Theorem gcdsupcl 12394

Description: Closure of the supremum used in defining gcd. A lemma for gcdval 12395 and gcdn0cl 12398. (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 9434	. . 3 |- ( ( ( X e. ZZ /\ Y e. ZZ ) /\ -. ( X = 0 /\ Y = 0 ) ) -> 1 e. ZZ )
2		breq1 4062	. . . 4 |- ( n = 1 -> ( n || X <-> 1 || X ) )
3		breq1 4062	. . . 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 12231	. . . . 5 |- ( X e. ZZ -> 1 || X )
6		1dvds 12231	. . . . 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 9720	. . . . . . 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 12224	. . . . . 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 12224	. . . . . 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	13, 16	dcand 935	. . . 4 |- ( ( ( X e. ZZ /\ Y e. ZZ ) /\ n e. ( ZZ>= ` 1 ) ) -> DECID ( n || X /\ n || Y ) )
18	17	adantlr 477	. . 3 |- ( ( ( ( X e. ZZ /\ Y e. ZZ ) /\ -. ( X = 0 /\ Y = 0 ) ) /\ n e. ( ZZ>= ` 1 ) ) -> DECID ( n || X /\ n || Y ) )
19		dvdsbnd 12392	. . . . . . 7 |- ( ( X e. ZZ /\ X =/= 0 ) -> E. j e. NN A. n e. ( ZZ>= ` j ) -. n || X )
20		nnuz 9719	. . . . . . . 8 |- NN = ( ZZ>= ` 1 )
21	20	rexeqi 2710	. . . . . . 7 |- ( E. j e. NN A. n e. ( ZZ>= ` j ) -. n || X <-> E. j e. ( ZZ>= ` 1 ) A. n e. ( ZZ>= ` j ) -. n || X )
22	19, 21	sylib 122	. . . . . 6 |- ( ( X e. ZZ /\ X =/= 0 ) -> E. j e. ( ZZ>= ` 1 ) A. n e. ( ZZ>= ` j ) -. n || X )
23		id 19	. . . . . . . . 9 |- ( -. n || X -> -. n || X )
24	23	intnanrd 934	. . . . . . . 8 |- ( -. n || X -> -. ( n || X /\ n || Y ) )
25	24	ralimi 2571	. . . . . . 7 |- ( A. n e. ( ZZ>= ` j ) -. n || X -> A. n e. ( ZZ>= ` j ) -. ( n || X /\ n || Y ) )
26	25	reximi 2605	. . . . . 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 ) )
27	22, 26	syl 14	. . . . 5 |- ( ( X e. ZZ /\ X =/= 0 ) -> E. j e. ( ZZ>= ` 1 ) A. n e. ( ZZ>= ` j ) -. ( n || X /\ n || Y ) )
28	27	ad4ant14 514	. . . 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 ) )
29		dvdsbnd 12392	. . . . . . 7 |- ( ( Y e. ZZ /\ Y =/= 0 ) -> E. j e. NN A. n e. ( ZZ>= ` j ) -. n || Y )
30	20	rexeqi 2710	. . . . . . 7 |- ( E. j e. NN A. n e. ( ZZ>= ` j ) -. n || Y <-> E. j e. ( ZZ>= ` 1 ) A. n e. ( ZZ>= ` j ) -. n || Y )
31	29, 30	sylib 122	. . . . . 6 |- ( ( Y e. ZZ /\ Y =/= 0 ) -> E. j e. ( ZZ>= ` 1 ) A. n e. ( ZZ>= ` j ) -. n || Y )
32		id 19	. . . . . . . . 9 |- ( -. n || Y -> -. n || Y )
33	32	intnand 933	. . . . . . . 8 |- ( -. n || Y -> -. ( n || X /\ n || Y ) )
34	33	ralimi 2571	. . . . . . 7 |- ( A. n e. ( ZZ>= ` j ) -. n || Y -> A. n e. ( ZZ>= ` j ) -. ( n || X /\ n || Y ) )
35	34	reximi 2605	. . . . . 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 ) )
36	31, 35	syl 14	. . . . 5 |- ( ( Y e. ZZ /\ Y =/= 0 ) -> E. j e. ( ZZ>= ` 1 ) A. n e. ( ZZ>= ` j ) -. ( n || X /\ n || Y ) )
37	36	ad4ant24 516	. . . 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 ) )
38		simpr 110	. . . . . 6 |- ( ( ( X e. ZZ /\ Y e. ZZ ) /\ -. ( X = 0 /\ Y = 0 ) ) -> -. ( X = 0 /\ Y = 0 ) )
39		simpll 527	. . . . . . . 8 |- ( ( ( X e. ZZ /\ Y e. ZZ ) /\ -. ( X = 0 /\ Y = 0 ) ) -> X e. ZZ )
40		0z 9418	. . . . . . . 8 |- 0 e. ZZ
41		zdceq 9483	. . . . . . . 8 |- ( ( X e. ZZ /\ 0 e. ZZ ) -> DECID X = 0 )
42	39, 40, 41	sylancl 413	. . . . . . 7 |- ( ( ( X e. ZZ /\ Y e. ZZ ) /\ -. ( X = 0 /\ Y = 0 ) ) -> DECID X = 0 )
43		ianordc 901	. . . . . . 7 |- (DECID X = 0 -> ( -. ( X = 0 /\ Y = 0 ) <-> ( -. X = 0 \/ -. Y = 0 ) ) )
44	42, 43	syl 14	. . . . . 6 |- ( ( ( X e. ZZ /\ Y e. ZZ ) /\ -. ( X = 0 /\ Y = 0 ) ) -> ( -. ( X = 0 /\ Y = 0 ) <-> ( -. X = 0 \/ -. Y = 0 ) ) )
45	38, 44	mpbid 147	. . . . 5 |- ( ( ( X e. ZZ /\ Y e. ZZ ) /\ -. ( X = 0 /\ Y = 0 ) ) -> ( -. X = 0 \/ -. Y = 0 ) )
46		df-ne 2379	. . . . . 6 |- ( X =/= 0 <-> -. X = 0 )
47		df-ne 2379	. . . . . 6 |- ( Y =/= 0 <-> -. Y = 0 )
48	46, 47	orbi12i 766	. . . . 5 |- ( ( X =/= 0 \/ Y =/= 0 ) <-> ( -. X = 0 \/ -. Y = 0 ) )
49	45, 48	sylibr 134	. . . 4 |- ( ( ( X e. ZZ /\ Y e. ZZ ) /\ -. ( X = 0 /\ Y = 0 ) ) -> ( X =/= 0 \/ Y =/= 0 ) )
50	28, 37, 49	mpjaodan 800	. . 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 ) )
51	1, 4, 8, 18, 50	zsupcl 10411	. 2 |- ( ( ( X e. ZZ /\ Y e. ZZ ) /\ -. ( X = 0 /\ Y = 0 ) ) -> sup ( { n e. ZZ | ( n || X /\ n || Y ) } , RR , < ) e. ( ZZ>= ` 1 ) )
52	51, 20	eleqtrrdi 2301	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 710  DECID wdc 836   = wceq 1373   e. wcel 2178   =/= wne 2378   A.wral 2486   E.wrex 2487   {crab 2490   class class class wbr 4059   ` cfv 5290   supcsup 7110   RRcr 7959   0cc0 7960   1c1 7961   < clt 8142   NNcn 9071   ZZcz 9407   ZZ>=cuz 9683   || cdvds 12213

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-iinf 4654  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-mulrcl 8059  ax-addcom 8060  ax-mulcom 8061  ax-addass 8062  ax-mulass 8063  ax-distr 8064  ax-i2m1 8065  ax-0lt1 8066  ax-1rid 8067  ax-0id 8068  ax-rnegex 8069  ax-precex 8070  ax-cnre 8071  ax-pre-ltirr 8072  ax-pre-ltwlin 8073  ax-pre-lttrn 8074  ax-pre-apti 8075  ax-pre-ltadd 8076  ax-pre-mulgt0 8077  ax-pre-mulext 8078  ax-arch 8079  ax-caucvg 8080

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rmo 2494  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-if 3580  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-tr 4159  df-id 4358  df-po 4361  df-iso 4362  df-iord 4431  df-on 4433  df-ilim 4434  df-suc 4436  df-iom 4657  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-recs 6414  df-frec 6500  df-sup 7112  df-pnf 8144  df-mnf 8145  df-xr 8146  df-ltxr 8147  df-le 8148  df-sub 8280  df-neg 8281  df-reap 8683  df-ap 8690  df-div 8781  df-inn 9072  df-2 9130  df-3 9131  df-4 9132  df-n0 9331  df-z 9408  df-uz 9684  df-q 9776  df-rp 9811  df-fz 10166  df-fzo 10300  df-fl 10450  df-mod 10505  df-seqfrec 10630  df-exp 10721  df-cj 11268  df-re 11269  df-im 11270  df-rsqrt 11424  df-abs 11425  df-dvds 12214

This theorem is referenced by:  gcdval  12395  gcdn0cl  12398