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Theorem fourierdlem2 40095
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 6655 . . . . . 6 (𝑚 = 𝑀 → (0...𝑚) = (0...𝑀))
21oveq2d 6663 . . . . 5 (𝑚 = 𝑀 → (ℝ ↑𝑚 (0...𝑚)) = (ℝ ↑𝑚 (0...𝑀)))
3 fveq2 6189 . . . . . . . 8 (𝑚 = 𝑀 → (𝑝𝑚) = (𝑝𝑀))
43eqeq1d 2623 . . . . . . 7 (𝑚 = 𝑀 → ((𝑝𝑚) = 𝐵 ↔ (𝑝𝑀) = 𝐵))
54anbi2d 740 . . . . . 6 (𝑚 = 𝑀 → (((𝑝‘0) = 𝐴 ∧ (𝑝𝑚) = 𝐵) ↔ ((𝑝‘0) = 𝐴 ∧ (𝑝𝑀) = 𝐵)))
6 oveq2 6655 . . . . . . 7 (𝑚 = 𝑀 → (0..^𝑚) = (0..^𝑀))
76raleqdv 3142 . . . . . 6 (𝑚 = 𝑀 → (∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)) ↔ ∀𝑖 ∈ (0..^𝑀)(𝑝𝑖) < (𝑝‘(𝑖 + 1))))
85, 7anbi12d 747 . . . . 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)))))
92, 8rabeqbidv 3193 . . . 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)))})
10 fourierdlem2.1 . . . 4 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑𝑚 (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})
11 ovex 6675 . . . . 5 (ℝ ↑𝑚 (0...𝑀)) ∈ V
1211rabex 4811 . . . 4 {𝑝 ∈ (ℝ ↑𝑚 (0...𝑀)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))} ∈ V
139, 10, 12fvmpt 6280 . . 3 (𝑀 ∈ ℕ → (𝑃𝑀) = {𝑝 ∈ (ℝ ↑𝑚 (0...𝑀)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})
1413eleq2d 2686 . 2 (𝑀 ∈ ℕ → (𝑄 ∈ (𝑃𝑀) ↔ 𝑄 ∈ {𝑝 ∈ (ℝ ↑𝑚 (0...𝑀)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))}))
15 fveq1 6188 . . . . . 6 (𝑝 = 𝑄 → (𝑝‘0) = (𝑄‘0))
1615eqeq1d 2623 . . . . 5 (𝑝 = 𝑄 → ((𝑝‘0) = 𝐴 ↔ (𝑄‘0) = 𝐴))
17 fveq1 6188 . . . . . 6 (𝑝 = 𝑄 → (𝑝𝑀) = (𝑄𝑀))
1817eqeq1d 2623 . . . . 5 (𝑝 = 𝑄 → ((𝑝𝑀) = 𝐵 ↔ (𝑄𝑀) = 𝐵))
1916, 18anbi12d 747 . . . 4 (𝑝 = 𝑄 → (((𝑝‘0) = 𝐴 ∧ (𝑝𝑀) = 𝐵) ↔ ((𝑄‘0) = 𝐴 ∧ (𝑄𝑀) = 𝐵)))
20 fveq1 6188 . . . . . 6 (𝑝 = 𝑄 → (𝑝𝑖) = (𝑄𝑖))
21 fveq1 6188 . . . . . 6 (𝑝 = 𝑄 → (𝑝‘(𝑖 + 1)) = (𝑄‘(𝑖 + 1)))
2220, 21breq12d 4664 . . . . 5 (𝑝 = 𝑄 → ((𝑝𝑖) < (𝑝‘(𝑖 + 1)) ↔ (𝑄𝑖) < (𝑄‘(𝑖 + 1))))
2322ralbidv 2985 . . . 4 (𝑝 = 𝑄 → (∀𝑖 ∈ (0..^𝑀)(𝑝𝑖) < (𝑝‘(𝑖 + 1)) ↔ ∀𝑖 ∈ (0..^𝑀)(𝑄𝑖) < (𝑄‘(𝑖 + 1))))
2419, 23anbi12d 747 . . 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)))))
2524elrab 3361 . 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)))))
2614, 25syl6bb 276 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 196  wa 384   = wceq 1482  wcel 1989  wral 2911  {crab 2915   class class class wbr 4651  cmpt 4727  cfv 5886  (class class class)co 6647  𝑚 cmap 7854  cr 9932  0cc0 9933  1c1 9934   + caddc 9936   < clt 10071  cn 11017  ...cfz 12323  ..^cfzo 12461
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pr 4904
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-sbc 3434  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-br 4652  df-opab 4711  df-mpt 4728  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-iota 5849  df-fun 5888  df-fv 5894  df-ov 6650
This theorem is referenced by:  fourierdlem11  40104  fourierdlem12  40105  fourierdlem13  40106  fourierdlem14  40107  fourierdlem15  40108  fourierdlem34  40127  fourierdlem37  40130  fourierdlem41  40134  fourierdlem48  40140  fourierdlem49  40141  fourierdlem50  40142  fourierdlem54  40146  fourierdlem63  40155  fourierdlem64  40156  fourierdlem65  40157  fourierdlem69  40161  fourierdlem70  40162  fourierdlem72  40164  fourierdlem74  40166  fourierdlem75  40167  fourierdlem76  40168  fourierdlem79  40171  fourierdlem81  40173  fourierdlem85  40177  fourierdlem88  40180  fourierdlem89  40181  fourierdlem90  40182  fourierdlem91  40183  fourierdlem92  40184  fourierdlem93  40185  fourierdlem94  40186  fourierdlem97  40189  fourierdlem102  40194  fourierdlem103  40195  fourierdlem104  40196  fourierdlem111  40203  fourierdlem113  40205  fourierdlem114  40206
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