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Mirrors > Home > ILE Home > Th. List > caucvgprlemk | GIF version |
Description: Lemma for caucvgpr 7490. Reciprocals of positive integers decrease as the positive integers increase. (Contributed by Jim Kingdon, 9-Oct-2020.) |
Ref | Expression |
---|---|
caucvgprlemk.jk | ⊢ (𝜑 → 𝐽 <N 𝐾) |
caucvgprlemk.jkq | ⊢ (𝜑 → (*Q‘[〈𝐽, 1o〉] ~Q ) <Q 𝑄) |
Ref | Expression |
---|---|
caucvgprlemk | ⊢ (𝜑 → (*Q‘[〈𝐾, 1o〉] ~Q ) <Q 𝑄) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | caucvgprlemk.jk | . . . 4 ⊢ (𝜑 → 𝐽 <N 𝐾) | |
2 | ltrelpi 7132 | . . . . . . 7 ⊢ <N ⊆ (N × N) | |
3 | 2 | brel 4591 | . . . . . 6 ⊢ (𝐽 <N 𝐾 → (𝐽 ∈ N ∧ 𝐾 ∈ N)) |
4 | 1, 3 | syl 14 | . . . . 5 ⊢ (𝜑 → (𝐽 ∈ N ∧ 𝐾 ∈ N)) |
5 | ltnnnq 7231 | . . . . 5 ⊢ ((𝐽 ∈ N ∧ 𝐾 ∈ N) → (𝐽 <N 𝐾 ↔ [〈𝐽, 1o〉] ~Q <Q [〈𝐾, 1o〉] ~Q )) | |
6 | 4, 5 | syl 14 | . . . 4 ⊢ (𝜑 → (𝐽 <N 𝐾 ↔ [〈𝐽, 1o〉] ~Q <Q [〈𝐾, 1o〉] ~Q )) |
7 | 1, 6 | mpbid 146 | . . 3 ⊢ (𝜑 → [〈𝐽, 1o〉] ~Q <Q [〈𝐾, 1o〉] ~Q ) |
8 | ltrnqi 7229 | . . 3 ⊢ ([〈𝐽, 1o〉] ~Q <Q [〈𝐾, 1o〉] ~Q → (*Q‘[〈𝐾, 1o〉] ~Q ) <Q (*Q‘[〈𝐽, 1o〉] ~Q )) | |
9 | 7, 8 | syl 14 | . 2 ⊢ (𝜑 → (*Q‘[〈𝐾, 1o〉] ~Q ) <Q (*Q‘[〈𝐽, 1o〉] ~Q )) |
10 | caucvgprlemk.jkq | . 2 ⊢ (𝜑 → (*Q‘[〈𝐽, 1o〉] ~Q ) <Q 𝑄) | |
11 | ltsonq 7206 | . . 3 ⊢ <Q Or Q | |
12 | ltrelnq 7173 | . . 3 ⊢ <Q ⊆ (Q × Q) | |
13 | 11, 12 | sotri 4934 | . 2 ⊢ (((*Q‘[〈𝐾, 1o〉] ~Q ) <Q (*Q‘[〈𝐽, 1o〉] ~Q ) ∧ (*Q‘[〈𝐽, 1o〉] ~Q ) <Q 𝑄) → (*Q‘[〈𝐾, 1o〉] ~Q ) <Q 𝑄) |
14 | 9, 10, 13 | syl2anc 408 | 1 ⊢ (𝜑 → (*Q‘[〈𝐾, 1o〉] ~Q ) <Q 𝑄) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∈ wcel 1480 〈cop 3530 class class class wbr 3929 ‘cfv 5123 1oc1o 6306 [cec 6427 Ncnpi 7080 <N clti 7083 ~Q ceq 7087 Qcnq 7088 *Qcrq 7092 <Q cltq 7093 |
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 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 |
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 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-eprel 4211 df-id 4215 df-po 4218 df-iso 4219 df-iord 4288 df-on 4290 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-irdg 6267 df-1o 6313 df-oadd 6317 df-omul 6318 df-er 6429 df-ec 6431 df-qs 6435 df-ni 7112 df-mi 7114 df-lti 7115 df-mpq 7153 df-enq 7155 df-nqqs 7156 df-mqqs 7158 df-1nqqs 7159 df-rq 7160 df-ltnqqs 7161 |
This theorem is referenced by: caucvgprlem1 7487 caucvgprlem2 7488 |
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