Theorem fourierdlem2 46234

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 7362	. . . . . 6 ⊢ (𝑚 = 𝑀 → (0...𝑚) = (0...𝑀))
2	1	oveq2d 7370	. . . . 5 ⊢ (𝑚 = 𝑀 → (ℝ ↑m (0...𝑚)) = (ℝ ↑m (0...𝑀)))
3		fveqeq2 6839	. . . . . . 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 7362	. . . . . . 7 ⊢ (𝑚 = 𝑀 → (0..^𝑚) = (0..^𝑀))
6	5	raleqdv 3293	. . . . . 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 3414	. . . 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 7387	. . . . 5 ⊢ (ℝ ↑m (0...𝑀)) ∈ V
11	10	rabex 5281	. . . 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 6937	. . 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 2819	. 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 6829	. . . . . 6 ⊢ (𝑝 = 𝑄 → (𝑝‘0) = (𝑄‘0))
15	14	eqeq1d 2735	. . . . 5 ⊢ (𝑝 = 𝑄 → ((𝑝‘0) = 𝐴 ↔ (𝑄‘0) = 𝐴))
16		fveq1 6829	. . . . . 6 ⊢ (𝑝 = 𝑄 → (𝑝‘𝑀) = (𝑄‘𝑀))
17	16	eqeq1d 2735	. . . . 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 6829	. . . . . 6 ⊢ (𝑝 = 𝑄 → (𝑝‘𝑖) = (𝑄‘𝑖))
20		fveq1 6829	. . . . . 6 ⊢ (𝑝 = 𝑄 → (𝑝‘(𝑖 + 1)) = (𝑄‘(𝑖 + 1)))
21	19, 20	breq12d 5108	. . . . 5 ⊢ (𝑝 = 𝑄 → ((𝑝‘𝑖) < (𝑝‘(𝑖 + 1)) ↔ (𝑄‘𝑖) < (𝑄‘(𝑖 + 1))))
22	21	ralbidv 3156	. . . 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 3643	. 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	bitrdi 287	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 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∀wral 3048  {crab 3396   class class class wbr 5095   ↦ cmpt 5176  ‘cfv 6488  (class class class)co 7354   ↑_m cmap 8758  ℝcr 11014  0cc0 11015  1c1 11016   + caddc 11018   < clt 11155  ℕcn 12134  ...cfz 13411  ..^cfzo 13558

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-iota 6444  df-fun 6490  df-fv 6496  df-ov 7357

This theorem is referenced by:  fourierdlem11  46243  fourierdlem12  46244  fourierdlem13  46245  fourierdlem14  46246  fourierdlem15  46247  fourierdlem34  46266  fourierdlem37  46269  fourierdlem41  46273  fourierdlem48  46279  fourierdlem49  46280  fourierdlem50  46281  fourierdlem54  46285  fourierdlem63  46294  fourierdlem64  46295  fourierdlem65  46296  fourierdlem69  46300  fourierdlem70  46301  fourierdlem72  46303  fourierdlem74  46305  fourierdlem75  46306  fourierdlem76  46307  fourierdlem79  46310  fourierdlem81  46312  fourierdlem85  46316  fourierdlem88  46319  fourierdlem89  46320  fourierdlem90  46321  fourierdlem91  46322  fourierdlem92  46323  fourierdlem93  46324  fourierdlem94  46325  fourierdlem97  46328  fourierdlem102  46333  fourierdlem103  46334  fourierdlem104  46335  fourierdlem111  46342  fourierdlem113  46344  fourierdlem114  46345