ovolunlem2 - Metamath Proof Explorer

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Theorem ovolunlem2 25668

Description: Lemma for ovolun 25669. (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 499	. . 3 ⊢ (𝜑 → 𝐴 ⊆ ℝ)
3	1	simprd 500	. . 3 ⊢ (𝜑 → (vol*‘𝐴) ∈ ℝ)
4		ovolun.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ ℝ⁺)
5	4	rphalfcld 13078	. . 3 ⊢ (𝜑 → (𝐶 / 2) ∈ ℝ⁺)
6		eqid 2762	. . . 4 ⊢ seq1( + , ((abs ∘ − ) ∘ 𝑔)) = seq1( + , ((abs ∘ − ) ∘ 𝑔))
7	6	ovolgelb 25650	. . 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 1397	. 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 499	. . 3 ⊢ (𝜑 → 𝐵 ⊆ ℝ)
11	9	simprd 500	. . 3 ⊢ (𝜑 → (vol*‘𝐵) ∈ ℝ)
12		eqid 2762	. . . 4 ⊢ seq1( + , ((abs ∘ − ) ∘ ℎ)) = seq1( + , ((abs ∘ − ) ∘ ℎ))
13	12	ovolgelb 25650	. . 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 1397	. 2 ⊢ (𝜑 → ∃ℎ ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐵 ⊆ ∪ ran ((,) ∘ ℎ) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ ℎ)), ℝ^, < ) ≤ ((vol‘𝐵) + (𝐶 / 2))))
15		reeanv 3236	. . 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 1150	. . . . . 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 1150	. . . . . 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 1150	. . . . . 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 2762	. . . . . 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 1217	. . . . . 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 1262	. . . . . 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 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))))) → sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ^, < ) ≤ ((vol‘𝐴) + (𝐶 / 2)))
23		simp2r 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))))) → ℎ ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ))
24		simp3rl 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))))) → 𝐵 ⊆ ∪ ran ((,) ∘ ℎ))
25		simp3rr 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))))) → sup(ran seq1( + , ((abs ∘ − ) ∘ ℎ)), ℝ^, < ) ≤ ((vol‘𝐵) + (𝐶 / 2)))
26		eqid 2762	. . . . . 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 25667	. . . . 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 1136	. . . 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 3220	. . 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 711	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 400 ∧ w3a 1102 ∈ wcel 2142 ∃wrex 3088 ∪ cun 3902 ∩ cin 3903 ⊆ wss 3904 ifcif 4486 ∪ cuni 4871 class class class wbr 5108 ↦ cmpt 5191 × cxp 5658 ran crn 5661 ∘ ccom 5664 ‘cfv 6536 (class class class)co 7412 ↑_m cmap 8822 supcsup 9398 ℝcr 11105 1c1 11107 + caddc 11109 ℝ^*cxr 11248 < clt 11249 ≤ cle 11250 − cmin 11447 / cdiv 11877 ℕcn 12239 2c2 12301 ℝ⁺crp 13022 (,)cioo 13378 seqcseq 14044 abscabs 15292 vol*covol 25632

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1824 ax-4 1838 ax-5 1939 ax-6 1996 ax-7 2037 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-sep 5256 ax-nul 5268 ax-pow 5335 ax-pr 5403 ax-un 7734 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 proof depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1103 df-3an 1104 df-tru 1572 df-fal 1582 df-ex 1809 df-nf 1813 df-sb 2096 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3368 df-reu 3369 df-rab 3416 df-v 3456 df-sbc 3744 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4487 df-pw 4563 df-sn 4589 df-pr 4591 df-op 4595 df-uni 4872 df-iun 4957 df-br 5109 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5555 df-eprel 5560 df-po 5568 df-so 5569 df-fr 5613 df-we 5615 df-xp 5666 df-rel 5667 df-cnv 5668 df-co 5669 df-dm 5670 df-rn 5671 df-res 5672 df-ima 5673 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7861 df-1st 7984 df-2nd 7985 df-frecs 8276 df-wrecs 8307 df-recs 8356 df-rdg 8395 df-er 8692 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-sup 9400 df-inf 9401 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11449 df-neg 11450 df-div 11878 df-nn 12240 df-2 12309 df-3 12310 df-n0 12511 df-z 12598 df-uz 12869 df-rp 13023 df-ioo 13382 df-ico 13384 df-fz 13542 df-fl 13832 df-seq 14045 df-exp 14105 df-cj 15157 df-re 15158 df-im 15159 df-sqrt 15293 df-abs 15294 df-ovol 25634

This theorem is used by: ovolun 25669

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