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Theorem exbtwnzlemex 9626
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 8835 . . . 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 2536 . . 3 (∃𝑚 ∈ ℤ ∃𝑗 ∈ ℤ (𝑚 < 𝐴𝐴 < 𝑗) ↔ (∃𝑚 ∈ ℤ 𝑚 < 𝐴 ∧ ∃𝑗 ∈ ℤ 𝐴 < 𝑗))
53, 4sylibr 132 . 2 (𝜑 → ∃𝑚 ∈ ℤ ∃𝑗 ∈ ℤ (𝑚 < 𝐴𝐴 < 𝑗))
6 simplrl 502 . . . . . 6 (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴𝐴 < 𝑗)) → 𝑚 ∈ ℤ)
76zred 8838 . . . . . . 7 (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴𝐴 < 𝑗)) → 𝑚 ∈ ℝ)
81ad2antrr 472 . . . . . . 7 (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴𝐴 < 𝑗)) → 𝐴 ∈ ℝ)
9 simprl 498 . . . . . . 7 (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴𝐴 < 𝑗)) → 𝑚 < 𝐴)
107, 8, 9ltled 7581 . . . . . 6 (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴𝐴 < 𝑗)) → 𝑚𝐴)
11 simprr 499 . . . . . . 7 (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴𝐴 < 𝑗)) → 𝐴 < 𝑗)
126zcnd 8839 . . . . . . . 8 (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴𝐴 < 𝑗)) → 𝑚 ∈ ℂ)
13 simplrr 503 . . . . . . . . 9 (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴𝐴 < 𝑗)) → 𝑗 ∈ ℤ)
1413zcnd 8839 . . . . . . . 8 (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴𝐴 < 𝑗)) → 𝑗 ∈ ℂ)
1512, 14pncan3d 7775 . . . . . . 7 (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴𝐴 < 𝑗)) → (𝑚 + (𝑗𝑚)) = 𝑗)
1611, 15breqtrrd 3863 . . . . . 6 (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴𝐴 < 𝑗)) → 𝐴 < (𝑚 + (𝑗𝑚)))
17 breq1 3840 . . . . . . . 8 (𝑦 = 𝑚 → (𝑦𝐴𝑚𝐴))
18 oveq1 5641 . . . . . . . . 9 (𝑦 = 𝑚 → (𝑦 + (𝑗𝑚)) = (𝑚 + (𝑗𝑚)))
1918breq2d 3849 . . . . . . . 8 (𝑦 = 𝑚 → (𝐴 < (𝑦 + (𝑗𝑚)) ↔ 𝐴 < (𝑚 + (𝑗𝑚))))
2017, 19anbi12d 457 . . . . . . 7 (𝑦 = 𝑚 → ((𝑦𝐴𝐴 < (𝑦 + (𝑗𝑚))) ↔ (𝑚𝐴𝐴 < (𝑚 + (𝑗𝑚)))))
2120rspcev 2722 . . . . . 6 ((𝑚 ∈ ℤ ∧ (𝑚𝐴𝐴 < (𝑚 + (𝑗𝑚)))) → ∃𝑦 ∈ ℤ (𝑦𝐴𝐴 < (𝑦 + (𝑗𝑚))))
226, 10, 16, 21syl12anc 1172 . . . . 5 (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴𝐴 < 𝑗)) → ∃𝑦 ∈ ℤ (𝑦𝐴𝐴 < (𝑦 + (𝑗𝑚))))
2313zred 8838 . . . . . . . 8 (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴𝐴 < 𝑗)) → 𝑗 ∈ ℝ)
247, 8, 23, 9, 11lttrd 7588 . . . . . . 7 (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴𝐴 < 𝑗)) → 𝑚 < 𝑗)
25 znnsub 8771 . . . . . . . 8 ((𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ) → (𝑚 < 𝑗 ↔ (𝑗𝑚) ∈ ℕ))
2625ad2antlr 473 . . . . . . 7 (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴𝐴 < 𝑗)) → (𝑚 < 𝑗 ↔ (𝑗𝑚) ∈ ℕ))
2724, 26mpbid 145 . . . . . 6 (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴𝐴 < 𝑗)) → (𝑗𝑚) ∈ ℕ)
28 exbtwnzlemex.tri . . . . . . . . . 10 ((𝜑𝑛 ∈ ℤ) → (𝑛𝐴𝐴 < 𝑛))
2928ralrimiva 2446 . . . . . . . . 9 (𝜑 → ∀𝑛 ∈ ℤ (𝑛𝐴𝐴 < 𝑛))
30 breq1 3840 . . . . . . . . . . 11 (𝑛 = 𝑎 → (𝑛𝐴𝑎𝐴))
31 breq2 3841 . . . . . . . . . . 11 (𝑛 = 𝑎 → (𝐴 < 𝑛𝐴 < 𝑎))
3230, 31orbi12d 742 . . . . . . . . . 10 (𝑛 = 𝑎 → ((𝑛𝐴𝐴 < 𝑛) ↔ (𝑎𝐴𝐴 < 𝑎)))
3332cbvralv 2590 . . . . . . . . 9 (∀𝑛 ∈ ℤ (𝑛𝐴𝐴 < 𝑛) ↔ ∀𝑎 ∈ ℤ (𝑎𝐴𝐴 < 𝑎))
3429, 33sylib 120 . . . . . . . 8 (𝜑 → ∀𝑎 ∈ ℤ (𝑎𝐴𝐴 < 𝑎))
3534ad2antrr 472 . . . . . . 7 (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴𝐴 < 𝑗)) → ∀𝑎 ∈ ℤ (𝑎𝐴𝐴 < 𝑎))
3635r19.21bi 2461 . . . . . 6 ((((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴𝐴 < 𝑗)) ∧ 𝑎 ∈ ℤ) → (𝑎𝐴𝐴 < 𝑎))
3727, 8, 36exbtwnzlemshrink 9625 . . . . 5 ((((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴𝐴 < 𝑗)) ∧ ∃𝑦 ∈ ℤ (𝑦𝐴𝐴 < (𝑦 + (𝑗𝑚)))) → ∃𝑥 ∈ ℤ (𝑥𝐴𝐴 < (𝑥 + 1)))
3822, 37mpdan 412 . . . 4 (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴𝐴 < 𝑗)) → ∃𝑥 ∈ ℤ (𝑥𝐴𝐴 < (𝑥 + 1)))
3938ex 113 . . 3 ((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → ((𝑚 < 𝐴𝐴 < 𝑗) → ∃𝑥 ∈ ℤ (𝑥𝐴𝐴 < (𝑥 + 1))))
4039rexlimdvva 2496 . 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 102  wb 103  wo 664  wcel 1438  wral 2359  wrex 2360   class class class wbr 3837  (class class class)co 5634  cr 7328  1c1 7330   + caddc 7332   < clt 7501  cle 7502  cmin 7632  cn 8394  cz 8720
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-cnex 7415  ax-resscn 7416  ax-1cn 7417  ax-1re 7418  ax-icn 7419  ax-addcl 7420  ax-addrcl 7421  ax-mulcl 7422  ax-addcom 7424  ax-addass 7426  ax-distr 7428  ax-i2m1 7429  ax-0lt1 7430  ax-0id 7432  ax-rnegex 7433  ax-cnre 7435  ax-pre-ltirr 7436  ax-pre-ltwlin 7437  ax-pre-lttrn 7438  ax-pre-ltadd 7440  ax-arch 7443
This theorem depends on definitions:  df-bi 115  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2839  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-br 3838  df-opab 3892  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-iota 4967  df-fun 5004  df-fv 5010  df-riota 5590  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-pnf 7503  df-mnf 7504  df-xr 7505  df-ltxr 7506  df-le 7507  df-sub 7634  df-neg 7635  df-inn 8395  df-n0 8644  df-z 8721
This theorem is referenced by:  qbtwnz  9628  apbtwnz  9646
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