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	⊢ (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 41862	. . . . . 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 222	. . . 4 ⊢ (𝜑 → (𝑃 ∈ (ℝ* ↑𝑚 (0...𝑀)) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑃‘𝑖) < (𝑃‘(𝑖 + 1))))
6	5	simpld 477	. . 3 ⊢ (𝜑 → 𝑃 ∈ (ℝ* ↑𝑚 (0...𝑀)))
7		elmapi 8045	. . 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	ffvelrnd 6523	1 ⊢ (𝜑 → (𝑃‘𝐼) ∈ ℝ*)

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  ℝ^*cxr 10265   < clt 10266  ℕcn 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