Theorem iccpartimp 47579

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 47578	. . 3 ⊢ (𝑀 ∈ ℕ → (𝑃 ∈ (RePart‘𝑀) ↔ (𝑃 ∈ (ℝ* ↑m (0...𝑀)) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑃‘𝑖) < (𝑃‘(𝑖 + 1)))))
2		fveq2 6831	. . . . . . 7 ⊢ (𝑖 = 𝐼 → (𝑃‘𝑖) = (𝑃‘𝐼))
3		fvoveq1 7378	. . . . . . 7 ⊢ (𝑖 = 𝐼 → (𝑃‘(𝑖 + 1)) = (𝑃‘(𝐼 + 1)))
4	2, 3	breq12d 5108	. . . . . 6 ⊢ (𝑖 = 𝐼 → ((𝑃‘𝑖) < (𝑃‘(𝑖 + 1)) ↔ (𝑃‘𝐼) < (𝑃‘(𝐼 + 1))))
5	4	rspccv 3570	. . . . 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 1541   ∈ wcel 2113  ∀wral 3048   class class class wbr 5095  ‘cfv 6489  (class class class)co 7355   ↑_m cmap 8759  0cc0 11017  1c1 11018   + caddc 11020  ℝ^*cxr 11156   < clt 11157  ℕcn 12136  ...cfz 13414  ..^cfzo 13561  RePartciccp 47575

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-iota 6445  df-fun 6491  df-fv 6497  df-ov 7358  df-iccp 47576

This theorem is referenced by:  iccpartgtprec  47582  iccpartipre  47583  iccpartiltu  47584  iccpartigtl  47585  iccpartlt  47586  iccpartgt  47589