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Theorem fourierdlem2 44910
Description: Membership in a partition. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Hypothesis
Ref Expression
fourierdlem2.1 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})
Assertion
Ref Expression
fourierdlem2 (𝑀 ∈ ℕ → (𝑄 ∈ (𝑃𝑀) ↔ (𝑄 ∈ (ℝ ↑m (0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄𝑖) < (𝑄‘(𝑖 + 1))))))
Distinct variable groups:   𝐴,𝑚,𝑝   𝐵,𝑚,𝑝   𝑖,𝑀,𝑚,𝑝   𝑄,𝑖,𝑝
Allowed substitution hints:   𝐴(𝑖)   𝐵(𝑖)   𝑃(𝑖,𝑚,𝑝)   𝑄(𝑚)

Proof of Theorem fourierdlem2
StepHypRef Expression
1 oveq2 7419 . . . . . 6 (𝑚 = 𝑀 → (0...𝑚) = (0...𝑀))
21oveq2d 7427 . . . . 5 (𝑚 = 𝑀 → (ℝ ↑m (0...𝑚)) = (ℝ ↑m (0...𝑀)))
3 fveqeq2 6900 . . . . . . 7 (𝑚 = 𝑀 → ((𝑝𝑚) = 𝐵 ↔ (𝑝𝑀) = 𝐵))
43anbi2d 629 . . . . . 6 (𝑚 = 𝑀 → (((𝑝‘0) = 𝐴 ∧ (𝑝𝑚) = 𝐵) ↔ ((𝑝‘0) = 𝐴 ∧ (𝑝𝑀) = 𝐵)))
5 oveq2 7419 . . . . . . 7 (𝑚 = 𝑀 → (0..^𝑚) = (0..^𝑀))
65raleqdv 3325 . . . . . 6 (𝑚 = 𝑀 → (∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)) ↔ ∀𝑖 ∈ (0..^𝑀)(𝑝𝑖) < (𝑝‘(𝑖 + 1))))
74, 6anbi12d 631 . . . . 5 (𝑚 = 𝑀 → ((((𝑝‘0) = 𝐴 ∧ (𝑝𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1))) ↔ (((𝑝‘0) = 𝐴 ∧ (𝑝𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))))
82, 7rabeqbidv 3449 . . . 4 (𝑚 = 𝑀 → {𝑝 ∈ (ℝ ↑m (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))} = {𝑝 ∈ (ℝ ↑m (0...𝑀)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})
9 fourierdlem2.1 . . . 4 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})
10 ovex 7444 . . . . 5 (ℝ ↑m (0...𝑀)) ∈ V
1110rabex 5332 . . . 4 {𝑝 ∈ (ℝ ↑m (0...𝑀)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))} ∈ V
128, 9, 11fvmpt 6998 . . 3 (𝑀 ∈ ℕ → (𝑃𝑀) = {𝑝 ∈ (ℝ ↑m (0...𝑀)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})
1312eleq2d 2819 . 2 (𝑀 ∈ ℕ → (𝑄 ∈ (𝑃𝑀) ↔ 𝑄 ∈ {𝑝 ∈ (ℝ ↑m (0...𝑀)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))}))
14 fveq1 6890 . . . . . 6 (𝑝 = 𝑄 → (𝑝‘0) = (𝑄‘0))
1514eqeq1d 2734 . . . . 5 (𝑝 = 𝑄 → ((𝑝‘0) = 𝐴 ↔ (𝑄‘0) = 𝐴))
16 fveq1 6890 . . . . . 6 (𝑝 = 𝑄 → (𝑝𝑀) = (𝑄𝑀))
1716eqeq1d 2734 . . . . 5 (𝑝 = 𝑄 → ((𝑝𝑀) = 𝐵 ↔ (𝑄𝑀) = 𝐵))
1815, 17anbi12d 631 . . . 4 (𝑝 = 𝑄 → (((𝑝‘0) = 𝐴 ∧ (𝑝𝑀) = 𝐵) ↔ ((𝑄‘0) = 𝐴 ∧ (𝑄𝑀) = 𝐵)))
19 fveq1 6890 . . . . . 6 (𝑝 = 𝑄 → (𝑝𝑖) = (𝑄𝑖))
20 fveq1 6890 . . . . . 6 (𝑝 = 𝑄 → (𝑝‘(𝑖 + 1)) = (𝑄‘(𝑖 + 1)))
2119, 20breq12d 5161 . . . . 5 (𝑝 = 𝑄 → ((𝑝𝑖) < (𝑝‘(𝑖 + 1)) ↔ (𝑄𝑖) < (𝑄‘(𝑖 + 1))))
2221ralbidv 3177 . . . 4 (𝑝 = 𝑄 → (∀𝑖 ∈ (0..^𝑀)(𝑝𝑖) < (𝑝‘(𝑖 + 1)) ↔ ∀𝑖 ∈ (0..^𝑀)(𝑄𝑖) < (𝑄‘(𝑖 + 1))))
2318, 22anbi12d 631 . . 3 (𝑝 = 𝑄 → ((((𝑝‘0) = 𝐴 ∧ (𝑝𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑝𝑖) < (𝑝‘(𝑖 + 1))) ↔ (((𝑄‘0) = 𝐴 ∧ (𝑄𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄𝑖) < (𝑄‘(𝑖 + 1)))))
2423elrab 3683 . 2 (𝑄 ∈ {𝑝 ∈ (ℝ ↑m (0...𝑀)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))} ↔ (𝑄 ∈ (ℝ ↑m (0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄𝑖) < (𝑄‘(𝑖 + 1)))))
2513, 24bitrdi 286 1 (𝑀 ∈ ℕ → (𝑄 ∈ (𝑃𝑀) ↔ (𝑄 ∈ (ℝ ↑m (0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄𝑖) < (𝑄‘(𝑖 + 1))))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wral 3061  {crab 3432   class class class wbr 5148  cmpt 5231  cfv 6543  (class class class)co 7411  m cmap 8822  cr 11111  0cc0 11112  1c1 11113   + caddc 11115   < clt 11250  cn 12214  ...cfz 13486  ..^cfzo 13629
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7414
This theorem is referenced by:  fourierdlem11  44919  fourierdlem12  44920  fourierdlem13  44921  fourierdlem14  44922  fourierdlem15  44923  fourierdlem34  44942  fourierdlem37  44945  fourierdlem41  44949  fourierdlem48  44955  fourierdlem49  44956  fourierdlem50  44957  fourierdlem54  44961  fourierdlem63  44970  fourierdlem64  44971  fourierdlem65  44972  fourierdlem69  44976  fourierdlem70  44977  fourierdlem72  44979  fourierdlem74  44981  fourierdlem75  44982  fourierdlem76  44983  fourierdlem79  44986  fourierdlem81  44988  fourierdlem85  44992  fourierdlem88  44995  fourierdlem89  44996  fourierdlem90  44997  fourierdlem91  44998  fourierdlem92  44999  fourierdlem93  45000  fourierdlem94  45001  fourierdlem97  45004  fourierdlem102  45009  fourierdlem103  45010  fourierdlem104  45011  fourierdlem111  45018  fourierdlem113  45020  fourierdlem114  45021
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