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Mirrors > Home > ILE Home > Th. List > caucvgprlemk | GIF version |
Description: Lemma for caucvgpr 6872. 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 6514 | . . . . . . 7 ⊢ <N ⊆ (N × N) | |
3 | 2 | brel 4410 | . . . . . 6 ⊢ (𝐽 <N 𝐾 → (𝐽 ∈ N ∧ 𝐾 ∈ N)) |
4 | 1, 3 | syl 14 | . . . . 5 ⊢ (𝜑 → (𝐽 ∈ N ∧ 𝐾 ∈ N)) |
5 | ltnnnq 6613 | . . . . 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 145 | . . 3 ⊢ (𝜑 → [〈𝐽, 1𝑜〉] ~Q <Q [〈𝐾, 1𝑜〉] ~Q ) |
8 | ltrnqi 6611 | . . 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 6588 | . . 3 ⊢ <Q Or Q | |
12 | ltrelnq 6555 | . . 3 ⊢ <Q ⊆ (Q × Q) | |
13 | 11, 12 | sotri 4740 | . 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 403 | 1 ⊢ (𝜑 → (*Q‘[〈𝐾, 1𝑜〉] ~Q ) <Q 𝑄) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ↔ wb 103 ∈ wcel 1433 〈cop 3401 class class class wbr 3785 ‘cfv 4922 1𝑜c1o 6017 [cec 6127 Ncnpi 6462 <N clti 6465 ~Q ceq 6469 Qcnq 6470 *Qcrq 6474 <Q cltq 6475 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-coll 3893 ax-sep 3896 ax-nul 3904 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-iinf 4329 |
This theorem depends on definitions: df-bi 115 df-dc 776 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-tr 3876 df-eprel 4044 df-id 4048 df-po 4051 df-iso 4052 df-iord 4121 df-on 4123 df-suc 4126 df-iom 4332 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-1st 5787 df-2nd 5788 df-recs 5943 df-irdg 5980 df-1o 6024 df-oadd 6028 df-omul 6029 df-er 6129 df-ec 6131 df-qs 6135 df-ni 6494 df-mi 6496 df-lti 6497 df-mpq 6535 df-enq 6537 df-nqqs 6538 df-mqqs 6540 df-1nqqs 6541 df-rq 6542 df-ltnqqs 6543 |
This theorem is referenced by: caucvgprlem1 6869 caucvgprlem2 6870 |
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