Theorem caucvgprprlemk 7624

Description: Lemma for caucvgprpr 7653. 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 7265	. . . . . 6 ⊢ <N ⊆ (N × N)
3	2	brel 4656	. . . . 5 ⊢ (𝐽 <N 𝐾 → (𝐽 ∈ N ∧ 𝐾 ∈ N))
4		ltnnnq 7364	. . . . 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 146	. . 3 ⊢ (𝜑 → [⟨𝐽, 1o⟩] ~Q <Q [⟨𝐾, 1o⟩] ~Q )
7		ltrnqi 7362	. . 3 ⊢ ([⟨𝐽, 1o⟩] ~Q <Q [⟨𝐾, 1o⟩] ~Q → (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q (*Q‘[⟨𝐽, 1o⟩] ~Q ))
8		ltnqpri 7535	. . 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 7537	. . 3 ⊢ <P Or P
12		ltrelpr 7446	. . 3 ⊢ <P ⊆ (P × P)
13	11, 12	sotri 4999	. 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 409	1 ⊢ (𝜑 → ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑢}⟩<P 𝑄)

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∈ wcel 2136  {cab 2151  ⟨cop 3579   class class class wbr 3982  ‘cfv 5188  1_oc1o 6377  [cec 6499  Ncnpi 7213   <_N clti 7216   ~_Q ceq 7220  *_Qcrq 7225   <_Q cltq 7226  Pcnp 7232  <_P cltp 7236

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-eprel 4267  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-irdg 6338  df-1o 6384  df-oadd 6388  df-omul 6389  df-er 6501  df-ec 6503  df-qs 6507  df-ni 7245  df-pli 7246  df-mi 7247  df-lti 7248  df-plpq 7285  df-mpq 7286  df-enq 7288  df-nqqs 7289  df-plqqs 7290  df-mqqs 7291  df-1nqqs 7292  df-rq 7293  df-ltnqqs 7294  df-inp 7407  df-iltp 7411

This theorem is referenced by:  caucvgprprlem1  7650  caucvgprprlem2  7651