Theorem modgcd 11975

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 9615	. . . . . . 7 |- ( M e. ZZ -> M e. QQ )
2	1	adantr 276	. . . . . 6 |- ( ( M e. ZZ /\ N e. NN ) -> M e. QQ )
3		nnq 9622	. . . . . . 7 |- ( N e. NN -> N e. QQ )
4	3	adantl 277	. . . . . 6 |- ( ( M e. ZZ /\ N e. NN ) -> N e. QQ )
5		nngt0 8933	. . . . . . 7 |- ( N e. NN -> 0 < N )
6	5	adantl 277	. . . . . 6 |- ( ( M e. ZZ /\ N e. NN ) -> 0 < N )
7		modqval 10310	. . . . . 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 9247	. . . . . . 7 |- ( M e. ZZ -> M e. CC )
10	9	adantr 276	. . . . . 6 |- ( ( M e. ZZ /\ N e. NN ) -> M e. CC )
11		nncn 8916	. . . . . . 7 |- ( N e. NN -> N e. CC )
12	11	adantl 277	. . . . . 6 |- ( ( M e. ZZ /\ N e. NN ) -> N e. CC )
13		znq 9613	. . . . . . . 8 |- ( ( M e. ZZ /\ N e. NN ) -> ( M / N ) e. QQ )
14	13	flqcld 10263	. . . . . . 7 |- ( ( M e. ZZ /\ N e. NN ) -> ( |_ ` ( M / N ) ) e. ZZ )
15	14	zcnd 9365	. . . . . 6 |- ( ( M e. ZZ /\ N e. NN ) -> ( |_ ` ( M / N ) ) e. CC )
16		mulneg1 8342	. . . . . . . . . . 11 |- ( ( ( |_ ` ( M / N ) ) e. CC /\ N e. CC ) -> ( -u ( |_ ` ( M / N ) ) x. N ) = -u ( ( |_ ` ( M / N ) ) x. N ) )
17		mulcom 7931	. . . . . . . . . . . 12 |- ( ( ( |_ ` ( M / N ) ) e. CC /\ N e. CC ) -> ( ( |_ ` ( M / N ) ) x. N ) = ( N x. ( |_ ` ( M / N ) ) ) )
18	17	negeqd 8142	. . . . . . . . . . 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 5885	. . . . . . 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 7929	. . . . . . . . 9 |- ( ( N e. CC /\ ( |_ ` ( M / N ) ) e. CC ) -> ( N x. ( |_ ` ( M / N ) ) ) e. CC )
24		negsub 8195	. . . . . . . . 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 5885	. . 3 |- ( ( M e. ZZ /\ N e. NN ) -> ( N gcd ( M mod N ) ) = ( N gcd ( M + ( -u ( |_ ` ( M / N ) ) x. N ) ) ) )
31	14	znegcld 9366	. . . 4 |- ( ( M e. ZZ /\ N e. NN ) -> -u ( |_ ` ( M / N ) ) e. ZZ )
32		nnz 9261	. . . . 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 11968	. . . 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 10330	. . . 4 |- ( ( M e. ZZ /\ N e. NN ) -> ( M mod N ) e. NN0 )
39	38	nn0zd 9362	. . 3 |- ( ( M e. ZZ /\ N e. NN ) -> ( M mod N ) e. ZZ )
40		gcdcom 11957	. . 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 11957	. . 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 4000   ` cfv 5212  (class class class)co 5869   CCcc 7800   0cc0 7802   + caddc 7805   x. cmul 7807   < clt 7982   - cmin 8118   -ucneg 8119   / cdiv 8618   NNcn 8908   ZZcz 9242   QQcq 9608   |_cfl 10254   mod cmo 10308   gcd cgcd 11926

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 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-mulrcl 7901  ax-addcom 7902  ax-mulcom 7903  ax-addass 7904  ax-mulass 7905  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-1rid 7909  ax-0id 7910  ax-rnegex 7911  ax-precex 7912  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-apti 7917  ax-pre-ltadd 7918  ax-pre-mulgt0 7919  ax-pre-mulext 7920  ax-arch 7921  ax-caucvg 7922

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 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4290  df-po 4293  df-iso 4294  df-iord 4363  df-on 4365  df-ilim 4366  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-frec 6386  df-sup 6977  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-sub 8120  df-neg 8121  df-reap 8522  df-ap 8529  df-div 8619  df-inn 8909  df-2 8967  df-3 8968  df-4 8969  df-n0 9166  df-z 9243  df-uz 9518  df-q 9609  df-rp 9641  df-fz 9996  df-fzo 10129  df-fl 10256  df-mod 10309  df-seqfrec 10432  df-exp 10506  df-cj 10835  df-re 10836  df-im 10837  df-rsqrt 10991  df-abs 10992  df-dvds 11779  df-gcd 11927

This theorem is referenced by:  eucalginv  12039  phimullem  12208  eulerthlem1  12210  eulerthlemth  12215  pockthlem  12337