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Mirrors > Home > MPE Home > Th. List > ovolsslem | Structured version Visualization version GIF version |
Description: Lemma for ovolss 25534. (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 4002 | . . . . . . . . 9 ⊢ (𝐴 ⊆ 𝐵 → (𝐵 ⊆ ∪ ran ((,) ∘ 𝑓) → 𝐴 ⊆ ∪ ran ((,) ∘ 𝑓))) | |
2 | 1 | ad2antrr 726 | . . . . . . . 8 ⊢ (((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℝ) ∧ 𝑦 ∈ ℝ*) → (𝐵 ⊆ ∪ ran ((,) ∘ 𝑓) → 𝐴 ⊆ ∪ ran ((,) ∘ 𝑓))) |
3 | 2 | anim1d 611 | . . . . . . 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 3168 | . . . . . 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 4086 | . . . . 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 4041 | . . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℝ) → 𝑁 ⊆ 𝑀) |
9 | sstr 4004 | . . . . 5 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℝ) → 𝐴 ⊆ ℝ) | |
10 | 7 | ovolval 25522 | . . . . . . . 8 ⊢ (𝐴 ⊆ ℝ → (vol*‘𝐴) = inf(𝑀, ℝ*, < )) |
11 | 10 | adantr 480 | . . . . . . 7 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑥 ∈ 𝑀) → (vol*‘𝐴) = inf(𝑀, ℝ*, < )) |
12 | 7 | ssrab3 4092 | . . . . . . . . 9 ⊢ 𝑀 ⊆ ℝ* |
13 | infxrlb 13373 | . . . . . . . . 9 ⊢ ((𝑀 ⊆ ℝ* ∧ 𝑥 ∈ 𝑀) → inf(𝑀, ℝ*, < ) ≤ 𝑥) | |
14 | 12, 13 | mpan 690 | . . . . . . . 8 ⊢ (𝑥 ∈ 𝑀 → inf(𝑀, ℝ*, < ) ≤ 𝑥) |
15 | 14 | adantl 481 | . . . . . . 7 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑥 ∈ 𝑀) → inf(𝑀, ℝ*, < ) ≤ 𝑥) |
16 | 11, 15 | eqbrtrd 5170 | . . . . . 6 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑥 ∈ 𝑀) → (vol*‘𝐴) ≤ 𝑥) |
17 | 16 | ralrimiva 3144 | . . . . 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 4064 | . . . 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 4092 | . . . 4 ⊢ 𝑁 ⊆ ℝ* |
22 | ovolcl 25527 | . . . . 5 ⊢ (𝐴 ⊆ ℝ → (vol*‘𝐴) ∈ ℝ*) | |
23 | 9, 22 | syl 17 | . . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℝ) → (vol*‘𝐴) ∈ ℝ*) |
24 | infxrgelb 13374 | . . . 4 ⊢ ((𝑁 ⊆ ℝ* ∧ (vol*‘𝐴) ∈ ℝ*) → ((vol*‘𝐴) ≤ inf(𝑁, ℝ*, < ) ↔ ∀𝑥 ∈ 𝑁 (vol*‘𝐴) ≤ 𝑥)) | |
25 | 21, 23, 24 | sylancr 587 | . . 3 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℝ) → ((vol*‘𝐴) ≤ inf(𝑁, ℝ*, < ) ↔ ∀𝑥 ∈ 𝑁 (vol*‘𝐴) ≤ 𝑥)) |
26 | 20, 25 | mpbird 257 | . 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℝ) → (vol*‘𝐴) ≤ inf(𝑁, ℝ*, < )) |
27 | 6 | ovolval 25522 | . . 3 ⊢ (𝐵 ⊆ ℝ → (vol*‘𝐵) = inf(𝑁, ℝ*, < )) |
28 | 27 | adantl 481 | . 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℝ) → (vol*‘𝐵) = inf(𝑁, ℝ*, < )) |
29 | 26, 28 | breqtrrd 5176 | 1 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℝ) → (vol*‘𝐴) ≤ (vol*‘𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∀wral 3059 ∃wrex 3068 {crab 3433 ∩ cin 3962 ⊆ wss 3963 ∪ cuni 4912 class class class wbr 5148 × cxp 5687 ran crn 5690 ∘ ccom 5693 ‘cfv 6563 (class class class)co 7431 ↑m cmap 8865 supcsup 9478 infcinf 9479 ℝcr 11152 1c1 11154 + caddc 11156 ℝ*cxr 11292 < clt 11293 ≤ cle 11294 − cmin 11490 ℕcn 12264 (,)cioo 13384 seqcseq 14039 abscabs 15270 vol*covol 25511 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-po 5597 df-so 5598 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-sup 9480 df-inf 9481 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-ovol 25513 |
This theorem is referenced by: ovolss 25534 |
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