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Theorem iccpartxr 47411
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 47408 . . . . . 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 8890 . . 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 7104 1 (𝜑 → (𝑃𝐼) ∈ ℝ*)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wcel 2107  wral 3060   class class class wbr 5142  wf 6556  cfv 6560  (class class class)co 7432  m cmap 8867  0cc0 11156  1c1 11157   + caddc 11159  *cxr 11295   < clt 11296  cn 12267  ...cfz 13548  ..^cfzo 13695  RePartciccp 47405
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-1st 8015  df-2nd 8016  df-map 8869  df-iccp 47406
This theorem is referenced by:  iccpartipre  47413  iccpartiltu  47414  iccpartigtl  47415  iccpartlt  47416  iccpartleu  47420  iccpartgel  47421  iccpartrn  47422  iccelpart  47425  iccpartiun  47426  icceuelpartlem  47427  icceuelpart  47428  iccpartdisj  47429  iccpartnel  47430  bgoldbtbndlem2  47798
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