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Theorem uniioombllem2a 25539

Description: Lemma for uniioombl 25546. (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 7029	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → (𝐹‘𝑧) ∈ ( ≤ ∩ (ℝ × ℝ)))
4	3	elin2d 4157	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → (𝐹‘𝑧) ∈ (ℝ × ℝ))
5		1st2nd2 7972	. . . . . . 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 6838	. . . . 5 ⊢ (((𝜑 ∧ 𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → ((,)‘(𝐹‘𝑧)) = ((,)‘⟨(1^st ‘(𝐹‘𝑧)), (2^nd ‘(𝐹‘𝑧))⟩))
8		df-ov 7361	. . . . 5 ⊢ ((1^st ‘(𝐹‘𝑧))(,)(2^nd ‘(𝐹‘𝑧))) = ((,)‘⟨(1^st ‘(𝐹‘𝑧)), (2^nd ‘(𝐹‘𝑧))⟩)
9	7, 8	eqtr4di 2789	. . . 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 7029	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → (𝐺‘𝐽) ∈ ( ≤ ∩ (ℝ × ℝ)))
12	11	elin2d 4157	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → (𝐺‘𝐽) ∈ (ℝ × ℝ))
13		1st2nd2 7972	. . . . . . . 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 6838	. . . . . 6 ⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → ((,)‘(𝐺‘𝐽)) = ((,)‘⟨(1^st ‘(𝐺‘𝐽)), (2^nd ‘(𝐺‘𝐽))⟩))
16		df-ov 7361	. . . . . 6 ⊢ ((1^st ‘(𝐺‘𝐽))(,)(2^nd ‘(𝐺‘𝐽))) = ((,)‘⟨(1^st ‘(𝐺‘𝐽)), (2^nd ‘(𝐺‘𝐽))⟩)
17	15, 16	eqtr4di 2789	. . . . 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 4173	. . 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 25423	. . . . . . 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 11182	. . . 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 11182	. . . 4 ⊢ (((𝜑 ∧ 𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → (2^nd ‘(𝐹‘𝑧)) ∈ ℝ^*)
26		ovolfcl 25423	. . . . . . . 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 11182	. . . . 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 11182	. . . . 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 13295	. . . 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 2771	. 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 13367	. 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 2844	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 2113 ∩ cin 3900 ⊆ wss 3901 ifcif 4479 ⟨cop 4586 ∪ cuni 4863 Disj wdisj 5065 class class class wbr 5098 × cxp 5622 ran crn 5625 ∘ ccom 5628 ⟶wf 6488 ‘cfv 6492 (class class class)co 7358 1^st c1st 7931 2^nd c2nd 7932 supcsup 9343 ℝcr 11025 1c1 11027 + caddc 11029 ℝ^*cxr 11165 < clt 11166 ≤ cle 11167 − cmin 11364 ℕcn 12145 ℝ⁺crp 12905 (,)cioo 13261 seqcseq 13924 abscabs 15157 vol*covol 25419

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 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-cnex 11082 ax-resscn 11083 ax-1cn 11084 ax-icn 11085 ax-addcl 11086 ax-addrcl 11087 ax-mulcl 11088 ax-mulrcl 11089 ax-mulcom 11090 ax-addass 11091 ax-mulass 11092 ax-distr 11093 ax-i2m1 11094 ax-1ne0 11095 ax-1rid 11096 ax-rnegex 11097 ax-rrecex 11098 ax-cnre 11099 ax-pre-lttri 11100 ax-pre-lttrn 11101 ax-pre-ltadd 11102 ax-pre-mulgt0 11103 ax-pre-sup 11104

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 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-op 4587 df-uni 4864 df-iun 4948 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7809 df-1st 7933 df-2nd 7934 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-er 8635 df-en 8884 df-dom 8885 df-sdom 8886 df-sup 9345 df-inf 9346 df-pnf 11168 df-mnf 11169 df-xr 11170 df-ltxr 11171 df-le 11172 df-sub 11366 df-neg 11367 df-div 11795 df-nn 12146 df-n0 12402 df-z 12489 df-uz 12752 df-q 12862 df-ioo 13265

This theorem is referenced by: uniioombllem2 25540

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