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Theorem caucvgprlemk 6763

Description: Lemma for caucvgpr 6780. Reciprocals of positive integers decrease as the positive integers increase. (Contributed by Jim Kingdon, 9-Oct-2020.)

**Hypotheses**
Ref	Expression
caucvgprlemk.jk	⊢ (𝜑 → 𝐽 <_N 𝐾)
caucvgprlemk.jkq	⊢ (𝜑 → (*_Q‘[⟨𝐽, 1_𝑜⟩] ~_Q ) <_Q 𝑄)

**Assertion**
Ref	Expression
caucvgprlemk	⊢ (𝜑 → (*_Q‘[⟨𝐾, 1_𝑜⟩] ~_Q ) <_Q 𝑄)

**Proof of Theorem caucvgprlemk**
Step	Hyp	Ref	Expression
1		caucvgprlemk.jk	. . . 4 ⊢ (𝜑 → 𝐽 <_N 𝐾)
2		ltrelpi 6422	. . . . . . 7 ⊢ <_N ⊆ (N × N)
3	2	brel 4392	. . . . . 6 ⊢ (𝐽 <_N 𝐾 → (𝐽 ∈ N ∧ 𝐾 ∈ N))
4	1, 3	syl 14	. . . . 5 ⊢ (𝜑 → (𝐽 ∈ N ∧ 𝐾 ∈ N))
5		ltnnnq 6521	. . . . 5 ⊢ ((𝐽 ∈ N ∧ 𝐾 ∈ N) → (𝐽 <_N 𝐾 ↔ [⟨𝐽, 1_𝑜⟩] ~_Q <_Q [⟨𝐾, 1_𝑜⟩] ~_Q ))
6	4, 5	syl 14	. . . 4 ⊢ (𝜑 → (𝐽 <_N 𝐾 ↔ [⟨𝐽, 1_𝑜⟩] ~_Q <_Q [⟨𝐾, 1_𝑜⟩] ~_Q ))
7	1, 6	mpbid 135	. . 3 ⊢ (𝜑 → [⟨𝐽, 1_𝑜⟩] ~_Q <_Q [⟨𝐾, 1_𝑜⟩] ~_Q )
8		ltrnqi 6519	. . 3 ⊢ ([⟨𝐽, 1_𝑜⟩] ~_Q <_Q [⟨𝐾, 1_𝑜⟩] ~_Q → (_Q‘[⟨𝐾, 1_𝑜⟩] ~_Q ) <_Q (_Q‘[⟨𝐽, 1_𝑜⟩] ~_Q ))
9	7, 8	syl 14	. 2 ⊢ (𝜑 → (_Q‘[⟨𝐾, 1_𝑜⟩] ~_Q ) <_Q (_Q‘[⟨𝐽, 1_𝑜⟩] ~_Q ))
10		caucvgprlemk.jkq	. 2 ⊢ (𝜑 → (*_Q‘[⟨𝐽, 1_𝑜⟩] ~_Q ) <_Q 𝑄)
11		ltsonq 6496	. . 3 ⊢ <_Q Or Q
12		ltrelnq 6463	. . 3 ⊢ <_Q ⊆ (Q × Q)
13	11, 12	sotri 4720	. 2 ⊢ (((_Q‘[⟨𝐾, 1_𝑜⟩] ~_Q ) <_Q (_Q‘[⟨𝐽, 1_𝑜⟩] ~_Q ) ∧ (_Q‘[⟨𝐽, 1_𝑜⟩] ~_Q ) <_Q 𝑄) → (_Q‘[⟨𝐾, 1_𝑜⟩] ~_Q ) <_Q 𝑄)
14	9, 10, 13	syl2anc 391	1 ⊢ (𝜑 → (*_Q‘[⟨𝐾, 1_𝑜⟩] ~_Q ) <_Q 𝑄)

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 ∈ wcel 1393 ⟨cop 3378 class class class wbr 3764 ‘cfv 4902 1_𝑜c1o 5994 [cec 6104 Ncnpi 6370 <_N clti 6373 ~_Q ceq 6377 Qcnq 6378 *_Qcrq 6382 <_Q cltq 6383

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-coll 3872 ax-sep 3875 ax-nul 3883 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 ax-iinf 4311

This theorem depends on definitions: df-bi 110 df-dc 743 df-3or 886 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-ral 2311 df-rex 2312 df-reu 2313 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-int 3616 df-iun 3659 df-br 3765 df-opab 3819 df-mpt 3820 df-tr 3855 df-eprel 4026 df-id 4030 df-po 4033 df-iso 4034 df-iord 4103 df-on 4105 df-suc 4108 df-iom 4314 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-fv 4910 df-ov 5515 df-oprab 5516 df-mpt2 5517 df-1st 5767 df-2nd 5768 df-recs 5920 df-irdg 5957 df-1o 6001 df-oadd 6005 df-omul 6006 df-er 6106 df-ec 6108 df-qs 6112 df-ni 6402 df-mi 6404 df-lti 6405 df-mpq 6443 df-enq 6445 df-nqqs 6446 df-mqqs 6448 df-1nqqs 6449 df-rq 6450 df-ltnqqs 6451

This theorem is referenced by: caucvgprlem1 6777 caucvgprlem2 6778

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