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Theorem iccpartimp 47404
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 47403 . . 3 (𝑀 ∈ ℕ → (𝑃 ∈ (RePart‘𝑀) ↔ (𝑃 ∈ (ℝ*m (0...𝑀)) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑃𝑖) < (𝑃‘(𝑖 + 1)))))
2 fveq2 6906 . . . . . . 7 (𝑖 = 𝐼 → (𝑃𝑖) = (𝑃𝐼))
3 fvoveq1 7454 . . . . . . 7 (𝑖 = 𝐼 → (𝑃‘(𝑖 + 1)) = (𝑃‘(𝐼 + 1)))
42, 3breq12d 5156 . . . . . 6 (𝑖 = 𝐼 → ((𝑃𝑖) < (𝑃‘(𝑖 + 1)) ↔ (𝑃𝐼) < (𝑃‘(𝐼 + 1))))
54rspccv 3619 . . . . 5 (∀𝑖 ∈ (0..^𝑀)(𝑃𝑖) < (𝑃‘(𝑖 + 1)) → (𝐼 ∈ (0..^𝑀) → (𝑃𝐼) < (𝑃‘(𝐼 + 1))))
65adantl 481 . . . 4 ((𝑃 ∈ (ℝ*m (0...𝑀)) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑃𝑖) < (𝑃‘(𝑖 + 1))) → (𝐼 ∈ (0..^𝑀) → (𝑃𝐼) < (𝑃‘(𝐼 + 1))))
7 simpl 482 . . . 4 ((𝑃 ∈ (ℝ*m (0...𝑀)) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑃𝑖) < (𝑃‘(𝑖 + 1))) → 𝑃 ∈ (ℝ*m (0...𝑀)))
86, 7jctild 525 . . 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, 8biimtrdi 253 . 2 (𝑀 ∈ ℕ → (𝑃 ∈ (RePart‘𝑀) → (𝐼 ∈ (0..^𝑀) → (𝑃 ∈ (ℝ*m (0...𝑀)) ∧ (𝑃𝐼) < (𝑃‘(𝐼 + 1))))))
1093imp 1111 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 395  w3a 1087   = wceq 1540  wcel 2108  wral 3061   class class class wbr 5143  cfv 6561  (class class class)co 7431  m cmap 8866  0cc0 11155  1c1 11156   + caddc 11158  *cxr 11294   < clt 11295  cn 12266  ...cfz 13547  ..^cfzo 13694  RePartciccp 47400
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-iccp 47401
This theorem is referenced by:  iccpartgtprec  47407  iccpartipre  47408  iccpartiltu  47409  iccpartigtl  47410  iccpartlt  47411  iccpartgt  47414
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