Theorem caucvgprprlemk 7685

Description: Lemma for caucvgprpr 7714. Reciprocals of positive integers decrease as the positive integers increase. (Contributed by Jim Kingdon, 28-Nov-2020.)

Hypotheses

Ref	Expression
caucvgprprlemk.jk	⊢ (𝜑 → 𝐽 <N 𝐾)
caucvgprprlemk.jkq	⊢ (𝜑 → ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐽, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐽, 1o⟩] ~Q ) <Q 𝑢}⟩<P 𝑄)

Assertion

Ref	Expression
caucvgprprlemk	⊢ (𝜑 → ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑢}⟩<P 𝑄)

Distinct variable groups:   𝐽,𝑙   𝑢,𝐽   𝐾,𝑙   𝑢,𝐾

Allowed substitution hints:   𝜑(𝑢,𝑙)   𝑄(𝑢,𝑙)

Proof of Theorem caucvgprprlemk

Step	Hyp	Ref	Expression
1		caucvgprprlemk.jk	. . . 4 ⊢ (𝜑 → 𝐽 <N 𝐾)
2		ltrelpi 7326	. . . . . 6 ⊢ <N ⊆ (N × N)
3	2	brel 4680	. . . . 5 ⊢ (𝐽 <N 𝐾 → (𝐽 ∈ N ∧ 𝐾 ∈ N))
4		ltnnnq 7425	. . . . 5 ⊢ ((𝐽 ∈ N ∧ 𝐾 ∈ N) → (𝐽 <N 𝐾 ↔ [⟨𝐽, 1o⟩] ~Q <Q [⟨𝐾, 1o⟩] ~Q ))
5	1, 3, 4	3syl 17	. . . 4 ⊢ (𝜑 → (𝐽 <N 𝐾 ↔ [⟨𝐽, 1o⟩] ~Q <Q [⟨𝐾, 1o⟩] ~Q ))
6	1, 5	mpbid 147	. . 3 ⊢ (𝜑 → [⟨𝐽, 1o⟩] ~Q <Q [⟨𝐾, 1o⟩] ~Q )
7		ltrnqi 7423	. . 3 ⊢ ([⟨𝐽, 1o⟩] ~Q <Q [⟨𝐾, 1o⟩] ~Q → (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q (*Q‘[⟨𝐽, 1o⟩] ~Q ))
8		ltnqpri 7596	. . 3 ⊢ ((*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q (*Q‘[⟨𝐽, 1o⟩] ~Q ) → ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑢}⟩<P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐽, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐽, 1o⟩] ~Q ) <Q 𝑢}⟩)
9	6, 7, 8	3syl 17	. 2 ⊢ (𝜑 → ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑢}⟩<P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐽, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐽, 1o⟩] ~Q ) <Q 𝑢}⟩)
10		caucvgprprlemk.jkq	. 2 ⊢ (𝜑 → ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐽, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐽, 1o⟩] ~Q ) <Q 𝑢}⟩<P 𝑄)
11		ltsopr 7598	. . 3 ⊢ <P Or P
12		ltrelpr 7507	. . 3 ⊢ <P ⊆ (P × P)
13	11, 12	sotri 5026	. 2 ⊢ ((⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑢}⟩<P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐽, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐽, 1o⟩] ~Q ) <Q 𝑢}⟩ ∧ ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐽, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐽, 1o⟩] ~Q ) <Q 𝑢}⟩<P 𝑄) → ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑢}⟩<P 𝑄)
14	9, 10, 13	syl2anc 411	1 ⊢ (𝜑 → ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑢}⟩<P 𝑄)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∈ wcel 2148  {cab 2163  ⟨cop 3597   class class class wbr 4005  ‘cfv 5218  1_oc1o 6413  [cec 6536  Ncnpi 7274   <_N clti 7277   ~_Q ceq 7281  *_Qcrq 7286   <_Q cltq 7287  Pcnp 7293  <_P cltp 7297

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-eprel 4291  df-id 4295  df-po 4298  df-iso 4299  df-iord 4368  df-on 4370  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-ov 5881  df-oprab 5882  df-mpo 5883  df-1st 6144  df-2nd 6145  df-recs 6309  df-irdg 6374  df-1o 6420  df-oadd 6424  df-omul 6425  df-er 6538  df-ec 6540  df-qs 6544  df-ni 7306  df-pli 7307  df-mi 7308  df-lti 7309  df-plpq 7346  df-mpq 7347  df-enq 7349  df-nqqs 7350  df-plqqs 7351  df-mqqs 7352  df-1nqqs 7353  df-rq 7354  df-ltnqqs 7355  df-inp 7468  df-iltp 7472

This theorem is referenced by:  caucvgprprlem1  7711  caucvgprprlem2  7712