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Mirrors > Home > ILE Home > Th. List > caucvgprlemk | GIF version |
Description: Lemma for caucvgpr 7483. 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 7125 | . . . . . . 7 ⊢ <N ⊆ (N × N) | |
3 | 2 | brel 4586 | . . . . . 6 ⊢ (𝐽 <N 𝐾 → (𝐽 ∈ N ∧ 𝐾 ∈ N)) |
4 | 1, 3 | syl 14 | . . . . 5 ⊢ (𝜑 → (𝐽 ∈ N ∧ 𝐾 ∈ N)) |
5 | ltnnnq 7224 | . . . . 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 7222 | . . 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 7199 | . . 3 ⊢ <Q Or Q | |
12 | ltrelnq 7166 | . . 3 ⊢ <Q ⊆ (Q × Q) | |
13 | 11, 12 | sotri 4929 | . 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 3525 class class class wbr 3924 ‘cfv 5118 1oc1o 6299 [cec 6420 Ncnpi 7073 <N clti 7076 ~Q ceq 7080 Qcnq 7081 *Qcrq 7085 <Q cltq 7086 |
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-coll 4038 ax-sep 4041 ax-nul 4049 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-iinf 4497 |
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-ral 2419 df-rex 2420 df-reu 2421 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-iun 3810 df-br 3925 df-opab 3985 df-mpt 3986 df-tr 4022 df-eprel 4206 df-id 4210 df-po 4213 df-iso 4214 df-iord 4283 df-on 4285 df-suc 4288 df-iom 4500 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 df-ov 5770 df-oprab 5771 df-mpo 5772 df-1st 6031 df-2nd 6032 df-recs 6195 df-irdg 6260 df-1o 6306 df-oadd 6310 df-omul 6311 df-er 6422 df-ec 6424 df-qs 6428 df-ni 7105 df-mi 7107 df-lti 7108 df-mpq 7146 df-enq 7148 df-nqqs 7149 df-mqqs 7151 df-1nqqs 7152 df-rq 7153 df-ltnqqs 7154 |
This theorem is referenced by: caucvgprlem1 7480 caucvgprlem2 7481 |
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