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Theorem iccpartimp 41678
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 41677 . . 3 (𝑀 ∈ ℕ → (𝑃 ∈ (RePart‘𝑀) ↔ (𝑃 ∈ (ℝ*𝑚 (0...𝑀)) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑃𝑖) < (𝑃‘(𝑖 + 1)))))
2 fveq2 6229 . . . . . . . 8 (𝑖 = 𝐼 → (𝑃𝑖) = (𝑃𝐼))
3 oveq1 6697 . . . . . . . . 9 (𝑖 = 𝐼 → (𝑖 + 1) = (𝐼 + 1))
43fveq2d 6233 . . . . . . . 8 (𝑖 = 𝐼 → (𝑃‘(𝑖 + 1)) = (𝑃‘(𝐼 + 1)))
52, 4breq12d 4698 . . . . . . 7 (𝑖 = 𝐼 → ((𝑃𝑖) < (𝑃‘(𝑖 + 1)) ↔ (𝑃𝐼) < (𝑃‘(𝐼 + 1))))
65rspcva 3338 . . . . . 6 ((𝐼 ∈ (0..^𝑀) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑃𝑖) < (𝑃‘(𝑖 + 1))) → (𝑃𝐼) < (𝑃‘(𝐼 + 1)))
76expcom 450 . . . . 5 (∀𝑖 ∈ (0..^𝑀)(𝑃𝑖) < (𝑃‘(𝑖 + 1)) → (𝐼 ∈ (0..^𝑀) → (𝑃𝐼) < (𝑃‘(𝐼 + 1))))
87adantl 481 . . . 4 ((𝑃 ∈ (ℝ*𝑚 (0...𝑀)) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑃𝑖) < (𝑃‘(𝑖 + 1))) → (𝐼 ∈ (0..^𝑀) → (𝑃𝐼) < (𝑃‘(𝐼 + 1))))
9 simpl 472 . . . 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 1275 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 383  w3a 1054   = wceq 1523  wcel 2030  wral 2941   class class class wbr 4685  cfv 5926  (class class class)co 6690  𝑚 cmap 7899  0cc0 9974  1c1 9975   + caddc 9977  *cxr 10111   < clt 10112  cn 11058  ...cfz 12364  ..^cfzo 12504  RePartciccp 41674
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fv 5934  df-ov 6693  df-iccp 41675
This theorem is referenced by:  iccpartgtprec  41681  iccpartipre  41682  iccpartiltu  41683  iccpartigtl  41684  iccpartlt  41685  iccpartgt  41688
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