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Mirrors > Home > ILE Home > Th. List > caucvgprlemk | Unicode version |
Description: Lemma for caucvgpr 7458. Reciprocals of positive integers decrease as the positive integers increase. (Contributed by Jim Kingdon, 9-Oct-2020.) |
Ref | Expression |
---|---|
caucvgprlemk.jk | |
caucvgprlemk.jkq |
Ref | Expression |
---|---|
caucvgprlemk |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | caucvgprlemk.jk | . . . 4 | |
2 | ltrelpi 7100 | . . . . . . 7 | |
3 | 2 | brel 4561 | . . . . . 6 |
4 | 1, 3 | syl 14 | . . . . 5 |
5 | ltnnnq 7199 | . . . . 5 | |
6 | 4, 5 | syl 14 | . . . 4 |
7 | 1, 6 | mpbid 146 | . . 3 |
8 | ltrnqi 7197 | . . 3 | |
9 | 7, 8 | syl 14 | . 2 |
10 | caucvgprlemk.jkq | . 2 | |
11 | ltsonq 7174 | . . 3 | |
12 | ltrelnq 7141 | . . 3 | |
13 | 11, 12 | sotri 4904 | . 2 |
14 | 9, 10, 13 | syl2anc 408 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wcel 1465 cop 3500 class class class wbr 3899 cfv 5093 c1o 6274 cec 6395 cnpi 7048 clti 7051 ceq 7055 cnq 7056 crq 7060 cltq 7061 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-coll 4013 ax-sep 4016 ax-nul 4024 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-iinf 4472 |
This theorem depends on definitions: df-bi 116 df-dc 805 df-3or 948 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-ral 2398 df-rex 2399 df-reu 2400 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-iun 3785 df-br 3900 df-opab 3960 df-mpt 3961 df-tr 3997 df-eprel 4181 df-id 4185 df-po 4188 df-iso 4189 df-iord 4258 df-on 4260 df-suc 4263 df-iom 4475 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 df-fv 5101 df-ov 5745 df-oprab 5746 df-mpo 5747 df-1st 6006 df-2nd 6007 df-recs 6170 df-irdg 6235 df-1o 6281 df-oadd 6285 df-omul 6286 df-er 6397 df-ec 6399 df-qs 6403 df-ni 7080 df-mi 7082 df-lti 7083 df-mpq 7121 df-enq 7123 df-nqqs 7124 df-mqqs 7126 df-1nqqs 7127 df-rq 7128 df-ltnqqs 7129 |
This theorem is referenced by: caucvgprlem1 7455 caucvgprlem2 7456 |
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