MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ovolsslem Structured version   Visualization version   GIF version

Theorem ovolsslem 25368
Description: Lemma for ovolss 25369. (Contributed by Mario Carneiro, 16-Mar-2014.) (Proof shortened by AV, 17-Sep-2020.)
Hypotheses
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*, < ))}
Assertion
Ref Expression
ovolsslem ((𝐴𝐵𝐵 ⊆ ℝ) → (vol*‘𝐴) ≤ (vol*‘𝐵))
Distinct variable groups:   𝑦,𝑓,𝐴   𝐵,𝑓,𝑦
Allowed substitution hints:   𝑀(𝑦,𝑓)   𝑁(𝑦,𝑓)

Proof of Theorem ovolsslem
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 sstr2 3984 . . . . . . . . 9 (𝐴𝐵 → (𝐵 ran ((,) ∘ 𝑓) → 𝐴 ran ((,) ∘ 𝑓)))
21ad2antrr 723 . . . . . . . 8 (((𝐴𝐵𝐵 ⊆ ℝ) ∧ 𝑦 ∈ ℝ*) → (𝐵 ran ((,) ∘ 𝑓) → 𝐴 ran ((,) ∘ 𝑓)))
32anim1d 610 . . . . . . 7 (((𝐴𝐵𝐵 ⊆ ℝ) ∧ 𝑦 ∈ ℝ*) → ((𝐵 ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < )) → (𝐴 ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))))
43reximdv 3164 . . . . . 6 (((𝐴𝐵𝐵 ⊆ 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*, < ))))
54ss2rabdv 4068 . . . . 5 ((𝐴𝐵𝐵 ⊆ 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*, < ))}
85, 6, 73sstr4g 4022 . . . 4 ((𝐴𝐵𝐵 ⊆ ℝ) → 𝑁𝑀)
9 sstr 3985 . . . . 5 ((𝐴𝐵𝐵 ⊆ ℝ) → 𝐴 ⊆ ℝ)
107ovolval 25357 . . . . . . . 8 (𝐴 ⊆ ℝ → (vol*‘𝐴) = inf(𝑀, ℝ*, < ))
1110adantr 480 . . . . . . 7 ((𝐴 ⊆ ℝ ∧ 𝑥𝑀) → (vol*‘𝐴) = inf(𝑀, ℝ*, < ))
127ssrab3 4075 . . . . . . . . 9 𝑀 ⊆ ℝ*
13 infxrlb 13319 . . . . . . . . 9 ((𝑀 ⊆ ℝ*𝑥𝑀) → inf(𝑀, ℝ*, < ) ≤ 𝑥)
1412, 13mpan 687 . . . . . . . 8 (𝑥𝑀 → inf(𝑀, ℝ*, < ) ≤ 𝑥)
1514adantl 481 . . . . . . 7 ((𝐴 ⊆ ℝ ∧ 𝑥𝑀) → inf(𝑀, ℝ*, < ) ≤ 𝑥)
1611, 15eqbrtrd 5163 . . . . . 6 ((𝐴 ⊆ ℝ ∧ 𝑥𝑀) → (vol*‘𝐴) ≤ 𝑥)
1716ralrimiva 3140 . . . . 5 (𝐴 ⊆ ℝ → ∀𝑥𝑀 (vol*‘𝐴) ≤ 𝑥)
189, 17syl 17 . . . 4 ((𝐴𝐵𝐵 ⊆ ℝ) → ∀𝑥𝑀 (vol*‘𝐴) ≤ 𝑥)
19 ssralv 4045 . . . 4 (𝑁𝑀 → (∀𝑥𝑀 (vol*‘𝐴) ≤ 𝑥 → ∀𝑥𝑁 (vol*‘𝐴) ≤ 𝑥))
208, 18, 19sylc 65 . . 3 ((𝐴𝐵𝐵 ⊆ ℝ) → ∀𝑥𝑁 (vol*‘𝐴) ≤ 𝑥)
216ssrab3 4075 . . . 4 𝑁 ⊆ ℝ*
22 ovolcl 25362 . . . . 5 (𝐴 ⊆ ℝ → (vol*‘𝐴) ∈ ℝ*)
239, 22syl 17 . . . 4 ((𝐴𝐵𝐵 ⊆ ℝ) → (vol*‘𝐴) ∈ ℝ*)
24 infxrgelb 13320 . . . 4 ((𝑁 ⊆ ℝ* ∧ (vol*‘𝐴) ∈ ℝ*) → ((vol*‘𝐴) ≤ inf(𝑁, ℝ*, < ) ↔ ∀𝑥𝑁 (vol*‘𝐴) ≤ 𝑥))
2521, 23, 24sylancr 586 . . 3 ((𝐴𝐵𝐵 ⊆ ℝ) → ((vol*‘𝐴) ≤ inf(𝑁, ℝ*, < ) ↔ ∀𝑥𝑁 (vol*‘𝐴) ≤ 𝑥))
2620, 25mpbird 257 . 2 ((𝐴𝐵𝐵 ⊆ ℝ) → (vol*‘𝐴) ≤ inf(𝑁, ℝ*, < ))
276ovolval 25357 . . 3 (𝐵 ⊆ ℝ → (vol*‘𝐵) = inf(𝑁, ℝ*, < ))
2827adantl 481 . 2 ((𝐴𝐵𝐵 ⊆ ℝ) → (vol*‘𝐵) = inf(𝑁, ℝ*, < ))
2926, 28breqtrrd 5169 1 ((𝐴𝐵𝐵 ⊆ ℝ) → (vol*‘𝐴) ≤ (vol*‘𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1533  wcel 2098  wral 3055  wrex 3064  {crab 3426  cin 3942  wss 3943   cuni 4902   class class class wbr 5141   × cxp 5667  ran crn 5670  ccom 5673  cfv 6537  (class class class)co 7405  m cmap 8822  supcsup 9437  infcinf 9438  cr 11111  1c1 11113   + caddc 11115  *cxr 11251   < clt 11252  cle 11253  cmin 11448  cn 12216  (,)cioo 13330  seqcseq 13972  abscabs 15187  vol*covol 25346
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189  ax-pre-sup 11190
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-po 5581  df-so 5582  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-sup 9439  df-inf 9440  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-ovol 25348
This theorem is referenced by:  ovolss  25369
  Copyright terms: Public domain W3C validator