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Theorem ovolunlem2 24567

Description: Lemma for ovolun 24568. (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 494	. . 3 ⊢ (𝜑 → 𝐴 ⊆ ℝ)
3	1	simprd 495	. . 3 ⊢ (𝜑 → (vol*‘𝐴) ∈ ℝ)
4		ovolun.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ ℝ⁺)
5	4	rphalfcld 12713	. . 3 ⊢ (𝜑 → (𝐶 / 2) ∈ ℝ⁺)
6		eqid 2738	. . . 4 ⊢ seq1( + , ((abs ∘ − ) ∘ 𝑔)) = seq1( + , ((abs ∘ − ) ∘ 𝑔))
7	6	ovolgelb 24549	. . 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 1369	. 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 494	. . 3 ⊢ (𝜑 → 𝐵 ⊆ ℝ)
11	9	simprd 495	. . 3 ⊢ (𝜑 → (vol*‘𝐵) ∈ ℝ)
12		eqid 2738	. . . 4 ⊢ seq1( + , ((abs ∘ − ) ∘ ℎ)) = seq1( + , ((abs ∘ − ) ∘ ℎ))
13	12	ovolgelb 24549	. . 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 1369	. 2 ⊢ (𝜑 → ∃ℎ ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐵 ⊆ ∪ ran ((,) ∘ ℎ) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ ℎ)), ℝ^, < ) ≤ ((vol‘𝐵) + (𝐶 / 2))))
15		reeanv 3292	. . 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 1131	. . . . . 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 1131	. . . . . 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 1131	. . . . . 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 2738	. . . . . 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 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))))) → 𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ))
21		simp3ll 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))))) → 𝐴 ⊆ ∪ ran ((,) ∘ 𝑔))
22		simp3lr 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))))) → sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ^, < ) ≤ ((vol‘𝐴) + (𝐶 / 2)))
23		simp2r 1198	. . . . . 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 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))))) → 𝐵 ⊆ ∪ ran ((,) ∘ ℎ))
25		simp3rr 1245	. . . . . 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 2738	. . . . . 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 24566	. . . . 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 1117	. . . 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 3221	. . 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 242	. 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 695	1 ⊢ (𝜑 → (vol‘(𝐴 ∪ 𝐵)) ≤ (((vol‘𝐴) + (vol*‘𝐵)) + 𝐶))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1085 ∈ wcel 2108 ∃wrex 3064 ∪ cun 3881 ∩ cin 3882 ⊆ wss 3883 ifcif 4456 ∪ cuni 4836 class class class wbr 5070 ↦ cmpt 5153 × cxp 5578 ran crn 5581 ∘ ccom 5584 ‘cfv 6418 (class class class)co 7255 ↑_m cmap 8573 supcsup 9129 ℝcr 10801 1c1 10803 + caddc 10805 ℝ^*cxr 10939 < clt 10940 ≤ cle 10941 − cmin 11135 / cdiv 11562 ℕcn 11903 2c2 11958 ℝ⁺crp 12659 (,)cioo 13008 seqcseq 13649 abscabs 14873 vol*covol 24531

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-1st 7804 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-er 8456 df-map 8575 df-en 8692 df-dom 8693 df-sdom 8694 df-sup 9131 df-inf 9132 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-n0 12164 df-z 12250 df-uz 12512 df-rp 12660 df-ioo 13012 df-ico 13014 df-fz 13169 df-fl 13440 df-seq 13650 df-exp 13711 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875 df-ovol 24533

This theorem is referenced by: ovolun 24568

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