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Theorem uniioombllem2a 25510

Description: Lemma for uniioombl 25517. (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	ffvelcdmda 7017	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → (𝐹‘𝑧) ∈ ( ≤ ∩ (ℝ × ℝ)))
4	3	elin2d 4152	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → (𝐹‘𝑧) ∈ (ℝ × ℝ))
5		1st2nd2 7960	. . . . . . 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 6826	. . . . 5 ⊢ (((𝜑 ∧ 𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → ((,)‘(𝐹‘𝑧)) = ((,)‘⟨(1^st ‘(𝐹‘𝑧)), (2^nd ‘(𝐹‘𝑧))⟩))
8		df-ov 7349	. . . . 5 ⊢ ((1^st ‘(𝐹‘𝑧))(,)(2^nd ‘(𝐹‘𝑧))) = ((,)‘⟨(1^st ‘(𝐹‘𝑧)), (2^nd ‘(𝐹‘𝑧))⟩)
9	7, 8	eqtr4di 2784	. . . 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	ffvelcdmda 7017	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → (𝐺‘𝐽) ∈ ( ≤ ∩ (ℝ × ℝ)))
12	11	elin2d 4152	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → (𝐺‘𝐽) ∈ (ℝ × ℝ))
13		1st2nd2 7960	. . . . . . . 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 6826	. . . . . 6 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → ((,)‘(𝐺‘𝐽)) = ((,)‘⟨(1^st ‘(𝐺‘𝐽)), (2^nd ‘(𝐺‘𝐽))⟩))
16		df-ov 7349	. . . . . 6 ⊢ ((1^st ‘(𝐺‘𝐽))(,)(2^nd ‘(𝐺‘𝐽))) = ((,)‘⟨(1^st ‘(𝐺‘𝐽)), (2^nd ‘(𝐺‘𝐽))⟩)
17	15, 16	eqtr4di 2784	. . . . 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 4168	. . 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 25394	. . . . . . 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 580	. . . . . 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 1142	. . . . 5 ⊢ (((𝜑 ∧ 𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → (1^st ‘(𝐹‘𝑧)) ∈ ℝ)
23	22	rexrd 11162	. . . 4 ⊢ (((𝜑 ∧ 𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → (1^st ‘(𝐹‘𝑧)) ∈ ℝ^*)
24	21	simp2d 1143	. . . . 5 ⊢ (((𝜑 ∧ 𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → (2^nd ‘(𝐹‘𝑧)) ∈ ℝ)
25	24	rexrd 11162	. . . 4 ⊢ (((𝜑 ∧ 𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → (2^nd ‘(𝐹‘𝑧)) ∈ ℝ^*)
26		ovolfcl 25394	. . . . . . . 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 580	. . . . . . 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 1142	. . . . . 6 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → (1^st ‘(𝐺‘𝐽)) ∈ ℝ)
29	28	rexrd 11162	. . . . 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 1143	. . . . . 6 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → (2^nd ‘(𝐺‘𝐽)) ∈ ℝ)
32	31	rexrd 11162	. . . . 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 13279	. . . 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 838	. . 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 2766	. 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 13351	. 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 2839	1 ⊢ (((𝜑 ∧ 𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽))) ∈ ran (,))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1541 ∈ wcel 2111 ∩ cin 3896 ⊆ wss 3897 ifcif 4472 ⟨cop 4579 ∪ cuni 4856 Disj wdisj 5056 class class class wbr 5089 × cxp 5612 ran crn 5615 ∘ ccom 5618 ⟶wf 6477 ‘cfv 6481 (class class class)co 7346 1^st c1st 7919 2^nd c2nd 7920 supcsup 9324 ℝcr 11005 1c1 11007 + caddc 11009 ℝ^*cxr 11145 < clt 11146 ≤ cle 11147 − cmin 11344 ℕcn 12125 ℝ⁺crp 12890 (,)cioo 13245 seqcseq 13908 abscabs 15141 vol*covol 25390

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7668 ax-cnex 11062 ax-resscn 11063 ax-1cn 11064 ax-icn 11065 ax-addcl 11066 ax-addrcl 11067 ax-mulcl 11068 ax-mulrcl 11069 ax-mulcom 11070 ax-addass 11071 ax-mulass 11072 ax-distr 11073 ax-i2m1 11074 ax-1ne0 11075 ax-1rid 11076 ax-rnegex 11077 ax-rrecex 11078 ax-cnre 11079 ax-pre-lttri 11080 ax-pre-lttrn 11081 ax-pre-ltadd 11082 ax-pre-mulgt0 11083 ax-pre-sup 11084

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4857 df-iun 4941 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-we 5569 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-om 7797 df-1st 7921 df-2nd 7922 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-er 8622 df-en 8870 df-dom 8871 df-sdom 8872 df-sup 9326 df-inf 9327 df-pnf 11148 df-mnf 11149 df-xr 11150 df-ltxr 11151 df-le 11152 df-sub 11346 df-neg 11347 df-div 11775 df-nn 12126 df-n0 12382 df-z 12469 df-uz 12733 df-q 12847 df-ioo 13249

This theorem is referenced by: uniioombllem2 25511

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