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Theorem iccpartrn 44851
Description: If there is a partition, then all intermediate points and bounds are contained in a closed interval of extended reals. (Contributed by AV, 14-Jul-2020.)
Hypotheses
Ref Expression
iccpartgtprec.m (𝜑𝑀 ∈ ℕ)
iccpartgtprec.p (𝜑𝑃 ∈ (RePart‘𝑀))
Assertion
Ref Expression
iccpartrn (𝜑 → ran 𝑃 ⊆ ((𝑃‘0)[,](𝑃𝑀)))

Proof of Theorem iccpartrn
Dummy variables 𝑖 𝑘 𝑝 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 iccpartgtprec.p . . . . 5 (𝜑𝑃 ∈ (RePart‘𝑀))
2 iccpartgtprec.m . . . . . . 7 (𝜑𝑀 ∈ ℕ)
3 iccpart 44837 . . . . . . 7 (𝑀 ∈ ℕ → (𝑃 ∈ (RePart‘𝑀) ↔ (𝑃 ∈ (ℝ*m (0...𝑀)) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑃𝑖) < (𝑃‘(𝑖 + 1)))))
42, 3syl 17 . . . . . 6 (𝜑 → (𝑃 ∈ (RePart‘𝑀) ↔ (𝑃 ∈ (ℝ*m (0...𝑀)) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑃𝑖) < (𝑃‘(𝑖 + 1)))))
5 elmapfn 8636 . . . . . . 7 (𝑃 ∈ (ℝ*m (0...𝑀)) → 𝑃 Fn (0...𝑀))
65adantr 481 . . . . . 6 ((𝑃 ∈ (ℝ*m (0...𝑀)) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑃𝑖) < (𝑃‘(𝑖 + 1))) → 𝑃 Fn (0...𝑀))
74, 6syl6bi 252 . . . . 5 (𝜑 → (𝑃 ∈ (RePart‘𝑀) → 𝑃 Fn (0...𝑀)))
81, 7mpd 15 . . . 4 (𝜑𝑃 Fn (0...𝑀))
9 fvelrnb 6827 . . . 4 (𝑃 Fn (0...𝑀) → (𝑝 ∈ ran 𝑃 ↔ ∃𝑖 ∈ (0...𝑀)(𝑃𝑖) = 𝑝))
108, 9syl 17 . . 3 (𝜑 → (𝑝 ∈ ran 𝑃 ↔ ∃𝑖 ∈ (0...𝑀)(𝑃𝑖) = 𝑝))
112adantr 481 . . . . . . 7 ((𝜑𝑖 ∈ (0...𝑀)) → 𝑀 ∈ ℕ)
121adantr 481 . . . . . . 7 ((𝜑𝑖 ∈ (0...𝑀)) → 𝑃 ∈ (RePart‘𝑀))
13 simpr 485 . . . . . . 7 ((𝜑𝑖 ∈ (0...𝑀)) → 𝑖 ∈ (0...𝑀))
1411, 12, 13iccpartxr 44840 . . . . . 6 ((𝜑𝑖 ∈ (0...𝑀)) → (𝑃𝑖) ∈ ℝ*)
152, 1iccpartgel 44850 . . . . . . . 8 (𝜑 → ∀𝑘 ∈ (0...𝑀)(𝑃‘0) ≤ (𝑃𝑘))
16 fveq2 6771 . . . . . . . . . . 11 (𝑘 = 𝑖 → (𝑃𝑘) = (𝑃𝑖))
1716breq2d 5091 . . . . . . . . . 10 (𝑘 = 𝑖 → ((𝑃‘0) ≤ (𝑃𝑘) ↔ (𝑃‘0) ≤ (𝑃𝑖)))
1817rspcva 3559 . . . . . . . . 9 ((𝑖 ∈ (0...𝑀) ∧ ∀𝑘 ∈ (0...𝑀)(𝑃‘0) ≤ (𝑃𝑘)) → (𝑃‘0) ≤ (𝑃𝑖))
1918expcom 414 . . . . . . . 8 (∀𝑘 ∈ (0...𝑀)(𝑃‘0) ≤ (𝑃𝑘) → (𝑖 ∈ (0...𝑀) → (𝑃‘0) ≤ (𝑃𝑖)))
2015, 19syl 17 . . . . . . 7 (𝜑 → (𝑖 ∈ (0...𝑀) → (𝑃‘0) ≤ (𝑃𝑖)))
2120imp 407 . . . . . 6 ((𝜑𝑖 ∈ (0...𝑀)) → (𝑃‘0) ≤ (𝑃𝑖))
222, 1iccpartleu 44849 . . . . . . . 8 (𝜑 → ∀𝑘 ∈ (0...𝑀)(𝑃𝑘) ≤ (𝑃𝑀))
2316breq1d 5089 . . . . . . . . . 10 (𝑘 = 𝑖 → ((𝑃𝑘) ≤ (𝑃𝑀) ↔ (𝑃𝑖) ≤ (𝑃𝑀)))
2423rspcva 3559 . . . . . . . . 9 ((𝑖 ∈ (0...𝑀) ∧ ∀𝑘 ∈ (0...𝑀)(𝑃𝑘) ≤ (𝑃𝑀)) → (𝑃𝑖) ≤ (𝑃𝑀))
2524expcom 414 . . . . . . . 8 (∀𝑘 ∈ (0...𝑀)(𝑃𝑘) ≤ (𝑃𝑀) → (𝑖 ∈ (0...𝑀) → (𝑃𝑖) ≤ (𝑃𝑀)))
2622, 25syl 17 . . . . . . 7 (𝜑 → (𝑖 ∈ (0...𝑀) → (𝑃𝑖) ≤ (𝑃𝑀)))
2726imp 407 . . . . . 6 ((𝜑𝑖 ∈ (0...𝑀)) → (𝑃𝑖) ≤ (𝑃𝑀))
28 nnnn0 12240 . . . . . . . . . . 11 (𝑀 ∈ ℕ → 𝑀 ∈ ℕ0)
29 0elfz 13352 . . . . . . . . . . 11 (𝑀 ∈ ℕ0 → 0 ∈ (0...𝑀))
302, 28, 293syl 18 . . . . . . . . . 10 (𝜑 → 0 ∈ (0...𝑀))
312, 1, 30iccpartxr 44840 . . . . . . . . 9 (𝜑 → (𝑃‘0) ∈ ℝ*)
32 nn0fz0 13353 . . . . . . . . . . . 12 (𝑀 ∈ ℕ0𝑀 ∈ (0...𝑀))
3328, 32sylib 217 . . . . . . . . . . 11 (𝑀 ∈ ℕ → 𝑀 ∈ (0...𝑀))
342, 33syl 17 . . . . . . . . . 10 (𝜑𝑀 ∈ (0...𝑀))
352, 1, 34iccpartxr 44840 . . . . . . . . 9 (𝜑 → (𝑃𝑀) ∈ ℝ*)
3631, 35jca 512 . . . . . . . 8 (𝜑 → ((𝑃‘0) ∈ ℝ* ∧ (𝑃𝑀) ∈ ℝ*))
3736adantr 481 . . . . . . 7 ((𝜑𝑖 ∈ (0...𝑀)) → ((𝑃‘0) ∈ ℝ* ∧ (𝑃𝑀) ∈ ℝ*))
38 elicc1 13122 . . . . . . 7 (((𝑃‘0) ∈ ℝ* ∧ (𝑃𝑀) ∈ ℝ*) → ((𝑃𝑖) ∈ ((𝑃‘0)[,](𝑃𝑀)) ↔ ((𝑃𝑖) ∈ ℝ* ∧ (𝑃‘0) ≤ (𝑃𝑖) ∧ (𝑃𝑖) ≤ (𝑃𝑀))))
3937, 38syl 17 . . . . . 6 ((𝜑𝑖 ∈ (0...𝑀)) → ((𝑃𝑖) ∈ ((𝑃‘0)[,](𝑃𝑀)) ↔ ((𝑃𝑖) ∈ ℝ* ∧ (𝑃‘0) ≤ (𝑃𝑖) ∧ (𝑃𝑖) ≤ (𝑃𝑀))))
4014, 21, 27, 39mpbir3and 1341 . . . . 5 ((𝜑𝑖 ∈ (0...𝑀)) → (𝑃𝑖) ∈ ((𝑃‘0)[,](𝑃𝑀)))
41 eleq1 2828 . . . . 5 ((𝑃𝑖) = 𝑝 → ((𝑃𝑖) ∈ ((𝑃‘0)[,](𝑃𝑀)) ↔ 𝑝 ∈ ((𝑃‘0)[,](𝑃𝑀))))
4240, 41syl5ibcom 244 . . . 4 ((𝜑𝑖 ∈ (0...𝑀)) → ((𝑃𝑖) = 𝑝𝑝 ∈ ((𝑃‘0)[,](𝑃𝑀))))
4342rexlimdva 3215 . . 3 (𝜑 → (∃𝑖 ∈ (0...𝑀)(𝑃𝑖) = 𝑝𝑝 ∈ ((𝑃‘0)[,](𝑃𝑀))))
4410, 43sylbid 239 . 2 (𝜑 → (𝑝 ∈ ran 𝑃𝑝 ∈ ((𝑃‘0)[,](𝑃𝑀))))
4544ssrdv 3932 1 (𝜑 → ran 𝑃 ⊆ ((𝑃‘0)[,](𝑃𝑀)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1086   = wceq 1542  wcel 2110  wral 3066  wrex 3067  wss 3892   class class class wbr 5079  ran crn 5591   Fn wfn 6427  cfv 6432  (class class class)co 7271  m cmap 8598  0cc0 10872  1c1 10873   + caddc 10875  *cxr 11009   < clt 11010  cle 11011  cn 11973  0cn0 12233  [,]cicc 13081  ...cfz 13238  ..^cfzo 13381  RePartciccp 44834
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-sep 5227  ax-nul 5234  ax-pow 5292  ax-pr 5356  ax-un 7582  ax-cnex 10928  ax-resscn 10929  ax-1cn 10930  ax-icn 10931  ax-addcl 10932  ax-addrcl 10933  ax-mulcl 10934  ax-mulrcl 10935  ax-mulcom 10936  ax-addass 10937  ax-mulass 10938  ax-distr 10939  ax-i2m1 10940  ax-1ne0 10941  ax-1rid 10942  ax-rnegex 10943  ax-rrecex 10944  ax-cnre 10945  ax-pre-lttri 10946  ax-pre-lttrn 10947  ax-pre-ltadd 10948  ax-pre-mulgt0 10949
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-nel 3052  df-ral 3071  df-rex 3072  df-reu 3073  df-rab 3075  df-v 3433  df-sbc 3721  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-pss 3911  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-iun 4932  df-br 5080  df-opab 5142  df-mpt 5163  df-tr 5197  df-id 5490  df-eprel 5496  df-po 5504  df-so 5505  df-fr 5545  df-we 5547  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-pred 6201  df-ord 6268  df-on 6269  df-lim 6270  df-suc 6271  df-iota 6390  df-fun 6434  df-fn 6435  df-f 6436  df-f1 6437  df-fo 6438  df-f1o 6439  df-fv 6440  df-riota 7228  df-ov 7274  df-oprab 7275  df-mpo 7276  df-om 7707  df-1st 7824  df-2nd 7825  df-frecs 8088  df-wrecs 8119  df-recs 8193  df-rdg 8232  df-er 8481  df-map 8600  df-en 8717  df-dom 8718  df-sdom 8719  df-pnf 11012  df-mnf 11013  df-xr 11014  df-ltxr 11015  df-le 11016  df-sub 11207  df-neg 11208  df-nn 11974  df-2 12036  df-n0 12234  df-z 12320  df-uz 12582  df-icc 13085  df-fz 13239  df-fzo 13382  df-iccp 44835
This theorem is referenced by:  iccpartf  44852
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