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| Description: Lemma for uniioombl 25625. (Contributed by Mario Carneiro, 25-Mar-2015.) | 
| 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)) | 
| Ref | Expression | 
|---|---|
| uniioombllem1 | ⊢ (𝜑 → sup(ran 𝑇, ℝ*, < ) ∈ ℝ) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | uniioombl.g | . . . . 5 ⊢ (𝜑 → 𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ))) | |
| 2 | eqid 2736 | . . . . . 6 ⊢ ((abs ∘ − ) ∘ 𝐺) = ((abs ∘ − ) ∘ 𝐺) | |
| 3 | uniioombl.t | . . . . . 6 ⊢ 𝑇 = seq1( + , ((abs ∘ − ) ∘ 𝐺)) | |
| 4 | 2, 3 | ovolsf 25508 | . . . . 5 ⊢ (𝐺:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝑇:ℕ⟶(0[,)+∞)) | 
| 5 | 1, 4 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑇:ℕ⟶(0[,)+∞)) | 
| 6 | 5 | frnd 6743 | . . 3 ⊢ (𝜑 → ran 𝑇 ⊆ (0[,)+∞)) | 
| 7 | rge0ssre 13497 | . . 3 ⊢ (0[,)+∞) ⊆ ℝ | |
| 8 | 6, 7 | sstrdi 3995 | . 2 ⊢ (𝜑 → ran 𝑇 ⊆ ℝ) | 
| 9 | 1nn 12278 | . . . . 5 ⊢ 1 ∈ ℕ | |
| 10 | 5 | fdmd 6745 | . . . . 5 ⊢ (𝜑 → dom 𝑇 = ℕ) | 
| 11 | 9, 10 | eleqtrrid 2847 | . . . 4 ⊢ (𝜑 → 1 ∈ dom 𝑇) | 
| 12 | 11 | ne0d 4341 | . . 3 ⊢ (𝜑 → dom 𝑇 ≠ ∅) | 
| 13 | dm0rn0 5934 | . . . 4 ⊢ (dom 𝑇 = ∅ ↔ ran 𝑇 = ∅) | |
| 14 | 13 | necon3bii 2992 | . . 3 ⊢ (dom 𝑇 ≠ ∅ ↔ ran 𝑇 ≠ ∅) | 
| 15 | 12, 14 | sylib 218 | . 2 ⊢ (𝜑 → ran 𝑇 ≠ ∅) | 
| 16 | icossxr 13473 | . . . . 5 ⊢ (0[,)+∞) ⊆ ℝ* | |
| 17 | 6, 16 | sstrdi 3995 | . . . 4 ⊢ (𝜑 → ran 𝑇 ⊆ ℝ*) | 
| 18 | supxrcl 13358 | . . . 4 ⊢ (ran 𝑇 ⊆ ℝ* → sup(ran 𝑇, ℝ*, < ) ∈ ℝ*) | |
| 19 | 17, 18 | syl 17 | . . 3 ⊢ (𝜑 → sup(ran 𝑇, ℝ*, < ) ∈ ℝ*) | 
| 20 | uniioombl.e | . . . . 5 ⊢ (𝜑 → (vol*‘𝐸) ∈ ℝ) | |
| 21 | uniioombl.c | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ ℝ+) | |
| 22 | 21 | rpred 13078 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℝ) | 
| 23 | 20, 22 | readdcld 11291 | . . . 4 ⊢ (𝜑 → ((vol*‘𝐸) + 𝐶) ∈ ℝ) | 
| 24 | 23 | rexrd 11312 | . . 3 ⊢ (𝜑 → ((vol*‘𝐸) + 𝐶) ∈ ℝ*) | 
| 25 | pnfxr 11316 | . . . 4 ⊢ +∞ ∈ ℝ* | |
| 26 | 25 | a1i 11 | . . 3 ⊢ (𝜑 → +∞ ∈ ℝ*) | 
| 27 | uniioombl.v | . . 3 ⊢ (𝜑 → sup(ran 𝑇, ℝ*, < ) ≤ ((vol*‘𝐸) + 𝐶)) | |
| 28 | 23 | ltpnfd 13164 | . . 3 ⊢ (𝜑 → ((vol*‘𝐸) + 𝐶) < +∞) | 
| 29 | 19, 24, 26, 27, 28 | xrlelttrd 13203 | . 2 ⊢ (𝜑 → sup(ran 𝑇, ℝ*, < ) < +∞) | 
| 30 | supxrbnd 13371 | . 2 ⊢ ((ran 𝑇 ⊆ ℝ ∧ ran 𝑇 ≠ ∅ ∧ sup(ran 𝑇, ℝ*, < ) < +∞) → sup(ran 𝑇, ℝ*, < ) ∈ ℝ) | |
| 31 | 8, 15, 29, 30 | syl3anc 1372 | 1 ⊢ (𝜑 → sup(ran 𝑇, ℝ*, < ) ∈ ℝ) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2107 ≠ wne 2939 ∩ cin 3949 ⊆ wss 3950 ∅c0 4332 ∪ cuni 4906 Disj wdisj 5109 class class class wbr 5142 × cxp 5682 dom cdm 5684 ran crn 5685 ∘ ccom 5688 ⟶wf 6556 ‘cfv 6560 (class class class)co 7432 supcsup 9481 ℝcr 11155 0cc0 11156 1c1 11157 + caddc 11159 +∞cpnf 11293 ℝ*cxr 11295 < clt 11296 ≤ cle 11297 − cmin 11493 ℕcn 12267 ℝ+crp 13035 (,)cioo 13388 [,)cico 13390 seqcseq 14043 abscabs 15274 vol*covol 25498 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 ax-pre-sup 11234 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-1st 8015 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-er 8746 df-en 8987 df-dom 8988 df-sdom 8989 df-sup 9483 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-div 11922 df-nn 12268 df-2 12330 df-3 12331 df-n0 12529 df-z 12616 df-uz 12880 df-rp 13036 df-ico 13394 df-fz 13549 df-seq 14044 df-exp 14104 df-cj 15139 df-re 15140 df-im 15141 df-sqrt 15275 df-abs 15276 | 
| This theorem is referenced by: uniioombllem3 25621 uniioombllem4 25622 uniioombllem5 25623 uniioombllem6 25624 | 
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