Theorem archpr 7705

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 7615. (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 7537	. . 3 ⊢ (𝐴 ∈ P → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)
2		prmu 7540	. . 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 7479	. . . 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 2763	. . . . . . . . 9 ⊢ 𝑧 ∈ V
12		breq1 4033	. . . . . . . . 9 ⊢ (𝑙 = 𝑧 → (𝑙 <Q [⟨𝑥, 1o⟩] ~Q ↔ 𝑧 <Q [⟨𝑥, 1o⟩] ~Q ))
13		ltnqex 7611	. . . . . . . . . 10 ⊢ {𝑙 ∣ 𝑙 <Q [⟨𝑥, 1o⟩] ~Q } ∈ V
14		gtnqex 7612	. . . . . . . . . 10 ⊢ {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢} ∈ V
15	13, 14	op1st 6201	. . . . . . . . 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 2909	. . . . . . . 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 2256	. . . . . . . . 9 ⊢ (𝑤 = 𝑧 → (𝑤 ∈ (2nd ‘𝐴) ↔ 𝑧 ∈ (2nd ‘𝐴)))
19		eleq1 2256	. . . . . . . . 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 2865	. . . . . . 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 7615	. . . . . . . 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 7568	. . . . . . 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 2596	. . 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 2616	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 2164  {cab 2179  ∃wrex 2473  ⟨cop 3622   class class class wbr 4030  ‘cfv 5255  1^st c1st 6193  2^nd c2nd 6194  1_oc1o 6464  [cec 6587  Ncnpi 7334   ~_Q ceq 7341  Qcnq 7342   <_Q cltq 7347  Pcnp 7353  <_P cltp 7357

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 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-iinf 4621

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 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-tr 4129  df-eprel 4321  df-id 4325  df-po 4328  df-iso 4329  df-iord 4398  df-on 4400  df-suc 4403  df-iom 4624  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1st 6195  df-2nd 6196  df-recs 6360  df-irdg 6425  df-1o 6471  df-oadd 6475  df-omul 6476  df-er 6589  df-ec 6591  df-qs 6595  df-ni 7366  df-pli 7367  df-mi 7368  df-lti 7369  df-plpq 7406  df-mpq 7407  df-enq 7409  df-nqqs 7410  df-plqqs 7411  df-mqqs 7412  df-1nqqs 7413  df-rq 7414  df-ltnqqs 7415  df-inp 7528  df-iltp 7532

This theorem is referenced by:  archsr  7844