Theorem archpr 7923

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 7833. (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 7755	. . 3 ⊢ (𝐴 ∈ P → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)
2		prmu 7758	. . 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 7697	. . . 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 531	. . . . . . . 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 533	. . . . . . . 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 2806	. . . . . . . . 9 ⊢ 𝑧 ∈ V
12		breq1 4096	. . . . . . . . 9 ⊢ (𝑙 = 𝑧 → (𝑙 <Q [⟨𝑥, 1o⟩] ~Q ↔ 𝑧 <Q [⟨𝑥, 1o⟩] ~Q ))
13		ltnqex 7829	. . . . . . . . . 10 ⊢ {𝑙 ∣ 𝑙 <Q [⟨𝑥, 1o⟩] ~Q } ∈ V
14		gtnqex 7830	. . . . . . . . . 10 ⊢ {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢} ∈ V
15	13, 14	op1st 6318	. . . . . . . . 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 2955	. . . . . . . 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 2294	. . . . . . . . 9 ⊢ (𝑤 = 𝑧 → (𝑤 ∈ (2nd ‘𝐴) ↔ 𝑧 ∈ (2nd ‘𝐴)))
19		eleq1 2294	. . . . . . . . 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 2911	. . . . . . 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 1272	. . . . . 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 535	. . . . . . 7 ⊢ ((((𝐴 ∈ P ∧ (𝑧 ∈ Q ∧ 𝑧 ∈ (2nd ‘𝐴))) ∧ 𝑥 ∈ N) ∧ 𝑧 <Q [⟨𝑥, 1o⟩] ~Q ) → 𝐴 ∈ P)
24		nnprlu 7833	. . . . . . . 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 7786	. . . . . . 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 2635	. . 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 2656	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 2202  {cab 2217  ∃wrex 2512  ⟨cop 3676   class class class wbr 4093  ‘cfv 5333  1^st c1st 6310  2^nd c2nd 6311  1_oc1o 6618  [cec 6743  Ncnpi 7552   ~_Q ceq 7559  Qcnq 7560   <_Q cltq 7565  Pcnp 7571  <_P cltp 7575

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-eprel 4392  df-id 4396  df-po 4399  df-iso 4400  df-iord 4469  df-on 4471  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-irdg 6579  df-1o 6625  df-oadd 6629  df-omul 6630  df-er 6745  df-ec 6747  df-qs 6751  df-ni 7584  df-pli 7585  df-mi 7586  df-lti 7587  df-plpq 7624  df-mpq 7625  df-enq 7627  df-nqqs 7628  df-plqqs 7629  df-mqqs 7630  df-1nqqs 7631  df-rq 7632  df-ltnqqs 7633  df-inp 7746  df-iltp 7750

This theorem is referenced by:  archsr  8062