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Theorem uniioombllem1 23349

Description: Lemma for uniioombl 23357. (Contributed by Mario Carneiro, 25-Mar-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
uniioombllem1	⊢ (𝜑 → sup(ran 𝑇, ℝ^*, < ) ∈ ℝ)

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

**Proof of Theorem uniioombllem1**
Step	Hyp	Ref	Expression
1		uniioombl.g	. . . . 5 ⊢ (𝜑 → 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
2		eqid 2622	. . . . . 6 ⊢ ((abs ∘ − ) ∘ 𝐺) = ((abs ∘ − ) ∘ 𝐺)
3		uniioombl.t	. . . . . 6 ⊢ 𝑇 = seq1( + , ((abs ∘ − ) ∘ 𝐺))
4	2, 3	ovolsf 23241	. . . . 5 ⊢ (𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝑇:ℕ⟶(0[,)+∞))
5	1, 4	syl 17	. . . 4 ⊢ (𝜑 → 𝑇:ℕ⟶(0[,)+∞))
6		frn 6053	. . . 4 ⊢ (𝑇:ℕ⟶(0[,)+∞) → ran 𝑇 ⊆ (0[,)+∞))
7	5, 6	syl 17	. . 3 ⊢ (𝜑 → ran 𝑇 ⊆ (0[,)+∞))
8		rge0ssre 12280	. . 3 ⊢ (0[,)+∞) ⊆ ℝ
9	7, 8	syl6ss 3615	. 2 ⊢ (𝜑 → ran 𝑇 ⊆ ℝ)
10		1nn 11031	. . . . 5 ⊢ 1 ∈ ℕ
11		fdm 6051	. . . . . 6 ⊢ (𝑇:ℕ⟶(0[,)+∞) → dom 𝑇 = ℕ)
12	5, 11	syl 17	. . . . 5 ⊢ (𝜑 → dom 𝑇 = ℕ)
13	10, 12	syl5eleqr 2708	. . . 4 ⊢ (𝜑 → 1 ∈ dom 𝑇)
14		ne0i 3921	. . . 4 ⊢ (1 ∈ dom 𝑇 → dom 𝑇 ≠ ∅)
15	13, 14	syl 17	. . 3 ⊢ (𝜑 → dom 𝑇 ≠ ∅)
16		dm0rn0 5342	. . . 4 ⊢ (dom 𝑇 = ∅ ↔ ran 𝑇 = ∅)
17	16	necon3bii 2846	. . 3 ⊢ (dom 𝑇 ≠ ∅ ↔ ran 𝑇 ≠ ∅)
18	15, 17	sylib 208	. 2 ⊢ (𝜑 → ran 𝑇 ≠ ∅)
19		icossxr 12258	. . . . 5 ⊢ (0[,)+∞) ⊆ ℝ^*
20	7, 19	syl6ss 3615	. . . 4 ⊢ (𝜑 → ran 𝑇 ⊆ ℝ^*)
21		supxrcl 12145	. . . 4 ⊢ (ran 𝑇 ⊆ ℝ^* → sup(ran 𝑇, ℝ^, < ) ∈ ℝ^)
22	20, 21	syl 17	. . 3 ⊢ (𝜑 → sup(ran 𝑇, ℝ^, < ) ∈ ℝ^)
23		uniioombl.e	. . . . 5 ⊢ (𝜑 → (vol*‘𝐸) ∈ ℝ)
24		uniioombl.c	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ ℝ⁺)
25	24	rpred 11872	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℝ)
26	23, 25	readdcld 10069	. . . 4 ⊢ (𝜑 → ((vol*‘𝐸) + 𝐶) ∈ ℝ)
27	26	rexrd 10089	. . 3 ⊢ (𝜑 → ((vol‘𝐸) + 𝐶) ∈ ℝ^)
28		pnfxr 10092	. . . 4 ⊢ +∞ ∈ ℝ^*
29	28	a1i 11	. . 3 ⊢ (𝜑 → +∞ ∈ ℝ^*)
30		uniioombl.v	. . 3 ⊢ (𝜑 → sup(ran 𝑇, ℝ^, < ) ≤ ((vol‘𝐸) + 𝐶))
31		ltpnf 11954	. . . 4 ⊢ (((vol‘𝐸) + 𝐶) ∈ ℝ → ((vol‘𝐸) + 𝐶) < +∞)
32	26, 31	syl 17	. . 3 ⊢ (𝜑 → ((vol*‘𝐸) + 𝐶) < +∞)
33	22, 27, 29, 30, 32	xrlelttrd 11991	. 2 ⊢ (𝜑 → sup(ran 𝑇, ℝ^*, < ) < +∞)
34		supxrbnd 12158	. 2 ⊢ ((ran 𝑇 ⊆ ℝ ∧ ran 𝑇 ≠ ∅ ∧ sup(ran 𝑇, ℝ^, < ) < +∞) → sup(ran 𝑇, ℝ^, < ) ∈ ℝ)
35	9, 18, 33, 34	syl3anc 1326	1 ⊢ (𝜑 → sup(ran 𝑇, ℝ^*, < ) ∈ ℝ)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∩ cin 3573 ⊆ wss 3574 ∅c0 3915 ∪ cuni 4436 Disj wdisj 4620 class class class wbr 4653 × cxp 5112 dom cdm 5114 ran crn 5115 ∘ ccom 5118 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 supcsup 8346 ℝcr 9935 0cc0 9936 1c1 9937 + caddc 9939 +∞cpnf 10071 ℝ^*cxr 10073 < clt 10074 ≤ cle 10075 − cmin 10266 ℕcn 11020 ℝ⁺crp 11832 (,)cioo 12175 [,)cico 12177 seqcseq 12801 abscabs 13974 vol*covol 23231

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-sup 8348 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-z 11378 df-uz 11688 df-rp 11833 df-ico 12181 df-fz 12327 df-seq 12802 df-exp 12861 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976

This theorem is referenced by: uniioombllem3 23353 uniioombllem4 23354 uniioombllem5 23355 uniioombllem6 23356

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