Theorem ovolsslem 25385

Description: Lemma for ovolss 25386. (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 3953	. . . . . . . . 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 3148	. . . . . 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 4039	. . . . 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 4000	. . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℝ) → 𝑁 ⊆ 𝑀)
9		sstr 3955	. . . . 5 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℝ) → 𝐴 ⊆ ℝ)
10	7	ovolval 25374	. . . . . . . 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 4045	. . . . . . . . 9 ⊢ 𝑀 ⊆ ℝ*
13		infxrlb 13295	. . . . . . . . 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 5129	. . . . . 6 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑥 ∈ 𝑀) → (vol*‘𝐴) ≤ 𝑥)
17	16	ralrimiva 3125	. . . . 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 4015	. . . 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 4045	. . . 4 ⊢ 𝑁 ⊆ ℝ*
22		ovolcl 25379	. . . . 5 ⊢ (𝐴 ⊆ ℝ → (vol*‘𝐴) ∈ ℝ*)
23	9, 22	syl 17	. . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℝ) → (vol*‘𝐴) ∈ ℝ*)
24		infxrgelb 13296	. . . 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 25374	. . 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 5135	1 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℝ) → (vol*‘𝐴) ≤ (vol*‘𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053  {crab 3405   ∩ cin 3913   ⊆ wss 3914  ∪ cuni 4871   class class class wbr 5107   × cxp 5636  ran crn 5639   ∘ ccom 5642  ‘cfv 6511  (class class class)co 7387   ↑_m cmap 8799  supcsup 9391  infcinf 9392  ℝcr 11067  1c1 11069   + caddc 11071  ℝ^*cxr 11207   < clt 11208   ≤ cle 11209   − cmin 11405  ℕcn 12186  (,)cioo 13306  seqcseq 13966  abscabs 15200  vol*covol 25363

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-cnex 11124  ax-resscn 11125  ax-1cn 11126  ax-icn 11127  ax-addcl 11128  ax-addrcl 11129  ax-mulcl 11130  ax-mulrcl 11131  ax-mulcom 11132  ax-addass 11133  ax-mulass 11134  ax-distr 11135  ax-i2m1 11136  ax-1ne0 11137  ax-1rid 11138  ax-rnegex 11139  ax-rrecex 11140  ax-cnre 11141  ax-pre-lttri 11142  ax-pre-lttrn 11143  ax-pre-ltadd 11144  ax-pre-mulgt0 11145  ax-pre-sup 11146

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-po 5546  df-so 5547  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-er 8671  df-en 8919  df-dom 8920  df-sdom 8921  df-sup 9393  df-inf 9394  df-pnf 11210  df-mnf 11211  df-xr 11212  df-ltxr 11213  df-le 11214  df-sub 11407  df-neg 11408  df-ovol 25365

This theorem is referenced by:  ovolss  25386