Theorem iccpartimp 41353

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 41352	. . 3 ⊢ (𝑀 ∈ ℕ → (𝑃 ∈ (RePart‘𝑀) ↔ (𝑃 ∈ (ℝ* ↑𝑚 (0...𝑀)) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑃‘𝑖) < (𝑃‘(𝑖 + 1)))))
2		fveq2 6191	. . . . . . . 8 ⊢ (𝑖 = 𝐼 → (𝑃‘𝑖) = (𝑃‘𝐼))
3		oveq1 6657	. . . . . . . . 9 ⊢ (𝑖 = 𝐼 → (𝑖 + 1) = (𝐼 + 1))
4	3	fveq2d 6195	. . . . . . . 8 ⊢ (𝑖 = 𝐼 → (𝑃‘(𝑖 + 1)) = (𝑃‘(𝐼 + 1)))
5	2, 4	breq12d 4666	. . . . . . 7 ⊢ (𝑖 = 𝐼 → ((𝑃‘𝑖) < (𝑃‘(𝑖 + 1)) ↔ (𝑃‘𝐼) < (𝑃‘(𝐼 + 1))))
6	5	rspcva 3307	. . . . . 6 ⊢ ((𝐼 ∈ (0..^𝑀) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑃‘𝑖) < (𝑃‘(𝑖 + 1))) → (𝑃‘𝐼) < (𝑃‘(𝐼 + 1)))
7	6	expcom 451	. . . . 5 ⊢ (∀𝑖 ∈ (0..^𝑀)(𝑃‘𝑖) < (𝑃‘(𝑖 + 1)) → (𝐼 ∈ (0..^𝑀) → (𝑃‘𝐼) < (𝑃‘(𝐼 + 1))))
8	7	adantl 482	. . . 4 ⊢ ((𝑃 ∈ (ℝ* ↑𝑚 (0...𝑀)) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑃‘𝑖) < (𝑃‘(𝑖 + 1))) → (𝐼 ∈ (0..^𝑀) → (𝑃‘𝐼) < (𝑃‘(𝐼 + 1))))
9		simpl 473	. . . 4 ⊢ ((𝑃 ∈ (ℝ* ↑𝑚 (0...𝑀)) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑃‘𝑖) < (𝑃‘(𝑖 + 1))) → 𝑃 ∈ (ℝ* ↑𝑚 (0...𝑀)))
10	8, 9	jctild 566	. . 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)))))
11	1, 10	syl6bi 243	. 2 ⊢ (𝑀 ∈ ℕ → (𝑃 ∈ (RePart‘𝑀) → (𝐼 ∈ (0..^𝑀) → (𝑃 ∈ (ℝ* ↑𝑚 (0...𝑀)) ∧ (𝑃‘𝐼) < (𝑃‘(𝐼 + 1))))))
12	11	3imp 1256	1 ⊢ ((𝑀 ∈ ℕ ∧ 𝑃 ∈ (RePart‘𝑀) ∧ 𝐼 ∈ (0..^𝑀)) → (𝑃 ∈ (ℝ* ↑𝑚 (0...𝑀)) ∧ (𝑃‘𝐼) < (𝑃‘(𝐼 + 1))))

Syntax hints:   → wi 4   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ∀wral 2912   class class class wbr 4653  ‘cfv 5888  (class class class)co 6650   ↑_𝑚 cmap 7857  0cc0 9936  1c1 9937   + caddc 9939  ℝ^*cxr 10073   < clt 10074  ℕcn 11020  ...cfz 12326  ..^cfzo 12465  RePartciccp 41349

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-iccp 41350

This theorem is referenced by:  iccpartgtprec  41356  iccpartipre  41357  iccpartiltu  41358  iccpartigtl  41359  iccpartlt  41360  iccpartgt  41363