Theorem ovolicc2lem1 25838

Description: Lemma for ovolicc2 25843. (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 4134	. . . . . 6 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
3		fss 6726	. . . . . 6 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)) → 𝐹:ℕ⟶(ℝ × ℝ))
4	1, 2, 3	sylancl 598	. . . . 5 ⊢ (𝜑 → 𝐹:ℕ⟶(ℝ × ℝ))
5		ovolicc2.8	. . . . . 6 ⊢ (𝜑 → 𝐺:𝑈⟶ℕ)
6	5	ffvelcdmda 7084	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → (𝐺‘𝑋) ∈ ℕ)
7		fvco3 6985	. . . . 5 ⊢ ((𝐹:ℕ⟶(ℝ × ℝ) ∧ (𝐺‘𝑋) ∈ ℕ) → (((,) ∘ 𝐹)‘(𝐺‘𝑋)) = ((,)‘(𝐹‘(𝐺‘𝑋))))
8	4, 6, 7	syl2an2r 698	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → (((,) ∘ 𝐹)‘(𝐺‘𝑋)) = ((,)‘(𝐹‘(𝐺‘𝑋))))
9		ovolicc2.9	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑈) → (((,) ∘ 𝐹)‘(𝐺‘𝑡)) = 𝑡)
10	9	ralrimiva 3154	. . . . 5 ⊢ (𝜑 → ∀𝑡 ∈ 𝑈 (((,) ∘ 𝐹)‘(𝐺‘𝑡)) = 𝑡)
11		2fveq3 6890	. . . . . . 7 ⊢ (𝑡 = 𝑋 → (((,) ∘ 𝐹)‘(𝐺‘𝑡)) = (((,) ∘ 𝐹)‘(𝐺‘𝑋)))
12		id 23	. . . . . . 7 ⊢ (𝑡 = 𝑋 → 𝑡 = 𝑋)
13	11, 12	eqeq12d 2776	. . . . . 6 ⊢ (𝑡 = 𝑋 → ((((,) ∘ 𝐹)‘(𝐺‘𝑡)) = 𝑡 ↔ (((,) ∘ 𝐹)‘(𝐺‘𝑋)) = 𝑋))
14	13	rspccva 3575	. . . . 5 ⊢ ((∀𝑡 ∈ 𝑈 (((,) ∘ 𝐹)‘(𝐺‘𝑡)) = 𝑡 ∧ 𝑋 ∈ 𝑈) → (((,) ∘ 𝐹)‘(𝐺‘𝑋)) = 𝑋)
15	10, 14	sylan 592	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → (((,) ∘ 𝐹)‘(𝐺‘𝑋)) = 𝑋)
16	4	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → 𝐹:ℕ⟶(ℝ × ℝ))
17	16, 6	ffvelcdmd 7085	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → (𝐹‘(𝐺‘𝑋)) ∈ (ℝ × ℝ))
18		1st2nd2 8040	. . . . . . 7 ⊢ ((𝐹‘(𝐺‘𝑋)) ∈ (ℝ × ℝ) → (𝐹‘(𝐺‘𝑋)) = ⟨(1st ‘(𝐹‘(𝐺‘𝑋))), (2nd ‘(𝐹‘(𝐺‘𝑋)))⟩)
19	17, 18	syl 18	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → (𝐹‘(𝐺‘𝑋)) = ⟨(1st ‘(𝐹‘(𝐺‘𝑋))), (2nd ‘(𝐹‘(𝐺‘𝑋)))⟩)
20	19	fveq2d 6889	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → ((,)‘(𝐹‘(𝐺‘𝑋))) = ((,)‘⟨(1st ‘(𝐹‘(𝐺‘𝑋))), (2nd ‘(𝐹‘(𝐺‘𝑋)))⟩))
21		df-ov 7423	. . . . 5 ⊢ ((1st ‘(𝐹‘(𝐺‘𝑋)))(,)(2nd ‘(𝐹‘(𝐺‘𝑋)))) = ((,)‘⟨(1st ‘(𝐹‘(𝐺‘𝑋))), (2nd ‘(𝐹‘(𝐺‘𝑋)))⟩)
22	20, 21	eqtr4di 2813	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → ((,)‘(𝐹‘(𝐺‘𝑋))) = ((1st ‘(𝐹‘(𝐺‘𝑋)))(,)(2nd ‘(𝐹‘(𝐺‘𝑋)))))
23	8, 15, 22	3eqtr3d 2803	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → 𝑋 = ((1st ‘(𝐹‘(𝐺‘𝑋)))(,)(2nd ‘(𝐹‘(𝐺‘𝑋)))))
24	23	eleq2d 2846	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → (𝑃 ∈ 𝑋 ↔ 𝑃 ∈ ((1st ‘(𝐹‘(𝐺‘𝑋)))(,)(2nd ‘(𝐹‘(𝐺‘𝑋))))))
25		xp1st 8033	. . . 4 ⊢ ((𝐹‘(𝐺‘𝑋)) ∈ (ℝ × ℝ) → (1st ‘(𝐹‘(𝐺‘𝑋))) ∈ ℝ)
26	17, 25	syl 18	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → (1st ‘(𝐹‘(𝐺‘𝑋))) ∈ ℝ)
27		xp2nd 8034	. . . 4 ⊢ ((𝐹‘(𝐺‘𝑋)) ∈ (ℝ × ℝ) → (2nd ‘(𝐹‘(𝐺‘𝑋))) ∈ ℝ)
28	17, 27	syl 18	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → (2nd ‘(𝐹‘(𝐺‘𝑋))) ∈ ℝ)
29		rexr 11355	. . . 4 ⊢ ((1st ‘(𝐹‘(𝐺‘𝑋))) ∈ ℝ → (1st ‘(𝐹‘(𝐺‘𝑋))) ∈ ℝ*)
30		rexr 11355	. . . 4 ⊢ ((2nd ‘(𝐹‘(𝐺‘𝑋))) ∈ ℝ → (2nd ‘(𝐹‘(𝐺‘𝑋))) ∈ ℝ*)
31		elioo2 13517	. . . 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 608	. . 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 596	. 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 282	1 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → (𝑃 ∈ 𝑋 ↔ (𝑃 ∈ ℝ ∧ (1st ‘(𝐹‘(𝐺‘𝑋))) < 𝑃 ∧ 𝑃 < (2nd ‘(𝐹‘(𝐺‘𝑋))))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144  ∀wral 3076   ⊆ wss 3895   ∩ cin 4089  𝒫 cpw 4556  ⟨cop 4589  ∪ cuni 4866   class class class wbr 5102   × cxp 5648  ran crn 5651   ∘ ccom 5654  ⟶wf 6534  ‘cfv 6538  (class class class)co 7420  1^st c1st 7999  2^nd c2nd 8000  Fincfn 8973  ℝcr 11199  1c1 11201   + caddc 11203  ℝ^*cxr 11342   < clt 11343   ≤ cle 11344   − cmin 11541  ℕcn 12335  (,)cioo 13476  [,]cicc 13479  seqcseq 14144  abscabs 15401

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-cnex 11256  ax-resscn 11257  ax-pre-lttri 11274  ax-pre-lttrn 11275

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-po 5558  df-so 5559  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1st 8001  df-2nd 8002  df-er 8717  df-en 8974  df-dom 8975  df-sdom 8976  df-pnf 11345  df-mnf 11346  df-xr 11347  df-ltxr 11348  df-le 11349  df-ioo 13480

This theorem is used by:  ovolicc2lem2  25839  ovolicc2lem3  25840  ovolicc2lem4  25841