ovolunlem2 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > ovolunlem2		Structured version Visualization version GIF version

Theorem ovolunlem2 24102

Description: Lemma for ovolun 24103. (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 498	. . 3 ⊢ (𝜑 → 𝐴 ⊆ ℝ)
3	1	simprd 499	. . 3 ⊢ (𝜑 → (vol*‘𝐴) ∈ ℝ)
4		ovolun.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ ℝ⁺)
5	4	rphalfcld 12431	. . 3 ⊢ (𝜑 → (𝐶 / 2) ∈ ℝ⁺)
6		eqid 2798	. . . 4 ⊢ seq1( + , ((abs ∘ − ) ∘ 𝑔)) = seq1( + , ((abs ∘ − ) ∘ 𝑔))
7	6	ovolgelb 24084	. . 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 1368	. 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 498	. . 3 ⊢ (𝜑 → 𝐵 ⊆ ℝ)
11	9	simprd 499	. . 3 ⊢ (𝜑 → (vol*‘𝐵) ∈ ℝ)
12		eqid 2798	. . . 4 ⊢ seq1( + , ((abs ∘ − ) ∘ ℎ)) = seq1( + , ((abs ∘ − ) ∘ ℎ))
13	12	ovolgelb 24084	. . 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 1368	. 2 ⊢ (𝜑 → ∃ℎ ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐵 ⊆ ∪ ran ((,) ∘ ℎ) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ ℎ)), ℝ^, < ) ≤ ((vol‘𝐵) + (𝐶 / 2))))
15		reeanv 3320	. . 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 1130	. . . . . 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 1130	. . . . . 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 1130	. . . . . 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 2798	. . . . . 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 1196	. . . . . 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 1241	. . . . . 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 1242	. . . . . 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 1197	. . . . . 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 1243	. . . . . 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 1244	. . . . . 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 2798	. . . . . 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 24101	. . . . 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 1116	. . . 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 3252	. . 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	syl5bir 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 698	1 ⊢ (𝜑 → (vol‘(𝐴 ∪ 𝐵)) ≤ (((vol‘𝐴) + (vol*‘𝐵)) + 𝐶))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1084 ∈ wcel 2111 ∃wrex 3107 ∪ cun 3879 ∩ cin 3880 ⊆ wss 3881 ifcif 4425 ∪ cuni 4800 class class class wbr 5030 ↦ cmpt 5110 × cxp 5517 ran crn 5520 ∘ ccom 5523 ‘cfv 6324 (class class class)co 7135 ↑_m cmap 8389 supcsup 8888 ℝcr 10525 1c1 10527 + caddc 10529 ℝ^*cxr 10663 < clt 10664 ≤ cle 10665 − cmin 10859 / cdiv 11286 ℕcn 11625 2c2 11680 ℝ⁺crp 12377 (,)cioo 12726 seqcseq 13364 abscabs 14585 vol*covol 24066

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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-sup 8890 df-inf 8891 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-ioo 12730 df-ico 12732 df-fz 12886 df-fl 13157 df-seq 13365 df-exp 13426 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-ovol 24068

This theorem is referenced by: ovolun 24103

W3C validator