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Theorem ovolunlem2 25486

Description: Lemma for ovolun 25487. (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 496	. . 3 ⊢ (𝜑 → 𝐴 ⊆ ℝ)
3	1	simprd 497	. . 3 ⊢ (𝜑 → (vol*‘𝐴) ∈ ℝ)
4		ovolun.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ ℝ⁺)
5	4	rphalfcld 12993	. . 3 ⊢ (𝜑 → (𝐶 / 2) ∈ ℝ⁺)
6		eqid 2741	. . . 4 ⊢ seq1( + , ((abs ∘ − ) ∘ 𝑔)) = seq1( + , ((abs ∘ − ) ∘ 𝑔))
7	6	ovolgelb 25468	. . 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 1380	. 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 496	. . 3 ⊢ (𝜑 → 𝐵 ⊆ ℝ)
11	9	simprd 497	. . 3 ⊢ (𝜑 → (vol*‘𝐵) ∈ ℝ)
12		eqid 2741	. . . 4 ⊢ seq1( + , ((abs ∘ − ) ∘ ℎ)) = seq1( + , ((abs ∘ − ) ∘ ℎ))
13	12	ovolgelb 25468	. . 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 1380	. 2 ⊢ (𝜑 → ∃ℎ ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐵 ⊆ ∪ ran ((,) ∘ ℎ) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ ℎ)), ℝ^, < ) ≤ ((vol‘𝐵) + (𝐶 / 2))))
15		reeanv 3213	. . 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 1140	. . . . . 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 1140	. . . . . 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 1140	. . . . . 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 2741	. . . . . 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 1207	. . . . . 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 1252	. . . . . 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 1253	. . . . . 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 1208	. . . . . 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 1254	. . . . . 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 1255	. . . . . 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 2741	. . . . . 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 25485	. . . . 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 1126	. . . 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 3197	. . 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 245	. 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 706	1 ⊢ (𝜑 → (vol‘(𝐴 ∪ 𝐵)) ≤ (((vol‘𝐴) + (vol*‘𝐵)) + 𝐶))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 397 ∧ w3a 1093 ∈ wcel 2121 ∃wrex 3065 ∪ cun 3882 ∩ cin 3883 ⊆ wss 3884 ifcif 4456 ∪ cuni 4840 class class class wbr 5074 ↦ cmpt 5155 × cxp 5618 ran crn 5621 ∘ ccom 5624 ‘cfv 6488 (class class class)co 7359 ↑_m cmap 8767 supcsup 9347 ℝcr 11033 1c1 11035 + caddc 11037 ℝ^*cxr 11174 < clt 11175 ≤ cle 11176 − cmin 11373 / cdiv 11803 ℕcn 12169 2c2 12231 ℝ⁺crp 12937 (,)cioo 13293 seqcseq 13958 abscabs 15191 vol*covol 25450

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1975 ax-7 2016 ax-8 2123 ax-9 2131 ax-10 2154 ax-11 2170 ax-12 2191 ax-ext 2713 ax-sep 5220 ax-nul 5230 ax-pow 5296 ax-pr 5364 ax-un 7681 ax-cnex 11090 ax-resscn 11091 ax-1cn 11092 ax-icn 11093 ax-addcl 11094 ax-addrcl 11095 ax-mulcl 11096 ax-mulrcl 11097 ax-mulcom 11098 ax-addass 11099 ax-mulass 11100 ax-distr 11101 ax-i2m1 11102 ax-1ne0 11103 ax-1rid 11104 ax-rnegex 11105 ax-rrecex 11106 ax-cnre 11107 ax-pre-lttri 11108 ax-pre-lttrn 11109 ax-pre-ltadd 11110 ax-pre-mulgt0 11111 ax-pre-sup 11112

This theorem depends on definitions: df-bi 209 df-an 398 df-or 855 df-3or 1094 df-3an 1095 df-tru 1551 df-fal 1561 df-ex 1788 df-nf 1792 df-sb 2075 df-mo 2545 df-eu 2575 df-clab 2720 df-cleq 2733 df-clel 2816 df-nfc 2890 df-ne 2937 df-nel 3041 df-ral 3056 df-rex 3066 df-rmo 3346 df-reu 3347 df-rab 3394 df-v 3435 df-sbc 3725 df-csb 3833 df-dif 3887 df-un 3889 df-in 3891 df-ss 3901 df-pss 3904 df-nul 4264 df-if 4457 df-pw 4533 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4841 df-iun 4925 df-br 5075 df-opab 5137 df-mpt 5156 df-tr 5182 df-id 5515 df-eprel 5520 df-po 5528 df-so 5529 df-fr 5573 df-we 5575 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-pred 6255 df-ord 6316 df-on 6317 df-lim 6318 df-suc 6319 df-iota 6444 df-fun 6490 df-fn 6491 df-f 6492 df-f1 6493 df-fo 6494 df-f1o 6495 df-fv 6496 df-riota 7316 df-ov 7362 df-oprab 7363 df-mpo 7364 df-om 7810 df-1st 7933 df-2nd 7934 df-frecs 8224 df-wrecs 8255 df-recs 8304 df-rdg 8343 df-er 8637 df-map 8769 df-en 8888 df-dom 8889 df-sdom 8890 df-sup 9349 df-inf 9350 df-pnf 11177 df-mnf 11178 df-xr 11179 df-ltxr 11180 df-le 11181 df-sub 11375 df-neg 11376 df-div 11804 df-nn 12170 df-2 12239 df-3 12240 df-n0 12433 df-z 12520 df-uz 12784 df-rp 12938 df-ioo 13297 df-ico 13299 df-fz 13457 df-fl 13746 df-seq 13959 df-exp 14019 df-cj 15056 df-re 15057 df-im 15058 df-sqrt 15192 df-abs 15193 df-ovol 25452

This theorem is referenced by: ovolun 25487

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