Theorem archpr 7667

Description: For any positive real, there is an integer that is greater than it. This is also known as the "archimedean property". The integer 𝑥 is embedded into the reals as described at nnprlu 7577. (Contributed by Jim Kingdon, 22-Apr-2020.)

Assertion

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

Distinct variable group:   𝐴,𝑙,𝑢,𝑥

Proof of Theorem archpr

Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		prop 7499	. . 3 ⊢ (𝐴 ∈ P → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)
2		prmu 7502	. . 3 ⊢ (⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P → ∃𝑧 ∈ Q 𝑧 ∈ (2nd ‘𝐴))
3	1, 2	syl 14	. 2 ⊢ (𝐴 ∈ P → ∃𝑧 ∈ Q 𝑧 ∈ (2nd ‘𝐴))
4		archnqq 7441	. . . 4 ⊢ (𝑧 ∈ Q → ∃𝑥 ∈ N 𝑧 <Q [⟨𝑥, 1o⟩] ~Q )
5	4	ad2antrl 490	. . 3 ⊢ ((𝐴 ∈ P ∧ (𝑧 ∈ Q ∧ 𝑧 ∈ (2nd ‘𝐴))) → ∃𝑥 ∈ N 𝑧 <Q [⟨𝑥, 1o⟩] ~Q )
6		simprl 529	. . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ (𝑧 ∈ Q ∧ 𝑧 ∈ (2nd ‘𝐴))) → 𝑧 ∈ Q)
7	6	ad2antrr 488	. . . . . . 7 ⊢ ((((𝐴 ∈ P ∧ (𝑧 ∈ Q ∧ 𝑧 ∈ (2nd ‘𝐴))) ∧ 𝑥 ∈ N) ∧ 𝑧 <Q [⟨𝑥, 1o⟩] ~Q ) → 𝑧 ∈ Q)
8		simprr 531	. . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ (𝑧 ∈ Q ∧ 𝑧 ∈ (2nd ‘𝐴))) → 𝑧 ∈ (2nd ‘𝐴))
9	8	ad2antrr 488	. . . . . . 7 ⊢ ((((𝐴 ∈ P ∧ (𝑧 ∈ Q ∧ 𝑧 ∈ (2nd ‘𝐴))) ∧ 𝑥 ∈ N) ∧ 𝑧 <Q [⟨𝑥, 1o⟩] ~Q ) → 𝑧 ∈ (2nd ‘𝐴))
10		simpr 110	. . . . . . . 8 ⊢ ((((𝐴 ∈ P ∧ (𝑧 ∈ Q ∧ 𝑧 ∈ (2nd ‘𝐴))) ∧ 𝑥 ∈ N) ∧ 𝑧 <Q [⟨𝑥, 1o⟩] ~Q ) → 𝑧 <Q [⟨𝑥, 1o⟩] ~Q )
11		vex 2755	. . . . . . . . 9 ⊢ 𝑧 ∈ V
12		breq1 4021	. . . . . . . . 9 ⊢ (𝑙 = 𝑧 → (𝑙 <Q [⟨𝑥, 1o⟩] ~Q ↔ 𝑧 <Q [⟨𝑥, 1o⟩] ~Q ))
13		ltnqex 7573	. . . . . . . . . 10 ⊢ {𝑙 ∣ 𝑙 <Q [⟨𝑥, 1o⟩] ~Q } ∈ V
14		gtnqex 7574	. . . . . . . . . 10 ⊢ {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢} ∈ V
15	13, 14	op1st 6166	. . . . . . . . 9 ⊢ (1st ‘⟨{𝑙 ∣ 𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩) = {𝑙 ∣ 𝑙 <Q [⟨𝑥, 1o⟩] ~Q }
16	11, 12, 15	elab2 2900	. . . . . . . 8 ⊢ (𝑧 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩) ↔ 𝑧 <Q [⟨𝑥, 1o⟩] ~Q )
17	10, 16	sylibr 134	. . . . . . 7 ⊢ ((((𝐴 ∈ P ∧ (𝑧 ∈ Q ∧ 𝑧 ∈ (2nd ‘𝐴))) ∧ 𝑥 ∈ N) ∧ 𝑧 <Q [⟨𝑥, 1o⟩] ~Q ) → 𝑧 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩))
18		eleq1 2252	. . . . . . . . 9 ⊢ (𝑤 = 𝑧 → (𝑤 ∈ (2nd ‘𝐴) ↔ 𝑧 ∈ (2nd ‘𝐴)))
19		eleq1 2252	. . . . . . . . 9 ⊢ (𝑤 = 𝑧 → (𝑤 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩) ↔ 𝑧 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩)))
20	18, 19	anbi12d 473	. . . . . . . 8 ⊢ (𝑤 = 𝑧 → ((𝑤 ∈ (2nd ‘𝐴) ∧ 𝑤 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩)) ↔ (𝑧 ∈ (2nd ‘𝐴) ∧ 𝑧 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩))))
21	20	rspcev 2856	. . . . . . 7 ⊢ ((𝑧 ∈ Q ∧ (𝑧 ∈ (2nd ‘𝐴) ∧ 𝑧 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩))) → ∃𝑤 ∈ Q (𝑤 ∈ (2nd ‘𝐴) ∧ 𝑤 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩)))
22	7, 9, 17, 21	syl12anc 1247	. . . . . 6 ⊢ ((((𝐴 ∈ P ∧ (𝑧 ∈ Q ∧ 𝑧 ∈ (2nd ‘𝐴))) ∧ 𝑥 ∈ N) ∧ 𝑧 <Q [⟨𝑥, 1o⟩] ~Q ) → ∃𝑤 ∈ Q (𝑤 ∈ (2nd ‘𝐴) ∧ 𝑤 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩)))
23		simplll 533	. . . . . . 7 ⊢ ((((𝐴 ∈ P ∧ (𝑧 ∈ Q ∧ 𝑧 ∈ (2nd ‘𝐴))) ∧ 𝑥 ∈ N) ∧ 𝑧 <Q [⟨𝑥, 1o⟩] ~Q ) → 𝐴 ∈ P)
24		nnprlu 7577	. . . . . . . 8 ⊢ (𝑥 ∈ N → ⟨{𝑙 ∣ 𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩ ∈ P)
25	24	ad2antlr 489	. . . . . . 7 ⊢ ((((𝐴 ∈ P ∧ (𝑧 ∈ Q ∧ 𝑧 ∈ (2nd ‘𝐴))) ∧ 𝑥 ∈ N) ∧ 𝑧 <Q [⟨𝑥, 1o⟩] ~Q ) → ⟨{𝑙 ∣ 𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩ ∈ P)
26		ltdfpr 7530	. . . . . . 7 ⊢ ((𝐴 ∈ P ∧ ⟨{𝑙 ∣ 𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩ ∈ P) → (𝐴<P ⟨{𝑙 ∣ 𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩ ↔ ∃𝑤 ∈ Q (𝑤 ∈ (2nd ‘𝐴) ∧ 𝑤 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩))))
27	23, 25, 26	syl2anc 411	. . . . . 6 ⊢ ((((𝐴 ∈ P ∧ (𝑧 ∈ Q ∧ 𝑧 ∈ (2nd ‘𝐴))) ∧ 𝑥 ∈ N) ∧ 𝑧 <Q [⟨𝑥, 1o⟩] ~Q ) → (𝐴<P ⟨{𝑙 ∣ 𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩ ↔ ∃𝑤 ∈ Q (𝑤 ∈ (2nd ‘𝐴) ∧ 𝑤 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩))))
28	22, 27	mpbird 167	. . . . 5 ⊢ ((((𝐴 ∈ P ∧ (𝑧 ∈ Q ∧ 𝑧 ∈ (2nd ‘𝐴))) ∧ 𝑥 ∈ N) ∧ 𝑧 <Q [⟨𝑥, 1o⟩] ~Q ) → 𝐴<P ⟨{𝑙 ∣ 𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩)
29	28	ex 115	. . . 4 ⊢ (((𝐴 ∈ P ∧ (𝑧 ∈ Q ∧ 𝑧 ∈ (2nd ‘𝐴))) ∧ 𝑥 ∈ N) → (𝑧 <Q [⟨𝑥, 1o⟩] ~Q → 𝐴<P ⟨{𝑙 ∣ 𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩))
30	29	reximdva 2592	. . 3 ⊢ ((𝐴 ∈ P ∧ (𝑧 ∈ Q ∧ 𝑧 ∈ (2nd ‘𝐴))) → (∃𝑥 ∈ N 𝑧 <Q [⟨𝑥, 1o⟩] ~Q → ∃𝑥 ∈ N 𝐴<P ⟨{𝑙 ∣ 𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩))
31	5, 30	mpd 13	. 2 ⊢ ((𝐴 ∈ P ∧ (𝑧 ∈ Q ∧ 𝑧 ∈ (2nd ‘𝐴))) → ∃𝑥 ∈ N 𝐴<P ⟨{𝑙 ∣ 𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩)
32	3, 31	rexlimddv 2612	1 ⊢ (𝐴 ∈ P → ∃𝑥 ∈ N 𝐴<P ⟨{𝑙 ∣ 𝑙 <Q [⟨𝑥, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢}⟩)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∈ wcel 2160  {cab 2175  ∃wrex 2469  ⟨cop 3610   class class class wbr 4018  ‘cfv 5232  1^st c1st 6158  2^nd c2nd 6159  1_oc1o 6429  [cec 6552  Ncnpi 7296   ~_Q ceq 7303  Qcnq 7304   <_Q cltq 7309  Pcnp 7315  <_P cltp 7319

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-iinf 4602

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-eprel 4304  df-id 4308  df-po 4311  df-iso 4312  df-iord 4381  df-on 4383  df-suc 4386  df-iom 4605  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5234  df-fn 5235  df-f 5236  df-f1 5237  df-fo 5238  df-f1o 5239  df-fv 5240  df-ov 5895  df-oprab 5896  df-mpo 5897  df-1st 6160  df-2nd 6161  df-recs 6325  df-irdg 6390  df-1o 6436  df-oadd 6440  df-omul 6441  df-er 6554  df-ec 6556  df-qs 6560  df-ni 7328  df-pli 7329  df-mi 7330  df-lti 7331  df-plpq 7368  df-mpq 7369  df-enq 7371  df-nqqs 7372  df-plqqs 7373  df-mqqs 7374  df-1nqqs 7375  df-rq 7376  df-ltnqqs 7377  df-inp 7490  df-iltp 7494

This theorem is referenced by:  archsr  7806