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Theorem exbtwnzlemex 10249
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 9371 . . . 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 9374 . . . . . . 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 8075 . . . . . 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 9375 . . . . . . . 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 9375 . . . . . . . 8 (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴𝐴 < 𝑗)) → 𝑗 ∈ ℂ)
1512, 14pncan3d 8270 . . . . . . 7 (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴𝐴 < 𝑗)) → (𝑚 + (𝑗𝑚)) = 𝑗)
1611, 15breqtrrd 4031 . . . . . 6 (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴𝐴 < 𝑗)) → 𝐴 < (𝑚 + (𝑗𝑚)))
17 breq1 4006 . . . . . . . 8 (𝑦 = 𝑚 → (𝑦𝐴𝑚𝐴))
18 oveq1 5881 . . . . . . . . 9 (𝑦 = 𝑚 → (𝑦 + (𝑗𝑚)) = (𝑚 + (𝑗𝑚)))
1918breq2d 4015 . . . . . . . 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 9374 . . . . . . . 8 (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴𝐴 < 𝑗)) → 𝑗 ∈ ℝ)
247, 8, 23, 9, 11lttrd 8082 . . . . . . 7 (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴𝐴 < 𝑗)) → 𝑚 < 𝑗)
25 znnsub 9303 . . . . . . . 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 4006 . . . . . . . . . . 11 (𝑛 = 𝑎 → (𝑛𝐴𝑎𝐴))
31 breq2 4007 . . . . . . . . . . 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 10248 . . . . 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 4003  (class class class)co 5874  cr 7809  1c1 7811   + caddc 7813   < clt 7991  cle 7992  cmin 8127  cn 8918  cz 9252
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 4121  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-cnex 7901  ax-resscn 7902  ax-1cn 7903  ax-1re 7904  ax-icn 7905  ax-addcl 7906  ax-addrcl 7907  ax-mulcl 7908  ax-addcom 7910  ax-addass 7912  ax-distr 7914  ax-i2m1 7915  ax-0lt1 7916  ax-0id 7918  ax-rnegex 7919  ax-cnre 7921  ax-pre-ltirr 7922  ax-pre-ltwlin 7923  ax-pre-lttrn 7924  ax-pre-ltadd 7926  ax-arch 7929
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 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-br 4004  df-opab 4065  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-iota 5178  df-fun 5218  df-fv 5224  df-riota 5830  df-ov 5877  df-oprab 5878  df-mpo 5879  df-pnf 7993  df-mnf 7994  df-xr 7995  df-ltxr 7996  df-le 7997  df-sub 8129  df-neg 8130  df-inn 8919  df-n0 9176  df-z 9253
This theorem is referenced by:  qbtwnz  10251  apbtwnz  10273
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