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Theorem iccpartrn 43874
 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 43860 . . . . . . 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 8426 . . . . . . 7 (𝑃 ∈ (ℝ*m (0...𝑀)) → 𝑃 Fn (0...𝑀))
65adantr 484 . . . . . 6 ((𝑃 ∈ (ℝ*m (0...𝑀)) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑃𝑖) < (𝑃‘(𝑖 + 1))) → 𝑃 Fn (0...𝑀))
74, 6syl6bi 256 . . . . 5 (𝜑 → (𝑃 ∈ (RePart‘𝑀) → 𝑃 Fn (0...𝑀)))
81, 7mpd 15 . . . 4 (𝜑𝑃 Fn (0...𝑀))
9 fvelrnb 6718 . . . 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 484 . . . . . . 7 ((𝜑𝑖 ∈ (0...𝑀)) → 𝑀 ∈ ℕ)
121adantr 484 . . . . . . 7 ((𝜑𝑖 ∈ (0...𝑀)) → 𝑃 ∈ (RePart‘𝑀))
13 simpr 488 . . . . . . 7 ((𝜑𝑖 ∈ (0...𝑀)) → 𝑖 ∈ (0...𝑀))
1411, 12, 13iccpartxr 43863 . . . . . 6 ((𝜑𝑖 ∈ (0...𝑀)) → (𝑃𝑖) ∈ ℝ*)
152, 1iccpartgel 43873 . . . . . . . 8 (𝜑 → ∀𝑘 ∈ (0...𝑀)(𝑃‘0) ≤ (𝑃𝑘))
16 fveq2 6662 . . . . . . . . . . 11 (𝑘 = 𝑖 → (𝑃𝑘) = (𝑃𝑖))
1716breq2d 5065 . . . . . . . . . 10 (𝑘 = 𝑖 → ((𝑃‘0) ≤ (𝑃𝑘) ↔ (𝑃‘0) ≤ (𝑃𝑖)))
1817rspcva 3608 . . . . . . . . 9 ((𝑖 ∈ (0...𝑀) ∧ ∀𝑘 ∈ (0...𝑀)(𝑃‘0) ≤ (𝑃𝑘)) → (𝑃‘0) ≤ (𝑃𝑖))
1918expcom 417 . . . . . . . 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 410 . . . . . 6 ((𝜑𝑖 ∈ (0...𝑀)) → (𝑃‘0) ≤ (𝑃𝑖))
222, 1iccpartleu 43872 . . . . . . . 8 (𝜑 → ∀𝑘 ∈ (0...𝑀)(𝑃𝑘) ≤ (𝑃𝑀))
2316breq1d 5063 . . . . . . . . . 10 (𝑘 = 𝑖 → ((𝑃𝑘) ≤ (𝑃𝑀) ↔ (𝑃𝑖) ≤ (𝑃𝑀)))
2423rspcva 3608 . . . . . . . . 9 ((𝑖 ∈ (0...𝑀) ∧ ∀𝑘 ∈ (0...𝑀)(𝑃𝑘) ≤ (𝑃𝑀)) → (𝑃𝑖) ≤ (𝑃𝑀))
2524expcom 417 . . . . . . . 8 (∀𝑘 ∈ (0...𝑀)(𝑃𝑘) ≤ (𝑃𝑀) → (𝑖 ∈ (0...𝑀) → (𝑃𝑖) ≤ (𝑃𝑀)))
2622, 25syl 17 . . . . . . 7 (𝜑 → (𝑖 ∈ (0...𝑀) → (𝑃𝑖) ≤ (𝑃𝑀)))
2726imp 410 . . . . . 6 ((𝜑𝑖 ∈ (0...𝑀)) → (𝑃𝑖) ≤ (𝑃𝑀))
28 nnnn0 11904 . . . . . . . . . . 11 (𝑀 ∈ ℕ → 𝑀 ∈ ℕ0)
29 0elfz 13011 . . . . . . . . . . 11 (𝑀 ∈ ℕ0 → 0 ∈ (0...𝑀))
302, 28, 293syl 18 . . . . . . . . . 10 (𝜑 → 0 ∈ (0...𝑀))
312, 1, 30iccpartxr 43863 . . . . . . . . 9 (𝜑 → (𝑃‘0) ∈ ℝ*)
32 nn0fz0 13012 . . . . . . . . . . . 12 (𝑀 ∈ ℕ0𝑀 ∈ (0...𝑀))
3328, 32sylib 221 . . . . . . . . . . 11 (𝑀 ∈ ℕ → 𝑀 ∈ (0...𝑀))
342, 33syl 17 . . . . . . . . . 10 (𝜑𝑀 ∈ (0...𝑀))
352, 1, 34iccpartxr 43863 . . . . . . . . 9 (𝜑 → (𝑃𝑀) ∈ ℝ*)
3631, 35jca 515 . . . . . . . 8 (𝜑 → ((𝑃‘0) ∈ ℝ* ∧ (𝑃𝑀) ∈ ℝ*))
3736adantr 484 . . . . . . 7 ((𝜑𝑖 ∈ (0...𝑀)) → ((𝑃‘0) ∈ ℝ* ∧ (𝑃𝑀) ∈ ℝ*))
38 elicc1 12782 . . . . . . 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 1339 . . . . 5 ((𝜑𝑖 ∈ (0...𝑀)) → (𝑃𝑖) ∈ ((𝑃‘0)[,](𝑃𝑀)))
41 eleq1 2903 . . . . 5 ((𝑃𝑖) = 𝑝 → ((𝑃𝑖) ∈ ((𝑃‘0)[,](𝑃𝑀)) ↔ 𝑝 ∈ ((𝑃‘0)[,](𝑃𝑀))))
4240, 41syl5ibcom 248 . . . 4 ((𝜑𝑖 ∈ (0...𝑀)) → ((𝑃𝑖) = 𝑝𝑝 ∈ ((𝑃‘0)[,](𝑃𝑀))))
4342rexlimdva 3277 . . 3 (𝜑 → (∃𝑖 ∈ (0...𝑀)(𝑃𝑖) = 𝑝𝑝 ∈ ((𝑃‘0)[,](𝑃𝑀))))
4410, 43sylbid 243 . 2 (𝜑 → (𝑝 ∈ ran 𝑃𝑝 ∈ ((𝑃‘0)[,](𝑃𝑀))))
4544ssrdv 3960 1 (𝜑 → ran 𝑃 ⊆ ((𝑃‘0)[,](𝑃𝑀)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2115  ∀wral 3133  ∃wrex 3134   ⊆ wss 3920   class class class wbr 5053  ran crn 5544   Fn wfn 6339  ‘cfv 6344  (class class class)co 7150   ↑m cmap 8403  0cc0 10536  1c1 10537   + caddc 10539  ℝ*cxr 10673   < clt 10674   ≤ cle 10675  ℕcn 11637  ℕ0cn0 11897  [,]cicc 12741  ...cfz 12897  ..^cfzo 13040  RePartciccp 43857 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7456  ax-cnex 10592  ax-resscn 10593  ax-1cn 10594  ax-icn 10595  ax-addcl 10596  ax-addrcl 10597  ax-mulcl 10598  ax-mulrcl 10599  ax-mulcom 10600  ax-addass 10601  ax-mulass 10602  ax-distr 10603  ax-i2m1 10604  ax-1ne0 10605  ax-1rid 10606  ax-rnegex 10607  ax-rrecex 10608  ax-cnre 10609  ax-pre-lttri 10610  ax-pre-lttrn 10611  ax-pre-ltadd 10612  ax-pre-mulgt0 10613 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3483  df-sbc 3760  df-csb 3868  df-dif 3923  df-un 3925  df-in 3927  df-ss 3937  df-pss 3939  df-nul 4278  df-if 4452  df-pw 4525  df-sn 4552  df-pr 4554  df-tp 4556  df-op 4558  df-uni 4826  df-iun 4908  df-br 5054  df-opab 5116  df-mpt 5134  df-tr 5160  df-id 5448  df-eprel 5453  df-po 5462  df-so 5463  df-fr 5502  df-we 5504  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-pred 6136  df-ord 6182  df-on 6183  df-lim 6184  df-suc 6185  df-iota 6303  df-fun 6346  df-fn 6347  df-f 6348  df-f1 6349  df-fo 6350  df-f1o 6351  df-fv 6352  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7576  df-1st 7685  df-2nd 7686  df-wrecs 7944  df-recs 8005  df-rdg 8043  df-er 8286  df-map 8405  df-en 8507  df-dom 8508  df-sdom 8509  df-pnf 10676  df-mnf 10677  df-xr 10678  df-ltxr 10679  df-le 10680  df-sub 10871  df-neg 10872  df-nn 11638  df-2 11700  df-n0 11898  df-z 11982  df-uz 12244  df-icc 12745  df-fz 12898  df-fzo 13041  df-iccp 43858 This theorem is referenced by:  iccpartf  43875
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