Theorem gcdsupcl 11961

Description: Closure of the supremum used in defining gcd. A lemma for gcdval 11962 and gcdn0cl 11965. (Contributed by Jim Kingdon, 11-Dec-2021.)

Assertion

Ref	Expression
gcdsupcl	|- ( ( ( X e. ZZ /\ Y e. ZZ ) /\ -. ( X = 0 /\ Y = 0 ) ) -> sup ( { n e. ZZ | ( n || X /\ n || Y ) } , RR , < ) e. NN )

Distinct variable groups:   n, X   n, Y

Proof of Theorem gcdsupcl

Dummy variable j is distinct from all other variables.

Step	Hyp	Ref	Expression
1		1zzd 9282	. . 3 |- ( ( ( X e. ZZ /\ Y e. ZZ ) /\ -. ( X = 0 /\ Y = 0 ) ) -> 1 e. ZZ )
2		breq1 4008	. . . 4 |- ( n = 1 -> ( n || X <-> 1 || X ) )
3		breq1 4008	. . . 4 |- ( n = 1 -> ( n || Y <-> 1 || Y ) )
4	2, 3	anbi12d 473	. . 3 |- ( n = 1 -> ( ( n || X /\ n || Y ) <-> ( 1 || X /\ 1 || Y ) ) )
5		1dvds 11814	. . . . 5 |- ( X e. ZZ -> 1 || X )
6		1dvds 11814	. . . . 5 |- ( Y e. ZZ -> 1 || Y )
7	5, 6	anim12i 338	. . . 4 |- ( ( X e. ZZ /\ Y e. ZZ ) -> ( 1 || X /\ 1 || Y ) )
8	7	adantr 276	. . 3 |- ( ( ( X e. ZZ /\ Y e. ZZ ) /\ -. ( X = 0 /\ Y = 0 ) ) -> ( 1 || X /\ 1 || Y ) )
9		elnnuz 9566	. . . . . . 7 |- ( n e. NN <-> n e. ( ZZ>= ` 1 ) )
10	9	biimpri 133	. . . . . 6 |- ( n e. ( ZZ>= ` 1 ) -> n e. NN )
11		simpll 527	. . . . . 6 |- ( ( ( X e. ZZ /\ Y e. ZZ ) /\ n e. ( ZZ>= ` 1 ) ) -> X e. ZZ )
12		dvdsdc 11807	. . . . . 6 |- ( ( n e. NN /\ X e. ZZ ) -> DECID n || X )
13	10, 11, 12	syl2an2 594	. . . . 5 |- ( ( ( X e. ZZ /\ Y e. ZZ ) /\ n e. ( ZZ>= ` 1 ) ) -> DECID n || X )
14		simplr 528	. . . . . 6 |- ( ( ( X e. ZZ /\ Y e. ZZ ) /\ n e. ( ZZ>= ` 1 ) ) -> Y e. ZZ )
15		dvdsdc 11807	. . . . . 6 |- ( ( n e. NN /\ Y e. ZZ ) -> DECID n || Y )
16	10, 14, 15	syl2an2 594	. . . . 5 |- ( ( ( X e. ZZ /\ Y e. ZZ ) /\ n e. ( ZZ>= ` 1 ) ) -> DECID n || Y )
17		dcan2 934	. . . . 5 |- (DECID n || X -> (DECID n || Y -> DECID ( n || X /\ n || Y ) ) )
18	13, 16, 17	sylc 62	. . . 4 |- ( ( ( X e. ZZ /\ Y e. ZZ ) /\ n e. ( ZZ>= ` 1 ) ) -> DECID ( n || X /\ n || Y ) )
19	18	adantlr 477	. . 3 |- ( ( ( ( X e. ZZ /\ Y e. ZZ ) /\ -. ( X = 0 /\ Y = 0 ) ) /\ n e. ( ZZ>= ` 1 ) ) -> DECID ( n || X /\ n || Y ) )
20		simplll 533	. . . . 5 |- ( ( ( ( X e. ZZ /\ Y e. ZZ ) /\ -. ( X = 0 /\ Y = 0 ) ) /\ X =/= 0 ) -> X e. ZZ )
21		dvdsbnd 11959	. . . . . . 7 |- ( ( X e. ZZ /\ X =/= 0 ) -> E. j e. NN A. n e. ( ZZ>= ` j ) -. n || X )
22		nnuz 9565	. . . . . . . 8 |- NN = ( ZZ>= ` 1 )
23	22	rexeqi 2678	. . . . . . 7 |- ( E. j e. NN A. n e. ( ZZ>= ` j ) -. n || X <-> E. j e. ( ZZ>= ` 1 ) A. n e. ( ZZ>= ` j ) -. n || X )
24	21, 23	sylib 122	. . . . . 6 |- ( ( X e. ZZ /\ X =/= 0 ) -> E. j e. ( ZZ>= ` 1 ) A. n e. ( ZZ>= ` j ) -. n || X )
25		id 19	. . . . . . . . 9 |- ( -. n || X -> -. n || X )
26	25	intnanrd 932	. . . . . . . 8 |- ( -. n || X -> -. ( n || X /\ n || Y ) )
27	26	ralimi 2540	. . . . . . 7 |- ( A. n e. ( ZZ>= ` j ) -. n || X -> A. n e. ( ZZ>= ` j ) -. ( n || X /\ n || Y ) )
28	27	reximi 2574	. . . . . 6 |- ( E. j e. ( ZZ>= ` 1 ) A. n e. ( ZZ>= ` j ) -. n || X -> E. j e. ( ZZ>= ` 1 ) A. n e. ( ZZ>= ` j ) -. ( n || X /\ n || Y ) )
29	24, 28	syl 14	. . . . 5 |- ( ( X e. ZZ /\ X =/= 0 ) -> E. j e. ( ZZ>= ` 1 ) A. n e. ( ZZ>= ` j ) -. ( n || X /\ n || Y ) )
30	20, 29	sylancom 420	. . . 4 |- ( ( ( ( X e. ZZ /\ Y e. ZZ ) /\ -. ( X = 0 /\ Y = 0 ) ) /\ X =/= 0 ) -> E. j e. ( ZZ>= ` 1 ) A. n e. ( ZZ>= ` j ) -. ( n || X /\ n || Y ) )
31		simpllr 534	. . . . 5 |- ( ( ( ( X e. ZZ /\ Y e. ZZ ) /\ -. ( X = 0 /\ Y = 0 ) ) /\ Y =/= 0 ) -> Y e. ZZ )
32		dvdsbnd 11959	. . . . . . 7 |- ( ( Y e. ZZ /\ Y =/= 0 ) -> E. j e. NN A. n e. ( ZZ>= ` j ) -. n || Y )
33	22	rexeqi 2678	. . . . . . 7 |- ( E. j e. NN A. n e. ( ZZ>= ` j ) -. n || Y <-> E. j e. ( ZZ>= ` 1 ) A. n e. ( ZZ>= ` j ) -. n || Y )
34	32, 33	sylib 122	. . . . . 6 |- ( ( Y e. ZZ /\ Y =/= 0 ) -> E. j e. ( ZZ>= ` 1 ) A. n e. ( ZZ>= ` j ) -. n || Y )
35		id 19	. . . . . . . . 9 |- ( -. n || Y -> -. n || Y )
36	35	intnand 931	. . . . . . . 8 |- ( -. n || Y -> -. ( n || X /\ n || Y ) )
37	36	ralimi 2540	. . . . . . 7 |- ( A. n e. ( ZZ>= ` j ) -. n || Y -> A. n e. ( ZZ>= ` j ) -. ( n || X /\ n || Y ) )
38	37	reximi 2574	. . . . . 6 |- ( E. j e. ( ZZ>= ` 1 ) A. n e. ( ZZ>= ` j ) -. n || Y -> E. j e. ( ZZ>= ` 1 ) A. n e. ( ZZ>= ` j ) -. ( n || X /\ n || Y ) )
39	34, 38	syl 14	. . . . 5 |- ( ( Y e. ZZ /\ Y =/= 0 ) -> E. j e. ( ZZ>= ` 1 ) A. n e. ( ZZ>= ` j ) -. ( n || X /\ n || Y ) )
40	31, 39	sylancom 420	. . . 4 |- ( ( ( ( X e. ZZ /\ Y e. ZZ ) /\ -. ( X = 0 /\ Y = 0 ) ) /\ Y =/= 0 ) -> E. j e. ( ZZ>= ` 1 ) A. n e. ( ZZ>= ` j ) -. ( n || X /\ n || Y ) )
41		simpr 110	. . . . . 6 |- ( ( ( X e. ZZ /\ Y e. ZZ ) /\ -. ( X = 0 /\ Y = 0 ) ) -> -. ( X = 0 /\ Y = 0 ) )
42		simpll 527	. . . . . . . 8 |- ( ( ( X e. ZZ /\ Y e. ZZ ) /\ -. ( X = 0 /\ Y = 0 ) ) -> X e. ZZ )
43		0z 9266	. . . . . . . 8 |- 0 e. ZZ
44		zdceq 9330	. . . . . . . 8 |- ( ( X e. ZZ /\ 0 e. ZZ ) -> DECID X = 0 )
45	42, 43, 44	sylancl 413	. . . . . . 7 |- ( ( ( X e. ZZ /\ Y e. ZZ ) /\ -. ( X = 0 /\ Y = 0 ) ) -> DECID X = 0 )
46		ianordc 899	. . . . . . 7 |- (DECID X = 0 -> ( -. ( X = 0 /\ Y = 0 ) <-> ( -. X = 0 \/ -. Y = 0 ) ) )
47	45, 46	syl 14	. . . . . 6 |- ( ( ( X e. ZZ /\ Y e. ZZ ) /\ -. ( X = 0 /\ Y = 0 ) ) -> ( -. ( X = 0 /\ Y = 0 ) <-> ( -. X = 0 \/ -. Y = 0 ) ) )
48	41, 47	mpbid 147	. . . . 5 |- ( ( ( X e. ZZ /\ Y e. ZZ ) /\ -. ( X = 0 /\ Y = 0 ) ) -> ( -. X = 0 \/ -. Y = 0 ) )
49		df-ne 2348	. . . . . 6 |- ( X =/= 0 <-> -. X = 0 )
50		df-ne 2348	. . . . . 6 |- ( Y =/= 0 <-> -. Y = 0 )
51	49, 50	orbi12i 764	. . . . 5 |- ( ( X =/= 0 \/ Y =/= 0 ) <-> ( -. X = 0 \/ -. Y = 0 ) )
52	48, 51	sylibr 134	. . . 4 |- ( ( ( X e. ZZ /\ Y e. ZZ ) /\ -. ( X = 0 /\ Y = 0 ) ) -> ( X =/= 0 \/ Y =/= 0 ) )
53	30, 40, 52	mpjaodan 798	. . 3 |- ( ( ( X e. ZZ /\ Y e. ZZ ) /\ -. ( X = 0 /\ Y = 0 ) ) -> E. j e. ( ZZ>= ` 1 ) A. n e. ( ZZ>= ` j ) -. ( n || X /\ n || Y ) )
54	1, 4, 8, 19, 53	zsupcl 11950	. 2 |- ( ( ( X e. ZZ /\ Y e. ZZ ) /\ -. ( X = 0 /\ Y = 0 ) ) -> sup ( { n e. ZZ | ( n || X /\ n || Y ) } , RR , < ) e. ( ZZ>= ` 1 ) )
55	54, 22	eleqtrrdi 2271	1 |- ( ( ( X e. ZZ /\ Y e. ZZ ) /\ -. ( X = 0 /\ Y = 0 ) ) -> sup ( { n e. ZZ | ( n || X /\ n || Y ) } , RR , < ) e. NN )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 708  DECID wdc 834   = wceq 1353   e. wcel 2148   =/= wne 2347   A.wral 2455   E.wrex 2456   {crab 2459   class class class wbr 4005   ` cfv 5218   supcsup 6983   RRcr 7812   0cc0 7813   1c1 7814   < clt 7994   NNcn 8921   ZZcz 9255   ZZ>=cuz 9530   || cdvds 11796

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-mulrcl 7912  ax-addcom 7913  ax-mulcom 7914  ax-addass 7915  ax-mulass 7916  ax-distr 7917  ax-i2m1 7918  ax-0lt1 7919  ax-1rid 7920  ax-0id 7921  ax-rnegex 7922  ax-precex 7923  ax-cnre 7924  ax-pre-ltirr 7925  ax-pre-ltwlin 7926  ax-pre-lttrn 7927  ax-pre-apti 7928  ax-pre-ltadd 7929  ax-pre-mulgt0 7930  ax-pre-mulext 7931  ax-arch 7932  ax-caucvg 7933

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-if 3537  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-po 4298  df-iso 4299  df-iord 4368  df-on 4370  df-ilim 4371  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-recs 6308  df-frec 6394  df-sup 6985  df-pnf 7996  df-mnf 7997  df-xr 7998  df-ltxr 7999  df-le 8000  df-sub 8132  df-neg 8133  df-reap 8534  df-ap 8541  df-div 8632  df-inn 8922  df-2 8980  df-3 8981  df-4 8982  df-n0 9179  df-z 9256  df-uz 9531  df-q 9622  df-rp 9656  df-fz 10011  df-fzo 10145  df-fl 10272  df-mod 10325  df-seqfrec 10448  df-exp 10522  df-cj 10853  df-re 10854  df-im 10855  df-rsqrt 11009  df-abs 11010  df-dvds 11797

This theorem is referenced by:  gcdval  11962  gcdn0cl  11965