Theorem gcdcllem3 15223

Description: Lemma for gcdn0cl 15224, gcddvds 15225 and dvdslegcd 15226. (Contributed by Paul Chapman, 21-Mar-2011.)

Hypotheses

Ref	Expression
gcdcllem2.1	|- S = { z e. ZZ | A. n e. { M , N } z || n }
gcdcllem2.2	|- R = { z e. ZZ | ( z || M /\ z || N ) }

Assertion

Ref	Expression
gcdcllem3	|- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) -> ( sup ( R , RR , < ) e. NN /\ ( sup ( R , RR , < ) || M /\ sup ( R , RR , < ) || N ) /\ ( ( K e. ZZ /\ K || M /\ K || N ) -> K <_ sup ( R , RR , < ) ) ) )

Distinct variable groups:   z, K   z, n, M   n, N, z

Allowed substitution hints:   R( z, n)   S( z, n)   K( n)

Proof of Theorem gcdcllem3

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		gcdcllem2.2	. . . . 5 |- R = { z e. ZZ | ( z || M /\ z || N ) }
2		ssrab2 3687	. . . . 5 |- { z e. ZZ | ( z || M /\ z || N ) } C_ ZZ
3	1, 2	eqsstri 3635	. . . 4 |- R C_ ZZ
4		prssi 4353	. . . . . . . 8 |- ( ( M e. ZZ /\ N e. ZZ ) -> { M , N } C_ ZZ )
5	4	adantr 481	. . . . . . 7 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) -> { M , N } C_ ZZ )
6		neorian 2888	. . . . . . . 8 |- ( ( M =/= 0 \/ N =/= 0 ) <-> -. ( M = 0 /\ N = 0 ) )
7		prid1g 4295	. . . . . . . . . . 11 |- ( M e. ZZ -> M e. { M , N } )
8		neeq1 2856	. . . . . . . . . . . 12 |- ( n = M -> ( n =/= 0 <-> M =/= 0 ) )
9	8	rspcev 3309	. . . . . . . . . . 11 |- ( ( M e. { M , N } /\ M =/= 0 ) -> E. n e. { M , N } n =/= 0 )
10	7, 9	sylan 488	. . . . . . . . . 10 |- ( ( M e. ZZ /\ M =/= 0 ) -> E. n e. { M , N } n =/= 0 )
11	10	adantlr 751	. . . . . . . . 9 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ M =/= 0 ) -> E. n e. { M , N } n =/= 0 )
12		prid2g 4296	. . . . . . . . . . 11 |- ( N e. ZZ -> N e. { M , N } )
13		neeq1 2856	. . . . . . . . . . . 12 |- ( n = N -> ( n =/= 0 <-> N =/= 0 ) )
14	13	rspcev 3309	. . . . . . . . . . 11 |- ( ( N e. { M , N } /\ N =/= 0 ) -> E. n e. { M , N } n =/= 0 )
15	12, 14	sylan 488	. . . . . . . . . 10 |- ( ( N e. ZZ /\ N =/= 0 ) -> E. n e. { M , N } n =/= 0 )
16	15	adantll 750	. . . . . . . . 9 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ N =/= 0 ) -> E. n e. { M , N } n =/= 0 )
17	11, 16	jaodan 826	. . . . . . . 8 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 \/ N =/= 0 ) ) -> E. n e. { M , N } n =/= 0 )
18	6, 17	sylan2br 493	. . . . . . 7 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) -> E. n e. { M , N } n =/= 0 )
19		gcdcllem2.1	. . . . . . . 8 |- S = { z e. ZZ | A. n e. { M , N } z || n }
20	19	gcdcllem1 15221	. . . . . . 7 |- ( ( { M , N } C_ ZZ /\ E. n e. { M , N } n =/= 0 ) -> ( S =/= (/) /\ E. x e. ZZ A. y e. S y <_ x ) )
21	5, 18, 20	syl2anc 693	. . . . . 6 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) -> ( S =/= (/) /\ E. x e. ZZ A. y e. S y <_ x ) )
22	19, 1	gcdcllem2 15222	. . . . . . . 8 |- ( ( M e. ZZ /\ N e. ZZ ) -> R = S )
23		neeq1 2856	. . . . . . . . 9 |- ( R = S -> ( R =/= (/) <-> S =/= (/) ) )
24		raleq 3138	. . . . . . . . . 10 |- ( R = S -> ( A. y e. R y <_ x <-> A. y e. S y <_ x ) )
25	24	rexbidv 3052	. . . . . . . . 9 |- ( R = S -> ( E. x e. ZZ A. y e. R y <_ x <-> E. x e. ZZ A. y e. S y <_ x ) )
26	23, 25	anbi12d 747	. . . . . . . 8 |- ( R = S -> ( ( R =/= (/) /\ E. x e. ZZ A. y e. R y <_ x ) <-> ( S =/= (/) /\ E. x e. ZZ A. y e. S y <_ x ) ) )
27	22, 26	syl 17	. . . . . . 7 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( R =/= (/) /\ E. x e. ZZ A. y e. R y <_ x ) <-> ( S =/= (/) /\ E. x e. ZZ A. y e. S y <_ x ) ) )
28	27	adantr 481	. . . . . 6 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) -> ( ( R =/= (/) /\ E. x e. ZZ A. y e. R y <_ x ) <-> ( S =/= (/) /\ E. x e. ZZ A. y e. S y <_ x ) ) )
29	21, 28	mpbird 247	. . . . 5 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) -> ( R =/= (/) /\ E. x e. ZZ A. y e. R y <_ x ) )
30		suprzcl2 11778	. . . . . 6 |- ( ( R C_ ZZ /\ R =/= (/) /\ E. x e. ZZ A. y e. R y <_ x ) -> sup ( R , RR , < ) e. R )
31	3, 30	mp3an1 1411	. . . . 5 |- ( ( R =/= (/) /\ E. x e. ZZ A. y e. R y <_ x ) -> sup ( R , RR , < ) e. R )
32	29, 31	syl 17	. . . 4 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) -> sup ( R , RR , < ) e. R )
33	3, 32	sseldi 3601	. . 3 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) -> sup ( R , RR , < ) e. ZZ )
34	3	a1i 11	. . . 4 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) -> R C_ ZZ )
35	29	simprd 479	. . . 4 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) -> E. x e. ZZ A. y e. R y <_ x )
36		1dvds 14996	. . . . . . 7 |- ( M e. ZZ -> 1 || M )
37		1dvds 14996	. . . . . . 7 |- ( N e. ZZ -> 1 || N )
38	36, 37	anim12i 590	. . . . . 6 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( 1 || M /\ 1 || N ) )
39		1z 11407	. . . . . . 7 |- 1 e. ZZ
40		breq1 4656	. . . . . . . . 9 |- ( z = 1 -> ( z || M <-> 1 || M ) )
41		breq1 4656	. . . . . . . . 9 |- ( z = 1 -> ( z || N <-> 1 || N ) )
42	40, 41	anbi12d 747	. . . . . . . 8 |- ( z = 1 -> ( ( z || M /\ z || N ) <-> ( 1 || M /\ 1 || N ) ) )
43	42, 1	elrab2 3366	. . . . . . 7 |- ( 1 e. R <-> ( 1 e. ZZ /\ ( 1 || M /\ 1 || N ) ) )
44	39, 43	mpbiran 953	. . . . . 6 |- ( 1 e. R <-> ( 1 || M /\ 1 || N ) )
45	38, 44	sylibr 224	. . . . 5 |- ( ( M e. ZZ /\ N e. ZZ ) -> 1 e. R )
46	45	adantr 481	. . . 4 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) -> 1 e. R )
47		suprzub 11779	. . . 4 |- ( ( R C_ ZZ /\ E. x e. ZZ A. y e. R y <_ x /\ 1 e. R ) -> 1 <_ sup ( R , RR , < ) )
48	34, 35, 46, 47	syl3anc 1326	. . 3 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) -> 1 <_ sup ( R , RR , < ) )
49		elnnz1 11403	. . 3 |- ( sup ( R , RR , < ) e. NN <-> ( sup ( R , RR , < ) e. ZZ /\ 1 <_ sup ( R , RR , < ) ) )
50	33, 48, 49	sylanbrc 698	. 2 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) -> sup ( R , RR , < ) e. NN )
51		breq1 4656	. . . . . 6 |- ( x = sup ( R , RR , < ) -> ( x || M <-> sup ( R , RR , < ) || M ) )
52		breq1 4656	. . . . . 6 |- ( x = sup ( R , RR , < ) -> ( x || N <-> sup ( R , RR , < ) || N ) )
53	51, 52	anbi12d 747	. . . . 5 |- ( x = sup ( R , RR , < ) -> ( ( x || M /\ x || N ) <-> ( sup ( R , RR , < ) || M /\ sup ( R , RR , < ) || N ) ) )
54		breq1 4656	. . . . . . . 8 |- ( z = x -> ( z || M <-> x || M ) )
55		breq1 4656	. . . . . . . 8 |- ( z = x -> ( z || N <-> x || N ) )
56	54, 55	anbi12d 747	. . . . . . 7 |- ( z = x -> ( ( z || M /\ z || N ) <-> ( x || M /\ x || N ) ) )
57	56	cbvrabv 3199	. . . . . 6 |- { z e. ZZ | ( z || M /\ z || N ) } = { x e. ZZ | ( x || M /\ x || N ) }
58	1, 57	eqtri 2644	. . . . 5 |- R = { x e. ZZ | ( x || M /\ x || N ) }
59	53, 58	elrab2 3366	. . . 4 |- ( sup ( R , RR , < ) e. R <-> ( sup ( R , RR , < ) e. ZZ /\ ( sup ( R , RR , < ) || M /\ sup ( R , RR , < ) || N ) ) )
60	32, 59	sylib 208	. . 3 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) -> ( sup ( R , RR , < ) e. ZZ /\ ( sup ( R , RR , < ) || M /\ sup ( R , RR , < ) || N ) ) )
61	60	simprd 479	. 2 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) -> ( sup ( R , RR , < ) || M /\ sup ( R , RR , < ) || N ) )
62		breq1 4656	. . . . . . 7 |- ( z = K -> ( z || M <-> K || M ) )
63		breq1 4656	. . . . . . 7 |- ( z = K -> ( z || N <-> K || N ) )
64	62, 63	anbi12d 747	. . . . . 6 |- ( z = K -> ( ( z || M /\ z || N ) <-> ( K || M /\ K || N ) ) )
65	64, 1	elrab2 3366	. . . . 5 |- ( K e. R <-> ( K e. ZZ /\ ( K || M /\ K || N ) ) )
66	65	biimpri 218	. . . 4 |- ( ( K e. ZZ /\ ( K || M /\ K || N ) ) -> K e. R )
67	66	3impb 1260	. . 3 |- ( ( K e. ZZ /\ K || M /\ K || N ) -> K e. R )
68		suprzub 11779	. . . . 5 |- ( ( R C_ ZZ /\ E. x e. ZZ A. y e. R y <_ x /\ K e. R ) -> K <_ sup ( R , RR , < ) )
69	68	3expia 1267	. . . 4 |- ( ( R C_ ZZ /\ E. x e. ZZ A. y e. R y <_ x ) -> ( K e. R -> K <_ sup ( R , RR , < ) ) )
70	3, 69	mpan 706	. . 3 |- ( E. x e. ZZ A. y e. R y <_ x -> ( K e. R -> K <_ sup ( R , RR , < ) ) )
71	35, 67, 70	syl2im 40	. 2 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) -> ( ( K e. ZZ /\ K || M /\ K || N ) -> K <_ sup ( R , RR , < ) ) )
72	50, 61, 71	3jca 1242	1 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) -> ( sup ( R , RR , < ) e. NN /\ ( sup ( R , RR , < ) || M /\ sup ( R , RR , < ) || N ) /\ ( ( K e. ZZ /\ K || M /\ K || N ) -> K <_ sup ( R , RR , < ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   {crab 2916   C_ wss 3574   (/)c0 3915   {cpr 4179   class class class wbr 4653   supcsup 8346   RRcr 9935   0cc0 9936   1c1 9937   < clt 10074   <_ cle 10075   NNcn 11020   ZZcz 11377   || cdvds 14983

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-dvds 14984

This theorem is referenced by:  gcdn0cl  15224  gcddvds  15225  dvdslegcd  15226