Theorem archpr 6741

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 6651. (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 6573	. . 3 ⊢ (𝐴 ∈ P → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)
2		prmu 6576	. . 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 6515	. . . 4 ⊢ (𝑧 ∈ Q → ∃𝑥 ∈ N 𝑧 <Q [⟨𝑥, 1𝑜⟩] ~Q )
5	4	ad2antrl 459	. . 3 ⊢ ((𝐴 ∈ P ∧ (𝑧 ∈ Q ∧ 𝑧 ∈ (2nd ‘𝐴))) → ∃𝑥 ∈ N 𝑧 <Q [⟨𝑥, 1𝑜⟩] ~Q )
6		simprl 483	. . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ (𝑧 ∈ Q ∧ 𝑧 ∈ (2nd ‘𝐴))) → 𝑧 ∈ Q)
7	6	ad2antrr 457	. . . . . . 7 ⊢ ((((𝐴 ∈ P ∧ (𝑧 ∈ Q ∧ 𝑧 ∈ (2nd ‘𝐴))) ∧ 𝑥 ∈ N) ∧ 𝑧 <Q [⟨𝑥, 1𝑜⟩] ~Q ) → 𝑧 ∈ Q)
8		simprr 484	. . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ (𝑧 ∈ Q ∧ 𝑧 ∈ (2nd ‘𝐴))) → 𝑧 ∈ (2nd ‘𝐴))
9	8	ad2antrr 457	. . . . . . 7 ⊢ ((((𝐴 ∈ P ∧ (𝑧 ∈ Q ∧ 𝑧 ∈ (2nd ‘𝐴))) ∧ 𝑥 ∈ N) ∧ 𝑧 <Q [⟨𝑥, 1𝑜⟩] ~Q ) → 𝑧 ∈ (2nd ‘𝐴))
10		simpr 103	. . . . . . . 8 ⊢ ((((𝐴 ∈ P ∧ (𝑧 ∈ Q ∧ 𝑧 ∈ (2nd ‘𝐴))) ∧ 𝑥 ∈ N) ∧ 𝑧 <Q [⟨𝑥, 1𝑜⟩] ~Q ) → 𝑧 <Q [⟨𝑥, 1𝑜⟩] ~Q )
11		vex 2560	. . . . . . . . 9 ⊢ 𝑧 ∈ V
12		breq1 3767	. . . . . . . . 9 ⊢ (𝑙 = 𝑧 → (𝑙 <Q [⟨𝑥, 1𝑜⟩] ~Q ↔ 𝑧 <Q [⟨𝑥, 1𝑜⟩] ~Q ))
13		ltnqex 6647	. . . . . . . . . 10 ⊢ {𝑙 ∣ 𝑙 <Q [⟨𝑥, 1𝑜⟩] ~Q } ∈ V
14		gtnqex 6648	. . . . . . . . . 10 ⊢ {𝑢 ∣ [⟨𝑥, 1𝑜⟩] ~Q <Q 𝑢} ∈ V
15	13, 14	op1st 5773	. . . . . . . . 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 2690	. . . . . . . 8 ⊢ (𝑧 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q [⟨𝑥, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1𝑜⟩] ~Q <Q 𝑢}⟩) ↔ 𝑧 <Q [⟨𝑥, 1𝑜⟩] ~Q )
17	10, 16	sylibr 137	. . . . . . 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 2100	. . . . . . . . 9 ⊢ (𝑤 = 𝑧 → (𝑤 ∈ (2nd ‘𝐴) ↔ 𝑧 ∈ (2nd ‘𝐴)))
19		eleq1 2100	. . . . . . . . 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 442	. . . . . . . 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 2656	. . . . . . 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 1133	. . . . . 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 485	. . . . . . 7 ⊢ ((((𝐴 ∈ P ∧ (𝑧 ∈ Q ∧ 𝑧 ∈ (2nd ‘𝐴))) ∧ 𝑥 ∈ N) ∧ 𝑧 <Q [⟨𝑥, 1𝑜⟩] ~Q ) → 𝐴 ∈ P)
24		nnprlu 6651	. . . . . . . 8 ⊢ (𝑥 ∈ N → ⟨{𝑙 ∣ 𝑙 <Q [⟨𝑥, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1𝑜⟩] ~Q <Q 𝑢}⟩ ∈ P)
25	24	ad2antlr 458	. . . . . . 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 6604	. . . . . . 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 391	. . . . . 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 156	. . . . 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 108	. . . 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 2421	. . 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 2437	1 ⊢ (𝐴 ∈ P → ∃𝑥 ∈ N 𝐴<P ⟨{𝑙 ∣ 𝑙 <Q [⟨𝑥, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1𝑜⟩] ~Q <Q 𝑢}⟩)

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   ∈ wcel 1393  {cab 2026  ∃wrex 2307  ⟨cop 3378   class class class wbr 3764  ‘cfv 4902  1^st c1st 5765  2^nd c2nd 5766  1_𝑜c1o 5994  [cec 6104  Ncnpi 6370   ~_Q ceq 6377  Qcnq 6378   <_Q cltq 6383  Pcnp 6389  <_P cltp 6393

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311

This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-1o 6001  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6402  df-pli 6403  df-mi 6404  df-lti 6405  df-plpq 6442  df-mpq 6443  df-enq 6445  df-nqqs 6446  df-plqqs 6447  df-mqqs 6448  df-1nqqs 6449  df-rq 6450  df-ltnqqs 6451  df-inp 6564  df-iltp 6568

This theorem is referenced by:  archsr  6866