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Theorem iccpartxr 41865
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 41862 . . . . . 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 477 . . 3 (𝜑𝑃 ∈ (ℝ*𝑚 (0...𝑀)))
7 elmapi 8045 . . 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 6523 1 (𝜑 → (𝑃𝐼) ∈ ℝ*)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wcel 2139  wral 3050   class class class wbr 4804  wf 6045  cfv 6049  (class class class)co 6813  𝑚 cmap 8023  0cc0 10128  1c1 10129   + caddc 10131  *cxr 10265   < clt 10266  cn 11212  ...cfz 12519  ..^cfzo 12659  RePartciccp 41859
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-1st 7333  df-2nd 7334  df-map 8025  df-iccp 41860
This theorem is referenced by:  iccpartipre  41867  iccpartiltu  41868  iccpartigtl  41869  iccpartlt  41870  iccpartleu  41874  iccpartgel  41875  iccpartrn  41876  iccelpart  41879  iccpartiun  41880  icceuelpartlem  41881  icceuelpart  41882  iccpartdisj  41883  iccpartnel  41884  bgoldbtbndlem2  42204
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