Theorem archrecpr 7777

Description: Archimedean principle for positive reals (reciprocal version). (Contributed by Jim Kingdon, 25-Nov-2020.)

Assertion

Ref	Expression
archrecpr	⊢ (𝐴 ∈ P → ∃𝑗 ∈ N ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑗, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q 𝑢}⟩<P 𝐴)

Distinct variable groups:   𝐴,𝑗   𝑗,𝑙,𝑢

Allowed substitution hints:   𝐴(𝑢,𝑙)

Proof of Theorem archrecpr

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		prop 7588	. . . 4 ⊢ (𝐴 ∈ P → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)
2		prml 7590	. . . 4 ⊢ (⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P → ∃𝑥 ∈ Q 𝑥 ∈ (1st ‘𝐴))
3	1, 2	syl 14	. . 3 ⊢ (𝐴 ∈ P → ∃𝑥 ∈ Q 𝑥 ∈ (1st ‘𝐴))
4		archrecnq 7776	. . . . 5 ⊢ (𝑥 ∈ Q → ∃𝑗 ∈ N (*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q 𝑥)
5	4	ad2antrl 490	. . . 4 ⊢ ((𝐴 ∈ P ∧ (𝑥 ∈ Q ∧ 𝑥 ∈ (1st ‘𝐴))) → ∃𝑗 ∈ N (*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q 𝑥)
6	1	ad2antrr 488	. . . . . 6 ⊢ (((𝐴 ∈ P ∧ (𝑥 ∈ Q ∧ 𝑥 ∈ (1st ‘𝐴))) ∧ 𝑗 ∈ N) → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)
7		simplrr 536	. . . . . 6 ⊢ (((𝐴 ∈ P ∧ (𝑥 ∈ Q ∧ 𝑥 ∈ (1st ‘𝐴))) ∧ 𝑗 ∈ N) → 𝑥 ∈ (1st ‘𝐴))
8		prcdnql 7597	. . . . . 6 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝑥 ∈ (1st ‘𝐴)) → ((*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q 𝑥 → (*Q‘[⟨𝑗, 1o⟩] ~Q ) ∈ (1st ‘𝐴)))
9	6, 7, 8	syl2anc 411	. . . . 5 ⊢ (((𝐴 ∈ P ∧ (𝑥 ∈ Q ∧ 𝑥 ∈ (1st ‘𝐴))) ∧ 𝑗 ∈ N) → ((*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q 𝑥 → (*Q‘[⟨𝑗, 1o⟩] ~Q ) ∈ (1st ‘𝐴)))
10	9	reximdva 2608	. . . 4 ⊢ ((𝐴 ∈ P ∧ (𝑥 ∈ Q ∧ 𝑥 ∈ (1st ‘𝐴))) → (∃𝑗 ∈ N (*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q 𝑥 → ∃𝑗 ∈ N (*Q‘[⟨𝑗, 1o⟩] ~Q ) ∈ (1st ‘𝐴)))
11	5, 10	mpd 13	. . 3 ⊢ ((𝐴 ∈ P ∧ (𝑥 ∈ Q ∧ 𝑥 ∈ (1st ‘𝐴))) → ∃𝑗 ∈ N (*Q‘[⟨𝑗, 1o⟩] ~Q ) ∈ (1st ‘𝐴))
12	3, 11	rexlimddv 2628	. 2 ⊢ (𝐴 ∈ P → ∃𝑗 ∈ N (*Q‘[⟨𝑗, 1o⟩] ~Q ) ∈ (1st ‘𝐴))
13		nnnq 7535	. . . . . 6 ⊢ (𝑗 ∈ N → [⟨𝑗, 1o⟩] ~Q ∈ Q)
14		recclnq 7505	. . . . . 6 ⊢ ([⟨𝑗, 1o⟩] ~Q ∈ Q → (*Q‘[⟨𝑗, 1o⟩] ~Q ) ∈ Q)
15	13, 14	syl 14	. . . . 5 ⊢ (𝑗 ∈ N → (*Q‘[⟨𝑗, 1o⟩] ~Q ) ∈ Q)
16	15	adantl 277	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝑗 ∈ N) → (*Q‘[⟨𝑗, 1o⟩] ~Q ) ∈ Q)
17		simpl 109	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝑗 ∈ N) → 𝐴 ∈ P)
18		nqprl 7664	. . . 4 ⊢ (((*Q‘[⟨𝑗, 1o⟩] ~Q ) ∈ Q ∧ 𝐴 ∈ P) → ((*Q‘[⟨𝑗, 1o⟩] ~Q ) ∈ (1st ‘𝐴) ↔ ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑗, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q 𝑢}⟩<P 𝐴))
19	16, 17, 18	syl2anc 411	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝑗 ∈ N) → ((*Q‘[⟨𝑗, 1o⟩] ~Q ) ∈ (1st ‘𝐴) ↔ ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑗, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q 𝑢}⟩<P 𝐴))
20	19	rexbidva 2503	. 2 ⊢ (𝐴 ∈ P → (∃𝑗 ∈ N (*Q‘[⟨𝑗, 1o⟩] ~Q ) ∈ (1st ‘𝐴) ↔ ∃𝑗 ∈ N ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑗, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q 𝑢}⟩<P 𝐴))
21	12, 20	mpbid 147	1 ⊢ (𝐴 ∈ P → ∃𝑗 ∈ N ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑗, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑗, 1o⟩] ~Q ) <Q 𝑢}⟩<P 𝐴)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∈ wcel 2176  {cab 2191  ∃wrex 2485  ⟨cop 3636   class class class wbr 4044  ‘cfv 5271  1^st c1st 6224  2^nd c2nd 6225  1_oc1o 6495  [cec 6618  Ncnpi 7385   ~_Q ceq 7392  Qcnq 7393  *_Qcrq 7397   <_Q cltq 7398  Pcnp 7404  <_P cltp 7408

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 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4159  ax-sep 4162  ax-nul 4170  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-iinf 4636

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-ral 2489  df-rex 2490  df-reu 2491  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-tr 4143  df-eprel 4336  df-id 4340  df-po 4343  df-iso 4344  df-iord 4413  df-on 4415  df-suc 4418  df-iom 4639  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5947  df-oprab 5948  df-mpo 5949  df-1st 6226  df-2nd 6227  df-recs 6391  df-irdg 6456  df-1o 6502  df-oadd 6506  df-omul 6507  df-er 6620  df-ec 6622  df-qs 6626  df-ni 7417  df-pli 7418  df-mi 7419  df-lti 7420  df-plpq 7457  df-mpq 7458  df-enq 7460  df-nqqs 7461  df-plqqs 7462  df-mqqs 7463  df-1nqqs 7464  df-rq 7465  df-ltnqqs 7466  df-inp 7579  df-iltp 7583

This theorem is referenced by:  caucvgprprlemlim  7824