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Theorem iccpartxr 46640
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 46637 . . . . . 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 8842 . . 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 7080 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 2098  βˆ€wral 3055   class class class wbr 5141  βŸΆwf 6532  β€˜cfv 6536  (class class class)co 7404   ↑m cmap 8819  0cc0 11109  1c1 11110   + caddc 11112  β„*cxr 11248   < clt 11249  β„•cn 12213  ...cfz 13487  ..^cfzo 13630  RePartciccp 46634
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544  df-ov 7407  df-oprab 7408  df-mpo 7409  df-1st 7971  df-2nd 7972  df-map 8821  df-iccp 46635
This theorem is referenced by:  iccpartipre  46642  iccpartiltu  46643  iccpartigtl  46644  iccpartlt  46645  iccpartleu  46649  iccpartgel  46650  iccpartrn  46651  iccelpart  46654  iccpartiun  46655  icceuelpartlem  46656  icceuelpart  46657  iccpartdisj  46658  iccpartnel  46659  bgoldbtbndlem2  47027
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