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Theorem rebtwn2z 9555
Description: A real number can be bounded by integers above and below which are two apart.

The proof starts by finding two integers which are less than and greater than the given real number. 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 weak linearity, and iterating until the range consists of integers which are two apart. (Contributed by Jim Kingdon, 13-Oct-2021.)

Assertion
Ref Expression
rebtwn2z (𝐴 ∈ ℝ → ∃𝑥 ∈ ℤ (𝑥 < 𝐴𝐴 < (𝑥 + 2)))
Distinct variable group:   𝑥,𝐴

Proof of Theorem rebtwn2z
Dummy variables 𝑚 𝑛 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 btwnz 8761 . . 3 (𝐴 ∈ ℝ → (∃𝑚 ∈ ℤ 𝑚 < 𝐴 ∧ ∃𝑛 ∈ ℤ 𝐴 < 𝑛))
2 reeanv 2529 . . 3 (∃𝑚 ∈ ℤ ∃𝑛 ∈ ℤ (𝑚 < 𝐴𝐴 < 𝑛) ↔ (∃𝑚 ∈ ℤ 𝑚 < 𝐴 ∧ ∃𝑛 ∈ ℤ 𝐴 < 𝑛))
31, 2sylibr 132 . 2 (𝐴 ∈ ℝ → ∃𝑚 ∈ ℤ ∃𝑛 ∈ ℤ (𝑚 < 𝐴𝐴 < 𝑛))
4 simpll 496 . . . . 5 (((𝐴 ∈ ℝ ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ (𝑚 < 𝐴𝐴 < 𝑛)) → 𝐴 ∈ ℝ)
5 simplrl 502 . . . . . . . . 9 (((𝐴 ∈ ℝ ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ (𝑚 < 𝐴𝐴 < 𝑛)) → 𝑚 ∈ ℤ)
65zred 8764 . . . . . . . 8 (((𝐴 ∈ ℝ ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ (𝑚 < 𝐴𝐴 < 𝑛)) → 𝑚 ∈ ℝ)
7 simplrr 503 . . . . . . . . 9 (((𝐴 ∈ ℝ ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ (𝑚 < 𝐴𝐴 < 𝑛)) → 𝑛 ∈ ℤ)
87zred 8764 . . . . . . . 8 (((𝐴 ∈ ℝ ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ (𝑚 < 𝐴𝐴 < 𝑛)) → 𝑛 ∈ ℝ)
9 simprl 498 . . . . . . . 8 (((𝐴 ∈ ℝ ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ (𝑚 < 𝐴𝐴 < 𝑛)) → 𝑚 < 𝐴)
10 simprr 499 . . . . . . . 8 (((𝐴 ∈ ℝ ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ (𝑚 < 𝐴𝐴 < 𝑛)) → 𝐴 < 𝑛)
116, 4, 8, 9, 10lttrd 7512 . . . . . . 7 (((𝐴 ∈ ℝ ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ (𝑚 < 𝐴𝐴 < 𝑛)) → 𝑚 < 𝑛)
12 znnsub 8697 . . . . . . . 8 ((𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (𝑚 < 𝑛 ↔ (𝑛𝑚) ∈ ℕ))
1312ad2antlr 473 . . . . . . 7 (((𝐴 ∈ ℝ ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ (𝑚 < 𝐴𝐴 < 𝑛)) → (𝑚 < 𝑛 ↔ (𝑛𝑚) ∈ ℕ))
1411, 13mpbid 145 . . . . . 6 (((𝐴 ∈ ℝ ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ (𝑚 < 𝐴𝐴 < 𝑛)) → (𝑛𝑚) ∈ ℕ)
15 elnnuz 8950 . . . . . . . 8 ((𝑛𝑚) ∈ ℕ ↔ (𝑛𝑚) ∈ (ℤ‘1))
16 eluzp1p1 8939 . . . . . . . 8 ((𝑛𝑚) ∈ (ℤ‘1) → ((𝑛𝑚) + 1) ∈ (ℤ‘(1 + 1)))
1715, 16sylbi 119 . . . . . . 7 ((𝑛𝑚) ∈ ℕ → ((𝑛𝑚) + 1) ∈ (ℤ‘(1 + 1)))
18 df-2 8375 . . . . . . . 8 2 = (1 + 1)
1918fveq2i 5256 . . . . . . 7 (ℤ‘2) = (ℤ‘(1 + 1))
2017, 19syl6eleqr 2176 . . . . . 6 ((𝑛𝑚) ∈ ℕ → ((𝑛𝑚) + 1) ∈ (ℤ‘2))
2114, 20syl 14 . . . . 5 (((𝐴 ∈ ℝ ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ (𝑚 < 𝐴𝐴 < 𝑛)) → ((𝑛𝑚) + 1) ∈ (ℤ‘2))
225zcnd 8765 . . . . . . . . 9 (((𝐴 ∈ ℝ ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ (𝑚 < 𝐴𝐴 < 𝑛)) → 𝑚 ∈ ℂ)
237zcnd 8765 . . . . . . . . 9 (((𝐴 ∈ ℝ ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ (𝑚 < 𝐴𝐴 < 𝑛)) → 𝑛 ∈ ℂ)
2422, 23pncan3d 7699 . . . . . . . 8 (((𝐴 ∈ ℝ ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ (𝑚 < 𝐴𝐴 < 𝑛)) → (𝑚 + (𝑛𝑚)) = 𝑛)
2524, 8eqeltrd 2159 . . . . . . 7 (((𝐴 ∈ ℝ ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ (𝑚 < 𝐴𝐴 < 𝑛)) → (𝑚 + (𝑛𝑚)) ∈ ℝ)
268, 6resubcld 7762 . . . . . . . . 9 (((𝐴 ∈ ℝ ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ (𝑚 < 𝐴𝐴 < 𝑛)) → (𝑛𝑚) ∈ ℝ)
27 1red 7406 . . . . . . . . 9 (((𝐴 ∈ ℝ ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ (𝑚 < 𝐴𝐴 < 𝑛)) → 1 ∈ ℝ)
2826, 27readdcld 7420 . . . . . . . 8 (((𝐴 ∈ ℝ ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ (𝑚 < 𝐴𝐴 < 𝑛)) → ((𝑛𝑚) + 1) ∈ ℝ)
296, 28readdcld 7420 . . . . . . 7 (((𝐴 ∈ ℝ ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ (𝑚 < 𝐴𝐴 < 𝑛)) → (𝑚 + ((𝑛𝑚) + 1)) ∈ ℝ)
3010, 24breqtrrd 3837 . . . . . . 7 (((𝐴 ∈ ℝ ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ (𝑚 < 𝐴𝐴 < 𝑛)) → 𝐴 < (𝑚 + (𝑛𝑚)))
3126ltp1d 8285 . . . . . . . 8 (((𝐴 ∈ ℝ ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ (𝑚 < 𝐴𝐴 < 𝑛)) → (𝑛𝑚) < ((𝑛𝑚) + 1))
3226, 28, 6, 31ltadd2dd 7803 . . . . . . 7 (((𝐴 ∈ ℝ ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ (𝑚 < 𝐴𝐴 < 𝑛)) → (𝑚 + (𝑛𝑚)) < (𝑚 + ((𝑛𝑚) + 1)))
334, 25, 29, 30, 32lttrd 7512 . . . . . 6 (((𝐴 ∈ ℝ ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ (𝑚 < 𝐴𝐴 < 𝑛)) → 𝐴 < (𝑚 + ((𝑛𝑚) + 1)))
34 breq1 3814 . . . . . . . 8 (𝑦 = 𝑚 → (𝑦 < 𝐴𝑚 < 𝐴))
35 oveq1 5598 . . . . . . . . 9 (𝑦 = 𝑚 → (𝑦 + ((𝑛𝑚) + 1)) = (𝑚 + ((𝑛𝑚) + 1)))
3635breq2d 3823 . . . . . . . 8 (𝑦 = 𝑚 → (𝐴 < (𝑦 + ((𝑛𝑚) + 1)) ↔ 𝐴 < (𝑚 + ((𝑛𝑚) + 1))))
3734, 36anbi12d 457 . . . . . . 7 (𝑦 = 𝑚 → ((𝑦 < 𝐴𝐴 < (𝑦 + ((𝑛𝑚) + 1))) ↔ (𝑚 < 𝐴𝐴 < (𝑚 + ((𝑛𝑚) + 1)))))
3837rspcev 2712 . . . . . 6 ((𝑚 ∈ ℤ ∧ (𝑚 < 𝐴𝐴 < (𝑚 + ((𝑛𝑚) + 1)))) → ∃𝑦 ∈ ℤ (𝑦 < 𝐴𝐴 < (𝑦 + ((𝑛𝑚) + 1))))
395, 9, 33, 38syl12anc 1168 . . . . 5 (((𝐴 ∈ ℝ ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ (𝑚 < 𝐴𝐴 < 𝑛)) → ∃𝑦 ∈ ℤ (𝑦 < 𝐴𝐴 < (𝑦 + ((𝑛𝑚) + 1))))
40 rebtwn2zlemshrink 9554 . . . . 5 ((𝐴 ∈ ℝ ∧ ((𝑛𝑚) + 1) ∈ (ℤ‘2) ∧ ∃𝑦 ∈ ℤ (𝑦 < 𝐴𝐴 < (𝑦 + ((𝑛𝑚) + 1)))) → ∃𝑥 ∈ ℤ (𝑥 < 𝐴𝐴 < (𝑥 + 2)))
414, 21, 39, 40syl3anc 1170 . . . 4 (((𝐴 ∈ ℝ ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ (𝑚 < 𝐴𝐴 < 𝑛)) → ∃𝑥 ∈ ℤ (𝑥 < 𝐴𝐴 < (𝑥 + 2)))
4241ex 113 . . 3 ((𝐴 ∈ ℝ ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) → ((𝑚 < 𝐴𝐴 < 𝑛) → ∃𝑥 ∈ ℤ (𝑥 < 𝐴𝐴 < (𝑥 + 2))))
4342rexlimdvva 2490 . 2 (𝐴 ∈ ℝ → (∃𝑚 ∈ ℤ ∃𝑛 ∈ ℤ (𝑚 < 𝐴𝐴 < 𝑛) → ∃𝑥 ∈ ℤ (𝑥 < 𝐴𝐴 < (𝑥 + 2))))
443, 43mpd 13 1 (𝐴 ∈ ℝ → ∃𝑥 ∈ ℤ (𝑥 < 𝐴𝐴 < (𝑥 + 2)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  wcel 1434  wrex 2354   class class class wbr 3811  cfv 4969  (class class class)co 5591  cr 7252  1c1 7254   + caddc 7256   < clt 7425  cmin 7556  cn 8316  2c2 8366  cz 8646  cuz 8914
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 3922  ax-pow 3974  ax-pr 4000  ax-un 4224  ax-setind 4316  ax-cnex 7339  ax-resscn 7340  ax-1cn 7341  ax-1re 7342  ax-icn 7343  ax-addcl 7344  ax-addrcl 7345  ax-mulcl 7346  ax-addcom 7348  ax-addass 7350  ax-distr 7352  ax-i2m1 7353  ax-0lt1 7354  ax-0id 7356  ax-rnegex 7357  ax-cnre 7359  ax-pre-ltirr 7360  ax-pre-ltwlin 7361  ax-pre-lttrn 7362  ax-pre-ltadd 7364  ax-arch 7367
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 2614  df-sbc 2827  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-int 3663  df-br 3812  df-opab 3866  df-mpt 3867  df-id 4084  df-xp 4407  df-rel 4408  df-cnv 4409  df-co 4410  df-dm 4411  df-rn 4412  df-res 4413  df-ima 4414  df-iota 4934  df-fun 4971  df-fn 4972  df-f 4973  df-fv 4977  df-riota 5547  df-ov 5594  df-oprab 5595  df-mpt2 5596  df-pnf 7427  df-mnf 7428  df-xr 7429  df-ltxr 7430  df-le 7431  df-sub 7558  df-neg 7559  df-inn 8317  df-2 8375  df-n0 8566  df-z 8647  df-uz 8915
This theorem is referenced by:  qbtwnre  9557
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