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Theorem iccpartrn 42997
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 42983 . . . . . . 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 8235 . . . . . . 7 (𝑃 ∈ (ℝ*𝑚 (0...𝑀)) → 𝑃 Fn (0...𝑀))
65adantr 473 . . . . . 6 ((𝑃 ∈ (ℝ*𝑚 (0...𝑀)) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑃𝑖) < (𝑃‘(𝑖 + 1))) → 𝑃 Fn (0...𝑀))
74, 6syl6bi 245 . . . . 5 (𝜑 → (𝑃 ∈ (RePart‘𝑀) → 𝑃 Fn (0...𝑀)))
81, 7mpd 15 . . . 4 (𝜑𝑃 Fn (0...𝑀))
9 fvelrnb 6561 . . . 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 473 . . . . . . 7 ((𝜑𝑖 ∈ (0...𝑀)) → 𝑀 ∈ ℕ)
121adantr 473 . . . . . . 7 ((𝜑𝑖 ∈ (0...𝑀)) → 𝑃 ∈ (RePart‘𝑀))
13 simpr 477 . . . . . . 7 ((𝜑𝑖 ∈ (0...𝑀)) → 𝑖 ∈ (0...𝑀))
1411, 12, 13iccpartxr 42986 . . . . . 6 ((𝜑𝑖 ∈ (0...𝑀)) → (𝑃𝑖) ∈ ℝ*)
152, 1iccpartgel 42996 . . . . . . . 8 (𝜑 → ∀𝑘 ∈ (0...𝑀)(𝑃‘0) ≤ (𝑃𝑘))
16 fveq2 6504 . . . . . . . . . . 11 (𝑘 = 𝑖 → (𝑃𝑘) = (𝑃𝑖))
1716breq2d 4946 . . . . . . . . . 10 (𝑘 = 𝑖 → ((𝑃‘0) ≤ (𝑃𝑘) ↔ (𝑃‘0) ≤ (𝑃𝑖)))
1817rspcva 3535 . . . . . . . . 9 ((𝑖 ∈ (0...𝑀) ∧ ∀𝑘 ∈ (0...𝑀)(𝑃‘0) ≤ (𝑃𝑘)) → (𝑃‘0) ≤ (𝑃𝑖))
1918expcom 406 . . . . . . . 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 398 . . . . . 6 ((𝜑𝑖 ∈ (0...𝑀)) → (𝑃‘0) ≤ (𝑃𝑖))
222, 1iccpartleu 42995 . . . . . . . 8 (𝜑 → ∀𝑘 ∈ (0...𝑀)(𝑃𝑘) ≤ (𝑃𝑀))
2316breq1d 4944 . . . . . . . . . 10 (𝑘 = 𝑖 → ((𝑃𝑘) ≤ (𝑃𝑀) ↔ (𝑃𝑖) ≤ (𝑃𝑀)))
2423rspcva 3535 . . . . . . . . 9 ((𝑖 ∈ (0...𝑀) ∧ ∀𝑘 ∈ (0...𝑀)(𝑃𝑘) ≤ (𝑃𝑀)) → (𝑃𝑖) ≤ (𝑃𝑀))
2524expcom 406 . . . . . . . 8 (∀𝑘 ∈ (0...𝑀)(𝑃𝑘) ≤ (𝑃𝑀) → (𝑖 ∈ (0...𝑀) → (𝑃𝑖) ≤ (𝑃𝑀)))
2622, 25syl 17 . . . . . . 7 (𝜑 → (𝑖 ∈ (0...𝑀) → (𝑃𝑖) ≤ (𝑃𝑀)))
2726imp 398 . . . . . 6 ((𝜑𝑖 ∈ (0...𝑀)) → (𝑃𝑖) ≤ (𝑃𝑀))
28 nnnn0 11721 . . . . . . . . . . 11 (𝑀 ∈ ℕ → 𝑀 ∈ ℕ0)
29 0elfz 12826 . . . . . . . . . . 11 (𝑀 ∈ ℕ0 → 0 ∈ (0...𝑀))
302, 28, 293syl 18 . . . . . . . . . 10 (𝜑 → 0 ∈ (0...𝑀))
312, 1, 30iccpartxr 42986 . . . . . . . . 9 (𝜑 → (𝑃‘0) ∈ ℝ*)
32 nn0fz0 12827 . . . . . . . . . . . 12 (𝑀 ∈ ℕ0𝑀 ∈ (0...𝑀))
3328, 32sylib 210 . . . . . . . . . . 11 (𝑀 ∈ ℕ → 𝑀 ∈ (0...𝑀))
342, 33syl 17 . . . . . . . . . 10 (𝜑𝑀 ∈ (0...𝑀))
352, 1, 34iccpartxr 42986 . . . . . . . . 9 (𝜑 → (𝑃𝑀) ∈ ℝ*)
3631, 35jca 504 . . . . . . . 8 (𝜑 → ((𝑃‘0) ∈ ℝ* ∧ (𝑃𝑀) ∈ ℝ*))
3736adantr 473 . . . . . . 7 ((𝜑𝑖 ∈ (0...𝑀)) → ((𝑃‘0) ∈ ℝ* ∧ (𝑃𝑀) ∈ ℝ*))
38 elicc1 12604 . . . . . . 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 1323 . . . . 5 ((𝜑𝑖 ∈ (0...𝑀)) → (𝑃𝑖) ∈ ((𝑃‘0)[,](𝑃𝑀)))
41 eleq1 2855 . . . . 5 ((𝑃𝑖) = 𝑝 → ((𝑃𝑖) ∈ ((𝑃‘0)[,](𝑃𝑀)) ↔ 𝑝 ∈ ((𝑃‘0)[,](𝑃𝑀))))
4240, 41syl5ibcom 237 . . . 4 ((𝜑𝑖 ∈ (0...𝑀)) → ((𝑃𝑖) = 𝑝𝑝 ∈ ((𝑃‘0)[,](𝑃𝑀))))
4342rexlimdva 3231 . . 3 (𝜑 → (∃𝑖 ∈ (0...𝑀)(𝑃𝑖) = 𝑝𝑝 ∈ ((𝑃‘0)[,](𝑃𝑀))))
4410, 43sylbid 232 . 2 (𝜑 → (𝑝 ∈ ran 𝑃𝑝 ∈ ((𝑃‘0)[,](𝑃𝑀))))
4544ssrdv 3866 1 (𝜑 → ran 𝑃 ⊆ ((𝑃‘0)[,](𝑃𝑀)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 387  w3a 1069   = wceq 1508  wcel 2051  wral 3090  wrex 3091  wss 3831   class class class wbr 4934  ran crn 5412   Fn wfn 6188  cfv 6193  (class class class)co 6982  𝑚 cmap 8212  0cc0 10341  1c1 10342   + caddc 10344  *cxr 10479   < clt 10480  cle 10481  cn 11445  0cn0 11713  [,]cicc 12563  ...cfz 12714  ..^cfzo 12855  RePartciccp 42980
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2752  ax-sep 5064  ax-nul 5071  ax-pow 5123  ax-pr 5190  ax-un 7285  ax-cnex 10397  ax-resscn 10398  ax-1cn 10399  ax-icn 10400  ax-addcl 10401  ax-addrcl 10402  ax-mulcl 10403  ax-mulrcl 10404  ax-mulcom 10405  ax-addass 10406  ax-mulass 10407  ax-distr 10408  ax-i2m1 10409  ax-1ne0 10410  ax-1rid 10411  ax-rnegex 10412  ax-rrecex 10413  ax-cnre 10414  ax-pre-lttri 10415  ax-pre-lttrn 10416  ax-pre-ltadd 10417  ax-pre-mulgt0 10418
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3or 1070  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2551  df-eu 2589  df-clab 2761  df-cleq 2773  df-clel 2848  df-nfc 2920  df-ne 2970  df-nel 3076  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3419  df-sbc 3684  df-csb 3789  df-dif 3834  df-un 3836  df-in 3838  df-ss 3845  df-pss 3847  df-nul 4182  df-if 4354  df-pw 4427  df-sn 4445  df-pr 4447  df-tp 4449  df-op 4451  df-uni 4718  df-iun 4799  df-br 4935  df-opab 4997  df-mpt 5014  df-tr 5036  df-id 5316  df-eprel 5321  df-po 5330  df-so 5331  df-fr 5370  df-we 5372  df-xp 5417  df-rel 5418  df-cnv 5419  df-co 5420  df-dm 5421  df-rn 5422  df-res 5423  df-ima 5424  df-pred 5991  df-ord 6037  df-on 6038  df-lim 6039  df-suc 6040  df-iota 6157  df-fun 6195  df-fn 6196  df-f 6197  df-f1 6198  df-fo 6199  df-f1o 6200  df-fv 6201  df-riota 6943  df-ov 6985  df-oprab 6986  df-mpo 6987  df-om 7403  df-1st 7507  df-2nd 7508  df-wrecs 7756  df-recs 7818  df-rdg 7856  df-er 8095  df-map 8214  df-en 8313  df-dom 8314  df-sdom 8315  df-pnf 10482  df-mnf 10483  df-xr 10484  df-ltxr 10485  df-le 10486  df-sub 10678  df-neg 10679  df-nn 11446  df-2 11509  df-n0 11714  df-z 11800  df-uz 12065  df-icc 12567  df-fz 12715  df-fzo 12856  df-iccp 42981
This theorem is referenced by:  iccpartf  42998
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