Theorem ovolicc2lem1 25445

Description: Lemma for ovolicc2 25450. (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 4185	. . . . . 6 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
3		fss 6667	. . . . . 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 7017	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → (𝐺‘𝑋) ∈ ℕ)
7		fvco3 6921	. . . . 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 3124	. . . . 5 ⊢ (𝜑 → ∀𝑡 ∈ 𝑈 (((,) ∘ 𝐹)‘(𝐺‘𝑡)) = 𝑡)
11		2fveq3 6827	. . . . . . 7 ⊢ (𝑡 = 𝑋 → (((,) ∘ 𝐹)‘(𝐺‘𝑡)) = (((,) ∘ 𝐹)‘(𝐺‘𝑋)))
12		id 22	. . . . . . 7 ⊢ (𝑡 = 𝑋 → 𝑡 = 𝑋)
13	11, 12	eqeq12d 2747	. . . . . 6 ⊢ (𝑡 = 𝑋 → ((((,) ∘ 𝐹)‘(𝐺‘𝑡)) = 𝑡 ↔ (((,) ∘ 𝐹)‘(𝐺‘𝑋)) = 𝑋))
14	13	rspccva 3571	. . . . 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 7018	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → (𝐹‘(𝐺‘𝑋)) ∈ (ℝ × ℝ))
18		1st2nd2 7960	. . . . . . 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 6826	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → ((,)‘(𝐹‘(𝐺‘𝑋))) = ((,)‘⟨(1st ‘(𝐹‘(𝐺‘𝑋))), (2nd ‘(𝐹‘(𝐺‘𝑋)))⟩))
21		df-ov 7349	. . . . 5 ⊢ ((1st ‘(𝐹‘(𝐺‘𝑋)))(,)(2nd ‘(𝐹‘(𝐺‘𝑋)))) = ((,)‘⟨(1st ‘(𝐹‘(𝐺‘𝑋))), (2nd ‘(𝐹‘(𝐺‘𝑋)))⟩)
22	20, 21	eqtr4di 2784	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → ((,)‘(𝐹‘(𝐺‘𝑋))) = ((1st ‘(𝐹‘(𝐺‘𝑋)))(,)(2nd ‘(𝐹‘(𝐺‘𝑋)))))
23	8, 15, 22	3eqtr3d 2774	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → 𝑋 = ((1st ‘(𝐹‘(𝐺‘𝑋)))(,)(2nd ‘(𝐹‘(𝐺‘𝑋)))))
24	23	eleq2d 2817	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → (𝑃 ∈ 𝑋 ↔ 𝑃 ∈ ((1st ‘(𝐹‘(𝐺‘𝑋)))(,)(2nd ‘(𝐹‘(𝐺‘𝑋))))))
25		xp1st 7953	. . . 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 7954	. . . 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 11158	. . . 4 ⊢ ((1st ‘(𝐹‘(𝐺‘𝑋))) ∈ ℝ → (1st ‘(𝐹‘(𝐺‘𝑋))) ∈ ℝ*)
30		rexr 11158	. . . 4 ⊢ ((2nd ‘(𝐹‘(𝐺‘𝑋))) ∈ ℝ → (2nd ‘(𝐹‘(𝐺‘𝑋))) ∈ ℝ*)
31		elioo2 13286	. . . 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 1541   ∈ wcel 2111  ∀wral 3047   ∩ cin 3896   ⊆ wss 3897  𝒫 cpw 4547  ⟨cop 4579  ∪ cuni 4856   class class class wbr 5089   × cxp 5612  ran crn 5615   ∘ ccom 5618  ⟶wf 6477  ‘cfv 6481  (class class class)co 7346  1^st c1st 7919  2^nd c2nd 7920  Fincfn 8869  ℝcr 11005  1c1 11007   + caddc 11009  ℝ^*cxr 11145   < clt 11146   ≤ cle 11147   − cmin 11344  ℕcn 12125  (,)cioo 13245  [,]cicc 13248  seqcseq 13908  abscabs 15141

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 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668  ax-cnex 11062  ax-resscn 11063  ax-pre-lttri 11080  ax-pre-lttrn 11081

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-po 5522  df-so 5523  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-1st 7921  df-2nd 7922  df-er 8622  df-en 8870  df-dom 8871  df-sdom 8872  df-pnf 11148  df-mnf 11149  df-xr 11150  df-ltxr 11151  df-le 11152  df-ioo 13249

This theorem is referenced by:  ovolicc2lem2  25446  ovolicc2lem3  25447  ovolicc2lem4  25448