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Theorem fzmaxdif 37865
Description: Bound on the difference between two integers constrained to two possibly overlapping finite ranges. (Contributed by Stefan O'Rear, 4-Oct-2014.)
Assertion
Ref Expression
fzmaxdif (((𝐶 ∈ ℤ ∧ 𝐴 ∈ (𝐵...𝐶)) ∧ (𝐹 ∈ ℤ ∧ 𝐷 ∈ (𝐸...𝐹)) ∧ (𝐶𝐸) ≤ (𝐹𝐵)) → (abs‘(𝐴𝐷)) ≤ (𝐹𝐵))

Proof of Theorem fzmaxdif
StepHypRef Expression
1 simp2r 1108 . . . . . 6 (((𝐶 ∈ ℤ ∧ 𝐴 ∈ (𝐵...𝐶)) ∧ (𝐹 ∈ ℤ ∧ 𝐷 ∈ (𝐸...𝐹)) ∧ (𝐶𝐸) ≤ (𝐹𝐵)) → 𝐷 ∈ (𝐸...𝐹))
2 elfzelz 12380 . . . . . 6 (𝐷 ∈ (𝐸...𝐹) → 𝐷 ∈ ℤ)
31, 2syl 17 . . . . 5 (((𝐶 ∈ ℤ ∧ 𝐴 ∈ (𝐵...𝐶)) ∧ (𝐹 ∈ ℤ ∧ 𝐷 ∈ (𝐸...𝐹)) ∧ (𝐶𝐸) ≤ (𝐹𝐵)) → 𝐷 ∈ ℤ)
43zred 11520 . . . 4 (((𝐶 ∈ ℤ ∧ 𝐴 ∈ (𝐵...𝐶)) ∧ (𝐹 ∈ ℤ ∧ 𝐷 ∈ (𝐸...𝐹)) ∧ (𝐶𝐸) ≤ (𝐹𝐵)) → 𝐷 ∈ ℝ)
5 simp2l 1107 . . . . . 6 (((𝐶 ∈ ℤ ∧ 𝐴 ∈ (𝐵...𝐶)) ∧ (𝐹 ∈ ℤ ∧ 𝐷 ∈ (𝐸...𝐹)) ∧ (𝐶𝐸) ≤ (𝐹𝐵)) → 𝐹 ∈ ℤ)
65zred 11520 . . . . 5 (((𝐶 ∈ ℤ ∧ 𝐴 ∈ (𝐵...𝐶)) ∧ (𝐹 ∈ ℤ ∧ 𝐷 ∈ (𝐸...𝐹)) ∧ (𝐶𝐸) ≤ (𝐹𝐵)) → 𝐹 ∈ ℝ)
7 simp1r 1106 . . . . . . 7 (((𝐶 ∈ ℤ ∧ 𝐴 ∈ (𝐵...𝐶)) ∧ (𝐹 ∈ ℤ ∧ 𝐷 ∈ (𝐸...𝐹)) ∧ (𝐶𝐸) ≤ (𝐹𝐵)) → 𝐴 ∈ (𝐵...𝐶))
8 elfzel1 12379 . . . . . . 7 (𝐴 ∈ (𝐵...𝐶) → 𝐵 ∈ ℤ)
97, 8syl 17 . . . . . 6 (((𝐶 ∈ ℤ ∧ 𝐴 ∈ (𝐵...𝐶)) ∧ (𝐹 ∈ ℤ ∧ 𝐷 ∈ (𝐸...𝐹)) ∧ (𝐶𝐸) ≤ (𝐹𝐵)) → 𝐵 ∈ ℤ)
109zred 11520 . . . . 5 (((𝐶 ∈ ℤ ∧ 𝐴 ∈ (𝐵...𝐶)) ∧ (𝐹 ∈ ℤ ∧ 𝐷 ∈ (𝐸...𝐹)) ∧ (𝐶𝐸) ≤ (𝐹𝐵)) → 𝐵 ∈ ℝ)
116, 10resubcld 10496 . . . 4 (((𝐶 ∈ ℤ ∧ 𝐴 ∈ (𝐵...𝐶)) ∧ (𝐹 ∈ ℤ ∧ 𝐷 ∈ (𝐸...𝐹)) ∧ (𝐶𝐸) ≤ (𝐹𝐵)) → (𝐹𝐵) ∈ ℝ)
124, 11resubcld 10496 . . 3 (((𝐶 ∈ ℤ ∧ 𝐴 ∈ (𝐵...𝐶)) ∧ (𝐹 ∈ ℤ ∧ 𝐷 ∈ (𝐸...𝐹)) ∧ (𝐶𝐸) ≤ (𝐹𝐵)) → (𝐷 − (𝐹𝐵)) ∈ ℝ)
13 elfzelz 12380 . . . . 5 (𝐴 ∈ (𝐵...𝐶) → 𝐴 ∈ ℤ)
147, 13syl 17 . . . 4 (((𝐶 ∈ ℤ ∧ 𝐴 ∈ (𝐵...𝐶)) ∧ (𝐹 ∈ ℤ ∧ 𝐷 ∈ (𝐸...𝐹)) ∧ (𝐶𝐸) ≤ (𝐹𝐵)) → 𝐴 ∈ ℤ)
1514zred 11520 . . 3 (((𝐶 ∈ ℤ ∧ 𝐴 ∈ (𝐵...𝐶)) ∧ (𝐹 ∈ ℤ ∧ 𝐷 ∈ (𝐸...𝐹)) ∧ (𝐶𝐸) ≤ (𝐹𝐵)) → 𝐴 ∈ ℝ)
16 elfzle2 12383 . . . . . 6 (𝐷 ∈ (𝐸...𝐹) → 𝐷𝐹)
171, 16syl 17 . . . . 5 (((𝐶 ∈ ℤ ∧ 𝐴 ∈ (𝐵...𝐶)) ∧ (𝐹 ∈ ℤ ∧ 𝐷 ∈ (𝐸...𝐹)) ∧ (𝐶𝐸) ≤ (𝐹𝐵)) → 𝐷𝐹)
184, 6, 11, 17lesub1dd 10681 . . . 4 (((𝐶 ∈ ℤ ∧ 𝐴 ∈ (𝐵...𝐶)) ∧ (𝐹 ∈ ℤ ∧ 𝐷 ∈ (𝐸...𝐹)) ∧ (𝐶𝐸) ≤ (𝐹𝐵)) → (𝐷 − (𝐹𝐵)) ≤ (𝐹 − (𝐹𝐵)))
196recnd 10106 . . . . 5 (((𝐶 ∈ ℤ ∧ 𝐴 ∈ (𝐵...𝐶)) ∧ (𝐹 ∈ ℤ ∧ 𝐷 ∈ (𝐸...𝐹)) ∧ (𝐶𝐸) ≤ (𝐹𝐵)) → 𝐹 ∈ ℂ)
2010recnd 10106 . . . . 5 (((𝐶 ∈ ℤ ∧ 𝐴 ∈ (𝐵...𝐶)) ∧ (𝐹 ∈ ℤ ∧ 𝐷 ∈ (𝐸...𝐹)) ∧ (𝐶𝐸) ≤ (𝐹𝐵)) → 𝐵 ∈ ℂ)
2119, 20nncand 10435 . . . 4 (((𝐶 ∈ ℤ ∧ 𝐴 ∈ (𝐵...𝐶)) ∧ (𝐹 ∈ ℤ ∧ 𝐷 ∈ (𝐸...𝐹)) ∧ (𝐶𝐸) ≤ (𝐹𝐵)) → (𝐹 − (𝐹𝐵)) = 𝐵)
2218, 21breqtrd 4711 . . 3 (((𝐶 ∈ ℤ ∧ 𝐴 ∈ (𝐵...𝐶)) ∧ (𝐹 ∈ ℤ ∧ 𝐷 ∈ (𝐸...𝐹)) ∧ (𝐶𝐸) ≤ (𝐹𝐵)) → (𝐷 − (𝐹𝐵)) ≤ 𝐵)
23 elfzle1 12382 . . . 4 (𝐴 ∈ (𝐵...𝐶) → 𝐵𝐴)
247, 23syl 17 . . 3 (((𝐶 ∈ ℤ ∧ 𝐴 ∈ (𝐵...𝐶)) ∧ (𝐹 ∈ ℤ ∧ 𝐷 ∈ (𝐸...𝐹)) ∧ (𝐶𝐸) ≤ (𝐹𝐵)) → 𝐵𝐴)
2512, 10, 15, 22, 24letrd 10232 . 2 (((𝐶 ∈ ℤ ∧ 𝐴 ∈ (𝐵...𝐶)) ∧ (𝐹 ∈ ℤ ∧ 𝐷 ∈ (𝐸...𝐹)) ∧ (𝐶𝐸) ≤ (𝐹𝐵)) → (𝐷 − (𝐹𝐵)) ≤ 𝐴)
26 simp1l 1105 . . . 4 (((𝐶 ∈ ℤ ∧ 𝐴 ∈ (𝐵...𝐶)) ∧ (𝐹 ∈ ℤ ∧ 𝐷 ∈ (𝐸...𝐹)) ∧ (𝐶𝐸) ≤ (𝐹𝐵)) → 𝐶 ∈ ℤ)
2726zred 11520 . . 3 (((𝐶 ∈ ℤ ∧ 𝐴 ∈ (𝐵...𝐶)) ∧ (𝐹 ∈ ℤ ∧ 𝐷 ∈ (𝐸...𝐹)) ∧ (𝐶𝐸) ≤ (𝐹𝐵)) → 𝐶 ∈ ℝ)
284, 11readdcld 10107 . . 3 (((𝐶 ∈ ℤ ∧ 𝐴 ∈ (𝐵...𝐶)) ∧ (𝐹 ∈ ℤ ∧ 𝐷 ∈ (𝐸...𝐹)) ∧ (𝐶𝐸) ≤ (𝐹𝐵)) → (𝐷 + (𝐹𝐵)) ∈ ℝ)
29 elfzle2 12383 . . . 4 (𝐴 ∈ (𝐵...𝐶) → 𝐴𝐶)
307, 29syl 17 . . 3 (((𝐶 ∈ ℤ ∧ 𝐴 ∈ (𝐵...𝐶)) ∧ (𝐹 ∈ ℤ ∧ 𝐷 ∈ (𝐸...𝐹)) ∧ (𝐶𝐸) ≤ (𝐹𝐵)) → 𝐴𝐶)
3127, 4resubcld 10496 . . . . . 6 (((𝐶 ∈ ℤ ∧ 𝐴 ∈ (𝐵...𝐶)) ∧ (𝐹 ∈ ℤ ∧ 𝐷 ∈ (𝐸...𝐹)) ∧ (𝐶𝐸) ≤ (𝐹𝐵)) → (𝐶𝐷) ∈ ℝ)
32 elfzel1 12379 . . . . . . . . 9 (𝐷 ∈ (𝐸...𝐹) → 𝐸 ∈ ℤ)
331, 32syl 17 . . . . . . . 8 (((𝐶 ∈ ℤ ∧ 𝐴 ∈ (𝐵...𝐶)) ∧ (𝐹 ∈ ℤ ∧ 𝐷 ∈ (𝐸...𝐹)) ∧ (𝐶𝐸) ≤ (𝐹𝐵)) → 𝐸 ∈ ℤ)
3433zred 11520 . . . . . . 7 (((𝐶 ∈ ℤ ∧ 𝐴 ∈ (𝐵...𝐶)) ∧ (𝐹 ∈ ℤ ∧ 𝐷 ∈ (𝐸...𝐹)) ∧ (𝐶𝐸) ≤ (𝐹𝐵)) → 𝐸 ∈ ℝ)
3527, 34resubcld 10496 . . . . . 6 (((𝐶 ∈ ℤ ∧ 𝐴 ∈ (𝐵...𝐶)) ∧ (𝐹 ∈ ℤ ∧ 𝐷 ∈ (𝐸...𝐹)) ∧ (𝐶𝐸) ≤ (𝐹𝐵)) → (𝐶𝐸) ∈ ℝ)
36 elfzle1 12382 . . . . . . . 8 (𝐷 ∈ (𝐸...𝐹) → 𝐸𝐷)
371, 36syl 17 . . . . . . 7 (((𝐶 ∈ ℤ ∧ 𝐴 ∈ (𝐵...𝐶)) ∧ (𝐹 ∈ ℤ ∧ 𝐷 ∈ (𝐸...𝐹)) ∧ (𝐶𝐸) ≤ (𝐹𝐵)) → 𝐸𝐷)
3834, 4, 27, 37lesub2dd 10682 . . . . . 6 (((𝐶 ∈ ℤ ∧ 𝐴 ∈ (𝐵...𝐶)) ∧ (𝐹 ∈ ℤ ∧ 𝐷 ∈ (𝐸...𝐹)) ∧ (𝐶𝐸) ≤ (𝐹𝐵)) → (𝐶𝐷) ≤ (𝐶𝐸))
39 simp3 1083 . . . . . 6 (((𝐶 ∈ ℤ ∧ 𝐴 ∈ (𝐵...𝐶)) ∧ (𝐹 ∈ ℤ ∧ 𝐷 ∈ (𝐸...𝐹)) ∧ (𝐶𝐸) ≤ (𝐹𝐵)) → (𝐶𝐸) ≤ (𝐹𝐵))
4031, 35, 11, 38, 39letrd 10232 . . . . 5 (((𝐶 ∈ ℤ ∧ 𝐴 ∈ (𝐵...𝐶)) ∧ (𝐹 ∈ ℤ ∧ 𝐷 ∈ (𝐸...𝐹)) ∧ (𝐶𝐸) ≤ (𝐹𝐵)) → (𝐶𝐷) ≤ (𝐹𝐵))
4127, 4, 11lesubaddd 10662 . . . . 5 (((𝐶 ∈ ℤ ∧ 𝐴 ∈ (𝐵...𝐶)) ∧ (𝐹 ∈ ℤ ∧ 𝐷 ∈ (𝐸...𝐹)) ∧ (𝐶𝐸) ≤ (𝐹𝐵)) → ((𝐶𝐷) ≤ (𝐹𝐵) ↔ 𝐶 ≤ ((𝐹𝐵) + 𝐷)))
4240, 41mpbid 222 . . . 4 (((𝐶 ∈ ℤ ∧ 𝐴 ∈ (𝐵...𝐶)) ∧ (𝐹 ∈ ℤ ∧ 𝐷 ∈ (𝐸...𝐹)) ∧ (𝐶𝐸) ≤ (𝐹𝐵)) → 𝐶 ≤ ((𝐹𝐵) + 𝐷))
4311recnd 10106 . . . . 5 (((𝐶 ∈ ℤ ∧ 𝐴 ∈ (𝐵...𝐶)) ∧ (𝐹 ∈ ℤ ∧ 𝐷 ∈ (𝐸...𝐹)) ∧ (𝐶𝐸) ≤ (𝐹𝐵)) → (𝐹𝐵) ∈ ℂ)
444recnd 10106 . . . . 5 (((𝐶 ∈ ℤ ∧ 𝐴 ∈ (𝐵...𝐶)) ∧ (𝐹 ∈ ℤ ∧ 𝐷 ∈ (𝐸...𝐹)) ∧ (𝐶𝐸) ≤ (𝐹𝐵)) → 𝐷 ∈ ℂ)
4543, 44addcomd 10276 . . . 4 (((𝐶 ∈ ℤ ∧ 𝐴 ∈ (𝐵...𝐶)) ∧ (𝐹 ∈ ℤ ∧ 𝐷 ∈ (𝐸...𝐹)) ∧ (𝐶𝐸) ≤ (𝐹𝐵)) → ((𝐹𝐵) + 𝐷) = (𝐷 + (𝐹𝐵)))
4642, 45breqtrd 4711 . . 3 (((𝐶 ∈ ℤ ∧ 𝐴 ∈ (𝐵...𝐶)) ∧ (𝐹 ∈ ℤ ∧ 𝐷 ∈ (𝐸...𝐹)) ∧ (𝐶𝐸) ≤ (𝐹𝐵)) → 𝐶 ≤ (𝐷 + (𝐹𝐵)))
4715, 27, 28, 30, 46letrd 10232 . 2 (((𝐶 ∈ ℤ ∧ 𝐴 ∈ (𝐵...𝐶)) ∧ (𝐹 ∈ ℤ ∧ 𝐷 ∈ (𝐸...𝐹)) ∧ (𝐶𝐸) ≤ (𝐹𝐵)) → 𝐴 ≤ (𝐷 + (𝐹𝐵)))
4815, 4, 11absdifled 14217 . 2 (((𝐶 ∈ ℤ ∧ 𝐴 ∈ (𝐵...𝐶)) ∧ (𝐹 ∈ ℤ ∧ 𝐷 ∈ (𝐸...𝐹)) ∧ (𝐶𝐸) ≤ (𝐹𝐵)) → ((abs‘(𝐴𝐷)) ≤ (𝐹𝐵) ↔ ((𝐷 − (𝐹𝐵)) ≤ 𝐴𝐴 ≤ (𝐷 + (𝐹𝐵)))))
4925, 47, 48mpbir2and 977 1 (((𝐶 ∈ ℤ ∧ 𝐴 ∈ (𝐵...𝐶)) ∧ (𝐹 ∈ ℤ ∧ 𝐷 ∈ (𝐸...𝐹)) ∧ (𝐶𝐸) ≤ (𝐹𝐵)) → (abs‘(𝐴𝐷)) ≤ (𝐹𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1054  wcel 2030   class class class wbr 4685  cfv 5926  (class class class)co 6690   + caddc 9977  cle 10113  cmin 10304  cz 11415  ...cfz 12364  abscabs 14018
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-sup 8389  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-n0 11331  df-z 11416  df-uz 11726  df-rp 11871  df-fz 12365  df-seq 12842  df-exp 12901  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020
This theorem is referenced by:  acongeq  37867
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