Theorem gcdass 12536

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 401	. . 3 |- ( ( ( N = 0 /\ M = 0 ) /\ P = 0 ) <-> ( N = 0 /\ ( M = 0 /\ P = 0 ) ) )
2		anass 401	. . . . . 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 2784	. . . 4 |- { x e. ZZ | ( ( x || N /\ x || M ) /\ x || P ) } = { x e. ZZ | ( x || N /\ ( x || M /\ x || P ) ) }
5	4	supeq1i 7155	. . 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 3626	. 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 12487	. . . . . 6 |- ( ( N e. ZZ /\ M e. ZZ ) -> ( N gcd M ) e. NN0 )
8	7	3adant3 1041	. . . . 5 |- ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) -> ( N gcd M ) e. NN0 )
9	8	nn0zd 9567	. . . 4 |- ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) -> ( N gcd M ) e. ZZ )
10		simp3 1023	. . . 4 |- ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) -> P e. ZZ )
11		gcdval 12480	. . . 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 411	. . 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 12498	. . . . . . 7 |- ( ( N e. ZZ /\ M e. ZZ ) -> ( ( N gcd M ) = 0 <-> ( N = 0 /\ M = 0 ) ) )
14	13	3adant3 1041	. . . . . 6 |- ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) -> ( ( N gcd M ) = 0 <-> ( N = 0 /\ M = 0 ) ) )
15	14	anbi1d 465	. . . . 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 141	. . . 4 |- ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) -> ( ( ( N = 0 /\ M = 0 ) /\ P = 0 ) <-> ( ( N gcd M ) = 0 /\ P = 0 ) ) )
17		simpr 110	. . . . . . . 8 |- ( ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) /\ x e. ZZ ) -> x e. ZZ )
18		simpl1 1024	. . . . . . . 8 |- ( ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) /\ x e. ZZ ) -> N e. ZZ )
19		simpl2 1025	. . . . . . . 8 |- ( ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) /\ x e. ZZ ) -> M e. ZZ )
20		dvdsgcdb 12534	. . . . . . . 8 |- ( ( x e. ZZ /\ N e. ZZ /\ M e. ZZ ) -> ( ( x || N /\ x || M ) <-> x || ( N gcd M ) ) )
21	17, 18, 19, 20	syl3anc 1271	. . . . . . 7 |- ( ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) /\ x e. ZZ ) -> ( ( x || N /\ x || M ) <-> x || ( N gcd M ) ) )
22	21	anbi1d 465	. . . . . 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 2787	. . . . 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 7154	. . . 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 3627	. . 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 2265	. 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 1021	. . . 4 |- ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) -> N e. ZZ )
28		gcdcl 12487	. . . . . 6 |- ( ( M e. ZZ /\ P e. ZZ ) -> ( M gcd P ) e. NN0 )
29	28	3adant1 1039	. . . . 5 |- ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) -> ( M gcd P ) e. NN0 )
30	29	nn0zd 9567	. . . 4 |- ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) -> ( M gcd P ) e. ZZ )
31		gcdval 12480	. . . 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 411	. . 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 12498	. . . . . . 7 |- ( ( M e. ZZ /\ P e. ZZ ) -> ( ( M gcd P ) = 0 <-> ( M = 0 /\ P = 0 ) ) )
34	33	3adant1 1039	. . . . . 6 |- ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) -> ( ( M gcd P ) = 0 <-> ( M = 0 /\ P = 0 ) ) )
35	34	anbi2d 464	. . . . 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 141	. . . 4 |- ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) -> ( ( N = 0 /\ ( M = 0 /\ P = 0 ) ) <-> ( N = 0 /\ ( M gcd P ) = 0 ) ) )
37		simpl3 1026	. . . . . . . 8 |- ( ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) /\ x e. ZZ ) -> P e. ZZ )
38		dvdsgcdb 12534	. . . . . . . 8 |- ( ( x e. ZZ /\ M e. ZZ /\ P e. ZZ ) -> ( ( x || M /\ x || P ) <-> x || ( M gcd P ) ) )
39	17, 19, 37, 38	syl3anc 1271	. . . . . . 7 |- ( ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) /\ x e. ZZ ) -> ( ( x || M /\ x || P ) <-> x || ( M gcd P ) ) )
40	39	anbi2d 464	. . . . . 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 2787	. . . . 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 7154	. . . 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 3627	. . 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 2265	. 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 2288	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 104   <-> wb 105   /\ w3a 1002   = wceq 1395   e. wcel 2200   {crab 2512   ifcif 3602   class class class wbr 4083  (class class class)co 6001   supcsup 7149   RRcr 7998   0cc0 7999   < clt 8181   NN0cn0 9369   ZZcz 9446   || cdvds 12298   gcd cgcd 12474

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-mulrcl 8098  ax-addcom 8099  ax-mulcom 8100  ax-addass 8101  ax-mulass 8102  ax-distr 8103  ax-i2m1 8104  ax-0lt1 8105  ax-1rid 8106  ax-0id 8107  ax-rnegex 8108  ax-precex 8109  ax-cnre 8110  ax-pre-ltirr 8111  ax-pre-ltwlin 8112  ax-pre-lttrn 8113  ax-pre-apti 8114  ax-pre-ltadd 8115  ax-pre-mulgt0 8116  ax-pre-mulext 8117  ax-arch 8118  ax-caucvg 8119

This theorem depends on definitions:  df-bi 117  df-stab 836  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-po 4387  df-iso 4388  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-1st 6286  df-2nd 6287  df-recs 6451  df-frec 6537  df-sup 7151  df-pnf 8183  df-mnf 8184  df-xr 8185  df-ltxr 8186  df-le 8187  df-sub 8319  df-neg 8320  df-reap 8722  df-ap 8729  df-div 8820  df-inn 9111  df-2 9169  df-3 9170  df-4 9171  df-n0 9370  df-z 9447  df-uz 9723  df-q 9815  df-rp 9850  df-fz 10205  df-fzo 10339  df-fl 10490  df-mod 10545  df-seqfrec 10670  df-exp 10761  df-cj 11353  df-re 11354  df-im 11355  df-rsqrt 11509  df-abs 11510  df-dvds 12299  df-gcd 12475

This theorem is referenced by:  rpmulgcd  12547  coprimeprodsq  12780