Theorem gcdass 12770

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 405	. . 3 |- ( ( ( N = 0 /\ M = 0 ) /\ P = 0 ) <-> ( N = 0 /\ ( M = 0 /\ P = 0 ) ) )
2		anass 405	. . . . . 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 2807	. . . 4 |- { x e. ZZ | ( ( x || N /\ x || M ) /\ x || P ) } = { x e. ZZ | ( x || N /\ ( x || M /\ x || P ) ) }
5	4	supeq1i 7318	. . 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 3661	. 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 12721	. . . . . 6 |- ( ( N e. ZZ /\ M e. ZZ ) -> ( N gcd M ) e. NN0 )
8	7	3adant3 1048	. . . . 5 |- ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) -> ( N gcd M ) e. NN0 )
9	8	nn0zd 9745	. . . 4 |- ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) -> ( N gcd M ) e. ZZ )
10		simp3 1030	. . . 4 |- ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) -> P e. ZZ )
11		gcdval 12714	. . . 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 415	. . 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 12732	. . . . . . 7 |- ( ( N e. ZZ /\ M e. ZZ ) -> ( ( N gcd M ) = 0 <-> ( N = 0 /\ M = 0 ) ) )
14	13	3adant3 1048	. . . . . 6 |- ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) -> ( ( N gcd M ) = 0 <-> ( N = 0 /\ M = 0 ) ) )
15	14	anbi1d 469	. . . . 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 1031	. . . . . . . 8 |- ( ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) /\ x e. ZZ ) -> N e. ZZ )
19		simpl2 1032	. . . . . . . 8 |- ( ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) /\ x e. ZZ ) -> M e. ZZ )
20		dvdsgcdb 12768	. . . . . . . 8 |- ( ( x e. ZZ /\ N e. ZZ /\ M e. ZZ ) -> ( ( x || N /\ x || M ) <-> x || ( N gcd M ) ) )
21	17, 18, 19, 20	syl3anc 1278	. . . . . . 7 |- ( ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) /\ x e. ZZ ) -> ( ( x || N /\ x || M ) <-> x || ( N gcd M ) ) )
22	21	anbi1d 469	. . . . . 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 2809	. . . . 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 7317	. . . 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 3662	. . 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 2274	. 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 1028	. . . 4 |- ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) -> N e. ZZ )
28		gcdcl 12721	. . . . . 6 |- ( ( M e. ZZ /\ P e. ZZ ) -> ( M gcd P ) e. NN0 )
29	28	3adant1 1046	. . . . 5 |- ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) -> ( M gcd P ) e. NN0 )
30	29	nn0zd 9745	. . . 4 |- ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) -> ( M gcd P ) e. ZZ )
31		gcdval 12714	. . . 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 415	. . 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 12732	. . . . . . 7 |- ( ( M e. ZZ /\ P e. ZZ ) -> ( ( M gcd P ) = 0 <-> ( M = 0 /\ P = 0 ) ) )
34	33	3adant1 1046	. . . . . 6 |- ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) -> ( ( M gcd P ) = 0 <-> ( M = 0 /\ P = 0 ) ) )
35	34	anbi2d 468	. . . . 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 1033	. . . . . . . 8 |- ( ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) /\ x e. ZZ ) -> P e. ZZ )
38		dvdsgcdb 12768	. . . . . . . 8 |- ( ( x e. ZZ /\ M e. ZZ /\ P e. ZZ ) -> ( ( x || M /\ x || P ) <-> x || ( M gcd P ) ) )
39	17, 19, 37, 38	syl3anc 1278	. . . . . . 7 |- ( ( ( N e. ZZ /\ M e. ZZ /\ P e. ZZ ) /\ x e. ZZ ) -> ( ( x || M /\ x || P ) <-> x || ( M gcd P ) ) )
40	39	anbi2d 468	. . . . . 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 2809	. . . . 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 7317	. . . 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 3662	. . 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 2274	. 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 2297	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 1009   = wceq 1402   e. wcel 2209   {crab 2532   ifcif 3635   class class class wbr 4125  (class class class)co 6075   supcsup 7312   RRcr 8168   0cc0 8169   < clt 8350   NN0cn0 9542   ZZcz 9623   || cdvds 12532   gcd cgcd 12708

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-mulrcl 8268  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-precex 8279  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285  ax-pre-mulgt0 8286  ax-pre-mulext 8287  ax-arch 8288  ax-caucvg 8289

This theorem depends on definitions:  df-bi 117  df-stab 843  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-po 4436  df-iso 4437  df-iord 4506  df-on 4508  df-ilim 4509  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-frec 6652  df-sup 7314  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-reap 8893  df-ap 8900  df-div 8993  df-inn 9284  df-2 9342  df-3 9343  df-4 9344  df-n0 9543  df-z 9624  df-uz 9901  df-q 9999  df-rp 10034  df-fz 10391  df-fzo 10528  df-fl 10683  df-mod 10738  df-seqfrec 10863  df-exp 10954  df-cj 11585  df-re 11586  df-im 11587  df-rsqrt 11742  df-abs 11743  df-dvds 12533  df-gcd 12709

This theorem is referenced by:  rpmulgcd  12781  coprimeprodsq  13014