ovolicc2lem1 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > ovolicc2lem1		Structured version Visualization version GIF version

Theorem ovolicc2lem1 25034

Description: Lemma for ovolicc2 25039. (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 4230	. . . . . 6 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
3		fss 6735	. . . . . 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 7087	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → (𝐺‘𝑋) ∈ ℕ)
7		fvco3 6991	. . . . 5 ⊢ ((𝐹:ℕ⟶(ℝ × ℝ) ∧ (𝐺‘𝑋) ∈ ℕ) → (((,) ∘ 𝐹)‘(𝐺‘𝑋)) = ((,)‘(𝐹‘(𝐺‘𝑋))))
8	4, 6, 7	syl2an2r 684	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → (((,) ∘ 𝐹)‘(𝐺‘𝑋)) = ((,)‘(𝐹‘(𝐺‘𝑋))))
9		ovolicc2.9	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑈) → (((,) ∘ 𝐹)‘(𝐺‘𝑡)) = 𝑡)
10	9	ralrimiva 3147	. . . . 5 ⊢ (𝜑 → ∀𝑡 ∈ 𝑈 (((,) ∘ 𝐹)‘(𝐺‘𝑡)) = 𝑡)
11		2fveq3 6897	. . . . . . 7 ⊢ (𝑡 = 𝑋 → (((,) ∘ 𝐹)‘(𝐺‘𝑡)) = (((,) ∘ 𝐹)‘(𝐺‘𝑋)))
12		id 22	. . . . . . 7 ⊢ (𝑡 = 𝑋 → 𝑡 = 𝑋)
13	11, 12	eqeq12d 2749	. . . . . 6 ⊢ (𝑡 = 𝑋 → ((((,) ∘ 𝐹)‘(𝐺‘𝑡)) = 𝑡 ↔ (((,) ∘ 𝐹)‘(𝐺‘𝑋)) = 𝑋))
14	13	rspccva 3612	. . . . 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 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → 𝐹:ℕ⟶(ℝ × ℝ))
17	16, 6	ffvelcdmd 7088	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → (𝐹‘(𝐺‘𝑋)) ∈ (ℝ × ℝ))
18		1st2nd2 8014	. . . . . . 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 6896	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → ((,)‘(𝐹‘(𝐺‘𝑋))) = ((,)‘⟨(1^st ‘(𝐹‘(𝐺‘𝑋))), (2^nd ‘(𝐹‘(𝐺‘𝑋)))⟩))
21		df-ov 7412	. . . . 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 2791	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → ((,)‘(𝐹‘(𝐺‘𝑋))) = ((1^st ‘(𝐹‘(𝐺‘𝑋)))(,)(2^nd ‘(𝐹‘(𝐺‘𝑋)))))
23	8, 15, 22	3eqtr3d 2781	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → 𝑋 = ((1^st ‘(𝐹‘(𝐺‘𝑋)))(,)(2^nd ‘(𝐹‘(𝐺‘𝑋)))))
24	23	eleq2d 2820	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → (𝑃 ∈ 𝑋 ↔ 𝑃 ∈ ((1^st ‘(𝐹‘(𝐺‘𝑋)))(,)(2^nd ‘(𝐹‘(𝐺‘𝑋))))))
25		xp1st 8007	. . . 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 8008	. . . 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 11260	. . . 4 ⊢ ((1^st ‘(𝐹‘(𝐺‘𝑋))) ∈ ℝ → (1^st ‘(𝐹‘(𝐺‘𝑋))) ∈ ℝ^*)
30		rexr 11260	. . . 4 ⊢ ((2^nd ‘(𝐹‘(𝐺‘𝑋))) ∈ ℝ → (2^nd ‘(𝐹‘(𝐺‘𝑋))) ∈ ℝ^*)
31		elioo2 13365	. . . 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 597	. . 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 585	. 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 279	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 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ∀wral 3062 ∩ cin 3948 ⊆ wss 3949 𝒫 cpw 4603 ⟨cop 4635 ∪ cuni 4909 class class class wbr 5149 × cxp 5675 ran crn 5678 ∘ ccom 5681 ⟶wf 6540 ‘cfv 6544 (class class class)co 7409 1^st c1st 7973 2^nd c2nd 7974 Fincfn 8939 ℝcr 11109 1c1 11111 + caddc 11113 ℝ^*cxr 11247 < clt 11248 ≤ cle 11249 − cmin 11444 ℕcn 12212 (,)cioo 13324 [,]cicc 13327 seqcseq 13966 abscabs 15181

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-pre-lttri 11184 ax-pre-lttrn 11185

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-po 5589 df-so 5590 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-ov 7412 df-oprab 7413 df-mpo 7414 df-1st 7975 df-2nd 7976 df-er 8703 df-en 8940 df-dom 8941 df-sdom 8942 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-ioo 13328

This theorem is referenced by: ovolicc2lem2 25035 ovolicc2lem3 25036 ovolicc2lem4 25037

W3C validator