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Theorem ovolsslem 23472
Description: Lemma for ovolss 23473. (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 3751 . . . . . . . . 9 (𝐴𝐵 → (𝐵 ran ((,) ∘ 𝑓) → 𝐴 ran ((,) ∘ 𝑓)))
21ad2antrr 764 . . . . . . . 8 (((𝐴𝐵𝐵 ⊆ ℝ) ∧ 𝑦 ∈ ℝ*) → (𝐵 ran ((,) ∘ 𝑓) → 𝐴 ran ((,) ∘ 𝑓)))
32anim1d 589 . . . . . . 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 3154 . . . . . 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 3824 . . . . 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 3787 . . . 4 ((𝐴𝐵𝐵 ⊆ ℝ) → 𝑁𝑀)
9 sstr 3752 . . . . 5 ((𝐴𝐵𝐵 ⊆ ℝ) → 𝐴 ⊆ ℝ)
107ovolval 23462 . . . . . . . 8 (𝐴 ⊆ ℝ → (vol*‘𝐴) = inf(𝑀, ℝ*, < ))
1110adantr 472 . . . . . . 7 ((𝐴 ⊆ ℝ ∧ 𝑥𝑀) → (vol*‘𝐴) = inf(𝑀, ℝ*, < ))
12 ssrab2 3828 . . . . . . . . . 10 {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ)(𝐴 ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))} ⊆ ℝ*
137, 12eqsstri 3776 . . . . . . . . 9 𝑀 ⊆ ℝ*
14 infxrlb 12377 . . . . . . . . 9 ((𝑀 ⊆ ℝ*𝑥𝑀) → inf(𝑀, ℝ*, < ) ≤ 𝑥)
1513, 14mpan 708 . . . . . . . 8 (𝑥𝑀 → inf(𝑀, ℝ*, < ) ≤ 𝑥)
1615adantl 473 . . . . . . 7 ((𝐴 ⊆ ℝ ∧ 𝑥𝑀) → inf(𝑀, ℝ*, < ) ≤ 𝑥)
1711, 16eqbrtrd 4826 . . . . . 6 ((𝐴 ⊆ ℝ ∧ 𝑥𝑀) → (vol*‘𝐴) ≤ 𝑥)
1817ralrimiva 3104 . . . . 5 (𝐴 ⊆ ℝ → ∀𝑥𝑀 (vol*‘𝐴) ≤ 𝑥)
199, 18syl 17 . . . 4 ((𝐴𝐵𝐵 ⊆ ℝ) → ∀𝑥𝑀 (vol*‘𝐴) ≤ 𝑥)
20 ssralv 3807 . . . 4 (𝑁𝑀 → (∀𝑥𝑀 (vol*‘𝐴) ≤ 𝑥 → ∀𝑥𝑁 (vol*‘𝐴) ≤ 𝑥))
218, 19, 20sylc 65 . . 3 ((𝐴𝐵𝐵 ⊆ ℝ) → ∀𝑥𝑁 (vol*‘𝐴) ≤ 𝑥)
22 ssrab2 3828 . . . . 5 {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ)(𝐵 ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))} ⊆ ℝ*
236, 22eqsstri 3776 . . . 4 𝑁 ⊆ ℝ*
24 ovolcl 23466 . . . . 5 (𝐴 ⊆ ℝ → (vol*‘𝐴) ∈ ℝ*)
259, 24syl 17 . . . 4 ((𝐴𝐵𝐵 ⊆ ℝ) → (vol*‘𝐴) ∈ ℝ*)
26 infxrgelb 12378 . . . 4 ((𝑁 ⊆ ℝ* ∧ (vol*‘𝐴) ∈ ℝ*) → ((vol*‘𝐴) ≤ inf(𝑁, ℝ*, < ) ↔ ∀𝑥𝑁 (vol*‘𝐴) ≤ 𝑥))
2723, 25, 26sylancr 698 . . 3 ((𝐴𝐵𝐵 ⊆ ℝ) → ((vol*‘𝐴) ≤ inf(𝑁, ℝ*, < ) ↔ ∀𝑥𝑁 (vol*‘𝐴) ≤ 𝑥))
2821, 27mpbird 247 . 2 ((𝐴𝐵𝐵 ⊆ ℝ) → (vol*‘𝐴) ≤ inf(𝑁, ℝ*, < ))
296ovolval 23462 . . 3 (𝐵 ⊆ ℝ → (vol*‘𝐵) = inf(𝑁, ℝ*, < ))
3029adantl 473 . 2 ((𝐴𝐵𝐵 ⊆ ℝ) → (vol*‘𝐵) = inf(𝑁, ℝ*, < ))
3128, 30breqtrrd 4832 1 ((𝐴𝐵𝐵 ⊆ ℝ) → (vol*‘𝐴) ≤ (vol*‘𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wcel 2139  wral 3050  wrex 3051  {crab 3054  cin 3714  wss 3715   cuni 4588   class class class wbr 4804   × cxp 5264  ran crn 5267  ccom 5270  cfv 6049  (class class class)co 6814  𝑚 cmap 8025  supcsup 8513  infcinf 8514  cr 10147  1c1 10149   + caddc 10151  *cxr 10285   < clt 10286  cle 10287  cmin 10478  cn 11232  (,)cioo 12388  seqcseq 13015  abscabs 14193  vol*covol 23451
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115  ax-cnex 10204  ax-resscn 10205  ax-1cn 10206  ax-icn 10207  ax-addcl 10208  ax-addrcl 10209  ax-mulcl 10210  ax-mulrcl 10211  ax-mulcom 10212  ax-addass 10213  ax-mulass 10214  ax-distr 10215  ax-i2m1 10216  ax-1ne0 10217  ax-1rid 10218  ax-rnegex 10219  ax-rrecex 10220  ax-cnre 10221  ax-pre-lttri 10222  ax-pre-lttrn 10223  ax-pre-ltadd 10224  ax-pre-mulgt0 10225  ax-pre-sup 10226
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-po 5187  df-so 5188  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6775  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-er 7913  df-en 8124  df-dom 8125  df-sdom 8126  df-sup 8515  df-inf 8516  df-pnf 10288  df-mnf 10289  df-xr 10290  df-ltxr 10291  df-le 10292  df-sub 10480  df-neg 10481  df-ovol 23453
This theorem is referenced by:  ovolss  23473
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