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Mirrors > Home > ILE Home > Th. List > sumdc | Unicode version |
Description: Decidability of a subset of upper integers. (Contributed by Jim Kingdon, 1-Jan-2022.) |
Ref | Expression |
---|---|
sumdc.m | |
sumdc.ss | |
sumdc.dc | DECID |
sumdc.n |
Ref | Expression |
---|---|
sumdc | DECID |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sumdc.dc | . . 3 DECID | |
2 | eleq1 2200 | . . . . 5 | |
3 | 2 | dcbid 823 | . . . 4 DECID DECID |
4 | 3 | rspcv 2780 | . . 3 DECID DECID |
5 | 1, 4 | mpan9 279 | . 2 DECID |
6 | sumdc.ss | . . . . . 6 | |
7 | 6 | ssneld 3094 | . . . . 5 |
8 | 7 | imp 123 | . . . 4 |
9 | 8 | olcd 723 | . . 3 |
10 | df-dc 820 | . . 3 DECID | |
11 | 9, 10 | sylibr 133 | . 2 DECID |
12 | sumdc.m | . . . 4 | |
13 | sumdc.n | . . . 4 | |
14 | eluzdc 9397 | . . . 4 DECID | |
15 | 12, 13, 14 | syl2anc 408 | . . 3 DECID |
16 | exmiddc 821 | . . 3 DECID | |
17 | 15, 16 | syl 14 | . 2 |
18 | 5, 11, 17 | mpjaodan 787 | 1 DECID |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wo 697 DECID wdc 819 wceq 1331 wcel 1480 wral 2414 wss 3066 cfv 5118 cz 9047 cuz 9319 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-cnex 7704 ax-resscn 7705 ax-1cn 7706 ax-1re 7707 ax-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-addcom 7713 ax-addass 7715 ax-distr 7717 ax-i2m1 7718 ax-0lt1 7719 ax-0id 7721 ax-rnegex 7722 ax-cnre 7724 ax-pre-ltirr 7725 ax-pre-ltwlin 7726 ax-pre-lttrn 7727 ax-pre-ltadd 7729 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-reu 2421 df-rab 2423 df-v 2683 df-sbc 2905 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-br 3925 df-opab 3985 df-mpt 3986 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-iota 5083 df-fun 5120 df-fv 5126 df-riota 5723 df-ov 5770 df-oprab 5771 df-mpo 5772 df-pnf 7795 df-mnf 7796 df-xr 7797 df-ltxr 7798 df-le 7799 df-sub 7928 df-neg 7929 df-inn 8714 df-n0 8971 df-z 9048 df-uz 9320 |
This theorem is referenced by: sumeq2 11121 prodeq2 11319 |
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