Theorem ovolicc2lem1 25491

Description: Lemma for ovolicc2 25496. (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 4192	. . . . . 6 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
3		fss 6688	. . . . . 6 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)) → 𝐹:ℕ⟶(ℝ × ℝ))
4	1, 2, 3	sylancl 587	. . . . 5 ⊢ (𝜑 → 𝐹:ℕ⟶(ℝ × ℝ))
5		ovolicc2.8	. . . . . 6 ⊢ (𝜑 → 𝐺:𝑈⟶ℕ)
6	5	ffvelcdmda 7040	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → (𝐺‘𝑋) ∈ ℕ)
7		fvco3 6943	. . . . 5 ⊢ ((𝐹:ℕ⟶(ℝ × ℝ) ∧ (𝐺‘𝑋) ∈ ℕ) → (((,) ∘ 𝐹)‘(𝐺‘𝑋)) = ((,)‘(𝐹‘(𝐺‘𝑋))))
8	4, 6, 7	syl2an2r 686	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → (((,) ∘ 𝐹)‘(𝐺‘𝑋)) = ((,)‘(𝐹‘(𝐺‘𝑋))))
9		ovolicc2.9	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑈) → (((,) ∘ 𝐹)‘(𝐺‘𝑡)) = 𝑡)
10	9	ralrimiva 3130	. . . . 5 ⊢ (𝜑 → ∀𝑡 ∈ 𝑈 (((,) ∘ 𝐹)‘(𝐺‘𝑡)) = 𝑡)
11		2fveq3 6849	. . . . . . 7 ⊢ (𝑡 = 𝑋 → (((,) ∘ 𝐹)‘(𝐺‘𝑡)) = (((,) ∘ 𝐹)‘(𝐺‘𝑋)))
12		id 22	. . . . . . 7 ⊢ (𝑡 = 𝑋 → 𝑡 = 𝑋)
13	11, 12	eqeq12d 2753	. . . . . 6 ⊢ (𝑡 = 𝑋 → ((((,) ∘ 𝐹)‘(𝐺‘𝑡)) = 𝑡 ↔ (((,) ∘ 𝐹)‘(𝐺‘𝑋)) = 𝑋))
14	13	rspccva 3577	. . . . 5 ⊢ ((∀𝑡 ∈ 𝑈 (((,) ∘ 𝐹)‘(𝐺‘𝑡)) = 𝑡 ∧ 𝑋 ∈ 𝑈) → (((,) ∘ 𝐹)‘(𝐺‘𝑋)) = 𝑋)
15	10, 14	sylan 581	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → (((,) ∘ 𝐹)‘(𝐺‘𝑋)) = 𝑋)
16	4	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → 𝐹:ℕ⟶(ℝ × ℝ))
17	16, 6	ffvelcdmd 7041	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → (𝐹‘(𝐺‘𝑋)) ∈ (ℝ × ℝ))
18		1st2nd2 7984	. . . . . . 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 6848	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → ((,)‘(𝐹‘(𝐺‘𝑋))) = ((,)‘⟨(1st ‘(𝐹‘(𝐺‘𝑋))), (2nd ‘(𝐹‘(𝐺‘𝑋)))⟩))
21		df-ov 7373	. . . . 5 ⊢ ((1st ‘(𝐹‘(𝐺‘𝑋)))(,)(2nd ‘(𝐹‘(𝐺‘𝑋)))) = ((,)‘⟨(1st ‘(𝐹‘(𝐺‘𝑋))), (2nd ‘(𝐹‘(𝐺‘𝑋)))⟩)
22	20, 21	eqtr4di 2790	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → ((,)‘(𝐹‘(𝐺‘𝑋))) = ((1st ‘(𝐹‘(𝐺‘𝑋)))(,)(2nd ‘(𝐹‘(𝐺‘𝑋)))))
23	8, 15, 22	3eqtr3d 2780	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → 𝑋 = ((1st ‘(𝐹‘(𝐺‘𝑋)))(,)(2nd ‘(𝐹‘(𝐺‘𝑋)))))
24	23	eleq2d 2823	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → (𝑃 ∈ 𝑋 ↔ 𝑃 ∈ ((1st ‘(𝐹‘(𝐺‘𝑋)))(,)(2nd ‘(𝐹‘(𝐺‘𝑋))))))
25		xp1st 7977	. . . 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 7978	. . . 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 11192	. . . 4 ⊢ ((1st ‘(𝐹‘(𝐺‘𝑋))) ∈ ℝ → (1st ‘(𝐹‘(𝐺‘𝑋))) ∈ ℝ*)
30		rexr 11192	. . . 4 ⊢ ((2nd ‘(𝐹‘(𝐺‘𝑋))) ∈ ℝ → (2nd ‘(𝐹‘(𝐺‘𝑋))) ∈ ℝ*)
31		elioo2 13316	. . . 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 597	. . 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 585	. 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 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3052   ∩ cin 3902   ⊆ wss 3903  𝒫 cpw 4556  ⟨cop 4588  ∪ cuni 4865   class class class wbr 5100   × cxp 5632  ran crn 5635   ∘ ccom 5638  ⟶wf 6498  ‘cfv 6502  (class class class)co 7370  1^st c1st 7943  2^nd c2nd 7944  Fincfn 8897  ℝcr 11039  1c1 11041   + caddc 11043  ℝ^*cxr 11179   < clt 11180   ≤ cle 11181   − cmin 11378  ℕcn 12159  (,)cioo 13275  [,]cicc 13278  seqcseq 13938  abscabs 15171

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5245  ax-nul 5255  ax-pow 5314  ax-pr 5381  ax-un 7692  ax-cnex 11096  ax-resscn 11097  ax-pre-lttri 11114  ax-pre-lttrn 11115

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5529  df-po 5542  df-so 5543  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6458  df-fun 6504  df-fn 6505  df-f 6506  df-f1 6507  df-fo 6508  df-f1o 6509  df-fv 6510  df-ov 7373  df-oprab 7374  df-mpo 7375  df-1st 7945  df-2nd 7946  df-er 8647  df-en 8898  df-dom 8899  df-sdom 8900  df-pnf 11182  df-mnf 11183  df-xr 11184  df-ltxr 11185  df-le 11186  df-ioo 13279

This theorem is referenced by:  ovolicc2lem2  25492  ovolicc2lem3  25493  ovolicc2lem4  25494