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Mirrors > Home > ILE Home > Th. List > gcdsupcl | Unicode version |
Description: Closure of the supremum used in defining . A lemma for gcdval 11907 and gcdn0cl 11910. (Contributed by Jim Kingdon, 11-Dec-2021.) |
Ref | Expression |
---|---|
gcdsupcl |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1zzd 9232 | . . 3 | |
2 | breq1 3990 | . . . 4 | |
3 | breq1 3990 | . . . 4 | |
4 | 2, 3 | anbi12d 470 | . . 3 |
5 | 1dvds 11760 | . . . . 5 | |
6 | 1dvds 11760 | . . . . 5 | |
7 | 5, 6 | anim12i 336 | . . . 4 |
8 | 7 | adantr 274 | . . 3 |
9 | elnnuz 9516 | . . . . . . 7 | |
10 | 9 | biimpri 132 | . . . . . 6 |
11 | simpll 524 | . . . . . 6 | |
12 | dvdsdc 11753 | . . . . . 6 DECID | |
13 | 10, 11, 12 | syl2an2 589 | . . . . 5 DECID |
14 | simplr 525 | . . . . . 6 | |
15 | dvdsdc 11753 | . . . . . 6 DECID | |
16 | 10, 14, 15 | syl2an2 589 | . . . . 5 DECID |
17 | dcan2 929 | . . . . 5 DECID DECID DECID | |
18 | 13, 16, 17 | sylc 62 | . . . 4 DECID |
19 | 18 | adantlr 474 | . . 3 DECID |
20 | simplll 528 | . . . . 5 | |
21 | dvdsbnd 11904 | . . . . . . 7 | |
22 | nnuz 9515 | . . . . . . . 8 | |
23 | 22 | rexeqi 2670 | . . . . . . 7 |
24 | 21, 23 | sylib 121 | . . . . . 6 |
25 | id 19 | . . . . . . . . 9 | |
26 | 25 | intnanrd 927 | . . . . . . . 8 |
27 | 26 | ralimi 2533 | . . . . . . 7 |
28 | 27 | reximi 2567 | . . . . . 6 |
29 | 24, 28 | syl 14 | . . . . 5 |
30 | 20, 29 | sylancom 418 | . . . 4 |
31 | simpllr 529 | . . . . 5 | |
32 | dvdsbnd 11904 | . . . . . . 7 | |
33 | 22 | rexeqi 2670 | . . . . . . 7 |
34 | 32, 33 | sylib 121 | . . . . . 6 |
35 | id 19 | . . . . . . . . 9 | |
36 | 35 | intnand 926 | . . . . . . . 8 |
37 | 36 | ralimi 2533 | . . . . . . 7 |
38 | 37 | reximi 2567 | . . . . . 6 |
39 | 34, 38 | syl 14 | . . . . 5 |
40 | 31, 39 | sylancom 418 | . . . 4 |
41 | simpr 109 | . . . . . 6 | |
42 | simpll 524 | . . . . . . . 8 | |
43 | 0z 9216 | . . . . . . . 8 | |
44 | zdceq 9280 | . . . . . . . 8 DECID | |
45 | 42, 43, 44 | sylancl 411 | . . . . . . 7 DECID |
46 | ianordc 894 | . . . . . . 7 DECID | |
47 | 45, 46 | syl 14 | . . . . . 6 |
48 | 41, 47 | mpbid 146 | . . . . 5 |
49 | df-ne 2341 | . . . . . 6 | |
50 | df-ne 2341 | . . . . . 6 | |
51 | 49, 50 | orbi12i 759 | . . . . 5 |
52 | 48, 51 | sylibr 133 | . . . 4 |
53 | 30, 40, 52 | mpjaodan 793 | . . 3 |
54 | 1, 4, 8, 19, 53 | zsupcl 11895 | . 2 |
55 | 54, 22 | eleqtrrdi 2264 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wo 703 DECID wdc 829 wceq 1348 wcel 2141 wne 2340 wral 2448 wrex 2449 crab 2452 class class class wbr 3987 cfv 5196 csup 6957 cr 7766 cc0 7767 c1 7768 clt 7947 cn 8871 cz 9205 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 |
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