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Theorem archpr 6613
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 6533. (Contributed by Jim Kingdon, 22-Apr-2020.)
Assertion
Ref Expression
archpr (A Px N A<P ⟨{𝑙𝑙 <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.
StepHypRef Expression
1 prop 6457 . . 3 (A P → ⟨(1stA), (2ndA)⟩ P)
2 prmu 6460 . . 3 (⟨(1stA), (2ndA)⟩ Pz Q z (2ndA))
31, 2syl 14 . 2 (A Pz Q z (2ndA))
4 archnqq 6400 . . . 4 (z Qx N z <Q [⟨x, 1𝑜⟩] ~Q )
54ad2antrl 459 . . 3 ((A P (z Q z (2ndA))) → x N z <Q [⟨x, 1𝑜⟩] ~Q )
6 simprl 483 . . . . . . . 8 ((A P (z Q z (2ndA))) → z Q)
76ad2antrr 457 . . . . . . 7 ((((A P (z Q z (2ndA))) x N) z <Q [⟨x, 1𝑜⟩] ~Q ) → z Q)
8 simprr 484 . . . . . . . 8 ((A P (z Q z (2ndA))) → z (2ndA))
98ad2antrr 457 . . . . . . 7 ((((A P (z Q z (2ndA))) x N) z <Q [⟨x, 1𝑜⟩] ~Q ) → z (2ndA))
10 simpr 103 . . . . . . . 8 ((((A P (z Q z (2ndA))) x N) z <Q [⟨x, 1𝑜⟩] ~Q ) → z <Q [⟨x, 1𝑜⟩] ~Q )
11 vex 2554 . . . . . . . . 9 z V
12 breq1 3758 . . . . . . . . 9 (𝑙 = z → (𝑙 <Q [⟨x, 1𝑜⟩] ~Qz <Q [⟨x, 1𝑜⟩] ~Q ))
13 ltnqex 6531 . . . . . . . . . 10 {𝑙𝑙 <Q [⟨x, 1𝑜⟩] ~Q } V
14 gtnqex 6532 . . . . . . . . . 10 {u ∣ [⟨x, 1𝑜⟩] ~Q <Q u} V
1513, 14op1st 5715 . . . . . . . . 9 (1st ‘⟨{𝑙𝑙 <Q [⟨x, 1𝑜⟩] ~Q }, {u ∣ [⟨x, 1𝑜⟩] ~Q <Q u}⟩) = {𝑙𝑙 <Q [⟨x, 1𝑜⟩] ~Q }
1611, 12, 15elab2 2684 . . . . . . . 8 (z (1st ‘⟨{𝑙𝑙 <Q [⟨x, 1𝑜⟩] ~Q }, {u ∣ [⟨x, 1𝑜⟩] ~Q <Q u}⟩) ↔ z <Q [⟨x, 1𝑜⟩] ~Q )
1710, 16sylibr 137 . . . . . . 7 ((((A P (z Q z (2ndA))) x N) z <Q [⟨x, 1𝑜⟩] ~Q ) → z (1st ‘⟨{𝑙𝑙 <Q [⟨x, 1𝑜⟩] ~Q }, {u ∣ [⟨x, 1𝑜⟩] ~Q <Q u}⟩))
18 eleq1 2097 . . . . . . . . 9 (w = z → (w (2ndA) ↔ z (2ndA)))
19 eleq1 2097 . . . . . . . . 9 (w = z → (w (1st ‘⟨{𝑙𝑙 <Q [⟨x, 1𝑜⟩] ~Q }, {u ∣ [⟨x, 1𝑜⟩] ~Q <Q u}⟩) ↔ z (1st ‘⟨{𝑙𝑙 <Q [⟨x, 1𝑜⟩] ~Q }, {u ∣ [⟨x, 1𝑜⟩] ~Q <Q u}⟩)))
2018, 19anbi12d 442 . . . . . . . 8 (w = z → ((w (2ndA) w (1st ‘⟨{𝑙𝑙 <Q [⟨x, 1𝑜⟩] ~Q }, {u ∣ [⟨x, 1𝑜⟩] ~Q <Q u}⟩)) ↔ (z (2ndA) z (1st ‘⟨{𝑙𝑙 <Q [⟨x, 1𝑜⟩] ~Q }, {u ∣ [⟨x, 1𝑜⟩] ~Q <Q u}⟩))))
2120rspcev 2650 . . . . . . 7 ((z Q (z (2ndA) z (1st ‘⟨{𝑙𝑙 <Q [⟨x, 1𝑜⟩] ~Q }, {u ∣ [⟨x, 1𝑜⟩] ~Q <Q u}⟩))) → w Q (w (2ndA) w (1st ‘⟨{𝑙𝑙 <Q [⟨x, 1𝑜⟩] ~Q }, {u ∣ [⟨x, 1𝑜⟩] ~Q <Q u}⟩)))
227, 9, 17, 21syl12anc 1132 . . . . . 6 ((((A P (z Q z (2ndA))) x N) z <Q [⟨x, 1𝑜⟩] ~Q ) → w Q (w (2ndA) w (1st ‘⟨{𝑙𝑙 <Q [⟨x, 1𝑜⟩] ~Q }, {u ∣ [⟨x, 1𝑜⟩] ~Q <Q u}⟩)))
23 simplll 485 . . . . . . 7 ((((A P (z Q z (2ndA))) x N) z <Q [⟨x, 1𝑜⟩] ~Q ) → A P)
24 nnprlu 6533 . . . . . . . 8 (x N → ⟨{𝑙𝑙 <Q [⟨x, 1𝑜⟩] ~Q }, {u ∣ [⟨x, 1𝑜⟩] ~Q <Q u}⟩ P)
2524ad2antlr 458 . . . . . . 7 ((((A P (z Q z (2ndA))) x N) z <Q [⟨x, 1𝑜⟩] ~Q ) → ⟨{𝑙𝑙 <Q [⟨x, 1𝑜⟩] ~Q }, {u ∣ [⟨x, 1𝑜⟩] ~Q <Q u}⟩ P)
26 ltdfpr 6488 . . . . . . 7 ((A P ⟨{𝑙𝑙 <Q [⟨x, 1𝑜⟩] ~Q }, {u ∣ [⟨x, 1𝑜⟩] ~Q <Q u}⟩ P) → (A<P ⟨{𝑙𝑙 <Q [⟨x, 1𝑜⟩] ~Q }, {u ∣ [⟨x, 1𝑜⟩] ~Q <Q u}⟩ ↔ w Q (w (2ndA) w (1st ‘⟨{𝑙𝑙 <Q [⟨x, 1𝑜⟩] ~Q }, {u ∣ [⟨x, 1𝑜⟩] ~Q <Q u}⟩))))
2723, 25, 26syl2anc 391 . . . . . 6 ((((A P (z Q z (2ndA))) x N) z <Q [⟨x, 1𝑜⟩] ~Q ) → (A<P ⟨{𝑙𝑙 <Q [⟨x, 1𝑜⟩] ~Q }, {u ∣ [⟨x, 1𝑜⟩] ~Q <Q u}⟩ ↔ w Q (w (2ndA) w (1st ‘⟨{𝑙𝑙 <Q [⟨x, 1𝑜⟩] ~Q }, {u ∣ [⟨x, 1𝑜⟩] ~Q <Q u}⟩))))
2822, 27mpbird 156 . . . . 5 ((((A P (z Q z (2ndA))) x N) z <Q [⟨x, 1𝑜⟩] ~Q ) → A<P ⟨{𝑙𝑙 <Q [⟨x, 1𝑜⟩] ~Q }, {u ∣ [⟨x, 1𝑜⟩] ~Q <Q u}⟩)
2928ex 108 . . . 4 (((A P (z Q z (2ndA))) x N) → (z <Q [⟨x, 1𝑜⟩] ~QA<P ⟨{𝑙𝑙 <Q [⟨x, 1𝑜⟩] ~Q }, {u ∣ [⟨x, 1𝑜⟩] ~Q <Q u}⟩))
3029reximdva 2415 . . 3 ((A P (z Q z (2ndA))) → (x N z <Q [⟨x, 1𝑜⟩] ~Qx N A<P ⟨{𝑙𝑙 <Q [⟨x, 1𝑜⟩] ~Q }, {u ∣ [⟨x, 1𝑜⟩] ~Q <Q u}⟩))
315, 30mpd 13 . 2 ((A P (z Q z (2ndA))) → x N A<P ⟨{𝑙𝑙 <Q [⟨x, 1𝑜⟩] ~Q }, {u ∣ [⟨x, 1𝑜⟩] ~Q <Q u}⟩)
323, 31rexlimddv 2431 1 (A Px N A<P ⟨{𝑙𝑙 <Q [⟨x, 1𝑜⟩] ~Q }, {u ∣ [⟨x, 1𝑜⟩] ~Q <Q u}⟩)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   wcel 1390  {cab 2023  wrex 2301  cop 3370   class class class wbr 3755  cfv 4845  1st c1st 5707  2nd c2nd 5708  1𝑜c1o 5933  [cec 6040  Ncnpi 6256   ~Q ceq 6263  Qcnq 6264   <Q cltq 6269  Pcnp 6275  <P cltp 6279
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-iinf 4254
This theorem depends on definitions:  df-bi 110  df-dc 742  df-3or 885  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-eprel 4017  df-id 4021  df-po 4024  df-iso 4025  df-iord 4069  df-on 4071  df-suc 4074  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-recs 5861  df-irdg 5897  df-1o 5940  df-oadd 5944  df-omul 5945  df-er 6042  df-ec 6044  df-qs 6048  df-ni 6288  df-pli 6289  df-mi 6290  df-lti 6291  df-plpq 6328  df-mpq 6329  df-enq 6331  df-nqqs 6332  df-plqqs 6333  df-mqqs 6334  df-1nqqs 6335  df-rq 6336  df-ltnqqs 6337  df-inp 6448  df-iltp 6452
This theorem is referenced by:  archsr  6668
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