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Mirrors > Home > MPE Home > Th. List > ovolsslem | Structured version Visualization version GIF version |
Description: Lemma for ovolss 25500. (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 3986 | . . . . . . . . 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 609 | . . . . . . 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 3160 | . . . . . 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 4070 | . . . . 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 4025 | . . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℝ) → 𝑁 ⊆ 𝑀) |
9 | sstr 3988 | . . . . 5 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℝ) → 𝐴 ⊆ ℝ) | |
10 | 7 | ovolval 25488 | . . . . . . . 8 ⊢ (𝐴 ⊆ ℝ → (vol*‘𝐴) = inf(𝑀, ℝ*, < )) |
11 | 10 | adantr 479 | . . . . . . 7 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑥 ∈ 𝑀) → (vol*‘𝐴) = inf(𝑀, ℝ*, < )) |
12 | 7 | ssrab3 4077 | . . . . . . . . 9 ⊢ 𝑀 ⊆ ℝ* |
13 | infxrlb 13359 | . . . . . . . . 9 ⊢ ((𝑀 ⊆ ℝ* ∧ 𝑥 ∈ 𝑀) → inf(𝑀, ℝ*, < ) ≤ 𝑥) | |
14 | 12, 13 | mpan 688 | . . . . . . . 8 ⊢ (𝑥 ∈ 𝑀 → inf(𝑀, ℝ*, < ) ≤ 𝑥) |
15 | 14 | adantl 480 | . . . . . . 7 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑥 ∈ 𝑀) → inf(𝑀, ℝ*, < ) ≤ 𝑥) |
16 | 11, 15 | eqbrtrd 5166 | . . . . . 6 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑥 ∈ 𝑀) → (vol*‘𝐴) ≤ 𝑥) |
17 | 16 | ralrimiva 3136 | . . . . 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 4048 | . . . 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 4077 | . . . 4 ⊢ 𝑁 ⊆ ℝ* |
22 | ovolcl 25493 | . . . . 5 ⊢ (𝐴 ⊆ ℝ → (vol*‘𝐴) ∈ ℝ*) | |
23 | 9, 22 | syl 17 | . . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℝ) → (vol*‘𝐴) ∈ ℝ*) |
24 | infxrgelb 13360 | . . . 4 ⊢ ((𝑁 ⊆ ℝ* ∧ (vol*‘𝐴) ∈ ℝ*) → ((vol*‘𝐴) ≤ inf(𝑁, ℝ*, < ) ↔ ∀𝑥 ∈ 𝑁 (vol*‘𝐴) ≤ 𝑥)) | |
25 | 21, 23, 24 | sylancr 585 | . . 3 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℝ) → ((vol*‘𝐴) ≤ inf(𝑁, ℝ*, < ) ↔ ∀𝑥 ∈ 𝑁 (vol*‘𝐴) ≤ 𝑥)) |
26 | 20, 25 | mpbird 256 | . 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℝ) → (vol*‘𝐴) ≤ inf(𝑁, ℝ*, < )) |
27 | 6 | ovolval 25488 | . . 3 ⊢ (𝐵 ⊆ ℝ → (vol*‘𝐵) = inf(𝑁, ℝ*, < )) |
28 | 27 | adantl 480 | . 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℝ) → (vol*‘𝐵) = inf(𝑁, ℝ*, < )) |
29 | 26, 28 | breqtrrd 5172 | 1 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℝ) → (vol*‘𝐴) ≤ (vol*‘𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1534 ∈ wcel 2099 ∀wral 3051 ∃wrex 3060 {crab 3420 ∩ cin 3946 ⊆ wss 3947 ∪ cuni 4906 class class class wbr 5144 × cxp 5671 ran crn 5674 ∘ ccom 5677 ‘cfv 6544 (class class class)co 7414 ↑m cmap 8845 supcsup 9474 infcinf 9475 ℝcr 11146 1c1 11148 + caddc 11150 ℝ*cxr 11286 < clt 11287 ≤ cle 11288 − cmin 11483 ℕcn 12256 (,)cioo 13370 seqcseq 14013 abscabs 15232 vol*covol 25477 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7736 ax-cnex 11203 ax-resscn 11204 ax-1cn 11205 ax-icn 11206 ax-addcl 11207 ax-addrcl 11208 ax-mulcl 11209 ax-mulrcl 11210 ax-mulcom 11211 ax-addass 11212 ax-mulass 11213 ax-distr 11214 ax-i2m1 11215 ax-1ne0 11216 ax-1rid 11217 ax-rnegex 11218 ax-rrecex 11219 ax-cnre 11220 ax-pre-lttri 11221 ax-pre-lttrn 11222 ax-pre-ltadd 11223 ax-pre-mulgt0 11224 ax-pre-sup 11225 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3365 df-reu 3366 df-rab 3421 df-v 3465 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4907 df-br 5145 df-opab 5207 df-mpt 5228 df-id 5571 df-po 5585 df-so 5586 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-er 8724 df-en 8965 df-dom 8966 df-sdom 8967 df-sup 9476 df-inf 9477 df-pnf 11289 df-mnf 11290 df-xr 11291 df-ltxr 11292 df-le 11293 df-sub 11485 df-neg 11486 df-ovol 25479 |
This theorem is referenced by: ovolss 25500 |
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