Theorem modgcd 12787

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 10036	. . . . . . 7 |- ( M e. ZZ -> M e. QQ )
2	1	adantr 276	. . . . . 6 |- ( ( M e. ZZ /\ N e. NN ) -> M e. QQ )
3		nnq 10043	. . . . . . 7 |- ( N e. NN -> N e. QQ )
4	3	adantl 277	. . . . . 6 |- ( ( M e. ZZ /\ N e. NN ) -> N e. QQ )
5		nngt0 9332	. . . . . . 7 |- ( N e. NN -> 0 < N )
6	5	adantl 277	. . . . . 6 |- ( ( M e. ZZ /\ N e. NN ) -> 0 < N )
7		modqval 10776	. . . . . 6 |- ( ( M e. QQ /\ N e. QQ /\ 0 < N ) -> ( M mod N ) = ( M - ( N x. ( |_ ` ( M / N ) ) ) ) )
8	2, 4, 6, 7	syl3anc 1278	. . . . 5 |- ( ( M e. ZZ /\ N e. NN ) -> ( M mod N ) = ( M - ( N x. ( |_ ` ( M / N ) ) ) ) )
9		zcn 9654	. . . . . . 7 |- ( M e. ZZ -> M e. CC )
10	9	adantr 276	. . . . . 6 |- ( ( M e. ZZ /\ N e. NN ) -> M e. CC )
11		nncn 9315	. . . . . . 7 |- ( N e. NN -> N e. CC )
12	11	adantl 277	. . . . . 6 |- ( ( M e. ZZ /\ N e. NN ) -> N e. CC )
13		znq 10034	. . . . . . . 8 |- ( ( M e. ZZ /\ N e. NN ) -> ( M / N ) e. QQ )
14	13	flqcld 10725	. . . . . . 7 |- ( ( M e. ZZ /\ N e. NN ) -> ( |_ ` ( M / N ) ) e. ZZ )
15	14	zcnd 9774	. . . . . 6 |- ( ( M e. ZZ /\ N e. NN ) -> ( |_ ` ( M / N ) ) e. CC )
16		mulneg1 8724	. . . . . . . . . . 11 |- ( ( ( |_ ` ( M / N ) ) e. CC /\ N e. CC ) -> ( -u ( |_ ` ( M / N ) ) x. N ) = -u ( ( |_ ` ( M / N ) ) x. N ) )
17		mulcom 8309	. . . . . . . . . . . 12 |- ( ( ( |_ ` ( M / N ) ) e. CC /\ N e. CC ) -> ( ( |_ ` ( M / N ) ) x. N ) = ( N x. ( |_ ` ( M / N ) ) ) )
18	17	negeqd 8523	. . . . . . . . . . 11 |- ( ( ( |_ ` ( M / N ) ) e. CC /\ N e. CC ) -> -u ( ( |_ ` ( M / N ) ) x. N ) = -u ( N x. ( |_ ` ( M / N ) ) ) )
19	16, 18	eqtrd 2271	. . . . . . . . . 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 1046	. . . . . . . 8 |- ( ( M e. CC /\ N e. CC /\ ( |_ ` ( M / N ) ) e. CC ) -> ( -u ( |_ ` ( M / N ) ) x. N ) = -u ( N x. ( |_ ` ( M / N ) ) ) )
22	21	oveq2d 6101	. . . . . . 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 8307	. . . . . . . . 9 |- ( ( N e. CC /\ ( |_ ` ( M / N ) ) e. CC ) -> ( N x. ( |_ ` ( M / N ) ) ) e. CC )
24		negsub 8576	. . . . . . . . 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 1230	. . . . . . 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 2271	. . . . . 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 1278	. . . . 5 |- ( ( M e. ZZ /\ N e. NN ) -> ( M + ( -u ( |_ ` ( M / N ) ) x. N ) ) = ( M - ( N x. ( |_ ` ( M / N ) ) ) ) )
29	8, 28	eqtr4d 2274	. . . 4 |- ( ( M e. ZZ /\ N e. NN ) -> ( M mod N ) = ( M + ( -u ( |_ ` ( M / N ) ) x. N ) ) )
30	29	oveq2d 6101	. . 3 |- ( ( M e. ZZ /\ N e. NN ) -> ( N gcd ( M mod N ) ) = ( N gcd ( M + ( -u ( |_ ` ( M / N ) ) x. N ) ) ) )
31	14	znegcld 9775	. . . 4 |- ( ( M e. ZZ /\ N e. NN ) -> -u ( |_ ` ( M / N ) ) e. ZZ )
32		nnz 9668	. . . . 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 12780	. . . 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 1278	. . 3 |- ( ( M e. ZZ /\ N e. NN ) -> ( N gcd M ) = ( N gcd ( M + ( -u ( |_ ` ( M / N ) ) x. N ) ) ) )
37	30, 36	eqtr4d 2274	. 2 |- ( ( M e. ZZ /\ N e. NN ) -> ( N gcd ( M mod N ) ) = ( N gcd M ) )
38		zmodcl 10796	. . . 4 |- ( ( M e. ZZ /\ N e. NN ) -> ( M mod N ) e. NN0 )
39	38	nn0zd 9771	. . 3 |- ( ( M e. ZZ /\ N e. NN ) -> ( M mod N ) e. ZZ )
40		gcdcom 12769	. . 3 |- ( ( N e. ZZ /\ ( M mod N ) e. ZZ ) -> ( N gcd ( M mod N ) ) = ( ( M mod N ) gcd N ) )
41	33, 39, 40	syl2anc 415	. 2 |- ( ( M e. ZZ /\ N e. NN ) -> ( N gcd ( M mod N ) ) = ( ( M mod N ) gcd N ) )
42		gcdcom 12769	. . 3 |- ( ( N e. ZZ /\ M e. ZZ ) -> ( N gcd M ) = ( M gcd N ) )
43	33, 34, 42	syl2anc 415	. 2 |- ( ( M e. ZZ /\ N e. NN ) -> ( N gcd M ) = ( M gcd N ) )
44	37, 41, 43	3eqtr3d 2279	1 |- ( ( M e. ZZ /\ N e. NN ) -> ( ( M mod N ) gcd N ) = ( M gcd N ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   /\ w3a 1009   = wceq 1402   e. wcel 2209   class class class wbr 4130   ` cfv 5377  (class class class)co 6085   CCcc 8178   0cc0 8180   + caddc 8183   x. cmul 8185   < clt 8361   - cmin 8499   -ucneg 8500   / cdiv 9005   NNcn 9307   ZZcz 9649   QQcq 10029   |_cfl 10714   mod cmo 10774   gcd cgcd 12749

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8271  ax-resscn 8272  ax-1cn 8273  ax-1re 8274  ax-icn 8275  ax-addcl 8276  ax-addrcl 8277  ax-mulcl 8278  ax-mulrcl 8279  ax-addcom 8280  ax-mulcom 8281  ax-addass 8282  ax-mulass 8283  ax-distr 8284  ax-i2m1 8285  ax-0lt1 8286  ax-1rid 8287  ax-0id 8288  ax-rnegex 8289  ax-precex 8290  ax-cnre 8291  ax-pre-ltirr 8292  ax-pre-ltwlin 8293  ax-pre-lttrn 8294  ax-pre-apti 8295  ax-pre-ltadd 8296  ax-pre-mulgt0 8297  ax-pre-mulext 8298  ax-arch 8299  ax-caucvg 8300

This proof depends on definitions:  df-bi 117  df-stab 843  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-frec 6662  df-sup 7325  df-pnf 8363  df-mnf 8364  df-xr 8365  df-ltxr 8366  df-le 8367  df-sub 8501  df-neg 8502  df-reap 8906  df-ap 8913  df-div 9006  df-inn 9308  df-2 9366  df-3 9367  df-4 9368  df-n0 9569  df-z 9650  df-uz 9932  df-q 10030  df-rp 10066  df-fz 10423  df-fzo 10561  df-fl 10716  df-mod 10775  df-seqfrec 10900  df-exp 10991  df-cj 11623  df-re 11624  df-im 11625  df-rsqrt 11780  df-abs 11781  df-dvds 12574  df-gcd 12750

This theorem is used by:  eucalginv  12853  phimullem  13026  eulerthlem1  13028  eulerthlemth  13033  pockthlem  13158  gcdmodi  13224