Theorem iccpartimp 47418

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)) ∧ (𝑃‘𝐼) < (𝑃‘(𝐼 + 1))))

Proof of Theorem iccpartimp

Dummy variable 𝑖 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		iccpart 47417	. . 3 ⊢ (𝑀 ∈ ℕ → (𝑃 ∈ (RePart‘𝑀) ↔ (𝑃 ∈ (ℝ* ↑m (0...𝑀)) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑃‘𝑖) < (𝑃‘(𝑖 + 1)))))
2		fveq2 6858	. . . . . . 7 ⊢ (𝑖 = 𝐼 → (𝑃‘𝑖) = (𝑃‘𝐼))
3		fvoveq1 7410	. . . . . . 7 ⊢ (𝑖 = 𝐼 → (𝑃‘(𝑖 + 1)) = (𝑃‘(𝐼 + 1)))
4	2, 3	breq12d 5120	. . . . . 6 ⊢ (𝑖 = 𝐼 → ((𝑃‘𝑖) < (𝑃‘(𝑖 + 1)) ↔ (𝑃‘𝐼) < (𝑃‘(𝐼 + 1))))
5	4	rspccv 3585	. . . . 5 ⊢ (∀𝑖 ∈ (0..^𝑀)(𝑃‘𝑖) < (𝑃‘(𝑖 + 1)) → (𝐼 ∈ (0..^𝑀) → (𝑃‘𝐼) < (𝑃‘(𝐼 + 1))))
6	5	adantl 481	. . . 4 ⊢ ((𝑃 ∈ (ℝ* ↑m (0...𝑀)) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑃‘𝑖) < (𝑃‘(𝑖 + 1))) → (𝐼 ∈ (0..^𝑀) → (𝑃‘𝐼) < (𝑃‘(𝐼 + 1))))
7		simpl 482	. . . 4 ⊢ ((𝑃 ∈ (ℝ* ↑m (0...𝑀)) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑃‘𝑖) < (𝑃‘(𝑖 + 1))) → 𝑃 ∈ (ℝ* ↑m (0...𝑀)))
8	6, 7	jctild 525	. . 3 ⊢ ((𝑃 ∈ (ℝ* ↑m (0...𝑀)) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑃‘𝑖) < (𝑃‘(𝑖 + 1))) → (𝐼 ∈ (0..^𝑀) → (𝑃 ∈ (ℝ* ↑m (0...𝑀)) ∧ (𝑃‘𝐼) < (𝑃‘(𝐼 + 1)))))
9	1, 8	biimtrdi 253	. 2 ⊢ (𝑀 ∈ ℕ → (𝑃 ∈ (RePart‘𝑀) → (𝐼 ∈ (0..^𝑀) → (𝑃 ∈ (ℝ* ↑m (0...𝑀)) ∧ (𝑃‘𝐼) < (𝑃‘(𝐼 + 1))))))
10	9	3imp 1110	1 ⊢ ((𝑀 ∈ ℕ ∧ 𝑃 ∈ (RePart‘𝑀) ∧ 𝐼 ∈ (0..^𝑀)) → (𝑃 ∈ (ℝ* ↑m (0...𝑀)) ∧ (𝑃‘𝐼) < (𝑃‘(𝐼 + 1))))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3044   class class class wbr 5107  ‘cfv 6511  (class class class)co 7387   ↑_m cmap 8799  0cc0 11068  1c1 11069   + caddc 11071  ℝ^*cxr 11207   < clt 11208  ℕcn 12186  ...cfz 13468  ..^cfzo 13615  RePartciccp 47414

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-iota 6464  df-fun 6513  df-fv 6519  df-ov 7390  df-iccp 47415

This theorem is referenced by:  iccpartgtprec  47421  iccpartipre  47422  iccpartiltu  47423  iccpartigtl  47424  iccpartlt  47425  iccpartgt  47428