Theorem modgcd 11994

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 9628	. . . . . . 7 |- ( M e. ZZ -> M e. QQ )
2	1	adantr 276	. . . . . 6 |- ( ( M e. ZZ /\ N e. NN ) -> M e. QQ )
3		nnq 9635	. . . . . . 7 |- ( N e. NN -> N e. QQ )
4	3	adantl 277	. . . . . 6 |- ( ( M e. ZZ /\ N e. NN ) -> N e. QQ )
5		nngt0 8946	. . . . . . 7 |- ( N e. NN -> 0 < N )
6	5	adantl 277	. . . . . 6 |- ( ( M e. ZZ /\ N e. NN ) -> 0 < N )
7		modqval 10326	. . . . . 6 |- ( ( M e. QQ /\ N e. QQ /\ 0 < N ) -> ( M mod N ) = ( M - ( N x. ( |_ ` ( M / N ) ) ) ) )
8	2, 4, 6, 7	syl3anc 1238	. . . . 5 |- ( ( M e. ZZ /\ N e. NN ) -> ( M mod N ) = ( M - ( N x. ( |_ ` ( M / N ) ) ) ) )
9		zcn 9260	. . . . . . 7 |- ( M e. ZZ -> M e. CC )
10	9	adantr 276	. . . . . 6 |- ( ( M e. ZZ /\ N e. NN ) -> M e. CC )
11		nncn 8929	. . . . . . 7 |- ( N e. NN -> N e. CC )
12	11	adantl 277	. . . . . 6 |- ( ( M e. ZZ /\ N e. NN ) -> N e. CC )
13		znq 9626	. . . . . . . 8 |- ( ( M e. ZZ /\ N e. NN ) -> ( M / N ) e. QQ )
14	13	flqcld 10279	. . . . . . 7 |- ( ( M e. ZZ /\ N e. NN ) -> ( |_ ` ( M / N ) ) e. ZZ )
15	14	zcnd 9378	. . . . . 6 |- ( ( M e. ZZ /\ N e. NN ) -> ( |_ ` ( M / N ) ) e. CC )
16		mulneg1 8354	. . . . . . . . . . 11 |- ( ( ( |_ ` ( M / N ) ) e. CC /\ N e. CC ) -> ( -u ( |_ ` ( M / N ) ) x. N ) = -u ( ( |_ ` ( M / N ) ) x. N ) )
17		mulcom 7942	. . . . . . . . . . . 12 |- ( ( ( |_ ` ( M / N ) ) e. CC /\ N e. CC ) -> ( ( |_ ` ( M / N ) ) x. N ) = ( N x. ( |_ ` ( M / N ) ) ) )
18	17	negeqd 8154	. . . . . . . . . . 11 |- ( ( ( |_ ` ( M / N ) ) e. CC /\ N e. CC ) -> -u ( ( |_ ` ( M / N ) ) x. N ) = -u ( N x. ( |_ ` ( M / N ) ) ) )
19	16, 18	eqtrd 2210	. . . . . . . . . 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 1015	. . . . . . . 8 |- ( ( M e. CC /\ N e. CC /\ ( |_ ` ( M / N ) ) e. CC ) -> ( -u ( |_ ` ( M / N ) ) x. N ) = -u ( N x. ( |_ ` ( M / N ) ) ) )
22	21	oveq2d 5893	. . . . . . 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 7940	. . . . . . . . 9 |- ( ( N e. CC /\ ( |_ ` ( M / N ) ) e. CC ) -> ( N x. ( |_ ` ( M / N ) ) ) e. CC )
24		negsub 8207	. . . . . . . . 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 1199	. . . . . . 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 2210	. . . . . 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 1238	. . . . 5 |- ( ( M e. ZZ /\ N e. NN ) -> ( M + ( -u ( |_ ` ( M / N ) ) x. N ) ) = ( M - ( N x. ( |_ ` ( M / N ) ) ) ) )
29	8, 28	eqtr4d 2213	. . . 4 |- ( ( M e. ZZ /\ N e. NN ) -> ( M mod N ) = ( M + ( -u ( |_ ` ( M / N ) ) x. N ) ) )
30	29	oveq2d 5893	. . 3 |- ( ( M e. ZZ /\ N e. NN ) -> ( N gcd ( M mod N ) ) = ( N gcd ( M + ( -u ( |_ ` ( M / N ) ) x. N ) ) ) )
31	14	znegcld 9379	. . . 4 |- ( ( M e. ZZ /\ N e. NN ) -> -u ( |_ ` ( M / N ) ) e. ZZ )
32		nnz 9274	. . . . 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 11987	. . . 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 1238	. . 3 |- ( ( M e. ZZ /\ N e. NN ) -> ( N gcd M ) = ( N gcd ( M + ( -u ( |_ ` ( M / N ) ) x. N ) ) ) )
37	30, 36	eqtr4d 2213	. 2 |- ( ( M e. ZZ /\ N e. NN ) -> ( N gcd ( M mod N ) ) = ( N gcd M ) )
38		zmodcl 10346	. . . 4 |- ( ( M e. ZZ /\ N e. NN ) -> ( M mod N ) e. NN0 )
39	38	nn0zd 9375	. . 3 |- ( ( M e. ZZ /\ N e. NN ) -> ( M mod N ) e. ZZ )
40		gcdcom 11976	. . 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 11976	. . 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 2218	1 |- ( ( M e. ZZ /\ N e. NN ) -> ( ( M mod N ) gcd N ) = ( M gcd N ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 978   = wceq 1353   e. wcel 2148   class class class wbr 4005   ` cfv 5218  (class class class)co 5877   CCcc 7811   0cc0 7813   + caddc 7816   x. cmul 7818   < clt 7994   - cmin 8130   -ucneg 8131   / cdiv 8631   NNcn 8921   ZZcz 9255   QQcq 9621   |_cfl 10270   mod cmo 10324   gcd cgcd 11945

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-mulrcl 7912  ax-addcom 7913  ax-mulcom 7914  ax-addass 7915  ax-mulass 7916  ax-distr 7917  ax-i2m1 7918  ax-0lt1 7919  ax-1rid 7920  ax-0id 7921  ax-rnegex 7922  ax-precex 7923  ax-cnre 7924  ax-pre-ltirr 7925  ax-pre-ltwlin 7926  ax-pre-lttrn 7927  ax-pre-apti 7928  ax-pre-ltadd 7929  ax-pre-mulgt0 7930  ax-pre-mulext 7931  ax-arch 7932  ax-caucvg 7933

This theorem depends on definitions:  df-bi 117  df-stab 831  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-if 3537  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-po 4298  df-iso 4299  df-iord 4368  df-on 4370  df-ilim 4371  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-recs 6308  df-frec 6394  df-sup 6985  df-pnf 7996  df-mnf 7997  df-xr 7998  df-ltxr 7999  df-le 8000  df-sub 8132  df-neg 8133  df-reap 8534  df-ap 8541  df-div 8632  df-inn 8922  df-2 8980  df-3 8981  df-4 8982  df-n0 9179  df-z 9256  df-uz 9531  df-q 9622  df-rp 9656  df-fz 10011  df-fzo 10145  df-fl 10272  df-mod 10325  df-seqfrec 10448  df-exp 10522  df-cj 10853  df-re 10854  df-im 10855  df-rsqrt 11009  df-abs 11010  df-dvds 11797  df-gcd 11946

This theorem is referenced by:  eucalginv  12058  phimullem  12227  eulerthlem1  12229  eulerthlemth  12234  pockthlem  12356