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Theorem fzmaxdif 39456
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 1192 . . . . . 6 (((𝐶 ∈ ℤ ∧ 𝐴 ∈ (𝐵...𝐶)) ∧ (𝐹 ∈ ℤ ∧ 𝐷 ∈ (𝐸...𝐹)) ∧ (𝐶𝐸) ≤ (𝐹𝐵)) → 𝐷 ∈ (𝐸...𝐹))
2 elfzelz 12896 . . . . . 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 12075 . . . 4 (((𝐶 ∈ ℤ ∧ 𝐴 ∈ (𝐵...𝐶)) ∧ (𝐹 ∈ ℤ ∧ 𝐷 ∈ (𝐸...𝐹)) ∧ (𝐶𝐸) ≤ (𝐹𝐵)) → 𝐷 ∈ ℝ)
5 simp2l 1191 . . . . . 6 (((𝐶 ∈ ℤ ∧ 𝐴 ∈ (𝐵...𝐶)) ∧ (𝐹 ∈ ℤ ∧ 𝐷 ∈ (𝐸...𝐹)) ∧ (𝐶𝐸) ≤ (𝐹𝐵)) → 𝐹 ∈ ℤ)
65zred 12075 . . . . 5 (((𝐶 ∈ ℤ ∧ 𝐴 ∈ (𝐵...𝐶)) ∧ (𝐹 ∈ ℤ ∧ 𝐷 ∈ (𝐸...𝐹)) ∧ (𝐶𝐸) ≤ (𝐹𝐵)) → 𝐹 ∈ ℝ)
7 simp1r 1190 . . . . . . 7 (((𝐶 ∈ ℤ ∧ 𝐴 ∈ (𝐵...𝐶)) ∧ (𝐹 ∈ ℤ ∧ 𝐷 ∈ (𝐸...𝐹)) ∧ (𝐶𝐸) ≤ (𝐹𝐵)) → 𝐴 ∈ (𝐵...𝐶))
8 elfzel1 12895 . . . . . . 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 12075 . . . . 5 (((𝐶 ∈ ℤ ∧ 𝐴 ∈ (𝐵...𝐶)) ∧ (𝐹 ∈ ℤ ∧ 𝐷 ∈ (𝐸...𝐹)) ∧ (𝐶𝐸) ≤ (𝐹𝐵)) → 𝐵 ∈ ℝ)
116, 10resubcld 11056 . . . 4 (((𝐶 ∈ ℤ ∧ 𝐴 ∈ (𝐵...𝐶)) ∧ (𝐹 ∈ ℤ ∧ 𝐷 ∈ (𝐸...𝐹)) ∧ (𝐶𝐸) ≤ (𝐹𝐵)) → (𝐹𝐵) ∈ ℝ)
124, 11resubcld 11056 . . 3 (((𝐶 ∈ ℤ ∧ 𝐴 ∈ (𝐵...𝐶)) ∧ (𝐹 ∈ ℤ ∧ 𝐷 ∈ (𝐸...𝐹)) ∧ (𝐶𝐸) ≤ (𝐹𝐵)) → (𝐷 − (𝐹𝐵)) ∈ ℝ)
13 elfzelz 12896 . . . . 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 12075 . . 3 (((𝐶 ∈ ℤ ∧ 𝐴 ∈ (𝐵...𝐶)) ∧ (𝐹 ∈ ℤ ∧ 𝐷 ∈ (𝐸...𝐹)) ∧ (𝐶𝐸) ≤ (𝐹𝐵)) → 𝐴 ∈ ℝ)
16 elfzle2 12899 . . . . . 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 11244 . . . 4 (((𝐶 ∈ ℤ ∧ 𝐴 ∈ (𝐵...𝐶)) ∧ (𝐹 ∈ ℤ ∧ 𝐷 ∈ (𝐸...𝐹)) ∧ (𝐶𝐸) ≤ (𝐹𝐵)) → (𝐷 − (𝐹𝐵)) ≤ (𝐹 − (𝐹𝐵)))
196recnd 10657 . . . . 5 (((𝐶 ∈ ℤ ∧ 𝐴 ∈ (𝐵...𝐶)) ∧ (𝐹 ∈ ℤ ∧ 𝐷 ∈ (𝐸...𝐹)) ∧ (𝐶𝐸) ≤ (𝐹𝐵)) → 𝐹 ∈ ℂ)
2010recnd 10657 . . . . 5 (((𝐶 ∈ ℤ ∧ 𝐴 ∈ (𝐵...𝐶)) ∧ (𝐹 ∈ ℤ ∧ 𝐷 ∈ (𝐸...𝐹)) ∧ (𝐶𝐸) ≤ (𝐹𝐵)) → 𝐵 ∈ ℂ)
2119, 20nncand 10990 . . . 4 (((𝐶 ∈ ℤ ∧ 𝐴 ∈ (𝐵...𝐶)) ∧ (𝐹 ∈ ℤ ∧ 𝐷 ∈ (𝐸...𝐹)) ∧ (𝐶𝐸) ≤ (𝐹𝐵)) → (𝐹 − (𝐹𝐵)) = 𝐵)
2218, 21breqtrd 5083 . . 3 (((𝐶 ∈ ℤ ∧ 𝐴 ∈ (𝐵...𝐶)) ∧ (𝐹 ∈ ℤ ∧ 𝐷 ∈ (𝐸...𝐹)) ∧ (𝐶𝐸) ≤ (𝐹𝐵)) → (𝐷 − (𝐹𝐵)) ≤ 𝐵)
23 elfzle1 12898 . . . 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 10785 . 2 (((𝐶 ∈ ℤ ∧ 𝐴 ∈ (𝐵...𝐶)) ∧ (𝐹 ∈ ℤ ∧ 𝐷 ∈ (𝐸...𝐹)) ∧ (𝐶𝐸) ≤ (𝐹𝐵)) → (𝐷 − (𝐹𝐵)) ≤ 𝐴)
26 simp1l 1189 . . . 4 (((𝐶 ∈ ℤ ∧ 𝐴 ∈ (𝐵...𝐶)) ∧ (𝐹 ∈ ℤ ∧ 𝐷 ∈ (𝐸...𝐹)) ∧ (𝐶𝐸) ≤ (𝐹𝐵)) → 𝐶 ∈ ℤ)
2726zred 12075 . . 3 (((𝐶 ∈ ℤ ∧ 𝐴 ∈ (𝐵...𝐶)) ∧ (𝐹 ∈ ℤ ∧ 𝐷 ∈ (𝐸...𝐹)) ∧ (𝐶𝐸) ≤ (𝐹𝐵)) → 𝐶 ∈ ℝ)
284, 11readdcld 10658 . . 3 (((𝐶 ∈ ℤ ∧ 𝐴 ∈ (𝐵...𝐶)) ∧ (𝐹 ∈ ℤ ∧ 𝐷 ∈ (𝐸...𝐹)) ∧ (𝐶𝐸) ≤ (𝐹𝐵)) → (𝐷 + (𝐹𝐵)) ∈ ℝ)
29 elfzle2 12899 . . . 4 (𝐴 ∈ (𝐵...𝐶) → 𝐴𝐶)
307, 29syl 17 . . 3 (((𝐶 ∈ ℤ ∧ 𝐴 ∈ (𝐵...𝐶)) ∧ (𝐹 ∈ ℤ ∧ 𝐷 ∈ (𝐸...𝐹)) ∧ (𝐶𝐸) ≤ (𝐹𝐵)) → 𝐴𝐶)
3127, 4resubcld 11056 . . . . . 6 (((𝐶 ∈ ℤ ∧ 𝐴 ∈ (𝐵...𝐶)) ∧ (𝐹 ∈ ℤ ∧ 𝐷 ∈ (𝐸...𝐹)) ∧ (𝐶𝐸) ≤ (𝐹𝐵)) → (𝐶𝐷) ∈ ℝ)
32 elfzel1 12895 . . . . . . . . 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 12075 . . . . . . 7 (((𝐶 ∈ ℤ ∧ 𝐴 ∈ (𝐵...𝐶)) ∧ (𝐹 ∈ ℤ ∧ 𝐷 ∈ (𝐸...𝐹)) ∧ (𝐶𝐸) ≤ (𝐹𝐵)) → 𝐸 ∈ ℝ)
3527, 34resubcld 11056 . . . . . 6 (((𝐶 ∈ ℤ ∧ 𝐴 ∈ (𝐵...𝐶)) ∧ (𝐹 ∈ ℤ ∧ 𝐷 ∈ (𝐸...𝐹)) ∧ (𝐶𝐸) ≤ (𝐹𝐵)) → (𝐶𝐸) ∈ ℝ)
36 elfzle1 12898 . . . . . . . 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 11245 . . . . . 6 (((𝐶 ∈ ℤ ∧ 𝐴 ∈ (𝐵...𝐶)) ∧ (𝐹 ∈ ℤ ∧ 𝐷 ∈ (𝐸...𝐹)) ∧ (𝐶𝐸) ≤ (𝐹𝐵)) → (𝐶𝐷) ≤ (𝐶𝐸))
39 simp3 1130 . . . . . 6 (((𝐶 ∈ ℤ ∧ 𝐴 ∈ (𝐵...𝐶)) ∧ (𝐹 ∈ ℤ ∧ 𝐷 ∈ (𝐸...𝐹)) ∧ (𝐶𝐸) ≤ (𝐹𝐵)) → (𝐶𝐸) ≤ (𝐹𝐵))
4031, 35, 11, 38, 39letrd 10785 . . . . 5 (((𝐶 ∈ ℤ ∧ 𝐴 ∈ (𝐵...𝐶)) ∧ (𝐹 ∈ ℤ ∧ 𝐷 ∈ (𝐸...𝐹)) ∧ (𝐶𝐸) ≤ (𝐹𝐵)) → (𝐶𝐷) ≤ (𝐹𝐵))
4127, 4, 11lesubaddd 11225 . . . . 5 (((𝐶 ∈ ℤ ∧ 𝐴 ∈ (𝐵...𝐶)) ∧ (𝐹 ∈ ℤ ∧ 𝐷 ∈ (𝐸...𝐹)) ∧ (𝐶𝐸) ≤ (𝐹𝐵)) → ((𝐶𝐷) ≤ (𝐹𝐵) ↔ 𝐶 ≤ ((𝐹𝐵) + 𝐷)))
4240, 41mpbid 233 . . . 4 (((𝐶 ∈ ℤ ∧ 𝐴 ∈ (𝐵...𝐶)) ∧ (𝐹 ∈ ℤ ∧ 𝐷 ∈ (𝐸...𝐹)) ∧ (𝐶𝐸) ≤ (𝐹𝐵)) → 𝐶 ≤ ((𝐹𝐵) + 𝐷))
4311recnd 10657 . . . . 5 (((𝐶 ∈ ℤ ∧ 𝐴 ∈ (𝐵...𝐶)) ∧ (𝐹 ∈ ℤ ∧ 𝐷 ∈ (𝐸...𝐹)) ∧ (𝐶𝐸) ≤ (𝐹𝐵)) → (𝐹𝐵) ∈ ℂ)
444recnd 10657 . . . . 5 (((𝐶 ∈ ℤ ∧ 𝐴 ∈ (𝐵...𝐶)) ∧ (𝐹 ∈ ℤ ∧ 𝐷 ∈ (𝐸...𝐹)) ∧ (𝐶𝐸) ≤ (𝐹𝐵)) → 𝐷 ∈ ℂ)
4543, 44addcomd 10830 . . . 4 (((𝐶 ∈ ℤ ∧ 𝐴 ∈ (𝐵...𝐶)) ∧ (𝐹 ∈ ℤ ∧ 𝐷 ∈ (𝐸...𝐹)) ∧ (𝐶𝐸) ≤ (𝐹𝐵)) → ((𝐹𝐵) + 𝐷) = (𝐷 + (𝐹𝐵)))
4642, 45breqtrd 5083 . . 3 (((𝐶 ∈ ℤ ∧ 𝐴 ∈ (𝐵...𝐶)) ∧ (𝐹 ∈ ℤ ∧ 𝐷 ∈ (𝐸...𝐹)) ∧ (𝐶𝐸) ≤ (𝐹𝐵)) → 𝐶 ≤ (𝐷 + (𝐹𝐵)))
4715, 27, 28, 30, 46letrd 10785 . 2 (((𝐶 ∈ ℤ ∧ 𝐴 ∈ (𝐵...𝐶)) ∧ (𝐹 ∈ ℤ ∧ 𝐷 ∈ (𝐸...𝐹)) ∧ (𝐶𝐸) ≤ (𝐹𝐵)) → 𝐴 ≤ (𝐷 + (𝐹𝐵)))
4815, 4, 11absdifled 14782 . 2 (((𝐶 ∈ ℤ ∧ 𝐴 ∈ (𝐵...𝐶)) ∧ (𝐹 ∈ ℤ ∧ 𝐷 ∈ (𝐸...𝐹)) ∧ (𝐶𝐸) ≤ (𝐹𝐵)) → ((abs‘(𝐴𝐷)) ≤ (𝐹𝐵) ↔ ((𝐷 − (𝐹𝐵)) ≤ 𝐴𝐴 ≤ (𝐷 + (𝐹𝐵)))))
4925, 47, 48mpbir2and 709 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 396  w3a 1079  wcel 2105   class class class wbr 5057  cfv 6348  (class class class)co 7145   + caddc 10528  cle 10664  cmin 10858  cz 11969  ...cfz 12880  abscabs 14581
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602  ax-pre-sup 10603
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-er 8278  df-en 8498  df-dom 8499  df-sdom 8500  df-sup 8894  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-div 11286  df-nn 11627  df-2 11688  df-3 11689  df-n0 11886  df-z 11970  df-uz 12232  df-rp 12378  df-fz 12881  df-seq 13358  df-exp 13418  df-cj 14446  df-re 14447  df-im 14448  df-sqrt 14582  df-abs 14583
This theorem is referenced by:  acongeq  39458
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