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Theorem iccpartxr 40683
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 40680 . . . . . 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 222 . . . 4 (𝜑 → (𝑃 ∈ (ℝ*𝑚 (0...𝑀)) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑃𝑖) < (𝑃‘(𝑖 + 1))))
65simpld 475 . . 3 (𝜑𝑃 ∈ (ℝ*𝑚 (0...𝑀)))
7 elmapi 7839 . . 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 6326 1 (𝜑 → (𝑃𝐼) ∈ ℝ*)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  wcel 1987  wral 2908   class class class wbr 4623  wf 5853  cfv 5857  (class class class)co 6615  𝑚 cmap 7817  0cc0 9896  1c1 9897   + caddc 9899  *cxr 10033   < clt 10034  cn 10980  ...cfz 12284  ..^cfzo 12422  RePartciccp 40677
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-fv 5865  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-1st 7128  df-2nd 7129  df-map 7819  df-iccp 40678
This theorem is referenced by:  iccpartipre  40685  iccpartiltu  40686  iccpartigtl  40687  iccpartlt  40688  iccpartleu  40692  iccpartgel  40693  iccpartrn  40694  iccelpart  40697  iccpartiun  40698  icceuelpartlem  40699  icceuelpart  40700  iccpartdisj  40701  iccpartnel  40702  bgoldbtbndlem2  41013
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