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Theorem uniioombllem2a 23156

Description: Lemma for uniioombl 23163. (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		inss2 3796	. . . . . . . 8 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
2		uniioombl.1	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
3	2	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
4	3	ffvelrnda 6267	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → (𝐹‘𝑧) ∈ ( ≤ ∩ (ℝ × ℝ)))
5	1, 4	sseldi 3566	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → (𝐹‘𝑧) ∈ (ℝ × ℝ))
6		1st2nd2 7096	. . . . . . 7 ⊢ ((𝐹‘𝑧) ∈ (ℝ × ℝ) → (𝐹‘𝑧) = ⟨(1^st ‘(𝐹‘𝑧)), (2^nd ‘(𝐹‘𝑧))⟩)
7	5, 6	syl 17	. . . . . 6 ⊢ (((𝜑 ∧ 𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → (𝐹‘𝑧) = ⟨(1^st ‘(𝐹‘𝑧)), (2^nd ‘(𝐹‘𝑧))⟩)
8	7	fveq2d 6107	. . . . 5 ⊢ (((𝜑 ∧ 𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → ((,)‘(𝐹‘𝑧)) = ((,)‘⟨(1^st ‘(𝐹‘𝑧)), (2^nd ‘(𝐹‘𝑧))⟩))
9		df-ov 6552	. . . . 5 ⊢ ((1^st ‘(𝐹‘𝑧))(,)(2^nd ‘(𝐹‘𝑧))) = ((,)‘⟨(1^st ‘(𝐹‘𝑧)), (2^nd ‘(𝐹‘𝑧))⟩)
10	8, 9	syl6eqr 2662	. . . 4 ⊢ (((𝜑 ∧ 𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → ((,)‘(𝐹‘𝑧)) = ((1^st ‘(𝐹‘𝑧))(,)(2^nd ‘(𝐹‘𝑧))))
11		uniioombl.g	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
12	11	ffvelrnda 6267	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → (𝐺‘𝐽) ∈ ( ≤ ∩ (ℝ × ℝ)))
13	1, 12	sseldi 3566	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → (𝐺‘𝐽) ∈ (ℝ × ℝ))
14		1st2nd2 7096	. . . . . . . 8 ⊢ ((𝐺‘𝐽) ∈ (ℝ × ℝ) → (𝐺‘𝐽) = ⟨(1^st ‘(𝐺‘𝐽)), (2^nd ‘(𝐺‘𝐽))⟩)
15	13, 14	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → (𝐺‘𝐽) = ⟨(1^st ‘(𝐺‘𝐽)), (2^nd ‘(𝐺‘𝐽))⟩)
16	15	fveq2d 6107	. . . . . 6 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → ((,)‘(𝐺‘𝐽)) = ((,)‘⟨(1^st ‘(𝐺‘𝐽)), (2^nd ‘(𝐺‘𝐽))⟩))
17		df-ov 6552	. . . . . 6 ⊢ ((1^st ‘(𝐺‘𝐽))(,)(2^nd ‘(𝐺‘𝐽))) = ((,)‘⟨(1^st ‘(𝐺‘𝐽)), (2^nd ‘(𝐺‘𝐽))⟩)
18	16, 17	syl6eqr 2662	. . . . 5 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → ((,)‘(𝐺‘𝐽)) = ((1^st ‘(𝐺‘𝐽))(,)(2^nd ‘(𝐺‘𝐽))))
19	18	adantr 480	. . . 4 ⊢ (((𝜑 ∧ 𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → ((,)‘(𝐺‘𝐽)) = ((1^st ‘(𝐺‘𝐽))(,)(2^nd ‘(𝐺‘𝐽))))
20	10, 19	ineq12d 3777	. . 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)))))
21		ovolfcl 23042	. . . . . . 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))))
22	3, 21	sylan 487	. . . . . 6 ⊢ (((𝜑 ∧ 𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → ((1^st ‘(𝐹‘𝑧)) ∈ ℝ ∧ (2^nd ‘(𝐹‘𝑧)) ∈ ℝ ∧ (1^st ‘(𝐹‘𝑧)) ≤ (2^nd ‘(𝐹‘𝑧))))
23	22	simp1d 1066	. . . . 5 ⊢ (((𝜑 ∧ 𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → (1^st ‘(𝐹‘𝑧)) ∈ ℝ)
24	23	rexrd 9968	. . . 4 ⊢ (((𝜑 ∧ 𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → (1^st ‘(𝐹‘𝑧)) ∈ ℝ^*)
25	22	simp2d 1067	. . . . 5 ⊢ (((𝜑 ∧ 𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → (2^nd ‘(𝐹‘𝑧)) ∈ ℝ)
26	25	rexrd 9968	. . . 4 ⊢ (((𝜑 ∧ 𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → (2^nd ‘(𝐹‘𝑧)) ∈ ℝ^*)
27		ovolfcl 23042	. . . . . . . 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))))
28	11, 27	sylan 487	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → ((1^st ‘(𝐺‘𝐽)) ∈ ℝ ∧ (2^nd ‘(𝐺‘𝐽)) ∈ ℝ ∧ (1^st ‘(𝐺‘𝐽)) ≤ (2^nd ‘(𝐺‘𝐽))))
29	28	simp1d 1066	. . . . . 6 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → (1^st ‘(𝐺‘𝐽)) ∈ ℝ)
30	29	rexrd 9968	. . . . 5 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → (1^st ‘(𝐺‘𝐽)) ∈ ℝ^*)
31	30	adantr 480	. . . 4 ⊢ (((𝜑 ∧ 𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → (1^st ‘(𝐺‘𝐽)) ∈ ℝ^*)
32	28	simp2d 1067	. . . . . 6 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → (2^nd ‘(𝐺‘𝐽)) ∈ ℝ)
33	32	rexrd 9968	. . . . 5 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → (2^nd ‘(𝐺‘𝐽)) ∈ ℝ^*)
34	33	adantr 480	. . . 4 ⊢ (((𝜑 ∧ 𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → (2^nd ‘(𝐺‘𝐽)) ∈ ℝ^*)
35		iooin 12080	. . . 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)))))
36	24, 26, 31, 34, 35	syl22anc 1319	. . 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)))))
37	20, 36	eqtrd 2644	. 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)))))
38		ioorebas 12146	. 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 (,)
39	37, 38	syl6eqel 2696	1 ⊢ (((𝜑 ∧ 𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽))) ∈ ran (,))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∩ cin 3539 ⊆ wss 3540 ifcif 4036 ⟨cop 4131 ∪ cuni 4372 Disj wdisj 4553 class class class wbr 4583 × cxp 5036 ran crn 5039 ∘ ccom 5042 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 1^st c1st 7057 2^nd c2nd 7058 supcsup 8229 ℝcr 9814 1c1 9816 + caddc 9818 ℝ^*cxr 9952 < clt 9953 ≤ cle 9954 − cmin 10145 ℕcn 10897 ℝ⁺crp 11708 (,)cioo 12046 seqcseq 12663 abscabs 13822 vol*covol 23038

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-sup 8231 df-inf 8232 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 df-q 11665 df-ioo 12050

This theorem is referenced by: uniioombllem2 23157

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