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Theorem uniioombllem2a 24651

Description: Lemma for uniioombl 24658. (Contributed by Mario Carneiro, 7-May-2015.)

**Hypotheses**
Ref	Expression
uniioombl.1	⊢ (𝜑 → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
uniioombl.2	⊢ (𝜑 → Disj 𝑥 ∈ ℕ ((,)‘(𝐹‘𝑥)))
uniioombl.3	⊢ 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))
uniioombl.a	⊢ 𝐴 = ∪ ran ((,) ∘ 𝐹)
uniioombl.e	⊢ (𝜑 → (vol*‘𝐸) ∈ ℝ)
uniioombl.c	⊢ (𝜑 → 𝐶 ∈ ℝ⁺)
uniioombl.g	⊢ (𝜑 → 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
uniioombl.s	⊢ (𝜑 → 𝐸 ⊆ ∪ ran ((,) ∘ 𝐺))
uniioombl.t	⊢ 𝑇 = seq1( + , ((abs ∘ − ) ∘ 𝐺))
uniioombl.v	⊢ (𝜑 → sup(ran 𝑇, ℝ^, < ) ≤ ((vol‘𝐸) + 𝐶))

**Assertion**
Ref	Expression
uniioombllem2a	⊢ (((𝜑 ∧ 𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽))) ∈ ran (,))

Distinct variable groups: 𝑥,𝑧,𝐹 𝑥,𝐺,𝑧 𝑥,𝐴,𝑧 𝑥,𝐶,𝑧 𝑥,𝐽,𝑧 𝜑,𝑥,𝑧 𝑥,𝑇,𝑧 Allowed substitution hints: 𝑆(𝑥,𝑧) 𝐸(𝑥,𝑧)

**Proof of Theorem uniioombllem2a**
Step	Hyp	Ref	Expression
1		uniioombl.1	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
2	1	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
3	2	ffvelrnda 6943	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → (𝐹‘𝑧) ∈ ( ≤ ∩ (ℝ × ℝ)))
4	3	elin2d 4129	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → (𝐹‘𝑧) ∈ (ℝ × ℝ))
5		1st2nd2 7843	. . . . . . 7 ⊢ ((𝐹‘𝑧) ∈ (ℝ × ℝ) → (𝐹‘𝑧) = ⟨(1^st ‘(𝐹‘𝑧)), (2^nd ‘(𝐹‘𝑧))⟩)
6	4, 5	syl 17	. . . . . 6 ⊢ (((𝜑 ∧ 𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → (𝐹‘𝑧) = ⟨(1^st ‘(𝐹‘𝑧)), (2^nd ‘(𝐹‘𝑧))⟩)
7	6	fveq2d 6760	. . . . 5 ⊢ (((𝜑 ∧ 𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → ((,)‘(𝐹‘𝑧)) = ((,)‘⟨(1^st ‘(𝐹‘𝑧)), (2^nd ‘(𝐹‘𝑧))⟩))
8		df-ov 7258	. . . . 5 ⊢ ((1^st ‘(𝐹‘𝑧))(,)(2^nd ‘(𝐹‘𝑧))) = ((,)‘⟨(1^st ‘(𝐹‘𝑧)), (2^nd ‘(𝐹‘𝑧))⟩)
9	7, 8	eqtr4di 2797	. . . 4 ⊢ (((𝜑 ∧ 𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → ((,)‘(𝐹‘𝑧)) = ((1^st ‘(𝐹‘𝑧))(,)(2^nd ‘(𝐹‘𝑧))))
10		uniioombl.g	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
11	10	ffvelrnda 6943	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → (𝐺‘𝐽) ∈ ( ≤ ∩ (ℝ × ℝ)))
12	11	elin2d 4129	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → (𝐺‘𝐽) ∈ (ℝ × ℝ))
13		1st2nd2 7843	. . . . . . . 8 ⊢ ((𝐺‘𝐽) ∈ (ℝ × ℝ) → (𝐺‘𝐽) = ⟨(1^st ‘(𝐺‘𝐽)), (2^nd ‘(𝐺‘𝐽))⟩)
14	12, 13	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → (𝐺‘𝐽) = ⟨(1^st ‘(𝐺‘𝐽)), (2^nd ‘(𝐺‘𝐽))⟩)
15	14	fveq2d 6760	. . . . . 6 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → ((,)‘(𝐺‘𝐽)) = ((,)‘⟨(1^st ‘(𝐺‘𝐽)), (2^nd ‘(𝐺‘𝐽))⟩))
16		df-ov 7258	. . . . . 6 ⊢ ((1^st ‘(𝐺‘𝐽))(,)(2^nd ‘(𝐺‘𝐽))) = ((,)‘⟨(1^st ‘(𝐺‘𝐽)), (2^nd ‘(𝐺‘𝐽))⟩)
17	15, 16	eqtr4di 2797	. . . . 5 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → ((,)‘(𝐺‘𝐽)) = ((1^st ‘(𝐺‘𝐽))(,)(2^nd ‘(𝐺‘𝐽))))
18	17	adantr 480	. . . 4 ⊢ (((𝜑 ∧ 𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → ((,)‘(𝐺‘𝐽)) = ((1^st ‘(𝐺‘𝐽))(,)(2^nd ‘(𝐺‘𝐽))))
19	9, 18	ineq12d 4144	. . 3 ⊢ (((𝜑 ∧ 𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽))) = (((1^st ‘(𝐹‘𝑧))(,)(2^nd ‘(𝐹‘𝑧))) ∩ ((1^st ‘(𝐺‘𝐽))(,)(2^nd ‘(𝐺‘𝐽)))))
20		ovolfcl 24535	. . . . . . 7 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑧 ∈ ℕ) → ((1^st ‘(𝐹‘𝑧)) ∈ ℝ ∧ (2^nd ‘(𝐹‘𝑧)) ∈ ℝ ∧ (1^st ‘(𝐹‘𝑧)) ≤ (2^nd ‘(𝐹‘𝑧))))
21	2, 20	sylan 579	. . . . . 6 ⊢ (((𝜑 ∧ 𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → ((1^st ‘(𝐹‘𝑧)) ∈ ℝ ∧ (2^nd ‘(𝐹‘𝑧)) ∈ ℝ ∧ (1^st ‘(𝐹‘𝑧)) ≤ (2^nd ‘(𝐹‘𝑧))))
22	21	simp1d 1140	. . . . 5 ⊢ (((𝜑 ∧ 𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → (1^st ‘(𝐹‘𝑧)) ∈ ℝ)
23	22	rexrd 10956	. . . 4 ⊢ (((𝜑 ∧ 𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → (1^st ‘(𝐹‘𝑧)) ∈ ℝ^*)
24	21	simp2d 1141	. . . . 5 ⊢ (((𝜑 ∧ 𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → (2^nd ‘(𝐹‘𝑧)) ∈ ℝ)
25	24	rexrd 10956	. . . 4 ⊢ (((𝜑 ∧ 𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → (2^nd ‘(𝐹‘𝑧)) ∈ ℝ^*)
26		ovolfcl 24535	. . . . . . . 8 ⊢ ((𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐽 ∈ ℕ) → ((1^st ‘(𝐺‘𝐽)) ∈ ℝ ∧ (2^nd ‘(𝐺‘𝐽)) ∈ ℝ ∧ (1^st ‘(𝐺‘𝐽)) ≤ (2^nd ‘(𝐺‘𝐽))))
27	10, 26	sylan 579	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → ((1^st ‘(𝐺‘𝐽)) ∈ ℝ ∧ (2^nd ‘(𝐺‘𝐽)) ∈ ℝ ∧ (1^st ‘(𝐺‘𝐽)) ≤ (2^nd ‘(𝐺‘𝐽))))
28	27	simp1d 1140	. . . . . 6 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → (1^st ‘(𝐺‘𝐽)) ∈ ℝ)
29	28	rexrd 10956	. . . . 5 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → (1^st ‘(𝐺‘𝐽)) ∈ ℝ^*)
30	29	adantr 480	. . . 4 ⊢ (((𝜑 ∧ 𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → (1^st ‘(𝐺‘𝐽)) ∈ ℝ^*)
31	27	simp2d 1141	. . . . . 6 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → (2^nd ‘(𝐺‘𝐽)) ∈ ℝ)
32	31	rexrd 10956	. . . . 5 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → (2^nd ‘(𝐺‘𝐽)) ∈ ℝ^*)
33	32	adantr 480	. . . 4 ⊢ (((𝜑 ∧ 𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → (2^nd ‘(𝐺‘𝐽)) ∈ ℝ^*)
34		iooin 13042	. . . 4 ⊢ ((((1^st ‘(𝐹‘𝑧)) ∈ ℝ^* ∧ (2^nd ‘(𝐹‘𝑧)) ∈ ℝ^) ∧ ((1^st ‘(𝐺‘𝐽)) ∈ ℝ^ ∧ (2^nd ‘(𝐺‘𝐽)) ∈ ℝ^*)) → (((1^st ‘(𝐹‘𝑧))(,)(2^nd ‘(𝐹‘𝑧))) ∩ ((1^st ‘(𝐺‘𝐽))(,)(2^nd ‘(𝐺‘𝐽)))) = (if((1^st ‘(𝐹‘𝑧)) ≤ (1^st ‘(𝐺‘𝐽)), (1^st ‘(𝐺‘𝐽)), (1^st ‘(𝐹‘𝑧)))(,)if((2^nd ‘(𝐹‘𝑧)) ≤ (2^nd ‘(𝐺‘𝐽)), (2^nd ‘(𝐹‘𝑧)), (2^nd ‘(𝐺‘𝐽)))))
35	23, 25, 30, 33, 34	syl22anc 835	. . 3 ⊢ (((𝜑 ∧ 𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → (((1^st ‘(𝐹‘𝑧))(,)(2^nd ‘(𝐹‘𝑧))) ∩ ((1^st ‘(𝐺‘𝐽))(,)(2^nd ‘(𝐺‘𝐽)))) = (if((1^st ‘(𝐹‘𝑧)) ≤ (1^st ‘(𝐺‘𝐽)), (1^st ‘(𝐺‘𝐽)), (1^st ‘(𝐹‘𝑧)))(,)if((2^nd ‘(𝐹‘𝑧)) ≤ (2^nd ‘(𝐺‘𝐽)), (2^nd ‘(𝐹‘𝑧)), (2^nd ‘(𝐺‘𝐽)))))
36	19, 35	eqtrd 2778	. 2 ⊢ (((𝜑 ∧ 𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽))) = (if((1^st ‘(𝐹‘𝑧)) ≤ (1^st ‘(𝐺‘𝐽)), (1^st ‘(𝐺‘𝐽)), (1^st ‘(𝐹‘𝑧)))(,)if((2^nd ‘(𝐹‘𝑧)) ≤ (2^nd ‘(𝐺‘𝐽)), (2^nd ‘(𝐹‘𝑧)), (2^nd ‘(𝐺‘𝐽)))))
37		ioorebas 13112	. 2 ⊢ (if((1^st ‘(𝐹‘𝑧)) ≤ (1^st ‘(𝐺‘𝐽)), (1^st ‘(𝐺‘𝐽)), (1^st ‘(𝐹‘𝑧)))(,)if((2^nd ‘(𝐹‘𝑧)) ≤ (2^nd ‘(𝐺‘𝐽)), (2^nd ‘(𝐹‘𝑧)), (2^nd ‘(𝐺‘𝐽)))) ∈ ran (,)
38	36, 37	eqeltrdi 2847	1 ⊢ (((𝜑 ∧ 𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽))) ∈ ran (,))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1085 = wceq 1539 ∈ wcel 2108 ∩ cin 3882 ⊆ wss 3883 ifcif 4456 ⟨cop 4564 ∪ cuni 4836 Disj wdisj 5035 class class class wbr 5070 × cxp 5578 ran crn 5581 ∘ ccom 5584 ⟶wf 6414 ‘cfv 6418 (class class class)co 7255 1^st c1st 7802 2^nd c2nd 7803 supcsup 9129 ℝcr 10801 1c1 10803 + caddc 10805 ℝ^*cxr 10939 < clt 10940 ≤ cle 10941 − cmin 11135 ℕcn 11903 ℝ⁺crp 12659 (,)cioo 13008 seqcseq 13649 abscabs 14873 vol*covol 24531

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-1st 7804 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-er 8456 df-en 8692 df-dom 8693 df-sdom 8694 df-sup 9131 df-inf 9132 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-n0 12164 df-z 12250 df-uz 12512 df-q 12618 df-ioo 13012

This theorem is referenced by: uniioombllem2 24652

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