Theorem iccpartimp 47984

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 47983	. . 3 ⊢ (𝑀 ∈ ℕ → (𝑃 ∈ (RePart‘𝑀) ↔ (𝑃 ∈ (ℝ* ↑m (0...𝑀)) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑃‘𝑖) < (𝑃‘(𝑖 + 1)))))
2		fveq2 6862	. . . . . . 7 ⊢ (𝑖 = 𝐼 → (𝑃‘𝑖) = (𝑃‘𝐼))
3		fvoveq1 7414	. . . . . . 7 ⊢ (𝑖 = 𝐼 → (𝑃‘(𝑖 + 1)) = (𝑃‘(𝐼 + 1)))
4	2, 3	breq12d 5110	. . . . . 6 ⊢ (𝑖 = 𝐼 → ((𝑃‘𝑖) < (𝑃‘(𝑖 + 1)) ↔ (𝑃‘𝐼) < (𝑃‘(𝐼 + 1))))
5	4	rspccv 3577	. . . . 5 ⊢ (∀𝑖 ∈ (0..^𝑀)(𝑃‘𝑖) < (𝑃‘(𝑖 + 1)) → (𝐼 ∈ (0..^𝑀) → (𝑃‘𝐼) < (𝑃‘(𝐼 + 1))))
6	5	adantl 485	. . . 4 ⊢ ((𝑃 ∈ (ℝ* ↑m (0...𝑀)) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑃‘𝑖) < (𝑃‘(𝑖 + 1))) → (𝐼 ∈ (0..^𝑀) → (𝑃‘𝐼) < (𝑃‘(𝐼 + 1))))
7		simpl 486	. . . 4 ⊢ ((𝑃 ∈ (ℝ* ↑m (0...𝑀)) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑃‘𝑖) < (𝑃‘(𝑖 + 1))) → 𝑃 ∈ (ℝ* ↑m (0...𝑀)))
8	6, 7	jctild 533	. . 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 255	. 2 ⊢ (𝑀 ∈ ℕ → (𝑃 ∈ (RePart‘𝑀) → (𝐼 ∈ (0..^𝑀) → (𝑃 ∈ (ℝ* ↑m (0...𝑀)) ∧ (𝑃‘𝐼) < (𝑃‘(𝐼 + 1))))))
10	9	3imp 1122	1 ⊢ ((𝑀 ∈ ℕ ∧ 𝑃 ∈ (RePart‘𝑀) ∧ 𝐼 ∈ (0..^𝑀)) → (𝑃 ∈ (ℝ* ↑m (0...𝑀)) ∧ (𝑃‘𝐼) < (𝑃‘(𝐼 + 1))))

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1097   = wceq 1559   ∈ wcel 2141  ∀wral 3075   class class class wbr 5097  ‘cfv 6516  (class class class)co 7391   ↑_m cmap 8802  0cc0 11067  1c1 11068   + caddc 11070  ℝ^*cxr 11209   < clt 11210  ℕcn 12204  ...cfz 13506  ..^cfzo 13653  RePartciccp 47980

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5243  ax-nul 5253  ax-pr 5387

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-opab 5160  df-mpt 5179  df-id 5538  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-iota 6472  df-fun 6518  df-fv 6524  df-ov 7394  df-iccp 47981

This theorem is referenced by:  iccpartgtprec  47987  iccpartipre  47988  iccpartiltu  47989  iccpartigtl  47990  iccpartlt  47991  iccpartgt  47994