Theorem iccpartxr 47456

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	⊢ (𝜑 → 𝑀 ∈ ℕ)
iccpartgtprec.p	⊢ (𝜑 → 𝑃 ∈ (RePart‘𝑀))
iccpartxr.i	⊢ (𝜑 → 𝐼 ∈ (0...𝑀))

Assertion

Ref	Expression
iccpartxr	⊢ (𝜑 → (𝑃‘𝐼) ∈ ℝ*)

Proof of Theorem iccpartxr

Dummy variable 𝑖 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		iccpartgtprec.p	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ (RePart‘𝑀))
2		iccpartgtprec.m	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℕ)
3		iccpart 47453	. . . . . 6 ⊢ (𝑀 ∈ ℕ → (𝑃 ∈ (RePart‘𝑀) ↔ (𝑃 ∈ (ℝ* ↑m (0...𝑀)) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑃‘𝑖) < (𝑃‘(𝑖 + 1)))))
4	2, 3	syl 17	. . . . 5 ⊢ (𝜑 → (𝑃 ∈ (RePart‘𝑀) ↔ (𝑃 ∈ (ℝ* ↑m (0...𝑀)) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑃‘𝑖) < (𝑃‘(𝑖 + 1)))))
5	1, 4	mpbid 232	. . . 4 ⊢ (𝜑 → (𝑃 ∈ (ℝ* ↑m (0...𝑀)) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑃‘𝑖) < (𝑃‘(𝑖 + 1))))
6	5	simpld 494	. . 3 ⊢ (𝜑 → 𝑃 ∈ (ℝ* ↑m (0...𝑀)))
7		elmapi 8773	. . 3 ⊢ (𝑃 ∈ (ℝ* ↑m (0...𝑀)) → 𝑃:(0...𝑀)⟶ℝ*)
8	6, 7	syl 17	. 2 ⊢ (𝜑 → 𝑃:(0...𝑀)⟶ℝ*)
9		iccpartxr.i	. 2 ⊢ (𝜑 → 𝐼 ∈ (0...𝑀))
10	8, 9	ffvelcdmd 7018	1 ⊢ (𝜑 → (𝑃‘𝐼) ∈ ℝ*)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∈ wcel 2111  ∀wral 3047   class class class wbr 5091  ⟶wf 6477  ‘cfv 6481  (class class class)co 7346   ↑_m cmap 8750  0cc0 11006  1c1 11007   + caddc 11009  ℝ^*cxr 11145   < clt 11146  ℕcn 12125  ...cfz 13407  ..^cfzo 13554  RePartciccp 47450

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-1st 7921  df-2nd 7922  df-map 8752  df-iccp 47451

This theorem is referenced by:  iccpartipre  47458  iccpartiltu  47459  iccpartigtl  47460  iccpartlt  47461  iccpartleu  47465  iccpartgel  47466  iccpartrn  47467  iccelpart  47470  iccpartiun  47471  icceuelpartlem  47472  icceuelpart  47473  iccpartdisj  47474  iccpartnel  47475  bgoldbtbndlem2  47843