Theorem gcdass 10404

Description: Associative law for gcd operator. Theorem 1.4(b) in [ApostolNT] p. 16. (Contributed by Scott Fenton, 2-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Assertion

Ref	Expression
gcdass	|- ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) -> ( ( N gcd M ) gcd P ) = ( N gcd ( M gcd P ) ) )

Proof of Theorem gcdass

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		anass 393	. . 3 |- ( ( ( N = 0 /\ M = 0 ) /\ P = 0 ) <-> ( N = 0 /\ ( M = 0 /\ P = 0 ) ) )
2		anass 393	. . . . . 6 |- ( ( ( x || N /\ x || M ) /\ x || P ) <-> ( x || N /\ ( x || M /\ x || P ) ) )
3	2	a1i 9	. . . . 5 |- ( x e. ZZ -> ( ( ( x || N /\ x || M ) /\ x || P ) <-> ( x || N /\ ( x || M /\ x || P ) ) ) )
4	3	rabbiia 2591	. . . 4 |- { x e. ZZ | ( ( x || N /\ x || M ) /\ x || P ) } = { x e. ZZ | ( x || N /\ ( x || M /\ x || P ) ) }
5	4	supeq1i 6401	. . 3 |- sup ( { x e. ZZ | ( ( x || N /\ x || M ) /\ x || P ) } , RR , < ) = sup ( { x e. ZZ | ( x || N /\ ( x || M /\ x || P ) ) } , RR , < )
6	1, 5	ifbieq2i 3372	. 2 |- if ( ( ( N = 0 /\ M = 0 ) /\ P = 0 ) , 0 , sup ( { x e. ZZ | ( ( x || N /\ x || M ) /\ x || P ) } , RR , < ) ) = if ( ( N = 0 /\ ( M = 0 /\ P = 0 ) ) , 0 , sup ( { x e. ZZ | ( x || N /\ ( x || M /\ x || P ) ) } , RR , < ) )
7		gcdcl 10358	. . . . . 6 |- ( ( N e. ZZ /\ M e. ZZ ) -> ( N gcd M ) e. NN0 )
8	7	3adant3 958	. . . . 5 |- ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) -> ( N gcd M ) e. NN0 )
9	8	nn0zd 8467	. . . 4 |- ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) -> ( N gcd M ) e. ZZ )
10		simp3 940	. . . 4 |- ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) -> P e. ZZ )
11		gcdval 10351	. . . 4 |- ( ( ( N gcd M ) e. ZZ /\ P e. ZZ ) -> ( ( N gcd M ) gcd P ) = if ( ( ( N gcd M ) = 0 /\ P = 0 ) , 0 , sup ( { x e. ZZ | ( x || ( N gcd M ) /\ x || P ) } , RR , < ) ) )
12	9, 10, 11	syl2anc 403	. . 3 |- ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) -> ( ( N gcd M ) gcd P ) = if ( ( ( N gcd M ) = 0 /\ P = 0 ) , 0 , sup ( { x e. ZZ | ( x || ( N gcd M ) /\ x || P ) } , RR , < ) ) )
13		gcdeq0 10368	. . . . . . 7 |- ( ( N e. ZZ /\ M e. ZZ ) -> ( ( N gcd M ) = 0 <-> ( N = 0 /\ M = 0 ) ) )
14	13	3adant3 958	. . . . . 6 |- ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) -> ( ( N gcd M ) = 0 <-> ( N = 0 /\ M = 0 ) ) )
15	14	anbi1d 452	. . . . 5 |- ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) -> ( ( ( N gcd M ) = 0 /\ P = 0 ) <-> ( ( N = 0 /\ M = 0 ) /\ P = 0 ) ) )
16	15	bicomd 139	. . . 4 |- ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) -> ( ( ( N = 0 /\ M = 0 ) /\ P = 0 ) <-> ( ( N gcd M ) = 0 /\ P = 0 ) ) )
17		simpr 108	. . . . . . . 8 |- ( ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) /\ x e. ZZ ) -> x e. ZZ )
18		simpl1 941	. . . . . . . 8 |- ( ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) /\ x e. ZZ ) -> N e. ZZ )
19		simpl2 942	. . . . . . . 8 |- ( ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) /\ x e. ZZ ) -> M e. ZZ )
20		dvdsgcdb 10402	. . . . . . . 8 |- ( ( x e. ZZ /\ N e. ZZ /\ M e. ZZ ) -> ( ( x || N /\ x || M ) <-> x || ( N gcd M ) ) )
21	17, 18, 19, 20	syl3anc 1169	. . . . . . 7 |- ( ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) /\ x e. ZZ ) -> ( ( x || N /\ x || M ) <-> x || ( N gcd M ) ) )
22	21	anbi1d 452	. . . . . 6 |- ( ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) /\ x e. ZZ ) -> ( ( ( x || N /\ x || M ) /\ x || P ) <-> ( x || ( N gcd M ) /\ x || P ) ) )
23	22	rabbidva 2592	. . . . 5 |- ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) -> { x e. ZZ | ( ( x || N /\ x || M ) /\ x || P ) } = { x e. ZZ | ( x || ( N gcd M ) /\ x || P ) } )
24	23	supeq1d 6400	. . . 4 |- ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) -> sup ( { x e. ZZ | ( ( x || N /\ x || M ) /\ x || P ) } , RR , < ) = sup ( { x e. ZZ | ( x || ( N gcd M ) /\ x || P ) } , RR , < ) )
25	16, 24	ifbieq2d 3373	. . 3 |- ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) -> if ( ( ( N = 0 /\ M = 0 ) /\ P = 0 ) , 0 , sup ( { x e. ZZ | ( ( x || N /\ x || M ) /\ x || P ) } , RR , < ) ) = if ( ( ( N gcd M ) = 0 /\ P = 0 ) , 0 , sup ( { x e. ZZ | ( x || ( N gcd M ) /\ x || P ) } , RR , < ) ) )
26	12, 25	eqtr4d 2116	. 2 |- ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) -> ( ( N gcd M ) gcd P ) = if ( ( ( N = 0 /\ M = 0 ) /\ P = 0 ) , 0 , sup ( { x e. ZZ | ( ( x || N /\ x || M ) /\ x || P ) } , RR , < ) ) )
27		simp1 938	. . . 4 |- ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) -> N e. ZZ )
28		gcdcl 10358	. . . . . 6 |- ( ( M e. ZZ /\ P e. ZZ ) -> ( M gcd P ) e. NN0 )
29	28	3adant1 956	. . . . 5 |- ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) -> ( M gcd P ) e. NN0 )
30	29	nn0zd 8467	. . . 4 |- ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) -> ( M gcd P ) e. ZZ )
31		gcdval 10351	. . . 4 |- ( ( N e. ZZ /\ ( M gcd P ) e. ZZ ) -> ( N gcd ( M gcd P ) ) = if ( ( N = 0 /\ ( M gcd P ) = 0 ) , 0 , sup ( { x e. ZZ | ( x || N /\ x || ( M gcd P ) ) } , RR , < ) ) )
32	27, 30, 31	syl2anc 403	. . 3 |- ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) -> ( N gcd ( M gcd P ) ) = if ( ( N = 0 /\ ( M gcd P ) = 0 ) , 0 , sup ( { x e. ZZ | ( x || N /\ x || ( M gcd P ) ) } , RR , < ) ) )
33		gcdeq0 10368	. . . . . . 7 |- ( ( M e. ZZ /\ P e. ZZ ) -> ( ( M gcd P ) = 0 <-> ( M = 0 /\ P = 0 ) ) )
34	33	3adant1 956	. . . . . 6 |- ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) -> ( ( M gcd P ) = 0 <-> ( M = 0 /\ P = 0 ) ) )
35	34	anbi2d 451	. . . . 5 |- ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) -> ( ( N = 0 /\ ( M gcd P ) = 0 ) <-> ( N = 0 /\ ( M = 0 /\ P = 0 ) ) ) )
36	35	bicomd 139	. . . 4 |- ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) -> ( ( N = 0 /\ ( M = 0 /\ P = 0 ) ) <-> ( N = 0 /\ ( M gcd P ) = 0 ) ) )
37		simpl3 943	. . . . . . . 8 |- ( ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) /\ x e. ZZ ) -> P e. ZZ )
38		dvdsgcdb 10402	. . . . . . . 8 |- ( ( x e. ZZ /\ M e. ZZ /\ P e. ZZ ) -> ( ( x || M /\ x || P ) <-> x || ( M gcd P ) ) )
39	17, 19, 37, 38	syl3anc 1169	. . . . . . 7 |- ( ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) /\ x e. ZZ ) -> ( ( x || M /\ x || P ) <-> x || ( M gcd P ) ) )
40	39	anbi2d 451	. . . . . 6 |- ( ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) /\ x e. ZZ ) -> ( ( x || N /\ ( x || M /\ x || P ) ) <-> ( x || N /\ x || ( M gcd P ) ) ) )
41	40	rabbidva 2592	. . . . 5 |- ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) -> { x e. ZZ | ( x || N /\ ( x || M /\ x || P ) ) } = { x e. ZZ | ( x || N /\ x || ( M gcd P ) ) } )
42	41	supeq1d 6400	. . . 4 |- ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) -> sup ( { x e. ZZ | ( x || N /\ ( x || M /\ x || P ) ) } , RR , < ) = sup ( { x e. ZZ | ( x || N /\ x || ( M gcd P ) ) } , RR , < ) )
43	36, 42	ifbieq2d 3373	. . 3 |- ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) -> if ( ( N = 0 /\ ( M = 0 /\ P = 0 ) ) , 0 , sup ( { x e. ZZ | ( x || N /\ ( x || M /\ x || P ) ) } , RR , < ) ) = if ( ( N = 0 /\ ( M gcd P ) = 0 ) , 0 , sup ( { x e. ZZ | ( x || N /\ x || ( M gcd P ) ) } , RR , < ) ) )
44	32, 43	eqtr4d 2116	. 2 |- ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) -> ( N gcd ( M gcd P ) ) = if ( ( N = 0 /\ ( M = 0 /\ P = 0 ) ) , 0 , sup ( { x e. ZZ | ( x || N /\ ( x || M /\ x || P ) ) } , RR , < ) ) )
45	6, 26, 44	3eqtr4a 2139	1 |- ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) -> ( ( N gcd M ) gcd P ) = ( N gcd ( M gcd P ) ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   /\ w3a 919   = wceq 1284   e. wcel 1433   {crab 2352   ifcif 3351   class class class wbr 3785  (class class class)co 5532   supcsup 6395   RRcr 6980   0cc0 6981   < clt 7153   NN0cn0 8288   ZZcz 8351   || cdvds 10195   gcd cgcd 10338

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-iinf 4329  ax-cnex 7067  ax-resscn 7068  ax-1cn 7069  ax-1re 7070  ax-icn 7071  ax-addcl 7072  ax-addrcl 7073  ax-mulcl 7074  ax-mulrcl 7075  ax-addcom 7076  ax-mulcom 7077  ax-addass 7078  ax-mulass 7079  ax-distr 7080  ax-i2m1 7081  ax-0lt1 7082  ax-1rid 7083  ax-0id 7084  ax-rnegex 7085  ax-precex 7086  ax-cnre 7087  ax-pre-ltirr 7088  ax-pre-ltwlin 7089  ax-pre-lttrn 7090  ax-pre-apti 7091  ax-pre-ltadd 7092  ax-pre-mulgt0 7093  ax-pre-mulext 7094  ax-arch 7095  ax-caucvg 7096

This theorem depends on definitions:  df-bi 115  df-dc 776  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-nel 2340  df-ral 2353  df-rex 2354  df-reu 2355  df-rmo 2356  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-if 3352  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-tr 3876  df-id 4048  df-po 4051  df-iso 4052  df-iord 4121  df-on 4123  df-suc 4126  df-iom 4332  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-riota 5488  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-1st 5787  df-2nd 5788  df-recs 5943  df-frec 6001  df-sup 6397  df-pnf 7155  df-mnf 7156  df-xr 7157  df-ltxr 7158  df-le 7159  df-sub 7281  df-neg 7282  df-reap 7675  df-ap 7682  df-div 7761  df-inn 8040  df-2 8098  df-3 8099  df-4 8100  df-n0 8289  df-z 8352  df-uz 8620  df-q 8705  df-rp 8735  df-fz 9030  df-fzo 9153  df-fl 9274  df-mod 9325  df-iseq 9432  df-iexp 9476  df-cj 9729  df-re 9730  df-im 9731  df-rsqrt 9884  df-abs 9885  df-dvds 10196  df-gcd 10339

This theorem is referenced by:  rpmulgcd  10415