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	⊢ (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 47408	. . . . . 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 8890	. . 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 7104	1 ⊢ (𝜑 → (𝑃‘𝐼) ∈ ℝ*)

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  ℝ^*cxr 11295   < clt 11296  ℕcn 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