Theorem iccpartxr 42195

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 42192	. . . . . 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 224	. . . 4 ⊢ (𝜑 → (𝑃 ∈ (ℝ* ↑𝑚 (0...𝑀)) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑃‘𝑖) < (𝑃‘(𝑖 + 1))))
6	5	simpld 489	. . 3 ⊢ (𝜑 → 𝑃 ∈ (ℝ* ↑𝑚 (0...𝑀)))
7		elmapi 8117	. . 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 6586	1 ⊢ (𝜑 → (𝑃‘𝐼) ∈ ℝ*)

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 385   ∈ wcel 2157  ∀wral 3089   class class class wbr 4843  ⟶wf 6097  ‘cfv 6101  (class class class)co 6878   ↑_𝑚 cmap 8095  0cc0 10224  1c1 10225   + caddc 10227  ℝ^*cxr 10362   < clt 10363  ℕcn 11312  ...cfz 12580  ..^cfzo 12720  RePartciccp 42189

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-fv 6109  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-1st 7401  df-2nd 7402  df-map 8097  df-iccp 42190

This theorem is referenced by:  iccpartipre  42197  iccpartiltu  42198  iccpartigtl  42199  iccpartlt  42200  iccpartleu  42204  iccpartgel  42205  iccpartrn  42206  iccelpart  42209  iccpartiun  42210  icceuelpartlem  42211  icceuelpart  42212  iccpartdisj  42213  iccpartnel  42214  bgoldbtbndlem2  42476