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Theorem iccpartxr 43570
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 43567 . . . . . 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 234 . . . 4 (𝜑 → (𝑃 ∈ (ℝ*m (0...𝑀)) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑃𝑖) < (𝑃‘(𝑖 + 1))))
65simpld 497 . . 3 (𝜑𝑃 ∈ (ℝ*m (0...𝑀)))
7 elmapi 8420 . . 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, 9ffvelrnd 6845 1 (𝜑 → (𝑃𝐼) ∈ ℝ*)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  wcel 2108  wral 3136   class class class wbr 5057  wf 6344  cfv 6348  (class class class)co 7148  m cmap 8398  0cc0 10529  1c1 10530   + caddc 10532  *cxr 10666   < clt 10667  cn 11630  ...cfz 12884  ..^cfzo 13025  RePartciccp 43564
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356  df-ov 7151  df-oprab 7152  df-mpo 7153  df-1st 7681  df-2nd 7682  df-map 8400  df-iccp 43565
This theorem is referenced by:  iccpartipre  43572  iccpartiltu  43573  iccpartigtl  43574  iccpartlt  43575  iccpartleu  43579  iccpartgel  43580  iccpartrn  43581  iccelpart  43584  iccpartiun  43585  icceuelpartlem  43586  icceuelpart  43587  iccpartdisj  43588  iccpartnel  43589  bgoldbtbndlem2  43962
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