Theorem gcdsupcl 11906

Description: Closure of the supremum used in defining gcd. A lemma for gcdval 11907 and gcdn0cl 11910. (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 9232	. . 3 |- ( ( ( X e. ZZ /\ Y e. ZZ ) /\ -. ( X = 0 /\ Y = 0 ) ) -> 1 e. ZZ )
2		breq1 3990	. . . 4 |- ( n = 1 -> ( n || X <-> 1 || X ) )
3		breq1 3990	. . . 4 |- ( n = 1 -> ( n || Y <-> 1 || Y ) )
4	2, 3	anbi12d 470	. . 3 |- ( n = 1 -> ( ( n || X /\ n || Y ) <-> ( 1 || X /\ 1 || Y ) ) )
5		1dvds 11760	. . . . 5 |- ( X e. ZZ -> 1 || X )
6		1dvds 11760	. . . . 5 |- ( Y e. ZZ -> 1 || Y )
7	5, 6	anim12i 336	. . . 4 |- ( ( X e. ZZ /\ Y e. ZZ ) -> ( 1 || X /\ 1 || Y ) )
8	7	adantr 274	. . 3 |- ( ( ( X e. ZZ /\ Y e. ZZ ) /\ -. ( X = 0 /\ Y = 0 ) ) -> ( 1 || X /\ 1 || Y ) )
9		elnnuz 9516	. . . . . . 7 |- ( n e. NN <-> n e. ( ZZ>= ` 1 ) )
10	9	biimpri 132	. . . . . 6 |- ( n e. ( ZZ>= ` 1 ) -> n e. NN )
11		simpll 524	. . . . . 6 |- ( ( ( X e. ZZ /\ Y e. ZZ ) /\ n e. ( ZZ>= ` 1 ) ) -> X e. ZZ )
12		dvdsdc 11753	. . . . . 6 |- ( ( n e. NN /\ X e. ZZ ) -> DECID n || X )
13	10, 11, 12	syl2an2 589	. . . . 5 |- ( ( ( X e. ZZ /\ Y e. ZZ ) /\ n e. ( ZZ>= ` 1 ) ) -> DECID n || X )
14		simplr 525	. . . . . 6 |- ( ( ( X e. ZZ /\ Y e. ZZ ) /\ n e. ( ZZ>= ` 1 ) ) -> Y e. ZZ )
15		dvdsdc 11753	. . . . . 6 |- ( ( n e. NN /\ Y e. ZZ ) -> DECID n || Y )
16	10, 14, 15	syl2an2 589	. . . . 5 |- ( ( ( X e. ZZ /\ Y e. ZZ ) /\ n e. ( ZZ>= ` 1 ) ) -> DECID n || Y )
17		dcan2 929	. . . . 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 474	. . 3 |- ( ( ( ( X e. ZZ /\ Y e. ZZ ) /\ -. ( X = 0 /\ Y = 0 ) ) /\ n e. ( ZZ>= ` 1 ) ) -> DECID ( n || X /\ n || Y ) )
20		simplll 528	. . . . 5 |- ( ( ( ( X e. ZZ /\ Y e. ZZ ) /\ -. ( X = 0 /\ Y = 0 ) ) /\ X =/= 0 ) -> X e. ZZ )
21		dvdsbnd 11904	. . . . . . 7 |- ( ( X e. ZZ /\ X =/= 0 ) -> E. j e. NN A. n e. ( ZZ>= ` j ) -. n || X )
22		nnuz 9515	. . . . . . . 8 |- NN = ( ZZ>= ` 1 )
23	22	rexeqi 2670	. . . . . . 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 121	. . . . . 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 927	. . . . . . . 8 |- ( -. n || X -> -. ( n || X /\ n || Y ) )
27	26	ralimi 2533	. . . . . . 7 |- ( A. n e. ( ZZ>= ` j ) -. n || X -> A. n e. ( ZZ>= ` j ) -. ( n || X /\ n || Y ) )
28	27	reximi 2567	. . . . . 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 418	. . . 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 529	. . . . 5 |- ( ( ( ( X e. ZZ /\ Y e. ZZ ) /\ -. ( X = 0 /\ Y = 0 ) ) /\ Y =/= 0 ) -> Y e. ZZ )
32		dvdsbnd 11904	. . . . . . 7 |- ( ( Y e. ZZ /\ Y =/= 0 ) -> E. j e. NN A. n e. ( ZZ>= ` j ) -. n || Y )
33	22	rexeqi 2670	. . . . . . 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 121	. . . . . 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 926	. . . . . . . 8 |- ( -. n || Y -> -. ( n || X /\ n || Y ) )
37	36	ralimi 2533	. . . . . . 7 |- ( A. n e. ( ZZ>= ` j ) -. n || Y -> A. n e. ( ZZ>= ` j ) -. ( n || X /\ n || Y ) )
38	37	reximi 2567	. . . . . 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 418	. . . 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 109	. . . . . 6 |- ( ( ( X e. ZZ /\ Y e. ZZ ) /\ -. ( X = 0 /\ Y = 0 ) ) -> -. ( X = 0 /\ Y = 0 ) )
42		simpll 524	. . . . . . . 8 |- ( ( ( X e. ZZ /\ Y e. ZZ ) /\ -. ( X = 0 /\ Y = 0 ) ) -> X e. ZZ )
43		0z 9216	. . . . . . . 8 |- 0 e. ZZ
44		zdceq 9280	. . . . . . . 8 |- ( ( X e. ZZ /\ 0 e. ZZ ) -> DECID X = 0 )
45	42, 43, 44	sylancl 411	. . . . . . 7 |- ( ( ( X e. ZZ /\ Y e. ZZ ) /\ -. ( X = 0 /\ Y = 0 ) ) -> DECID X = 0 )
46		ianordc 894	. . . . . . 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 146	. . . . 5 |- ( ( ( X e. ZZ /\ Y e. ZZ ) /\ -. ( X = 0 /\ Y = 0 ) ) -> ( -. X = 0 \/ -. Y = 0 ) )
49		df-ne 2341	. . . . . 6 |- ( X =/= 0 <-> -. X = 0 )
50		df-ne 2341	. . . . . 6 |- ( Y =/= 0 <-> -. Y = 0 )
51	49, 50	orbi12i 759	. . . . 5 |- ( ( X =/= 0 \/ Y =/= 0 ) <-> ( -. X = 0 \/ -. Y = 0 ) )
52	48, 51	sylibr 133	. . . 4 |- ( ( ( X e. ZZ /\ Y e. ZZ ) /\ -. ( X = 0 /\ Y = 0 ) ) -> ( X =/= 0 \/ Y =/= 0 ) )
53	30, 40, 52	mpjaodan 793	. . 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 11895	. 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 2264	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 103   <-> wb 104   \/ wo 703  DECID wdc 829   = wceq 1348   e. wcel 2141   =/= wne 2340   A.wral 2448   E.wrex 2449   {crab 2452   class class class wbr 3987   ` cfv 5196   supcsup 6957   RRcr 7766   0cc0 7767   1c1 7768   < clt 7947   NNcn 8871   ZZcz 9205   ZZ>=cuz 9480   || cdvds 11742

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4102  ax-sep 4105  ax-nul 4113  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-iinf 4570  ax-cnex 7858  ax-resscn 7859  ax-1cn 7860  ax-1re 7861  ax-icn 7862  ax-addcl 7863  ax-addrcl 7864  ax-mulcl 7865  ax-mulrcl 7866  ax-addcom 7867  ax-mulcom 7868  ax-addass 7869  ax-mulass 7870  ax-distr 7871  ax-i2m1 7872  ax-0lt1 7873  ax-1rid 7874  ax-0id 7875  ax-rnegex 7876  ax-precex 7877  ax-cnre 7878  ax-pre-ltirr 7879  ax-pre-ltwlin 7880  ax-pre-lttrn 7881  ax-pre-apti 7882  ax-pre-ltadd 7883  ax-pre-mulgt0 7884  ax-pre-mulext 7885  ax-arch 7886  ax-caucvg 7887

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3526  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-int 3830  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-tr 4086  df-id 4276  df-po 4279  df-iso 4280  df-iord 4349  df-on 4351  df-ilim 4352  df-suc 4354  df-iom 4573  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-riota 5807  df-ov 5854  df-oprab 5855  df-mpo 5856  df-1st 6117  df-2nd 6118  df-recs 6282  df-frec 6368  df-sup 6959  df-pnf 7949  df-mnf 7950  df-xr 7951  df-ltxr 7952  df-le 7953  df-sub 8085  df-neg 8086  df-reap 8487  df-ap 8494  df-div 8583  df-inn 8872  df-2 8930  df-3 8931  df-4 8932  df-n0 9129  df-z 9206  df-uz 9481  df-q 9572  df-rp 9604  df-fz 9959  df-fzo 10092  df-fl 10219  df-mod 10272  df-seqfrec 10395  df-exp 10469  df-cj 10799  df-re 10800  df-im 10801  df-rsqrt 10955  df-abs 10956  df-dvds 11743

This theorem is referenced by:  gcdval  11907  gcdn0cl  11910