Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  iccpartxr Structured version   Visualization version   GIF version

Theorem iccpartxr 47344
Description: If there is a partition, then all intermediate points and bounds are extended real numbers. (Contributed by AV, 11-Jul-2020.)
Hypotheses
Ref Expression
iccpartgtprec.m (𝜑𝑀 ∈ ℕ)
iccpartgtprec.p (𝜑𝑃 ∈ (RePart‘𝑀))
iccpartxr.i (𝜑𝐼 ∈ (0...𝑀))
Assertion
Ref Expression
iccpartxr (𝜑 → (𝑃𝐼) ∈ ℝ*)

Proof of Theorem iccpartxr
Dummy variable 𝑖 is distinct from all other variables.
StepHypRef Expression
1 iccpartgtprec.p . . . . 5 (𝜑𝑃 ∈ (RePart‘𝑀))
2 iccpartgtprec.m . . . . . 6 (𝜑𝑀 ∈ ℕ)
3 iccpart 47341 . . . . . 6 (𝑀 ∈ ℕ → (𝑃 ∈ (RePart‘𝑀) ↔ (𝑃 ∈ (ℝ*m (0...𝑀)) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑃𝑖) < (𝑃‘(𝑖 + 1)))))
42, 3syl 17 . . . . 5 (𝜑 → (𝑃 ∈ (RePart‘𝑀) ↔ (𝑃 ∈ (ℝ*m (0...𝑀)) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑃𝑖) < (𝑃‘(𝑖 + 1)))))
51, 4mpbid 232 . . . 4 (𝜑 → (𝑃 ∈ (ℝ*m (0...𝑀)) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑃𝑖) < (𝑃‘(𝑖 + 1))))
65simpld 494 . . 3 (𝜑𝑃 ∈ (ℝ*m (0...𝑀)))
7 elmapi 8888 . . 3 (𝑃 ∈ (ℝ*m (0...𝑀)) → 𝑃:(0...𝑀)⟶ℝ*)
86, 7syl 17 . 2 (𝜑𝑃:(0...𝑀)⟶ℝ*)
9 iccpartxr.i . 2 (𝜑𝐼 ∈ (0...𝑀))
108, 9ffvelcdmd 7105 1 (𝜑 → (𝑃𝐼) ∈ ℝ*)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wcel 2106  wral 3059   class class class wbr 5148  wf 6559  cfv 6563  (class class class)co 7431  m cmap 8865  0cc0 11153  1c1 11154   + caddc 11156  *cxr 11292   < clt 11293  cn 12264  ...cfz 13544  ..^cfzo 13691  RePartciccp 47338
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-map 8867  df-iccp 47339
This theorem is referenced by:  iccpartipre  47346  iccpartiltu  47347  iccpartigtl  47348  iccpartlt  47349  iccpartleu  47353  iccpartgel  47354  iccpartrn  47355  iccelpart  47358  iccpartiun  47359  icceuelpartlem  47360  icceuelpart  47361  iccpartdisj  47362  iccpartnel  47363  bgoldbtbndlem2  47731
  Copyright terms: Public domain W3C validator