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Theorem exbtwnzlemex 9406
 Description: Existence of an integer so that a given real number is between the integer and its successor. The real number must satisfy the 𝑛 ≤ 𝐴 ∨ 𝐴 < 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 𝑛 ≤ 𝐴 ∨ 𝐴 < 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 8617 . . . 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 2528 . . 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 8620 . . . . . . 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 7365 . . . . . 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 8621 . . . . . . . 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 8621 . . . . . . . 8 (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴𝐴 < 𝑗)) → 𝑗 ∈ ℂ)
1512, 14pncan3d 7559 . . . . . . 7 (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴𝐴 < 𝑗)) → (𝑚 + (𝑗𝑚)) = 𝑗)
1611, 15breqtrrd 3831 . . . . . 6 (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴𝐴 < 𝑗)) → 𝐴 < (𝑚 + (𝑗𝑚)))
17 breq1 3808 . . . . . . . 8 (𝑦 = 𝑚 → (𝑦𝐴𝑚𝐴))
18 oveq1 5571 . . . . . . . . 9 (𝑦 = 𝑚 → (𝑦 + (𝑗𝑚)) = (𝑚 + (𝑗𝑚)))
1918breq2d 3817 . . . . . . . 8 (𝑦 = 𝑚 → (𝐴 < (𝑦 + (𝑗𝑚)) ↔ 𝐴 < (𝑚 + (𝑗𝑚))))
2017, 19anbi12d 457 . . . . . . 7 (𝑦 = 𝑚 → ((𝑦𝐴𝐴 < (𝑦 + (𝑗𝑚))) ↔ (𝑚𝐴𝐴 < (𝑚 + (𝑗𝑚)))))
2120rspcev 2710 . . . . . 6 ((𝑚 ∈ ℤ ∧ (𝑚𝐴𝐴 < (𝑚 + (𝑗𝑚)))) → ∃𝑦 ∈ ℤ (𝑦𝐴𝐴 < (𝑦 + (𝑗𝑚))))
226, 10, 16, 21syl12anc 1168 . . . . 5 (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴𝐴 < 𝑗)) → ∃𝑦 ∈ ℤ (𝑦𝐴𝐴 < (𝑦 + (𝑗𝑚))))
2313zred 8620 . . . . . . . 8 (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴𝐴 < 𝑗)) → 𝑗 ∈ ℝ)
247, 8, 23, 9, 11lttrd 7372 . . . . . . 7 (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴𝐴 < 𝑗)) → 𝑚 < 𝑗)
25 znnsub 8553 . . . . . . . 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 2439 . . . . . . . . 9 (𝜑 → ∀𝑛 ∈ ℤ (𝑛𝐴𝐴 < 𝑛))
30 breq1 3808 . . . . . . . . . . 11 (𝑛 = 𝑎 → (𝑛𝐴𝑎𝐴))
31 breq2 3809 . . . . . . . . . . 11 (𝑛 = 𝑎 → (𝐴 < 𝑛𝐴 < 𝑎))
3230, 31orbi12d 740 . . . . . . . . . 10 (𝑛 = 𝑎 → ((𝑛𝐴𝐴 < 𝑛) ↔ (𝑎𝐴𝐴 < 𝑎)))
3332cbvralv 2582 . . . . . . . . 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 2454 . . . . . 6 ((((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑗 ∈ ℤ)) ∧ (𝑚 < 𝐴𝐴 < 𝑗)) ∧ 𝑎 ∈ ℤ) → (𝑎𝐴𝐴 < 𝑎))
3727, 8, 36exbtwnzlemshrink 9405 . . . . 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 2489 . 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 662   ∈ wcel 1434  ∀wral 2353  ∃wrex 2354   class class class wbr 3805  (class class class)co 5564  ℝcr 7112  1c1 7114   + caddc 7116   < clt 7285   ≤ cle 7286   − cmin 7416  ℕcn 8176  ℤcz 8502 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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-setind 4308  ax-cnex 7199  ax-resscn 7200  ax-1cn 7201  ax-1re 7202  ax-icn 7203  ax-addcl 7204  ax-addrcl 7205  ax-mulcl 7206  ax-addcom 7208  ax-addass 7210  ax-distr 7212  ax-i2m1 7213  ax-0lt1 7214  ax-0id 7216  ax-rnegex 7217  ax-cnre 7219  ax-pre-ltirr 7220  ax-pre-ltwlin 7221  ax-pre-lttrn 7222  ax-pre-ltadd 7224  ax-arch 7227 This theorem depends on definitions:  df-bi 115  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2612  df-sbc 2825  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-int 3657  df-br 3806  df-opab 3860  df-id 4076  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-iota 4917  df-fun 4954  df-fv 4960  df-riota 5520  df-ov 5567  df-oprab 5568  df-mpt2 5569  df-pnf 7287  df-mnf 7288  df-xr 7289  df-ltxr 7290  df-le 7291  df-sub 7418  df-neg 7419  df-inn 8177  df-n0 8426  df-z 8503 This theorem is referenced by:  qbtwnz  9408  apbtwnz  9426
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