ovolunlem2 - Metamath Proof Explorer

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Theorem ovolunlem2 25780

Description: Lemma for ovolun 25781. (Contributed by Mario Carneiro, 12-Jun-2014.)

**Hypotheses**
Ref	Expression
ovolun.a	⊢ (𝜑 → (𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ))
ovolun.b	⊢ (𝜑 → (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ))
ovolun.c	⊢ (𝜑 → 𝐶 ∈ ℝ⁺)

**Assertion**
Ref	Expression
ovolunlem2	⊢ (𝜑 → (vol‘(𝐴 ∪ 𝐵)) ≤ (((vol‘𝐴) + (vol*‘𝐵)) + 𝐶))

Proof of Theorem ovolunlem2 Dummy variables 𝑔 ℎ 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ovolun.a	. . . 4 ⊢ (𝜑 → (𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ))
2	1	simpld 500	. . 3 ⊢ (𝜑 → 𝐴 ⊆ ℝ)
3	1	simprd 501	. . 3 ⊢ (𝜑 → (vol*‘𝐴) ∈ ℝ)
4		ovolun.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ ℝ⁺)
5	4	rphalfcld 13145	. . 3 ⊢ (𝜑 → (𝐶 / 2) ∈ ℝ⁺)
6		eqid 2760	. . . 4 ⊢ seq1( + , ((abs ∘ − ) ∘ 𝑔)) = seq1( + , ((abs ∘ − ) ∘ 𝑔))
7	6	ovolgelb 25762	. . 3 ⊢ ((𝐴 ⊆ ℝ ∧ (vol‘𝐴) ∈ ℝ ∧ (𝐶 / 2) ∈ ℝ⁺) → ∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ^, < ) ≤ ((vol*‘𝐴) + (𝐶 / 2))))
8	2, 3, 5, 7	syl3anc 1398	. 2 ⊢ (𝜑 → ∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ^, < ) ≤ ((vol‘𝐴) + (𝐶 / 2))))
9		ovolun.b	. . . 4 ⊢ (𝜑 → (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ))
10	9	simpld 500	. . 3 ⊢ (𝜑 → 𝐵 ⊆ ℝ)
11	9	simprd 501	. . 3 ⊢ (𝜑 → (vol*‘𝐵) ∈ ℝ)
12		eqid 2760	. . . 4 ⊢ seq1( + , ((abs ∘ − ) ∘ ℎ)) = seq1( + , ((abs ∘ − ) ∘ ℎ))
13	12	ovolgelb 25762	. . 3 ⊢ ((𝐵 ⊆ ℝ ∧ (vol‘𝐵) ∈ ℝ ∧ (𝐶 / 2) ∈ ℝ⁺) → ∃ℎ ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐵 ⊆ ∪ ran ((,) ∘ ℎ) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ ℎ)), ℝ^, < ) ≤ ((vol*‘𝐵) + (𝐶 / 2))))
14	10, 11, 5, 13	syl3anc 1398	. 2 ⊢ (𝜑 → ∃ℎ ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐵 ⊆ ∪ ran ((,) ∘ ℎ) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ ℎ)), ℝ^, < ) ≤ ((vol‘𝐵) + (𝐶 / 2))))
15		reeanv 3234	. . 3 ⊢ (∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)∃ℎ ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)((𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ^, < ) ≤ ((vol‘𝐴) + (𝐶 / 2))) ∧ (𝐵 ⊆ ∪ ran ((,) ∘ ℎ) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ ℎ)), ℝ^, < ) ≤ ((vol‘𝐵) + (𝐶 / 2)))) ↔ (∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ^, < ) ≤ ((vol‘𝐴) + (𝐶 / 2))) ∧ ∃ℎ ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐵 ⊆ ∪ ran ((,) ∘ ℎ) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ ℎ)), ℝ^, < ) ≤ ((vol‘𝐵) + (𝐶 / 2)))))
16	1	3ad2ant1 1151	. . . . . 6 ⊢ ((𝜑 ∧ (𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ ℎ ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)) ∧ ((𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ^, < ) ≤ ((vol‘𝐴) + (𝐶 / 2))) ∧ (𝐵 ⊆ ∪ ran ((,) ∘ ℎ) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ ℎ)), ℝ^, < ) ≤ ((vol‘𝐵) + (𝐶 / 2))))) → (𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ))
17	9	3ad2ant1 1151	. . . . . 6 ⊢ ((𝜑 ∧ (𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ ℎ ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)) ∧ ((𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ^, < ) ≤ ((vol‘𝐴) + (𝐶 / 2))) ∧ (𝐵 ⊆ ∪ ran ((,) ∘ ℎ) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ ℎ)), ℝ^, < ) ≤ ((vol‘𝐵) + (𝐶 / 2))))) → (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ))
18	4	3ad2ant1 1151	. . . . . 6 ⊢ ((𝜑 ∧ (𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ ℎ ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)) ∧ ((𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ^, < ) ≤ ((vol‘𝐴) + (𝐶 / 2))) ∧ (𝐵 ⊆ ∪ ran ((,) ∘ ℎ) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ ℎ)), ℝ^, < ) ≤ ((vol‘𝐵) + (𝐶 / 2))))) → 𝐶 ∈ ℝ⁺)
19		eqid 2760	. . . . . 6 ⊢ seq1( + , ((abs ∘ − ) ∘ (𝑛 ∈ ℕ ↦ if((𝑛 / 2) ∈ ℕ, (ℎ‘(𝑛 / 2)), (𝑔‘((𝑛 + 1) / 2)))))) = seq1( + , ((abs ∘ − ) ∘ (𝑛 ∈ ℕ ↦ if((𝑛 / 2) ∈ ℕ, (ℎ‘(𝑛 / 2)), (𝑔‘((𝑛 + 1) / 2))))))
20		simp2l 1218	. . . . . 6 ⊢ ((𝜑 ∧ (𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ ℎ ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)) ∧ ((𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ^, < ) ≤ ((vol‘𝐴) + (𝐶 / 2))) ∧ (𝐵 ⊆ ∪ ran ((,) ∘ ℎ) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ ℎ)), ℝ^, < ) ≤ ((vol‘𝐵) + (𝐶 / 2))))) → 𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ))
21		simp3ll 1263	. . . . . 6 ⊢ ((𝜑 ∧ (𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ ℎ ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)) ∧ ((𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ^, < ) ≤ ((vol‘𝐴) + (𝐶 / 2))) ∧ (𝐵 ⊆ ∪ ran ((,) ∘ ℎ) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ ℎ)), ℝ^, < ) ≤ ((vol‘𝐵) + (𝐶 / 2))))) → 𝐴 ⊆ ∪ ran ((,) ∘ 𝑔))
22		simp3lr 1264	. . . . . 6 ⊢ ((𝜑 ∧ (𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ ℎ ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)) ∧ ((𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ^, < ) ≤ ((vol‘𝐴) + (𝐶 / 2))) ∧ (𝐵 ⊆ ∪ ran ((,) ∘ ℎ) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ ℎ)), ℝ^, < ) ≤ ((vol‘𝐵) + (𝐶 / 2))))) → sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ^, < ) ≤ ((vol‘𝐴) + (𝐶 / 2)))
23		simp2r 1219	. . . . . 6 ⊢ ((𝜑 ∧ (𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ ℎ ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)) ∧ ((𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ^, < ) ≤ ((vol‘𝐴) + (𝐶 / 2))) ∧ (𝐵 ⊆ ∪ ran ((,) ∘ ℎ) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ ℎ)), ℝ^, < ) ≤ ((vol‘𝐵) + (𝐶 / 2))))) → ℎ ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ))
24		simp3rl 1265	. . . . . 6 ⊢ ((𝜑 ∧ (𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ ℎ ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)) ∧ ((𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ^, < ) ≤ ((vol‘𝐴) + (𝐶 / 2))) ∧ (𝐵 ⊆ ∪ ran ((,) ∘ ℎ) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ ℎ)), ℝ^, < ) ≤ ((vol‘𝐵) + (𝐶 / 2))))) → 𝐵 ⊆ ∪ ran ((,) ∘ ℎ))
25		simp3rr 1266	. . . . . 6 ⊢ ((𝜑 ∧ (𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ ℎ ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)) ∧ ((𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ^, < ) ≤ ((vol‘𝐴) + (𝐶 / 2))) ∧ (𝐵 ⊆ ∪ ran ((,) ∘ ℎ) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ ℎ)), ℝ^, < ) ≤ ((vol‘𝐵) + (𝐶 / 2))))) → sup(ran seq1( + , ((abs ∘ − ) ∘ ℎ)), ℝ^, < ) ≤ ((vol‘𝐵) + (𝐶 / 2)))
26		eqid 2760	. . . . . 6 ⊢ (𝑛 ∈ ℕ ↦ if((𝑛 / 2) ∈ ℕ, (ℎ‘(𝑛 / 2)), (𝑔‘((𝑛 + 1) / 2)))) = (𝑛 ∈ ℕ ↦ if((𝑛 / 2) ∈ ℕ, (ℎ‘(𝑛 / 2)), (𝑔‘((𝑛 + 1) / 2))))
27	16, 17, 18, 6, 12, 19, 20, 21, 22, 23, 24, 25, 26	ovolunlem1 25779	. . . . 5 ⊢ ((𝜑 ∧ (𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ ℎ ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)) ∧ ((𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ^, < ) ≤ ((vol‘𝐴) + (𝐶 / 2))) ∧ (𝐵 ⊆ ∪ ran ((,) ∘ ℎ) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ ℎ)), ℝ^, < ) ≤ ((vol‘𝐵) + (𝐶 / 2))))) → (vol‘(𝐴 ∪ 𝐵)) ≤ (((vol‘𝐴) + (vol*‘𝐵)) + 𝐶))
28	27	3exp 1137	. . . 4 ⊢ (𝜑 → ((𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ ℎ ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)) → (((𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ^, < ) ≤ ((vol‘𝐴) + (𝐶 / 2))) ∧ (𝐵 ⊆ ∪ ran ((,) ∘ ℎ) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ ℎ)), ℝ^, < ) ≤ ((vol‘𝐵) + (𝐶 / 2)))) → (vol‘(𝐴 ∪ 𝐵)) ≤ (((vol‘𝐴) + (vol*‘𝐵)) + 𝐶))))
29	28	rexlimdvv 3218	. . 3 ⊢ (𝜑 → (∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)∃ℎ ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)((𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ^, < ) ≤ ((vol‘𝐴) + (𝐶 / 2))) ∧ (𝐵 ⊆ ∪ ran ((,) ∘ ℎ) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ ℎ)), ℝ^, < ) ≤ ((vol‘𝐵) + (𝐶 / 2)))) → (vol‘(𝐴 ∪ 𝐵)) ≤ (((vol‘𝐴) + (vol*‘𝐵)) + 𝐶)))
30	15, 29	biimtrrid 246	. 2 ⊢ (𝜑 → ((∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ^, < ) ≤ ((vol‘𝐴) + (𝐶 / 2))) ∧ ∃ℎ ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐵 ⊆ ∪ ran ((,) ∘ ℎ) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ ℎ)), ℝ^, < ) ≤ ((vol‘𝐵) + (𝐶 / 2)))) → (vol‘(𝐴 ∪ 𝐵)) ≤ (((vol‘𝐴) + (vol*‘𝐵)) + 𝐶)))
31	8, 14, 30	mp2and 712	1 ⊢ (𝜑 → (vol‘(𝐴 ∪ 𝐵)) ≤ (((vol‘𝐴) + (vol*‘𝐵)) + 𝐶))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 ∧ w3a 1103 ∈ wcel 2145 ∃wrex 3086 ∪ cun 3896 ∩ cin 3897 ⊆ wss 3898 ifcif 4481 ∪ cuni 4866 class class class wbr 5102 ↦ cmpt 5185 × cxp 5645 ran crn 5648 ∘ ccom 5651 ‘cfv 6527 (class class class)co 7408 ↑_m cmap 8825 supcsup 9410 ℝcr 11170 1c1 11172 + caddc 11174 ℝ^*cxr 11313 < clt 11314 ≤ cle 11315 − cmin 11512 / cdiv 11942 ℕcn 12304 2c2 12366 ℝ⁺crp 13089 (,)cioo 13445 seqcseq 14112 abscabs 15368 vol*covol 25744

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7734 ax-cnex 11227 ax-resscn 11228 ax-1cn 11229 ax-icn 11230 ax-addcl 11231 ax-addrcl 11232 ax-mulcl 11233 ax-mulrcl 11234 ax-mulcom 11235 ax-addass 11236 ax-mulass 11237 ax-distr 11238 ax-i2m1 11239 ax-1ne0 11240 ax-1rid 11241 ax-rnegex 11242 ax-rrecex 11243 ax-cnre 11244 ax-pre-lttri 11245 ax-pre-lttrn 11246 ax-pre-ltadd 11247 ax-pre-mulgt0 11248 ax-pre-sup 11249

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4584 df-pr 4586 df-op 4590 df-uni 4867 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5542 df-eprel 5547 df-po 5555 df-so 5556 df-fr 5600 df-we 5602 df-xp 5653 df-rel 5654 df-cnv 5655 df-co 5656 df-dm 5657 df-rn 5658 df-res 5659 df-ima 5660 df-pred 6293 df-ord 6354 df-on 6355 df-lim 6356 df-suc 6357 df-iota 6483 df-fun 6529 df-fn 6530 df-f 6531 df-f1 6532 df-fo 6533 df-f1o 6534 df-fv 6535 df-riota 7365 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7861 df-1st 7984 df-2nd 7985 df-frecs 8277 df-wrecs 8308 df-recs 8357 df-rdg 8396 df-er 8695 df-map 8827 df-en 8952 df-dom 8953 df-sdom 8954 df-sup 9412 df-inf 9413 df-pnf 11316 df-mnf 11317 df-xr 11318 df-ltxr 11319 df-le 11320 df-sub 11514 df-neg 11515 df-div 11943 df-nn 12305 df-2 12374 df-3 12375 df-n0 12576 df-z 12663 df-uz 12935 df-rp 13090 df-ioo 13449 df-ico 13451 df-fz 13609 df-fl 13900 df-seq 14113 df-exp 14173 df-cj 15233 df-re 15234 df-im 15235 df-sqrt 15369 df-abs 15370 df-ovol 25746

This theorem is used by: ovolun 25781

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