Theorem ovolsslem 24992

Description: Lemma for ovolss 24993. (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	⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 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.

Step	Hyp	Ref	Expression
1		sstr2 3988	. . . . . . . . 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 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 3170	. . . . . 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 4072	. . . . 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 4026	. . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℝ) → 𝑁 ⊆ 𝑀)
9		sstr 3989	. . . . 5 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℝ) → 𝐴 ⊆ ℝ)
10	7	ovolval 24981	. . . . . . . 8 ⊢ (𝐴 ⊆ ℝ → (vol*‘𝐴) = inf(𝑀, ℝ*, < ))
11	10	adantr 481	. . . . . . 7 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑥 ∈ 𝑀) → (vol*‘𝐴) = inf(𝑀, ℝ*, < ))
12	7	ssrab3 4079	. . . . . . . . 9 ⊢ 𝑀 ⊆ ℝ*
13		infxrlb 13309	. . . . . . . . 9 ⊢ ((𝑀 ⊆ ℝ* ∧ 𝑥 ∈ 𝑀) → inf(𝑀, ℝ*, < ) ≤ 𝑥)
14	12, 13	mpan 688	. . . . . . . 8 ⊢ (𝑥 ∈ 𝑀 → inf(𝑀, ℝ*, < ) ≤ 𝑥)
15	14	adantl 482	. . . . . . 7 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑥 ∈ 𝑀) → inf(𝑀, ℝ*, < ) ≤ 𝑥)
16	11, 15	eqbrtrd 5169	. . . . . 6 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑥 ∈ 𝑀) → (vol*‘𝐴) ≤ 𝑥)
17	16	ralrimiva 3146	. . . . 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 4049	. . . 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 4079	. . . 4 ⊢ 𝑁 ⊆ ℝ*
22		ovolcl 24986	. . . . 5 ⊢ (𝐴 ⊆ ℝ → (vol*‘𝐴) ∈ ℝ*)
23	9, 22	syl 17	. . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℝ) → (vol*‘𝐴) ∈ ℝ*)
24		infxrgelb 13310	. . . 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 256	. 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℝ) → (vol*‘𝐴) ≤ inf(𝑁, ℝ*, < ))
27	6	ovolval 24981	. . 3 ⊢ (𝐵 ⊆ ℝ → (vol*‘𝐵) = inf(𝑁, ℝ*, < ))
28	27	adantl 482	. 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℝ) → (vol*‘𝐵) = inf(𝑁, ℝ*, < ))
29	26, 28	breqtrrd 5175	1 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℝ) → (vol*‘𝐴) ≤ (vol*‘𝐵))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ∀wral 3061  ∃wrex 3070  {crab 3432   ∩ cin 3946   ⊆ wss 3947  ∪ cuni 4907   class class class wbr 5147   × cxp 5673  ran crn 5676   ∘ ccom 5679  ‘cfv 6540  (class class class)co 7405   ↑_m cmap 8816  supcsup 9431  infcinf 9432  ℝcr 11105  1c1 11107   + caddc 11109  ℝ^*cxr 11243   < clt 11244   ≤ cle 11245   − cmin 11440  ℕcn 12208  (,)cioo 13320  seqcseq 13962  abscabs 15177  vol*covol 24970

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183  ax-pre-sup 11184

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-po 5587  df-so 5588  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-er 8699  df-en 8936  df-dom 8937  df-sdom 8938  df-sup 9433  df-inf 9434  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-ovol 24972

This theorem is referenced by:  ovolss  24993