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Theorem fourierdlem2 45635
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 7427 . . . . . 6 (𝑚 = 𝑀 → (0...𝑚) = (0...𝑀))
21oveq2d 7435 . . . . 5 (𝑚 = 𝑀 → (ℝ ↑m (0...𝑚)) = (ℝ ↑m (0...𝑀)))
3 fveqeq2 6905 . . . . . . 7 (𝑚 = 𝑀 → ((𝑝𝑚) = 𝐵 ↔ (𝑝𝑀) = 𝐵))
43anbi2d 628 . . . . . 6 (𝑚 = 𝑀 → (((𝑝‘0) = 𝐴 ∧ (𝑝𝑚) = 𝐵) ↔ ((𝑝‘0) = 𝐴 ∧ (𝑝𝑀) = 𝐵)))
5 oveq2 7427 . . . . . . 7 (𝑚 = 𝑀 → (0..^𝑚) = (0..^𝑀))
65raleqdv 3314 . . . . . 6 (𝑚 = 𝑀 → (∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)) ↔ ∀𝑖 ∈ (0..^𝑀)(𝑝𝑖) < (𝑝‘(𝑖 + 1))))
74, 6anbi12d 630 . . . . 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 3436 . . . 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 7452 . . . . 5 (ℝ ↑m (0...𝑀)) ∈ V
1110rabex 5335 . . . 4 {𝑝 ∈ (ℝ ↑m (0...𝑀)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))} ∈ V
128, 9, 11fvmpt 7004 . . 3 (𝑀 ∈ ℕ → (𝑃𝑀) = {𝑝 ∈ (ℝ ↑m (0...𝑀)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})
1312eleq2d 2811 . 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 6895 . . . . . 6 (𝑝 = 𝑄 → (𝑝‘0) = (𝑄‘0))
1514eqeq1d 2727 . . . . 5 (𝑝 = 𝑄 → ((𝑝‘0) = 𝐴 ↔ (𝑄‘0) = 𝐴))
16 fveq1 6895 . . . . . 6 (𝑝 = 𝑄 → (𝑝𝑀) = (𝑄𝑀))
1716eqeq1d 2727 . . . . 5 (𝑝 = 𝑄 → ((𝑝𝑀) = 𝐵 ↔ (𝑄𝑀) = 𝐵))
1815, 17anbi12d 630 . . . 4 (𝑝 = 𝑄 → (((𝑝‘0) = 𝐴 ∧ (𝑝𝑀) = 𝐵) ↔ ((𝑄‘0) = 𝐴 ∧ (𝑄𝑀) = 𝐵)))
19 fveq1 6895 . . . . . 6 (𝑝 = 𝑄 → (𝑝𝑖) = (𝑄𝑖))
20 fveq1 6895 . . . . . 6 (𝑝 = 𝑄 → (𝑝‘(𝑖 + 1)) = (𝑄‘(𝑖 + 1)))
2119, 20breq12d 5162 . . . . 5 (𝑝 = 𝑄 → ((𝑝𝑖) < (𝑝‘(𝑖 + 1)) ↔ (𝑄𝑖) < (𝑄‘(𝑖 + 1))))
2221ralbidv 3167 . . . 4 (𝑝 = 𝑄 → (∀𝑖 ∈ (0..^𝑀)(𝑝𝑖) < (𝑝‘(𝑖 + 1)) ↔ ∀𝑖 ∈ (0..^𝑀)(𝑄𝑖) < (𝑄‘(𝑖 + 1))))
2318, 22anbi12d 630 . . 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 3679 . 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 394   = wceq 1533  wcel 2098  wral 3050  {crab 3418   class class class wbr 5149  cmpt 5232  cfv 6549  (class class class)co 7419  m cmap 8845  cr 11139  0cc0 11140  1c1 11141   + caddc 11143   < clt 11280  cn 12245  ...cfz 13519  ..^cfzo 13662
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-iota 6501  df-fun 6551  df-fv 6557  df-ov 7422
This theorem is referenced by:  fourierdlem11  45644  fourierdlem12  45645  fourierdlem13  45646  fourierdlem14  45647  fourierdlem15  45648  fourierdlem34  45667  fourierdlem37  45670  fourierdlem41  45674  fourierdlem48  45680  fourierdlem49  45681  fourierdlem50  45682  fourierdlem54  45686  fourierdlem63  45695  fourierdlem64  45696  fourierdlem65  45697  fourierdlem69  45701  fourierdlem70  45702  fourierdlem72  45704  fourierdlem74  45706  fourierdlem75  45707  fourierdlem76  45708  fourierdlem79  45711  fourierdlem81  45713  fourierdlem85  45717  fourierdlem88  45720  fourierdlem89  45721  fourierdlem90  45722  fourierdlem91  45723  fourierdlem92  45724  fourierdlem93  45725  fourierdlem94  45726  fourierdlem97  45729  fourierdlem102  45734  fourierdlem103  45735  fourierdlem104  45736  fourierdlem111  45743  fourierdlem113  45745  fourierdlem114  45746
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