Theorem gcdass 11703

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 398	. . 3 |- ( ( ( N = 0 /\ M = 0 ) /\ P = 0 ) <-> ( N = 0 /\ ( M = 0 /\ P = 0 ) ) )
2		anass 398	. . . . . 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 2671	. . . 4 |- { x e. ZZ | ( ( x || N /\ x || M ) /\ x || P ) } = { x e. ZZ | ( x || N /\ ( x || M /\ x || P ) ) }
5	4	supeq1i 6875	. . 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 3495	. 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 11655	. . . . . 6 |- ( ( N e. ZZ /\ M e. ZZ ) -> ( N gcd M ) e. NN0 )
8	7	3adant3 1001	. . . . 5 |- ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) -> ( N gcd M ) e. NN0 )
9	8	nn0zd 9171	. . . 4 |- ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) -> ( N gcd M ) e. ZZ )
10		simp3 983	. . . 4 |- ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) -> P e. ZZ )
11		gcdval 11648	. . . 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 408	. . 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 11665	. . . . . . 7 |- ( ( N e. ZZ /\ M e. ZZ ) -> ( ( N gcd M ) = 0 <-> ( N = 0 /\ M = 0 ) ) )
14	13	3adant3 1001	. . . . . 6 |- ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) -> ( ( N gcd M ) = 0 <-> ( N = 0 /\ M = 0 ) ) )
15	14	anbi1d 460	. . . . 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 140	. . . 4 |- ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) -> ( ( ( N = 0 /\ M = 0 ) /\ P = 0 ) <-> ( ( N gcd M ) = 0 /\ P = 0 ) ) )
17		simpr 109	. . . . . . . 8 |- ( ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) /\ x e. ZZ ) -> x e. ZZ )
18		simpl1 984	. . . . . . . 8 |- ( ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) /\ x e. ZZ ) -> N e. ZZ )
19		simpl2 985	. . . . . . . 8 |- ( ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) /\ x e. ZZ ) -> M e. ZZ )
20		dvdsgcdb 11701	. . . . . . . 8 |- ( ( x e. ZZ /\ N e. ZZ /\ M e. ZZ ) -> ( ( x || N /\ x || M ) <-> x || ( N gcd M ) ) )
21	17, 18, 19, 20	syl3anc 1216	. . . . . . 7 |- ( ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) /\ x e. ZZ ) -> ( ( x || N /\ x || M ) <-> x || ( N gcd M ) ) )
22	21	anbi1d 460	. . . . . 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 2674	. . . . 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 6874	. . . 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 3496	. . 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 2175	. 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 981	. . . 4 |- ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) -> N e. ZZ )
28		gcdcl 11655	. . . . . 6 |- ( ( M e. ZZ /\ P e. ZZ ) -> ( M gcd P ) e. NN0 )
29	28	3adant1 999	. . . . 5 |- ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) -> ( M gcd P ) e. NN0 )
30	29	nn0zd 9171	. . . 4 |- ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) -> ( M gcd P ) e. ZZ )
31		gcdval 11648	. . . 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 408	. . 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 11665	. . . . . . 7 |- ( ( M e. ZZ /\ P e. ZZ ) -> ( ( M gcd P ) = 0 <-> ( M = 0 /\ P = 0 ) ) )
34	33	3adant1 999	. . . . . 6 |- ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) -> ( ( M gcd P ) = 0 <-> ( M = 0 /\ P = 0 ) ) )
35	34	anbi2d 459	. . . . 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 140	. . . 4 |- ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) -> ( ( N = 0 /\ ( M = 0 /\ P = 0 ) ) <-> ( N = 0 /\ ( M gcd P ) = 0 ) ) )
37		simpl3 986	. . . . . . . 8 |- ( ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) /\ x e. ZZ ) -> P e. ZZ )
38		dvdsgcdb 11701	. . . . . . . 8 |- ( ( x e. ZZ /\ M e. ZZ /\ P e. ZZ ) -> ( ( x || M /\ x || P ) <-> x || ( M gcd P ) ) )
39	17, 19, 37, 38	syl3anc 1216	. . . . . . 7 |- ( ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) /\ x e. ZZ ) -> ( ( x || M /\ x || P ) <-> x || ( M gcd P ) ) )
40	39	anbi2d 459	. . . . . 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 2674	. . . . 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 6874	. . . 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 3496	. . 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 2175	. 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 2198	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 103   <-> wb 104   /\ w3a 962   = wceq 1331   e. wcel 1480   {crab 2420   ifcif 3474   class class class wbr 3929  (class class class)co 5774   supcsup 6869   RRcr 7619   0cc0 7620   < clt 7800   NN0cn0 8977   ZZcz 9054   || cdvds 11493   gcd cgcd 11635

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7711  ax-resscn 7712  ax-1cn 7713  ax-1re 7714  ax-icn 7715  ax-addcl 7716  ax-addrcl 7717  ax-mulcl 7718  ax-mulrcl 7719  ax-addcom 7720  ax-mulcom 7721  ax-addass 7722  ax-mulass 7723  ax-distr 7724  ax-i2m1 7725  ax-0lt1 7726  ax-1rid 7727  ax-0id 7728  ax-rnegex 7729  ax-precex 7730  ax-cnre 7731  ax-pre-ltirr 7732  ax-pre-ltwlin 7733  ax-pre-lttrn 7734  ax-pre-apti 7735  ax-pre-ltadd 7736  ax-pre-mulgt0 7737  ax-pre-mulext 7738  ax-arch 7739  ax-caucvg 7740

This theorem depends on definitions:  df-bi 116  df-stab 816  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-frec 6288  df-sup 6871  df-pnf 7802  df-mnf 7803  df-xr 7804  df-ltxr 7805  df-le 7806  df-sub 7935  df-neg 7936  df-reap 8337  df-ap 8344  df-div 8433  df-inn 8721  df-2 8779  df-3 8780  df-4 8781  df-n0 8978  df-z 9055  df-uz 9327  df-q 9412  df-rp 9442  df-fz 9791  df-fzo 9920  df-fl 10043  df-mod 10096  df-seqfrec 10219  df-exp 10293  df-cj 10614  df-re 10615  df-im 10616  df-rsqrt 10770  df-abs 10771  df-dvds 11494  df-gcd 11636

This theorem is referenced by:  rpmulgcd  11714