 8246  ax-addass 8247  ax-mulass 8248  ax-distr 8249  ax-i2m1 8250  ax-0lt1 8251  ax-1rid 8252  ax-0id 8253  ax-rnegex 8254  ax-precex 8255  ax-cnre 8256  ax-pre-ltirr 8257  ax-pre-ltwlin 8258  ax-pre-lttrn 8259  ax-pre-apti 8260  ax-pre-ltadd 8261  ax-pre-mulgt0 8262  ax-pre-mulext 8263  ax-arch 8264  ax-caucvg 8265

This theorem depends on definitions:  df-bi 117  df-stab 839  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3626  df-pw 3677  df-sn 3701  df-pr 3702  df-op 3704  df-uni 3921  df-int 3956  df-iun 3999  df-br 4116  df-opab 4178  df-mpt 4179  df-tr 4215  df-id 4420  df-po 4423  df-iso 4424  df-iord 4493  df-on 4495  df-ilim 4496  df-suc 4498  df-iom 4720  df-xp 4762  df-rel 4763  df-cnv 4764  df-co 4765  df-dm 4766  df-rn 4767  df-res 4768  df-ima 4769  df-iota 5319  df-fun 5361  df-fn 5362  df-f 5363  df-f1 5364  df-fo 5365  df-f1o 5366  df-fv 5367  df-riota 6013  df-ov 6063  df-oprab 6064  df-mpo 6065  df-1st 6349  df-2nd 6350  df-recs 6551  df-frec 6637  df-sup 7290  df-pnf 8328  df-mnf 8329  df-xr 8330  df-ltxr 8331  df-le 8332  df-sub 8465  df-neg 8466  df-reap 8869  df-ap 8876  df-div 8969  df-inn 9260  df-2 9318  df-3 9319  df-4 9320  df-n0 9519  df-z 9600  df-uz 9877  df-q 9975  df-rp 10010  df-fz 10367  df-fzo 10504  df-fl 10659  df-mod 10714  df-seqfrec 10839  df-exp 10930  df-cj 11557  df-re 11558  df-im 11559  df-rsqrt 11714  df-abs 11715  df-dvds 12505  df-gcd 12681

This theorem is referenced by:  eucalginv  12784  phimullem  12953  eulerthlem1  12955  eulerthlemth  12960  pockthlem  13085  gcdmodi  13150