ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  exbtwnzlemex GIF version

Theorem exbtwnzlemex 10223
Description: Existence of an integer so that a given real number is between the integer and its successor. The real number must satisfy the 𝑛𝐴𝐴 < 𝑛 hypothesis. For example either a rational number or a number which is irrational (in the sense of being apart from any rational number) will meet this condition.

The proof starts by finding two integers which are less than and greater than 𝐴. Then this range can be shrunk by choosing an integer in between the endpoints of the range and then deciding which half of the range to keep based on the 𝑛𝐴𝐴 < 𝑛 hypothesis, and iterating until the range consists of two consecutive integers. (Contributed by Jim Kingdon, 8-Oct-2021.)

Hypotheses
Ref Expression
exbtwnzlemex.a (𝜑𝐴 ∈ ℝ)
exbtwnzlemex.tri ((𝜑𝑛 ∈ ℤ) → (𝑛𝐴𝐴 < 𝑛))
Assertion
Ref Expression
exbtwnzlemex (𝜑 → ∃𝑥 ∈ ℤ (𝑥𝐴𝐴 < (𝑥 + 1)))
Distinct variable groups:   𝐴,𝑛   𝑥,𝐴   𝜑,𝑛
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem exbtwnzlemex
Dummy variables 𝑎 𝑗 𝑚 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 exbtwnzlemex.a . . . 4 (𝜑𝐴 ∈ ℝ)
2 btwnz 9348 . . . 4 (𝐴 ∈ ℝ → (∃𝑚 ∈ ℤ 𝑚 < 𝐴 ∧ ∃𝑗 ∈ ℤ 𝐴 < 𝑗))
31, 2syl 14 . . 3 (𝜑 → (∃𝑚 ∈ ℤ 𝑚 < 𝐴 ∧ ∃𝑗 ∈ ℤ 𝐴 < 𝑗))
4 reeanv 2646 . . 3 (∃𝑚 ∈ ℤ ∃𝑗 ∈ ℤ (𝑚 < 𝐴𝐴 < 𝑗) ↔ (∃𝑚 ∈ ℤ 𝑚 < 𝐴 ∧ ∃𝑗 ∈ ℤ 𝐴 < 𝑗))
53, 4sylibr 134 . 2 (𝜑 → ∃𝑚 ∈ ℤ ∃𝑗 ∈ ℤ (𝑚 < 𝐴𝐴 < 𝑗))
6 simplrl 535 . . . . . 6 (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴𝐴 < 𝑗)) → 𝑚 ∈ ℤ)
76zred 9351 . . . . . . 7 (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴𝐴 < 𝑗)) → 𝑚 ∈ ℝ)
81ad2antrr 488 . . . . . . 7 (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴𝐴 < 𝑗)) → 𝐴 ∈ ℝ)
9 simprl 529 . . . . . . 7 (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴𝐴 < 𝑗)) → 𝑚 < 𝐴)
107, 8, 9ltled 8053 . . . . . 6 (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴𝐴 < 𝑗)) → 𝑚𝐴)
11 simprr 531 . . . . . . 7 (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴𝐴 < 𝑗)) → 𝐴 < 𝑗)
126zcnd 9352 . . . . . . . 8 (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴𝐴 < 𝑗)) → 𝑚 ∈ ℂ)
13 simplrr 536 . . . . . . . . 9 (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴𝐴 < 𝑗)) → 𝑗 ∈ ℤ)
1413zcnd 9352 . . . . . . . 8 (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴𝐴 < 𝑗)) → 𝑗 ∈ ℂ)
1512, 14pncan3d 8248 . . . . . . 7 (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴𝐴 < 𝑗)) → (𝑚 + (𝑗𝑚)) = 𝑗)
1611, 15breqtrrd 4028 . . . . . 6 (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴𝐴 < 𝑗)) → 𝐴 < (𝑚 + (𝑗𝑚)))
17 breq1 4003 . . . . . . . 8 (𝑦 = 𝑚 → (𝑦𝐴𝑚𝐴))
18 oveq1 5875 . . . . . . . . 9 (𝑦 = 𝑚 → (𝑦 + (𝑗𝑚)) = (𝑚 + (𝑗𝑚)))
1918breq2d 4012 . . . . . . . 8 (𝑦 = 𝑚 → (𝐴 < (𝑦 + (𝑗𝑚)) ↔ 𝐴 < (𝑚 + (𝑗𝑚))))
2017, 19anbi12d 473 . . . . . . 7 (𝑦 = 𝑚 → ((𝑦𝐴𝐴 < (𝑦 + (𝑗𝑚))) ↔ (𝑚𝐴𝐴 < (𝑚 + (𝑗𝑚)))))
2120rspcev 2841 . . . . . 6 ((𝑚 ∈ ℤ ∧ (𝑚𝐴𝐴 < (𝑚 + (𝑗𝑚)))) → ∃𝑦 ∈ ℤ (𝑦𝐴𝐴 < (𝑦 + (𝑗𝑚))))
226, 10, 16, 21syl12anc 1236 . . . . 5 (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴𝐴 < 𝑗)) → ∃𝑦 ∈ ℤ (𝑦𝐴𝐴 < (𝑦 + (𝑗𝑚))))
2313zred 9351 . . . . . . . 8 (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴𝐴 < 𝑗)) → 𝑗 ∈ ℝ)
247, 8, 23, 9, 11lttrd 8060 . . . . . . 7 (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴𝐴 < 𝑗)) → 𝑚 < 𝑗)
25 znnsub 9280 . . . . . . . 8 ((𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ) → (𝑚 < 𝑗 ↔ (𝑗𝑚) ∈ ℕ))
2625ad2antlr 489 . . . . . . 7 (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴𝐴 < 𝑗)) → (𝑚 < 𝑗 ↔ (𝑗𝑚) ∈ ℕ))
2724, 26mpbid 147 . . . . . 6 (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴𝐴 < 𝑗)) → (𝑗𝑚) ∈ ℕ)
28 exbtwnzlemex.tri . . . . . . . . . 10 ((𝜑𝑛 ∈ ℤ) → (𝑛𝐴𝐴 < 𝑛))
2928ralrimiva 2550 . . . . . . . . 9 (𝜑 → ∀𝑛 ∈ ℤ (𝑛𝐴𝐴 < 𝑛))
30 breq1 4003 . . . . . . . . . . 11 (𝑛 = 𝑎 → (𝑛𝐴𝑎𝐴))
31 breq2 4004 . . . . . . . . . . 11 (𝑛 = 𝑎 → (𝐴 < 𝑛𝐴 < 𝑎))
3230, 31orbi12d 793 . . . . . . . . . 10 (𝑛 = 𝑎 → ((𝑛𝐴𝐴 < 𝑛) ↔ (𝑎𝐴𝐴 < 𝑎)))
3332cbvralv 2703 . . . . . . . . 9 (∀𝑛 ∈ ℤ (𝑛𝐴𝐴 < 𝑛) ↔ ∀𝑎 ∈ ℤ (𝑎𝐴𝐴 < 𝑎))
3429, 33sylib 122 . . . . . . . 8 (𝜑 → ∀𝑎 ∈ ℤ (𝑎𝐴𝐴 < 𝑎))
3534ad2antrr 488 . . . . . . 7 (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴𝐴 < 𝑗)) → ∀𝑎 ∈ ℤ (𝑎𝐴𝐴 < 𝑎))
3635r19.21bi 2565 . . . . . 6 ((((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴𝐴 < 𝑗)) ∧ 𝑎 ∈ ℤ) → (𝑎𝐴𝐴 < 𝑎))
3727, 8, 36exbtwnzlemshrink 10222 . . . . 5 ((((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴𝐴 < 𝑗)) ∧ ∃𝑦 ∈ ℤ (𝑦𝐴𝐴 < (𝑦 + (𝑗𝑚)))) → ∃𝑥 ∈ ℤ (𝑥𝐴𝐴 < (𝑥 + 1)))
3822, 37mpdan 421 . . . 4 (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴𝐴 < 𝑗)) → ∃𝑥 ∈ ℤ (𝑥𝐴𝐴 < (𝑥 + 1)))
3938ex 115 . . 3 ((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → ((𝑚 < 𝐴𝐴 < 𝑗) → ∃𝑥 ∈ ℤ (𝑥𝐴𝐴 < (𝑥 + 1))))
4039rexlimdvva 2602 . 2 (𝜑 → (∃𝑚 ∈ ℤ ∃𝑗 ∈ ℤ (𝑚 < 𝐴𝐴 < 𝑗) → ∃𝑥 ∈ ℤ (𝑥𝐴𝐴 < (𝑥 + 1))))
415, 40mpd 13 1 (𝜑 → ∃𝑥 ∈ ℤ (𝑥𝐴𝐴 < (𝑥 + 1)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  wo 708  wcel 2148  wral 2455  wrex 2456   class class class wbr 4000  (class class class)co 5868  cr 7788  1c1 7790   + caddc 7792   < clt 7969  cle 7970  cmin 8105  cn 8895  cz 9229
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4205  ax-un 4429  ax-setind 4532  ax-cnex 7880  ax-resscn 7881  ax-1cn 7882  ax-1re 7883  ax-icn 7884  ax-addcl 7885  ax-addrcl 7886  ax-mulcl 7887  ax-addcom 7889  ax-addass 7891  ax-distr 7893  ax-i2m1 7894  ax-0lt1 7895  ax-0id 7897  ax-rnegex 7898  ax-cnre 7900  ax-pre-ltirr 7901  ax-pre-ltwlin 7902  ax-pre-lttrn 7903  ax-pre-ltadd 7905  ax-arch 7908
This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-br 4001  df-opab 4062  df-id 4289  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-iota 5173  df-fun 5213  df-fv 5219  df-riota 5824  df-ov 5871  df-oprab 5872  df-mpo 5873  df-pnf 7971  df-mnf 7972  df-xr 7973  df-ltxr 7974  df-le 7975  df-sub 8107  df-neg 8108  df-inn 8896  df-n0 9153  df-z 9230
This theorem is referenced by:  qbtwnz  10225  apbtwnz  10247
  Copyright terms: Public domain W3C validator