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Theorem iccpartxr 45701
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 (πœ‘ β†’ 𝑀 ∈ β„•)
iccpartgtprec.p (πœ‘ β†’ 𝑃 ∈ (RePartβ€˜π‘€))
iccpartxr.i (πœ‘ β†’ 𝐼 ∈ (0...𝑀))
Assertion
Ref Expression
iccpartxr (πœ‘ β†’ (π‘ƒβ€˜πΌ) ∈ ℝ*)

Proof of Theorem iccpartxr
Dummy variable 𝑖 is distinct from all other variables.
StepHypRef Expression
1 iccpartgtprec.p . . . . 5 (πœ‘ β†’ 𝑃 ∈ (RePartβ€˜π‘€))
2 iccpartgtprec.m . . . . . 6 (πœ‘ β†’ 𝑀 ∈ β„•)
3 iccpart 45698 . . . . . 6 (𝑀 ∈ β„• β†’ (𝑃 ∈ (RePartβ€˜π‘€) ↔ (𝑃 ∈ (ℝ* ↑m (0...𝑀)) ∧ βˆ€π‘– ∈ (0..^𝑀)(π‘ƒβ€˜π‘–) < (π‘ƒβ€˜(𝑖 + 1)))))
42, 3syl 17 . . . . 5 (πœ‘ β†’ (𝑃 ∈ (RePartβ€˜π‘€) ↔ (𝑃 ∈ (ℝ* ↑m (0...𝑀)) ∧ βˆ€π‘– ∈ (0..^𝑀)(π‘ƒβ€˜π‘–) < (π‘ƒβ€˜(𝑖 + 1)))))
51, 4mpbid 231 . . . 4 (πœ‘ β†’ (𝑃 ∈ (ℝ* ↑m (0...𝑀)) ∧ βˆ€π‘– ∈ (0..^𝑀)(π‘ƒβ€˜π‘–) < (π‘ƒβ€˜(𝑖 + 1))))
65simpld 496 . . 3 (πœ‘ β†’ 𝑃 ∈ (ℝ* ↑m (0...𝑀)))
7 elmapi 8793 . . 3 (𝑃 ∈ (ℝ* ↑m (0...𝑀)) β†’ 𝑃:(0...𝑀)βŸΆβ„*)
86, 7syl 17 . 2 (πœ‘ β†’ 𝑃:(0...𝑀)βŸΆβ„*)
9 iccpartxr.i . 2 (πœ‘ β†’ 𝐼 ∈ (0...𝑀))
108, 9ffvelcdmd 7040 1 (πœ‘ β†’ (π‘ƒβ€˜πΌ) ∈ ℝ*)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∈ wcel 2107  βˆ€wral 3061   class class class wbr 5109  βŸΆwf 6496  β€˜cfv 6500  (class class class)co 7361   ↑m cmap 8771  0cc0 11059  1c1 11060   + caddc 11062  β„*cxr 11196   < clt 11197  β„•cn 12161  ...cfz 13433  ..^cfzo 13576  RePartciccp 45695
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-fv 6508  df-ov 7364  df-oprab 7365  df-mpo 7366  df-1st 7925  df-2nd 7926  df-map 8773  df-iccp 45696
This theorem is referenced by:  iccpartipre  45703  iccpartiltu  45704  iccpartigtl  45705  iccpartlt  45706  iccpartleu  45710  iccpartgel  45711  iccpartrn  45712  iccelpart  45715  iccpartiun  45716  icceuelpartlem  45717  icceuelpart  45718  iccpartdisj  45719  iccpartnel  45720  bgoldbtbndlem2  46088
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