Theorem fourierdlem2 42393

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 ∧ (𝑝‘𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))})

Assertion

Ref	Expression
fourierdlem2	⊢ (𝑀 ∈ ℕ → (𝑄 ∈ (𝑃‘𝑀) ↔ (𝑄 ∈ (ℝ ↑m (0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (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

Step	Hyp	Ref	Expression
1		oveq2 7163	. . . . . 6 ⊢ (𝑚 = 𝑀 → (0...𝑚) = (0...𝑀))
2	1	oveq2d 7171	. . . . 5 ⊢ (𝑚 = 𝑀 → (ℝ ↑m (0...𝑚)) = (ℝ ↑m (0...𝑀)))
3		fveqeq2 6678	. . . . . . 7 ⊢ (𝑚 = 𝑀 → ((𝑝‘𝑚) = 𝐵 ↔ (𝑝‘𝑀) = 𝐵))
4	3	anbi2d 630	. . . . . 6 ⊢ (𝑚 = 𝑀 → (((𝑝‘0) = 𝐴 ∧ (𝑝‘𝑚) = 𝐵) ↔ ((𝑝‘0) = 𝐴 ∧ (𝑝‘𝑀) = 𝐵)))
5		oveq2 7163	. . . . . . 7 ⊢ (𝑚 = 𝑀 → (0..^𝑚) = (0..^𝑀))
6	5	raleqdv 3415	. . . . . 6 ⊢ (𝑚 = 𝑀 → (∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)) ↔ ∀𝑖 ∈ (0..^𝑀)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1))))
7	4, 6	anbi12d 632	. . . . 5 ⊢ (𝑚 = 𝑀 → ((((𝑝‘0) = 𝐴 ∧ (𝑝‘𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1))) ↔ (((𝑝‘0) = 𝐴 ∧ (𝑝‘𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))))
8	2, 7	rabeqbidv 3485	. . . 4 ⊢ (𝑚 = 𝑀 → {𝑝 ∈ (ℝ ↑m (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝‘𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))} = {𝑝 ∈ (ℝ ↑m (0...𝑀)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝‘𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))})
9		fourierdlem2.1	. . . 4 ⊢ 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝‘𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))})
10		ovex 7188	. . . . 5 ⊢ (ℝ ↑m (0...𝑀)) ∈ V
11	10	rabex 5234	. . . 4 ⊢ {𝑝 ∈ (ℝ ↑m (0...𝑀)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝‘𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))} ∈ V
12	8, 9, 11	fvmpt 6767	. . 3 ⊢ (𝑀 ∈ ℕ → (𝑃‘𝑀) = {𝑝 ∈ (ℝ ↑m (0...𝑀)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝‘𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))})
13	12	eleq2d 2898	. 2 ⊢ (𝑀 ∈ ℕ → (𝑄 ∈ (𝑃‘𝑀) ↔ 𝑄 ∈ {𝑝 ∈ (ℝ ↑m (0...𝑀)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝‘𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))}))
14		fveq1 6668	. . . . . 6 ⊢ (𝑝 = 𝑄 → (𝑝‘0) = (𝑄‘0))
15	14	eqeq1d 2823	. . . . 5 ⊢ (𝑝 = 𝑄 → ((𝑝‘0) = 𝐴 ↔ (𝑄‘0) = 𝐴))
16		fveq1 6668	. . . . . 6 ⊢ (𝑝 = 𝑄 → (𝑝‘𝑀) = (𝑄‘𝑀))
17	16	eqeq1d 2823	. . . . 5 ⊢ (𝑝 = 𝑄 → ((𝑝‘𝑀) = 𝐵 ↔ (𝑄‘𝑀) = 𝐵))
18	15, 17	anbi12d 632	. . . 4 ⊢ (𝑝 = 𝑄 → (((𝑝‘0) = 𝐴 ∧ (𝑝‘𝑀) = 𝐵) ↔ ((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵)))
19		fveq1 6668	. . . . . 6 ⊢ (𝑝 = 𝑄 → (𝑝‘𝑖) = (𝑄‘𝑖))
20		fveq1 6668	. . . . . 6 ⊢ (𝑝 = 𝑄 → (𝑝‘(𝑖 + 1)) = (𝑄‘(𝑖 + 1)))
21	19, 20	breq12d 5078	. . . . 5 ⊢ (𝑝 = 𝑄 → ((𝑝‘𝑖) < (𝑝‘(𝑖 + 1)) ↔ (𝑄‘𝑖) < (𝑄‘(𝑖 + 1))))
22	21	ralbidv 3197	. . . 4 ⊢ (𝑝 = 𝑄 → (∀𝑖 ∈ (0..^𝑀)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)) ↔ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1))))
23	18, 22	anbi12d 632	. . 3 ⊢ (𝑝 = 𝑄 → ((((𝑝‘0) = 𝐴 ∧ (𝑝‘𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1))) ↔ (((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))))
24	23	elrab 3679	. 2 ⊢ (𝑄 ∈ {𝑝 ∈ (ℝ ↑m (0...𝑀)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝‘𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))} ↔ (𝑄 ∈ (ℝ ↑m (0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))))
25	13, 24	syl6bb 289	1 ⊢ (𝑀 ∈ ℕ → (𝑄 ∈ (𝑃‘𝑀) ↔ (𝑄 ∈ (ℝ ↑m (0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1))))))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1533   ∈ wcel 2110  ∀wral 3138  {crab 3142   class class class wbr 5065   ↦ cmpt 5145  ‘cfv 6354  (class class class)co 7155   ↑_m cmap 8405  ℝcr 10535  0cc0 10536  1c1 10537   + caddc 10539   < clt 10674  ℕcn 11637  ...cfz 12891  ..^cfzo 13032

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pr 5329

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-br 5066  df-opab 5128  df-mpt 5146  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-iota 6313  df-fun 6356  df-fv 6362  df-ov 7158

This theorem is referenced by:  fourierdlem11  42402  fourierdlem12  42403  fourierdlem13  42404  fourierdlem14  42405  fourierdlem15  42406  fourierdlem34  42425  fourierdlem37  42428  fourierdlem41  42432  fourierdlem48  42438  fourierdlem49  42439  fourierdlem50  42440  fourierdlem54  42444  fourierdlem63  42453  fourierdlem64  42454  fourierdlem65  42455  fourierdlem69  42459  fourierdlem70  42460  fourierdlem72  42462  fourierdlem74  42464  fourierdlem75  42465  fourierdlem76  42466  fourierdlem79  42469  fourierdlem81  42471  fourierdlem85  42475  fourierdlem88  42478  fourierdlem89  42479  fourierdlem90  42480  fourierdlem91  42481  fourierdlem92  42482  fourierdlem93  42483  fourierdlem94  42484  fourierdlem97  42487  fourierdlem102  42492  fourierdlem103  42493  fourierdlem104  42494  fourierdlem111  42501  fourierdlem113  42503  fourierdlem114  42504