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Mirrors > Home > MPE Home > Th. List > ovolsslem | Structured version Visualization version GIF version |
Description: Lemma for ovolss 24088. (Contributed by Mario Carneiro, 16-Mar-2014.) (Proof shortened by AV, 17-Sep-2020.) |
Ref | Expression |
---|---|
ovolss.1 | ⊢ 𝑀 = {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))} |
ovolss.2 | ⊢ 𝑁 = {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐵 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))} |
Ref | Expression |
---|---|
ovolsslem | ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℝ) → (vol*‘𝐴) ≤ (vol*‘𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sstr2 3976 | . . . . . . . . 9 ⊢ (𝐴 ⊆ 𝐵 → (𝐵 ⊆ ∪ ran ((,) ∘ 𝑓) → 𝐴 ⊆ ∪ ran ((,) ∘ 𝑓))) | |
2 | 1 | ad2antrr 724 | . . . . . . . 8 ⊢ (((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℝ) ∧ 𝑦 ∈ ℝ*) → (𝐵 ⊆ ∪ ran ((,) ∘ 𝑓) → 𝐴 ⊆ ∪ ran ((,) ∘ 𝑓))) |
3 | 2 | anim1d 612 | . . . . . . 7 ⊢ (((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℝ) ∧ 𝑦 ∈ ℝ*) → ((𝐵 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < )) → (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < )))) |
4 | 3 | reximdv 3275 | . . . . . 6 ⊢ (((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℝ) ∧ 𝑦 ∈ ℝ*) → (∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐵 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < )) → ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < )))) |
5 | 4 | ss2rabdv 4054 | . . . . 5 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℝ) → {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐵 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))} ⊆ {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))}) |
6 | ovolss.2 | . . . . 5 ⊢ 𝑁 = {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐵 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))} | |
7 | ovolss.1 | . . . . 5 ⊢ 𝑀 = {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))} | |
8 | 5, 6, 7 | 3sstr4g 4014 | . . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℝ) → 𝑁 ⊆ 𝑀) |
9 | sstr 3977 | . . . . 5 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℝ) → 𝐴 ⊆ ℝ) | |
10 | 7 | ovolval 24076 | . . . . . . . 8 ⊢ (𝐴 ⊆ ℝ → (vol*‘𝐴) = inf(𝑀, ℝ*, < )) |
11 | 10 | adantr 483 | . . . . . . 7 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑥 ∈ 𝑀) → (vol*‘𝐴) = inf(𝑀, ℝ*, < )) |
12 | 7 | ssrab3 4059 | . . . . . . . . 9 ⊢ 𝑀 ⊆ ℝ* |
13 | infxrlb 12730 | . . . . . . . . 9 ⊢ ((𝑀 ⊆ ℝ* ∧ 𝑥 ∈ 𝑀) → inf(𝑀, ℝ*, < ) ≤ 𝑥) | |
14 | 12, 13 | mpan 688 | . . . . . . . 8 ⊢ (𝑥 ∈ 𝑀 → inf(𝑀, ℝ*, < ) ≤ 𝑥) |
15 | 14 | adantl 484 | . . . . . . 7 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑥 ∈ 𝑀) → inf(𝑀, ℝ*, < ) ≤ 𝑥) |
16 | 11, 15 | eqbrtrd 5090 | . . . . . 6 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑥 ∈ 𝑀) → (vol*‘𝐴) ≤ 𝑥) |
17 | 16 | ralrimiva 3184 | . . . . 5 ⊢ (𝐴 ⊆ ℝ → ∀𝑥 ∈ 𝑀 (vol*‘𝐴) ≤ 𝑥) |
18 | 9, 17 | syl 17 | . . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℝ) → ∀𝑥 ∈ 𝑀 (vol*‘𝐴) ≤ 𝑥) |
19 | ssralv 4035 | . . . 4 ⊢ (𝑁 ⊆ 𝑀 → (∀𝑥 ∈ 𝑀 (vol*‘𝐴) ≤ 𝑥 → ∀𝑥 ∈ 𝑁 (vol*‘𝐴) ≤ 𝑥)) | |
20 | 8, 18, 19 | sylc 65 | . . 3 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℝ) → ∀𝑥 ∈ 𝑁 (vol*‘𝐴) ≤ 𝑥) |
21 | 6 | ssrab3 4059 | . . . 4 ⊢ 𝑁 ⊆ ℝ* |
22 | ovolcl 24081 | . . . . 5 ⊢ (𝐴 ⊆ ℝ → (vol*‘𝐴) ∈ ℝ*) | |
23 | 9, 22 | syl 17 | . . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℝ) → (vol*‘𝐴) ∈ ℝ*) |
24 | infxrgelb 12731 | . . . 4 ⊢ ((𝑁 ⊆ ℝ* ∧ (vol*‘𝐴) ∈ ℝ*) → ((vol*‘𝐴) ≤ inf(𝑁, ℝ*, < ) ↔ ∀𝑥 ∈ 𝑁 (vol*‘𝐴) ≤ 𝑥)) | |
25 | 21, 23, 24 | sylancr 589 | . . 3 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℝ) → ((vol*‘𝐴) ≤ inf(𝑁, ℝ*, < ) ↔ ∀𝑥 ∈ 𝑁 (vol*‘𝐴) ≤ 𝑥)) |
26 | 20, 25 | mpbird 259 | . 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℝ) → (vol*‘𝐴) ≤ inf(𝑁, ℝ*, < )) |
27 | 6 | ovolval 24076 | . . 3 ⊢ (𝐵 ⊆ ℝ → (vol*‘𝐵) = inf(𝑁, ℝ*, < )) |
28 | 27 | adantl 484 | . 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℝ) → (vol*‘𝐵) = inf(𝑁, ℝ*, < )) |
29 | 26, 28 | breqtrrd 5096 | 1 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℝ) → (vol*‘𝐴) ≤ (vol*‘𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3140 ∃wrex 3141 {crab 3144 ∩ cin 3937 ⊆ wss 3938 ∪ cuni 4840 class class class wbr 5068 × cxp 5555 ran crn 5558 ∘ ccom 5561 ‘cfv 6357 (class class class)co 7158 ↑m cmap 8408 supcsup 8906 infcinf 8907 ℝcr 10538 1c1 10540 + caddc 10542 ℝ*cxr 10676 < clt 10677 ≤ cle 10678 − cmin 10872 ℕcn 11640 (,)cioo 12741 seqcseq 13372 abscabs 14595 vol*covol 24065 |
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 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-po 5476 df-so 5477 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-sup 8908 df-inf 8909 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-ovol 24067 |
This theorem is referenced by: ovolss 24088 |
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