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Theorem ovolicc2lem1 25033

Description: Lemma for ovolicc2 25038. (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 ∧ (1^st ‘(𝐹‘(𝐺‘𝑋))) < 𝑃 ∧ 𝑃 < (2^nd ‘(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 4229	. . . . . 6 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
3		fss 6734	. . . . . 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 7086	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → (𝐺‘𝑋) ∈ ℕ)
7		fvco3 6990	. . . . 5 ⊢ ((𝐹:ℕ⟶(ℝ × ℝ) ∧ (𝐺‘𝑋) ∈ ℕ) → (((,) ∘ 𝐹)‘(𝐺‘𝑋)) = ((,)‘(𝐹‘(𝐺‘𝑋))))
8	4, 6, 7	syl2an2r 683	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → (((,) ∘ 𝐹)‘(𝐺‘𝑋)) = ((,)‘(𝐹‘(𝐺‘𝑋))))
9		ovolicc2.9	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑈) → (((,) ∘ 𝐹)‘(𝐺‘𝑡)) = 𝑡)
10	9	ralrimiva 3146	. . . . 5 ⊢ (𝜑 → ∀𝑡 ∈ 𝑈 (((,) ∘ 𝐹)‘(𝐺‘𝑡)) = 𝑡)
11		2fveq3 6896	. . . . . . 7 ⊢ (𝑡 = 𝑋 → (((,) ∘ 𝐹)‘(𝐺‘𝑡)) = (((,) ∘ 𝐹)‘(𝐺‘𝑋)))
12		id 22	. . . . . . 7 ⊢ (𝑡 = 𝑋 → 𝑡 = 𝑋)
13	11, 12	eqeq12d 2748	. . . . . 6 ⊢ (𝑡 = 𝑋 → ((((,) ∘ 𝐹)‘(𝐺‘𝑡)) = 𝑡 ↔ (((,) ∘ 𝐹)‘(𝐺‘𝑋)) = 𝑋))
14	13	rspccva 3611	. . . . 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 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → 𝐹:ℕ⟶(ℝ × ℝ))
17	16, 6	ffvelcdmd 7087	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → (𝐹‘(𝐺‘𝑋)) ∈ (ℝ × ℝ))
18		1st2nd2 8013	. . . . . . 7 ⊢ ((𝐹‘(𝐺‘𝑋)) ∈ (ℝ × ℝ) → (𝐹‘(𝐺‘𝑋)) = ⟨(1^st ‘(𝐹‘(𝐺‘𝑋))), (2^nd ‘(𝐹‘(𝐺‘𝑋)))⟩)
19	17, 18	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → (𝐹‘(𝐺‘𝑋)) = ⟨(1^st ‘(𝐹‘(𝐺‘𝑋))), (2^nd ‘(𝐹‘(𝐺‘𝑋)))⟩)
20	19	fveq2d 6895	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → ((,)‘(𝐹‘(𝐺‘𝑋))) = ((,)‘⟨(1^st ‘(𝐹‘(𝐺‘𝑋))), (2^nd ‘(𝐹‘(𝐺‘𝑋)))⟩))
21		df-ov 7411	. . . . 5 ⊢ ((1^st ‘(𝐹‘(𝐺‘𝑋)))(,)(2^nd ‘(𝐹‘(𝐺‘𝑋)))) = ((,)‘⟨(1^st ‘(𝐹‘(𝐺‘𝑋))), (2^nd ‘(𝐹‘(𝐺‘𝑋)))⟩)
22	20, 21	eqtr4di 2790	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → ((,)‘(𝐹‘(𝐺‘𝑋))) = ((1^st ‘(𝐹‘(𝐺‘𝑋)))(,)(2^nd ‘(𝐹‘(𝐺‘𝑋)))))
23	8, 15, 22	3eqtr3d 2780	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → 𝑋 = ((1^st ‘(𝐹‘(𝐺‘𝑋)))(,)(2^nd ‘(𝐹‘(𝐺‘𝑋)))))
24	23	eleq2d 2819	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → (𝑃 ∈ 𝑋 ↔ 𝑃 ∈ ((1^st ‘(𝐹‘(𝐺‘𝑋)))(,)(2^nd ‘(𝐹‘(𝐺‘𝑋))))))
25		xp1st 8006	. . . 4 ⊢ ((𝐹‘(𝐺‘𝑋)) ∈ (ℝ × ℝ) → (1^st ‘(𝐹‘(𝐺‘𝑋))) ∈ ℝ)
26	17, 25	syl 17	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → (1^st ‘(𝐹‘(𝐺‘𝑋))) ∈ ℝ)
27		xp2nd 8007	. . . 4 ⊢ ((𝐹‘(𝐺‘𝑋)) ∈ (ℝ × ℝ) → (2^nd ‘(𝐹‘(𝐺‘𝑋))) ∈ ℝ)
28	17, 27	syl 17	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → (2^nd ‘(𝐹‘(𝐺‘𝑋))) ∈ ℝ)
29		rexr 11259	. . . 4 ⊢ ((1^st ‘(𝐹‘(𝐺‘𝑋))) ∈ ℝ → (1^st ‘(𝐹‘(𝐺‘𝑋))) ∈ ℝ^*)
30		rexr 11259	. . . 4 ⊢ ((2^nd ‘(𝐹‘(𝐺‘𝑋))) ∈ ℝ → (2^nd ‘(𝐹‘(𝐺‘𝑋))) ∈ ℝ^*)
31		elioo2 13364	. . . 4 ⊢ (((1^st ‘(𝐹‘(𝐺‘𝑋))) ∈ ℝ^* ∧ (2^nd ‘(𝐹‘(𝐺‘𝑋))) ∈ ℝ^*) → (𝑃 ∈ ((1^st ‘(𝐹‘(𝐺‘𝑋)))(,)(2^nd ‘(𝐹‘(𝐺‘𝑋)))) ↔ (𝑃 ∈ ℝ ∧ (1^st ‘(𝐹‘(𝐺‘𝑋))) < 𝑃 ∧ 𝑃 < (2^nd ‘(𝐹‘(𝐺‘𝑋))))))
32	29, 30, 31	syl2an 596	. . 3 ⊢ (((1^st ‘(𝐹‘(𝐺‘𝑋))) ∈ ℝ ∧ (2^nd ‘(𝐹‘(𝐺‘𝑋))) ∈ ℝ) → (𝑃 ∈ ((1^st ‘(𝐹‘(𝐺‘𝑋)))(,)(2^nd ‘(𝐹‘(𝐺‘𝑋)))) ↔ (𝑃 ∈ ℝ ∧ (1^st ‘(𝐹‘(𝐺‘𝑋))) < 𝑃 ∧ 𝑃 < (2^nd ‘(𝐹‘(𝐺‘𝑋))))))
33	26, 28, 32	syl2anc 584	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → (𝑃 ∈ ((1^st ‘(𝐹‘(𝐺‘𝑋)))(,)(2^nd ‘(𝐹‘(𝐺‘𝑋)))) ↔ (𝑃 ∈ ℝ ∧ (1^st ‘(𝐹‘(𝐺‘𝑋))) < 𝑃 ∧ 𝑃 < (2^nd ‘(𝐹‘(𝐺‘𝑋))))))
34	24, 33	bitrd 278	1 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → (𝑃 ∈ 𝑋 ↔ (𝑃 ∈ ℝ ∧ (1^st ‘(𝐹‘(𝐺‘𝑋))) < 𝑃 ∧ 𝑃 < (2^nd ‘(𝐹‘(𝐺‘𝑋))))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ∀wral 3061 ∩ cin 3947 ⊆ wss 3948 𝒫 cpw 4602 ⟨cop 4634 ∪ cuni 4908 class class class wbr 5148 × cxp 5674 ran crn 5677 ∘ ccom 5680 ⟶wf 6539 ‘cfv 6543 (class class class)co 7408 1^st c1st 7972 2^nd c2nd 7973 Fincfn 8938 ℝcr 11108 1c1 11110 + caddc 11112 ℝ^*cxr 11246 < clt 11247 ≤ cle 11248 − cmin 11443 ℕcn 12211 (,)cioo 13323 [,]cicc 13326 seqcseq 13965 abscabs 15180

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-cnex 11165 ax-resscn 11166 ax-pre-lttri 11183 ax-pre-lttrn 11184

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-po 5588 df-so 5589 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7411 df-oprab 7412 df-mpo 7413 df-1st 7974 df-2nd 7975 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-ioo 13327

This theorem is referenced by: ovolicc2lem2 25034 ovolicc2lem3 25035 ovolicc2lem4 25036

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