uniioombllem2a - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > uniioombllem2a		Structured version Visualization version GIF version

Theorem uniioombllem2a 25710

Description: Lemma for uniioombl 25717. (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 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
3	2	ffvelcdmda 7080	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → (𝐹‘𝑧) ∈ ( ≤ ∩ (ℝ × ℝ)))
4	3	elin2d 4166	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → (𝐹‘𝑧) ∈ (ℝ × ℝ))
5		1st2nd2 8025	. . . . . . 7 ⊢ ((𝐹‘𝑧) ∈ (ℝ × ℝ) → (𝐹‘𝑧) = ⟨(1^st ‘(𝐹‘𝑧)), (2^nd ‘(𝐹‘𝑧))⟩)
6	4, 5	syl 18	. . . . . 6 ⊢ (((𝜑 ∧ 𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → (𝐹‘𝑧) = ⟨(1^st ‘(𝐹‘𝑧)), (2^nd ‘(𝐹‘𝑧))⟩)
7	6	fveq2d 6886	. . . . 5 ⊢ (((𝜑 ∧ 𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → ((,)‘(𝐹‘𝑧)) = ((,)‘⟨(1^st ‘(𝐹‘𝑧)), (2^nd ‘(𝐹‘𝑧))⟩))
8		df-ov 7414	. . . . 5 ⊢ ((1^st ‘(𝐹‘𝑧))(,)(2^nd ‘(𝐹‘𝑧))) = ((,)‘⟨(1^st ‘(𝐹‘𝑧)), (2^nd ‘(𝐹‘𝑧))⟩)
9	7, 8	eqtr4di 2822	. . . 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 7080	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → (𝐺‘𝐽) ∈ ( ≤ ∩ (ℝ × ℝ)))
12	11	elin2d 4166	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → (𝐺‘𝐽) ∈ (ℝ × ℝ))
13		1st2nd2 8025	. . . . . . . 8 ⊢ ((𝐺‘𝐽) ∈ (ℝ × ℝ) → (𝐺‘𝐽) = ⟨(1^st ‘(𝐺‘𝐽)), (2^nd ‘(𝐺‘𝐽))⟩)
14	12, 13	syl 18	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → (𝐺‘𝐽) = ⟨(1^st ‘(𝐺‘𝐽)), (2^nd ‘(𝐺‘𝐽))⟩)
15	14	fveq2d 6886	. . . . . 6 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → ((,)‘(𝐺‘𝐽)) = ((,)‘⟨(1^st ‘(𝐺‘𝐽)), (2^nd ‘(𝐺‘𝐽))⟩))
16		df-ov 7414	. . . . . 6 ⊢ ((1^st ‘(𝐺‘𝐽))(,)(2^nd ‘(𝐺‘𝐽))) = ((,)‘⟨(1^st ‘(𝐺‘𝐽)), (2^nd ‘(𝐺‘𝐽))⟩)
17	15, 16	eqtr4di 2822	. . . . 5 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → ((,)‘(𝐺‘𝐽)) = ((1^st ‘(𝐺‘𝐽))(,)(2^nd ‘(𝐺‘𝐽))))
18	17	adantr 485	. . . 4 ⊢ (((𝜑 ∧ 𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → ((,)‘(𝐺‘𝐽)) = ((1^st ‘(𝐺‘𝐽))(,)(2^nd ‘(𝐺‘𝐽))))
19	9, 18	ineq12d 4182	. . 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 25594	. . . . . . 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 591	. . . . . 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 1158	. . . . 5 ⊢ (((𝜑 ∧ 𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → (1^st ‘(𝐹‘𝑧)) ∈ ℝ)
23	22	rexrd 11259	. . . 4 ⊢ (((𝜑 ∧ 𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → (1^st ‘(𝐹‘𝑧)) ∈ ℝ^*)
24	21	simp2d 1159	. . . . 5 ⊢ (((𝜑 ∧ 𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → (2^nd ‘(𝐹‘𝑧)) ∈ ℝ)
25	24	rexrd 11259	. . . 4 ⊢ (((𝜑 ∧ 𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → (2^nd ‘(𝐹‘𝑧)) ∈ ℝ^*)
26		ovolfcl 25594	. . . . . . . 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 591	. . . . . . 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 1158	. . . . . 6 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → (1^st ‘(𝐺‘𝐽)) ∈ ℝ)
29	28	rexrd 11259	. . . . 5 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → (1^st ‘(𝐺‘𝐽)) ∈ ℝ^*)
30	29	adantr 485	. . . 4 ⊢ (((𝜑 ∧ 𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → (1^st ‘(𝐺‘𝐽)) ∈ ℝ^*)
31	27	simp2d 1159	. . . . . 6 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → (2^nd ‘(𝐺‘𝐽)) ∈ ℝ)
32	31	rexrd 11259	. . . . 5 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → (2^nd ‘(𝐺‘𝐽)) ∈ ℝ^*)
33	32	adantr 485	. . . 4 ⊢ (((𝜑 ∧ 𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → (2^nd ‘(𝐺‘𝐽)) ∈ ℝ^*)
34		iooin 13406	. . . 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 851	. . 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 2804	. 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 13478	. 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 2877	1 ⊢ (((𝜑 ∧ 𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽))) ∈ ran (,))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 400 ∧ w3a 1101 = wceq 1567 ∈ wcel 2149 ∩ cin 3912 ⊆ wss 3913 ifcif 4492 ⟨cop 4600 ∪ cuni 4876 Disj wdisj 5080 class class class wbr 5113 × cxp 5660 ran crn 5663 ∘ ccom 5666 ⟶wf 6533 ‘cfv 6537 (class class class)co 7411 1^st c1st 7984 2^nd c2nd 7985 supcsup 9400 ℝcr 11099 1c1 11101 + caddc 11103 ℝ^*cxr 11242 < clt 11243 ≤ cle 11244 − cmin 11441 ℕcn 12233 ℝ⁺crp 13016 (,)cioo 13372 seqcseq 14037 abscabs 15285 vol*covol 25590

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-sep 5261 ax-nul 5271 ax-pow 5337 ax-pr 5405 ax-un 7733 ax-cnex 11156 ax-resscn 11157 ax-1cn 11158 ax-icn 11159 ax-addcl 11160 ax-addrcl 11161 ax-mulcl 11162 ax-mulrcl 11163 ax-mulcom 11164 ax-addass 11165 ax-mulass 11166 ax-distr 11167 ax-i2m1 11168 ax-1ne0 11169 ax-1rid 11170 ax-rnegex 11171 ax-rrecex 11172 ax-cnre 11173 ax-pre-lttri 11174 ax-pre-lttrn 11175 ax-pre-ltadd 11176 ax-pre-mulgt0 11177 ax-pre-sup 11178

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-rmo 3376 df-reu 3377 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3933 df-nul 4295 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-op 4601 df-uni 4877 df-iun 4962 df-br 5114 df-opab 5178 df-mpt 5197 df-tr 5223 df-id 5557 df-eprel 5562 df-po 5570 df-so 5571 df-fr 5615 df-we 5617 df-xp 5668 df-rel 5669 df-cnv 5670 df-co 5671 df-dm 5672 df-rn 5673 df-res 5674 df-ima 5675 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7368 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7863 df-1st 7986 df-2nd 7987 df-frecs 8278 df-wrecs 8309 df-recs 8358 df-rdg 8397 df-er 8694 df-en 8944 df-dom 8945 df-sdom 8946 df-sup 9402 df-inf 9403 df-pnf 11245 df-mnf 11246 df-xr 11247 df-ltxr 11248 df-le 11249 df-sub 11443 df-neg 11444 df-div 11872 df-nn 12234 df-n0 12505 df-z 12592 df-uz 12863 df-q 12973 df-ioo 13376

This theorem is referenced by: uniioombllem2 25711

W3C validator