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Theorem iccpartxr 46759
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 46756 . . . . . 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 494 . . 3 (πœ‘ β†’ 𝑃 ∈ (ℝ* ↑m (0...𝑀)))
7 elmapi 8868 . . 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 7095 1 (πœ‘ β†’ (π‘ƒβ€˜πΌ) ∈ ℝ*)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∈ wcel 2099  βˆ€wral 3058   class class class wbr 5148  βŸΆwf 6544  β€˜cfv 6548  (class class class)co 7420   ↑m cmap 8845  0cc0 11139  1c1 11140   + caddc 11142  β„*cxr 11278   < clt 11279  β„•cn 12243  ...cfz 13517  ..^cfzo 13660  RePartciccp 46753
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-fv 6556  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1st 7993  df-2nd 7994  df-map 8847  df-iccp 46754
This theorem is referenced by:  iccpartipre  46761  iccpartiltu  46762  iccpartigtl  46763  iccpartlt  46764  iccpartleu  46768  iccpartgel  46769  iccpartrn  46770  iccelpart  46773  iccpartiun  46774  icceuelpartlem  46775  icceuelpart  46776  iccpartdisj  46777  iccpartnel  46778  bgoldbtbndlem2  47146
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