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Theorem iccpartimp 40672
Description: Implications for a class being a partition. (Contributed by AV, 11-Jul-2020.)
Assertion
Ref Expression
iccpartimp ((𝑀 ∈ ℕ ∧ 𝑃 ∈ (RePart‘𝑀) ∧ 𝐼 ∈ (0..^𝑀)) → (𝑃 ∈ (ℝ*𝑚 (0...𝑀)) ∧ (𝑃𝐼) < (𝑃‘(𝐼 + 1))))

Proof of Theorem iccpartimp
Dummy variable 𝑖 is distinct from all other variables.
StepHypRef Expression
1 iccpart 40671 . . 3 (𝑀 ∈ ℕ → (𝑃 ∈ (RePart‘𝑀) ↔ (𝑃 ∈ (ℝ*𝑚 (0...𝑀)) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑃𝑖) < (𝑃‘(𝑖 + 1)))))
2 fveq2 6153 . . . . . . . 8 (𝑖 = 𝐼 → (𝑃𝑖) = (𝑃𝐼))
3 oveq1 6617 . . . . . . . . 9 (𝑖 = 𝐼 → (𝑖 + 1) = (𝐼 + 1))
43fveq2d 6157 . . . . . . . 8 (𝑖 = 𝐼 → (𝑃‘(𝑖 + 1)) = (𝑃‘(𝐼 + 1)))
52, 4breq12d 4631 . . . . . . 7 (𝑖 = 𝐼 → ((𝑃𝑖) < (𝑃‘(𝑖 + 1)) ↔ (𝑃𝐼) < (𝑃‘(𝐼 + 1))))
65rspcva 3296 . . . . . 6 ((𝐼 ∈ (0..^𝑀) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑃𝑖) < (𝑃‘(𝑖 + 1))) → (𝑃𝐼) < (𝑃‘(𝐼 + 1)))
76expcom 451 . . . . 5 (∀𝑖 ∈ (0..^𝑀)(𝑃𝑖) < (𝑃‘(𝑖 + 1)) → (𝐼 ∈ (0..^𝑀) → (𝑃𝐼) < (𝑃‘(𝐼 + 1))))
87adantl 482 . . . 4 ((𝑃 ∈ (ℝ*𝑚 (0...𝑀)) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑃𝑖) < (𝑃‘(𝑖 + 1))) → (𝐼 ∈ (0..^𝑀) → (𝑃𝐼) < (𝑃‘(𝐼 + 1))))
9 simpl 473 . . . 4 ((𝑃 ∈ (ℝ*𝑚 (0...𝑀)) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑃𝑖) < (𝑃‘(𝑖 + 1))) → 𝑃 ∈ (ℝ*𝑚 (0...𝑀)))
108, 9jctild 565 . . 3 ((𝑃 ∈ (ℝ*𝑚 (0...𝑀)) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑃𝑖) < (𝑃‘(𝑖 + 1))) → (𝐼 ∈ (0..^𝑀) → (𝑃 ∈ (ℝ*𝑚 (0...𝑀)) ∧ (𝑃𝐼) < (𝑃‘(𝐼 + 1)))))
111, 10syl6bi 243 . 2 (𝑀 ∈ ℕ → (𝑃 ∈ (RePart‘𝑀) → (𝐼 ∈ (0..^𝑀) → (𝑃 ∈ (ℝ*𝑚 (0...𝑀)) ∧ (𝑃𝐼) < (𝑃‘(𝐼 + 1))))))
12113imp 1254 1 ((𝑀 ∈ ℕ ∧ 𝑃 ∈ (RePart‘𝑀) ∧ 𝐼 ∈ (0..^𝑀)) → (𝑃 ∈ (ℝ*𝑚 (0...𝑀)) ∧ (𝑃𝐼) < (𝑃‘(𝐼 + 1))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907   class class class wbr 4618  cfv 5852  (class class class)co 6610  𝑚 cmap 7809  0cc0 9887  1c1 9888   + caddc 9890  *cxr 10024   < clt 10025  cn 10971  ...cfz 12275  ..^cfzo 12413  RePartciccp 40668
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-iota 5815  df-fun 5854  df-fv 5860  df-ov 6613  df-iccp 40669
This theorem is referenced by:  iccpartgtprec  40675  iccpartipre  40676  iccpartiltu  40677  iccpartigtl  40678  iccpartlt  40679  iccpartgt  40682
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