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Theorem iccpartimp 43757
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 43756 . . 3 (𝑀 ∈ ℕ → (𝑃 ∈ (RePart‘𝑀) ↔ (𝑃 ∈ (ℝ*m (0...𝑀)) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑃𝑖) < (𝑃‘(𝑖 + 1)))))
2 fveq2 6643 . . . . . . 7 (𝑖 = 𝐼 → (𝑃𝑖) = (𝑃𝐼))
3 fvoveq1 7153 . . . . . . 7 (𝑖 = 𝐼 → (𝑃‘(𝑖 + 1)) = (𝑃‘(𝐼 + 1)))
42, 3breq12d 5052 . . . . . 6 (𝑖 = 𝐼 → ((𝑃𝑖) < (𝑃‘(𝑖 + 1)) ↔ (𝑃𝐼) < (𝑃‘(𝐼 + 1))))
54rspccv 3597 . . . . 5 (∀𝑖 ∈ (0..^𝑀)(𝑃𝑖) < (𝑃‘(𝑖 + 1)) → (𝐼 ∈ (0..^𝑀) → (𝑃𝐼) < (𝑃‘(𝐼 + 1))))
65adantl 485 . . . 4 ((𝑃 ∈ (ℝ*m (0...𝑀)) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑃𝑖) < (𝑃‘(𝑖 + 1))) → (𝐼 ∈ (0..^𝑀) → (𝑃𝐼) < (𝑃‘(𝐼 + 1))))
7 simpl 486 . . . 4 ((𝑃 ∈ (ℝ*m (0...𝑀)) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑃𝑖) < (𝑃‘(𝑖 + 1))) → 𝑃 ∈ (ℝ*m (0...𝑀)))
86, 7jctild 529 . . 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)))))
91, 8syl6bi 256 . 2 (𝑀 ∈ ℕ → (𝑃 ∈ (RePart‘𝑀) → (𝐼 ∈ (0..^𝑀) → (𝑃 ∈ (ℝ*m (0...𝑀)) ∧ (𝑃𝐼) < (𝑃‘(𝐼 + 1))))))
1093imp 1108 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 399  w3a 1084   = wceq 1538  wcel 2115  wral 3126   class class class wbr 5039  cfv 6328  (class class class)co 7130  m cmap 8381  0cc0 10514  1c1 10515   + caddc 10517  *cxr 10651   < clt 10652  cn 11615  ...cfz 12875  ..^cfzo 13016  RePartciccp 43753
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pr 5303
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-sbc 3750  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-br 5040  df-opab 5102  df-mpt 5120  df-id 5433  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-iota 6287  df-fun 6330  df-fv 6336  df-ov 7133  df-iccp 43754
This theorem is referenced by:  iccpartgtprec  43760  iccpartipre  43761  iccpartiltu  43762  iccpartigtl  43763  iccpartlt  43764  iccpartgt  43767
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