Theorem ovolicc2lem1 25418

Description: Lemma for ovolicc2 25423. (Contributed by Mario Carneiro, 14-Jun-2014.)

Hypotheses

Ref	Expression
ovolicc.1	⊢ (𝜑 → 𝐴 ∈ ℝ)
ovolicc.2	⊢ (𝜑 → 𝐵 ∈ ℝ)
ovolicc.3	⊢ (𝜑 → 𝐴 ≤ 𝐵)
ovolicc2.4	⊢ 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))
ovolicc2.5	⊢ (𝜑 → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
ovolicc2.6	⊢ (𝜑 → 𝑈 ∈ (𝒫 ran ((,) ∘ 𝐹) ∩ Fin))
ovolicc2.7	⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ∪ 𝑈)
ovolicc2.8	⊢ (𝜑 → 𝐺:𝑈⟶ℕ)
ovolicc2.9	⊢ ((𝜑 ∧ 𝑡 ∈ 𝑈) → (((,) ∘ 𝐹)‘(𝐺‘𝑡)) = 𝑡)

Assertion

Ref	Expression
ovolicc2lem1	⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → (𝑃 ∈ 𝑋 ↔ (𝑃 ∈ ℝ ∧ (1st ‘(𝐹‘(𝐺‘𝑋))) < 𝑃 ∧ 𝑃 < (2nd ‘(𝐹‘(𝐺‘𝑋))))))

Distinct variable groups:   𝑡,𝐴   𝑡,𝐵   𝑡,𝐹   𝑡,𝐺   𝜑,𝑡   𝑡,𝑈   𝑡,𝑋

Allowed substitution hints:   𝑃(𝑡)   𝑆(𝑡)

Proof of Theorem ovolicc2lem1

Step	Hyp	Ref	Expression
1		ovolicc2.5	. . . . . 6 ⊢ (𝜑 → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
2		inss2 4201	. . . . . 6 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
3		fss 6704	. . . . . 6 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)) → 𝐹:ℕ⟶(ℝ × ℝ))
4	1, 2, 3	sylancl 586	. . . . 5 ⊢ (𝜑 → 𝐹:ℕ⟶(ℝ × ℝ))
5		ovolicc2.8	. . . . . 6 ⊢ (𝜑 → 𝐺:𝑈⟶ℕ)
6	5	ffvelcdmda 7056	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → (𝐺‘𝑋) ∈ ℕ)
7		fvco3 6960	. . . . 5 ⊢ ((𝐹:ℕ⟶(ℝ × ℝ) ∧ (𝐺‘𝑋) ∈ ℕ) → (((,) ∘ 𝐹)‘(𝐺‘𝑋)) = ((,)‘(𝐹‘(𝐺‘𝑋))))
8	4, 6, 7	syl2an2r 685	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → (((,) ∘ 𝐹)‘(𝐺‘𝑋)) = ((,)‘(𝐹‘(𝐺‘𝑋))))
9		ovolicc2.9	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑈) → (((,) ∘ 𝐹)‘(𝐺‘𝑡)) = 𝑡)
10	9	ralrimiva 3125	. . . . 5 ⊢ (𝜑 → ∀𝑡 ∈ 𝑈 (((,) ∘ 𝐹)‘(𝐺‘𝑡)) = 𝑡)
11		2fveq3 6863	. . . . . . 7 ⊢ (𝑡 = 𝑋 → (((,) ∘ 𝐹)‘(𝐺‘𝑡)) = (((,) ∘ 𝐹)‘(𝐺‘𝑋)))
12		id 22	. . . . . . 7 ⊢ (𝑡 = 𝑋 → 𝑡 = 𝑋)
13	11, 12	eqeq12d 2745	. . . . . 6 ⊢ (𝑡 = 𝑋 → ((((,) ∘ 𝐹)‘(𝐺‘𝑡)) = 𝑡 ↔ (((,) ∘ 𝐹)‘(𝐺‘𝑋)) = 𝑋))
14	13	rspccva 3587	. . . . 5 ⊢ ((∀𝑡 ∈ 𝑈 (((,) ∘ 𝐹)‘(𝐺‘𝑡)) = 𝑡 ∧ 𝑋 ∈ 𝑈) → (((,) ∘ 𝐹)‘(𝐺‘𝑋)) = 𝑋)
15	10, 14	sylan 580	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → (((,) ∘ 𝐹)‘(𝐺‘𝑋)) = 𝑋)
16	4	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → 𝐹:ℕ⟶(ℝ × ℝ))
17	16, 6	ffvelcdmd 7057	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → (𝐹‘(𝐺‘𝑋)) ∈ (ℝ × ℝ))
18		1st2nd2 8007	. . . . . . 7 ⊢ ((𝐹‘(𝐺‘𝑋)) ∈ (ℝ × ℝ) → (𝐹‘(𝐺‘𝑋)) = ⟨(1st ‘(𝐹‘(𝐺‘𝑋))), (2nd ‘(𝐹‘(𝐺‘𝑋)))⟩)
19	17, 18	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → (𝐹‘(𝐺‘𝑋)) = ⟨(1st ‘(𝐹‘(𝐺‘𝑋))), (2nd ‘(𝐹‘(𝐺‘𝑋)))⟩)
20	19	fveq2d 6862	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → ((,)‘(𝐹‘(𝐺‘𝑋))) = ((,)‘⟨(1st ‘(𝐹‘(𝐺‘𝑋))), (2nd ‘(𝐹‘(𝐺‘𝑋)))⟩))
21		df-ov 7390	. . . . 5 ⊢ ((1st ‘(𝐹‘(𝐺‘𝑋)))(,)(2nd ‘(𝐹‘(𝐺‘𝑋)))) = ((,)‘⟨(1st ‘(𝐹‘(𝐺‘𝑋))), (2nd ‘(𝐹‘(𝐺‘𝑋)))⟩)
22	20, 21	eqtr4di 2782	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → ((,)‘(𝐹‘(𝐺‘𝑋))) = ((1st ‘(𝐹‘(𝐺‘𝑋)))(,)(2nd ‘(𝐹‘(𝐺‘𝑋)))))
23	8, 15, 22	3eqtr3d 2772	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → 𝑋 = ((1st ‘(𝐹‘(𝐺‘𝑋)))(,)(2nd ‘(𝐹‘(𝐺‘𝑋)))))
24	23	eleq2d 2814	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → (𝑃 ∈ 𝑋 ↔ 𝑃 ∈ ((1st ‘(𝐹‘(𝐺‘𝑋)))(,)(2nd ‘(𝐹‘(𝐺‘𝑋))))))
25		xp1st 8000	. . . 4 ⊢ ((𝐹‘(𝐺‘𝑋)) ∈ (ℝ × ℝ) → (1st ‘(𝐹‘(𝐺‘𝑋))) ∈ ℝ)
26	17, 25	syl 17	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → (1st ‘(𝐹‘(𝐺‘𝑋))) ∈ ℝ)
27		xp2nd 8001	. . . 4 ⊢ ((𝐹‘(𝐺‘𝑋)) ∈ (ℝ × ℝ) → (2nd ‘(𝐹‘(𝐺‘𝑋))) ∈ ℝ)
28	17, 27	syl 17	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → (2nd ‘(𝐹‘(𝐺‘𝑋))) ∈ ℝ)
29		rexr 11220	. . . 4 ⊢ ((1st ‘(𝐹‘(𝐺‘𝑋))) ∈ ℝ → (1st ‘(𝐹‘(𝐺‘𝑋))) ∈ ℝ*)
30		rexr 11220	. . . 4 ⊢ ((2nd ‘(𝐹‘(𝐺‘𝑋))) ∈ ℝ → (2nd ‘(𝐹‘(𝐺‘𝑋))) ∈ ℝ*)
31		elioo2 13347	. . . 4 ⊢ (((1st ‘(𝐹‘(𝐺‘𝑋))) ∈ ℝ* ∧ (2nd ‘(𝐹‘(𝐺‘𝑋))) ∈ ℝ*) → (𝑃 ∈ ((1st ‘(𝐹‘(𝐺‘𝑋)))(,)(2nd ‘(𝐹‘(𝐺‘𝑋)))) ↔ (𝑃 ∈ ℝ ∧ (1st ‘(𝐹‘(𝐺‘𝑋))) < 𝑃 ∧ 𝑃 < (2nd ‘(𝐹‘(𝐺‘𝑋))))))
32	29, 30, 31	syl2an 596	. . 3 ⊢ (((1st ‘(𝐹‘(𝐺‘𝑋))) ∈ ℝ ∧ (2nd ‘(𝐹‘(𝐺‘𝑋))) ∈ ℝ) → (𝑃 ∈ ((1st ‘(𝐹‘(𝐺‘𝑋)))(,)(2nd ‘(𝐹‘(𝐺‘𝑋)))) ↔ (𝑃 ∈ ℝ ∧ (1st ‘(𝐹‘(𝐺‘𝑋))) < 𝑃 ∧ 𝑃 < (2nd ‘(𝐹‘(𝐺‘𝑋))))))
33	26, 28, 32	syl2anc 584	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → (𝑃 ∈ ((1st ‘(𝐹‘(𝐺‘𝑋)))(,)(2nd ‘(𝐹‘(𝐺‘𝑋)))) ↔ (𝑃 ∈ ℝ ∧ (1st ‘(𝐹‘(𝐺‘𝑋))) < 𝑃 ∧ 𝑃 < (2nd ‘(𝐹‘(𝐺‘𝑋))))))
34	24, 33	bitrd 279	1 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → (𝑃 ∈ 𝑋 ↔ (𝑃 ∈ ℝ ∧ (1st ‘(𝐹‘(𝐺‘𝑋))) < 𝑃 ∧ 𝑃 < (2nd ‘(𝐹‘(𝐺‘𝑋))))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3044   ∩ cin 3913   ⊆ wss 3914  𝒫 cpw 4563  ⟨cop 4595  ∪ cuni 4871   class class class wbr 5107   × cxp 5636  ran crn 5639   ∘ ccom 5642  ⟶wf 6507  ‘cfv 6511  (class class class)co 7387  1^st c1st 7966  2^nd c2nd 7967  Fincfn 8918  ℝcr 11067  1c1 11069   + caddc 11071  ℝ^*cxr 11207   < clt 11208   ≤ cle 11209   − cmin 11405  ℕcn 12186  (,)cioo 13306  [,]cicc 13309  seqcseq 13966  abscabs 15200

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-cnex 11124  ax-resscn 11125  ax-pre-lttri 11142  ax-pre-lttrn 11143

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-po 5546  df-so 5547  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-1st 7968  df-2nd 7969  df-er 8671  df-en 8919  df-dom 8920  df-sdom 8921  df-pnf 11210  df-mnf 11211  df-xr 11212  df-ltxr 11213  df-le 11214  df-ioo 13310

This theorem is referenced by:  ovolicc2lem2  25419  ovolicc2lem3  25420  ovolicc2lem4  25421