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Theorem exbtwnzlemex 10057
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 9193 . . . 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 2603 . . 3 (∃𝑚 ∈ ℤ ∃𝑗 ∈ ℤ (𝑚 < 𝐴𝐴 < 𝑗) ↔ (∃𝑚 ∈ ℤ 𝑚 < 𝐴 ∧ ∃𝑗 ∈ ℤ 𝐴 < 𝑗))
53, 4sylibr 133 . 2 (𝜑 → ∃𝑚 ∈ ℤ ∃𝑗 ∈ ℤ (𝑚 < 𝐴𝐴 < 𝑗))
6 simplrl 525 . . . . . 6 (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴𝐴 < 𝑗)) → 𝑚 ∈ ℤ)
76zred 9196 . . . . . . 7 (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴𝐴 < 𝑗)) → 𝑚 ∈ ℝ)
81ad2antrr 480 . . . . . . 7 (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴𝐴 < 𝑗)) → 𝐴 ∈ ℝ)
9 simprl 521 . . . . . . 7 (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴𝐴 < 𝑗)) → 𝑚 < 𝐴)
107, 8, 9ltled 7904 . . . . . 6 (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴𝐴 < 𝑗)) → 𝑚𝐴)
11 simprr 522 . . . . . . 7 (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴𝐴 < 𝑗)) → 𝐴 < 𝑗)
126zcnd 9197 . . . . . . . 8 (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴𝐴 < 𝑗)) → 𝑚 ∈ ℂ)
13 simplrr 526 . . . . . . . . 9 (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴𝐴 < 𝑗)) → 𝑗 ∈ ℤ)
1413zcnd 9197 . . . . . . . 8 (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴𝐴 < 𝑗)) → 𝑗 ∈ ℂ)
1512, 14pncan3d 8099 . . . . . . 7 (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴𝐴 < 𝑗)) → (𝑚 + (𝑗𝑚)) = 𝑗)
1611, 15breqtrrd 3963 . . . . . 6 (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴𝐴 < 𝑗)) → 𝐴 < (𝑚 + (𝑗𝑚)))
17 breq1 3939 . . . . . . . 8 (𝑦 = 𝑚 → (𝑦𝐴𝑚𝐴))
18 oveq1 5788 . . . . . . . . 9 (𝑦 = 𝑚 → (𝑦 + (𝑗𝑚)) = (𝑚 + (𝑗𝑚)))
1918breq2d 3948 . . . . . . . 8 (𝑦 = 𝑚 → (𝐴 < (𝑦 + (𝑗𝑚)) ↔ 𝐴 < (𝑚 + (𝑗𝑚))))
2017, 19anbi12d 465 . . . . . . 7 (𝑦 = 𝑚 → ((𝑦𝐴𝐴 < (𝑦 + (𝑗𝑚))) ↔ (𝑚𝐴𝐴 < (𝑚 + (𝑗𝑚)))))
2120rspcev 2792 . . . . . 6 ((𝑚 ∈ ℤ ∧ (𝑚𝐴𝐴 < (𝑚 + (𝑗𝑚)))) → ∃𝑦 ∈ ℤ (𝑦𝐴𝐴 < (𝑦 + (𝑗𝑚))))
226, 10, 16, 21syl12anc 1215 . . . . 5 (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴𝐴 < 𝑗)) → ∃𝑦 ∈ ℤ (𝑦𝐴𝐴 < (𝑦 + (𝑗𝑚))))
2313zred 9196 . . . . . . . 8 (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴𝐴 < 𝑗)) → 𝑗 ∈ ℝ)
247, 8, 23, 9, 11lttrd 7911 . . . . . . 7 (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴𝐴 < 𝑗)) → 𝑚 < 𝑗)
25 znnsub 9128 . . . . . . . 8 ((𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ) → (𝑚 < 𝑗 ↔ (𝑗𝑚) ∈ ℕ))
2625ad2antlr 481 . . . . . . 7 (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴𝐴 < 𝑗)) → (𝑚 < 𝑗 ↔ (𝑗𝑚) ∈ ℕ))
2724, 26mpbid 146 . . . . . 6 (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴𝐴 < 𝑗)) → (𝑗𝑚) ∈ ℕ)
28 exbtwnzlemex.tri . . . . . . . . . 10 ((𝜑𝑛 ∈ ℤ) → (𝑛𝐴𝐴 < 𝑛))
2928ralrimiva 2508 . . . . . . . . 9 (𝜑 → ∀𝑛 ∈ ℤ (𝑛𝐴𝐴 < 𝑛))
30 breq1 3939 . . . . . . . . . . 11 (𝑛 = 𝑎 → (𝑛𝐴𝑎𝐴))
31 breq2 3940 . . . . . . . . . . 11 (𝑛 = 𝑎 → (𝐴 < 𝑛𝐴 < 𝑎))
3230, 31orbi12d 783 . . . . . . . . . 10 (𝑛 = 𝑎 → ((𝑛𝐴𝐴 < 𝑛) ↔ (𝑎𝐴𝐴 < 𝑎)))
3332cbvralv 2657 . . . . . . . . 9 (∀𝑛 ∈ ℤ (𝑛𝐴𝐴 < 𝑛) ↔ ∀𝑎 ∈ ℤ (𝑎𝐴𝐴 < 𝑎))
3429, 33sylib 121 . . . . . . . 8 (𝜑 → ∀𝑎 ∈ ℤ (𝑎𝐴𝐴 < 𝑎))
3534ad2antrr 480 . . . . . . 7 (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴𝐴 < 𝑗)) → ∀𝑎 ∈ ℤ (𝑎𝐴𝐴 < 𝑎))
3635r19.21bi 2523 . . . . . 6 ((((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴𝐴 < 𝑗)) ∧ 𝑎 ∈ ℤ) → (𝑎𝐴𝐴 < 𝑎))
3727, 8, 36exbtwnzlemshrink 10056 . . . . 5 ((((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴𝐴 < 𝑗)) ∧ ∃𝑦 ∈ ℤ (𝑦𝐴𝐴 < (𝑦 + (𝑗𝑚)))) → ∃𝑥 ∈ ℤ (𝑥𝐴𝐴 < (𝑥 + 1)))
3822, 37mpdan 418 . . . 4 (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴𝐴 < 𝑗)) → ∃𝑥 ∈ ℤ (𝑥𝐴𝐴 < (𝑥 + 1)))
3938ex 114 . . 3 ((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → ((𝑚 < 𝐴𝐴 < 𝑗) → ∃𝑥 ∈ ℤ (𝑥𝐴𝐴 < (𝑥 + 1))))
4039rexlimdvva 2560 . 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 103  wb 104  wo 698  wcel 1481  wral 2417  wrex 2418   class class class wbr 3936  (class class class)co 5781  cr 7642  1c1 7644   + caddc 7646   < clt 7823  cle 7824  cmin 7956  cn 8743  cz 9077
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-pow 4105  ax-pr 4138  ax-un 4362  ax-setind 4459  ax-cnex 7734  ax-resscn 7735  ax-1cn 7736  ax-1re 7737  ax-icn 7738  ax-addcl 7739  ax-addrcl 7740  ax-mulcl 7741  ax-addcom 7743  ax-addass 7745  ax-distr 7747  ax-i2m1 7748  ax-0lt1 7749  ax-0id 7751  ax-rnegex 7752  ax-cnre 7754  ax-pre-ltirr 7755  ax-pre-ltwlin 7756  ax-pre-lttrn 7757  ax-pre-ltadd 7759  ax-arch 7762
This theorem depends on definitions:  df-bi 116  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2913  df-dif 3077  df-un 3079  df-in 3081  df-ss 3088  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-int 3779  df-br 3937  df-opab 3997  df-id 4222  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-iota 5095  df-fun 5132  df-fv 5138  df-riota 5737  df-ov 5784  df-oprab 5785  df-mpo 5786  df-pnf 7825  df-mnf 7826  df-xr 7827  df-ltxr 7828  df-le 7829  df-sub 7958  df-neg 7959  df-inn 8744  df-n0 9001  df-z 9078
This theorem is referenced by:  qbtwnz  10059  apbtwnz  10077
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