Theorem iccpartxr 47581

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 47578	. . . . . 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 8782	. . 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 7027	1 ⊢ (𝜑 → (𝑃‘𝐼) ∈ ℝ*)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∈ wcel 2113  ∀wral 3048   class class class wbr 5095  ⟶wf 6485  ‘cfv 6489  (class class class)co 7355   ↑_m cmap 8759  0cc0 11017  1c1 11018   + caddc 11020  ℝ^*cxr 11156   < clt 11157  ℕcn 12136  ...cfz 13414  ..^cfzo 13561  RePartciccp 47575

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 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7677

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 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-fv 6497  df-ov 7358  df-oprab 7359  df-mpo 7360  df-1st 7930  df-2nd 7931  df-map 8761  df-iccp 47576

This theorem is referenced by:  iccpartipre  47583  iccpartiltu  47584  iccpartigtl  47585  iccpartlt  47586  iccpartleu  47590  iccpartgel  47591  iccpartrn  47592  iccelpart  47595  iccpartiun  47596  icceuelpartlem  47597  icceuelpart  47598  iccpartdisj  47599  iccpartnel  47600  bgoldbtbndlem2  47968