Theorem iccpart 47892

Description: A special partition. Corresponds to fourierdlem2 46559 in GS's mathbox. (Contributed by AV, 9-Jul-2020.)

Assertion

Ref	Expression
iccpart	⊢ (𝑀 ∈ ℕ → (𝑃 ∈ (RePart‘𝑀) ↔ (𝑃 ∈ (ℝ* ↑m (0...𝑀)) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑃‘𝑖) < (𝑃‘(𝑖 + 1)))))

Distinct variable groups:   𝑖,𝑀   𝑃,𝑖

Proof of Theorem iccpart

Dummy variable 𝑝 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		iccpval 47891	. . 3 ⊢ (𝑀 ∈ ℕ → (RePart‘𝑀) = {𝑝 ∈ (ℝ* ↑m (0...𝑀)) ∣ ∀𝑖 ∈ (0..^𝑀)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1))})
2	1	eleq2d 2823	. 2 ⊢ (𝑀 ∈ ℕ → (𝑃 ∈ (RePart‘𝑀) ↔ 𝑃 ∈ {𝑝 ∈ (ℝ* ↑m (0...𝑀)) ∣ ∀𝑖 ∈ (0..^𝑀)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1))}))
3		fveq1 6835	. . . . 5 ⊢ (𝑝 = 𝑃 → (𝑝‘𝑖) = (𝑃‘𝑖))
4		fveq1 6835	. . . . 5 ⊢ (𝑝 = 𝑃 → (𝑝‘(𝑖 + 1)) = (𝑃‘(𝑖 + 1)))
5	3, 4	breq12d 5099	. . . 4 ⊢ (𝑝 = 𝑃 → ((𝑝‘𝑖) < (𝑝‘(𝑖 + 1)) ↔ (𝑃‘𝑖) < (𝑃‘(𝑖 + 1))))
6	5	ralbidv 3161	. . 3 ⊢ (𝑝 = 𝑃 → (∀𝑖 ∈ (0..^𝑀)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)) ↔ ∀𝑖 ∈ (0..^𝑀)(𝑃‘𝑖) < (𝑃‘(𝑖 + 1))))
7	6	elrab 3635	. 2 ⊢ (𝑃 ∈ {𝑝 ∈ (ℝ* ↑m (0...𝑀)) ∣ ∀𝑖 ∈ (0..^𝑀)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1))} ↔ (𝑃 ∈ (ℝ* ↑m (0...𝑀)) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑃‘𝑖) < (𝑃‘(𝑖 + 1))))
8	2, 7	bitrdi 287	1 ⊢ (𝑀 ∈ ℕ → (𝑃 ∈ (RePart‘𝑀) ↔ (𝑃 ∈ (ℝ* ↑m (0...𝑀)) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑃‘𝑖) < (𝑃‘(𝑖 + 1)))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052  {crab 3390   class class class wbr 5086  ‘cfv 6494  (class class class)co 7362   ↑_m cmap 8768  0cc0 11033  1c1 11034   + caddc 11036  ℝ^*cxr 11173   < clt 11174  ℕcn 12169  ...cfz 13456  ..^cfzo 13603  RePartciccp 47889

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5232  ax-nul 5242  ax-pr 5372

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5521  df-xp 5632  df-rel 5633  df-cnv 5634  df-co 5635  df-dm 5636  df-iota 6450  df-fun 6496  df-fv 6502  df-ov 7365  df-iccp 47890

This theorem is referenced by:  iccpartimp  47893  iccpartres  47894  iccpartxr  47895  iccpartrn  47906  iccpartf  47907  iccpartnel  47914