Theorem archpr 7727

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 7637. (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 7559	. . 3 ⊢ (𝐴 ∈ P → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)
2		prmu 7562	. . 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 7501	. . . 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 2766	. . . . . . . . 9 ⊢ 𝑧 ∈ V
12		breq1 4037	. . . . . . . . 9 ⊢ (𝑙 = 𝑧 → (𝑙 <Q [⟨𝑥, 1o⟩] ~Q ↔ 𝑧 <Q [⟨𝑥, 1o⟩] ~Q ))
13		ltnqex 7633	. . . . . . . . . 10 ⊢ {𝑙 ∣ 𝑙 <Q [⟨𝑥, 1o⟩] ~Q } ∈ V
14		gtnqex 7634	. . . . . . . . . 10 ⊢ {𝑢 ∣ [⟨𝑥, 1o⟩] ~Q <Q 𝑢} ∈ V
15	13, 14	op1st 6213	. . . . . . . . 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 2912	. . . . . . . 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 2259	. . . . . . . . 9 ⊢ (𝑤 = 𝑧 → (𝑤 ∈ (2nd ‘𝐴) ↔ 𝑧 ∈ (2nd ‘𝐴)))
19		eleq1 2259	. . . . . . . . 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 2868	. . . . . . 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 7637	. . . . . . . 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 7590	. . . . . . 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 2599	. . 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 2619	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 2167  {cab 2182  ∃wrex 2476  ⟨cop 3626   class class class wbr 4034  ‘cfv 5259  1^st c1st 6205  2^nd c2nd 6206  1_oc1o 6476  [cec 6599  Ncnpi 7356   ~_Q ceq 7363  Qcnq 7364   <_Q cltq 7369  Pcnp 7375  <_P cltp 7379

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625

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 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-eprel 4325  df-id 4329  df-po 4332  df-iso 4333  df-iord 4402  df-on 4404  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-recs 6372  df-irdg 6437  df-1o 6483  df-oadd 6487  df-omul 6488  df-er 6601  df-ec 6603  df-qs 6607  df-ni 7388  df-pli 7389  df-mi 7390  df-lti 7391  df-plpq 7428  df-mpq 7429  df-enq 7431  df-nqqs 7432  df-plqqs 7433  df-mqqs 7434  df-1nqqs 7435  df-rq 7436  df-ltnqqs 7437  df-inp 7550  df-iltp 7554

This theorem is referenced by:  archsr  7866