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Theorem ovolsslem 23059
 Description: Lemma for ovolss 23060. (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 3575 . . . . . . . . 9 (𝐴𝐵 → (𝐵 ran ((,) ∘ 𝑓) → 𝐴 ran ((,) ∘ 𝑓)))
21ad2antrr 758 . . . . . . . 8 (((𝐴𝐵𝐵 ⊆ ℝ) ∧ 𝑦 ∈ ℝ*) → (𝐵 ran ((,) ∘ 𝑓) → 𝐴 ran ((,) ∘ 𝑓)))
32anim1d 586 . . . . . . 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 2999 . . . . . 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 3646 . . . . 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 3609 . . . 4 ((𝐴𝐵𝐵 ⊆ ℝ) → 𝑁𝑀)
9 sstr 3576 . . . . 5 ((𝐴𝐵𝐵 ⊆ ℝ) → 𝐴 ⊆ ℝ)
107ovolval 23049 . . . . . . . 8 (𝐴 ⊆ ℝ → (vol*‘𝐴) = inf(𝑀, ℝ*, < ))
1110adantr 480 . . . . . . 7 ((𝐴 ⊆ ℝ ∧ 𝑥𝑀) → (vol*‘𝐴) = inf(𝑀, ℝ*, < ))
12 ssrab2 3650 . . . . . . . . . 10 {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ)(𝐴 ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))} ⊆ ℝ*
137, 12eqsstri 3598 . . . . . . . . 9 𝑀 ⊆ ℝ*
14 infxrlb 12036 . . . . . . . . 9 ((𝑀 ⊆ ℝ*𝑥𝑀) → inf(𝑀, ℝ*, < ) ≤ 𝑥)
1513, 14mpan 702 . . . . . . . 8 (𝑥𝑀 → inf(𝑀, ℝ*, < ) ≤ 𝑥)
1615adantl 481 . . . . . . 7 ((𝐴 ⊆ ℝ ∧ 𝑥𝑀) → inf(𝑀, ℝ*, < ) ≤ 𝑥)
1711, 16eqbrtrd 4605 . . . . . 6 ((𝐴 ⊆ ℝ ∧ 𝑥𝑀) → (vol*‘𝐴) ≤ 𝑥)
1817ralrimiva 2949 . . . . 5 (𝐴 ⊆ ℝ → ∀𝑥𝑀 (vol*‘𝐴) ≤ 𝑥)
199, 18syl 17 . . . 4 ((𝐴𝐵𝐵 ⊆ ℝ) → ∀𝑥𝑀 (vol*‘𝐴) ≤ 𝑥)
20 ssralv 3629 . . . 4 (𝑁𝑀 → (∀𝑥𝑀 (vol*‘𝐴) ≤ 𝑥 → ∀𝑥𝑁 (vol*‘𝐴) ≤ 𝑥))
218, 19, 20sylc 63 . . 3 ((𝐴𝐵𝐵 ⊆ ℝ) → ∀𝑥𝑁 (vol*‘𝐴) ≤ 𝑥)
22 ssrab2 3650 . . . . 5 {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ)(𝐵 ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))} ⊆ ℝ*
236, 22eqsstri 3598 . . . 4 𝑁 ⊆ ℝ*
24 ovolcl 23053 . . . . 5 (𝐴 ⊆ ℝ → (vol*‘𝐴) ∈ ℝ*)
259, 24syl 17 . . . 4 ((𝐴𝐵𝐵 ⊆ ℝ) → (vol*‘𝐴) ∈ ℝ*)
26 infxrgelb 12037 . . . 4 ((𝑁 ⊆ ℝ* ∧ (vol*‘𝐴) ∈ ℝ*) → ((vol*‘𝐴) ≤ inf(𝑁, ℝ*, < ) ↔ ∀𝑥𝑁 (vol*‘𝐴) ≤ 𝑥))
2723, 25, 26sylancr 694 . . 3 ((𝐴𝐵𝐵 ⊆ ℝ) → ((vol*‘𝐴) ≤ inf(𝑁, ℝ*, < ) ↔ ∀𝑥𝑁 (vol*‘𝐴) ≤ 𝑥))
2821, 27mpbird 246 . 2 ((𝐴𝐵𝐵 ⊆ ℝ) → (vol*‘𝐴) ≤ inf(𝑁, ℝ*, < ))
296ovolval 23049 . . 3 (𝐵 ⊆ ℝ → (vol*‘𝐵) = inf(𝑁, ℝ*, < ))
3029adantl 481 . 2 ((𝐴𝐵𝐵 ⊆ ℝ) → (vol*‘𝐵) = inf(𝑁, ℝ*, < ))
3128, 30breqtrrd 4611 1 ((𝐴𝐵𝐵 ⊆ ℝ) → (vol*‘𝐴) ≤ (vol*‘𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  {crab 2900   ∩ cin 3539   ⊆ wss 3540  ∪ cuni 4372   class class class wbr 4583   × cxp 5036  ran crn 5039   ∘ ccom 5042  ‘cfv 5804  (class class class)co 6549   ↑𝑚 cmap 7744  supcsup 8229  infcinf 8230  ℝcr 9814  1c1 9816   + caddc 9818  ℝ*cxr 9952   < clt 9953   ≤ cle 9954   − cmin 10145  ℕcn 10897  (,)cioo 12046  seqcseq 12663  abscabs 13822  vol*covol 23038 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-po 4959  df-so 4960  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-sup 8231  df-inf 8232  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-ovol 23040 This theorem is referenced by:  ovolss  23060
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