Theorem hoidmv1lelem3 43232

Description: The dimensional volume of a 1-dimensional half-open interval is less than or equal the generalized sum of the dimensional volumes of countable half-open intervals that cover it. This is the nonempty, finite generalized sum, sub case in Lemma 114B of [Fremlin1] p. 23. (Contributed by Glauco Siliprandi, 21-Nov-2020.)

Hypotheses

Ref	Expression
hoidmv1lelem3.a	⊢ (𝜑 → 𝐴 ∈ ℝ)
hoidmv1lelem3.b	⊢ (𝜑 → 𝐵 ∈ ℝ)
hoidmv1lelem3.l	⊢ (𝜑 → 𝐴 < 𝐵)
hoidmv1lelem3.c	⊢ (𝜑 → 𝐶:ℕ⟶ℝ)
hoidmv1lelem3.d	⊢ (𝜑 → 𝐷:ℕ⟶ℝ)
hoidmv1lelem3.x	⊢ (𝜑 → (𝐴[,)𝐵) ⊆ ∪ 𝑗 ∈ ℕ ((𝐶‘𝑗)[,)(𝐷‘𝑗)))
hoidmv1lelem3.r	⊢ (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)(𝐷‘𝑗))))) ∈ ℝ)
hoidmv1lelem3.u	⊢ 𝑈 = {𝑧 ∈ (𝐴[,]𝐵) ∣ (𝑧 − 𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧)))))}
hoidmv1lelem3.s	⊢ 𝑆 = sup(𝑈, ℝ, < )

Assertion

Ref	Expression
hoidmv1lelem3	⊢ (𝜑 → (𝐵 − 𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)(𝐷‘𝑗))))))

Distinct variable groups:   𝐴,𝑗,𝑧   𝐵,𝑗,𝑧   𝐶,𝑗,𝑧   𝐷,𝑗,𝑧   𝑆,𝑗,𝑧   𝑈,𝑗,𝑧   𝜑,𝑗,𝑧

Proof of Theorem hoidmv1lelem3

Dummy variables 𝑦 𝑖 𝑢 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		hoidmv1lelem3.b	. . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ)
2		hoidmv1lelem3.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ)
3	1, 2	resubcld 11057	. 2 ⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℝ)
4		nnex 11631	. . . . . . 7 ⊢ ℕ ∈ V
5	4	a1i 11	. . . . . 6 ⊢ (𝜑 → ℕ ∈ V)
6		icossicc 12814	. . . . . . . 8 ⊢ (0[,)+∞) ⊆ (0[,]+∞)
7		0xr 10677	. . . . . . . . . 10 ⊢ 0 ∈ ℝ*
8	7	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 0 ∈ ℝ*)
9		pnfxr 10684	. . . . . . . . . 10 ⊢ +∞ ∈ ℝ*
10	9	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → +∞ ∈ ℝ*)
11		hoidmv1lelem3.c	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐶:ℕ⟶ℝ)
12	11	ffvelrnda 6828	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐶‘𝑗) ∈ ℝ)
13		hoidmv1lelem3.d	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐷:ℕ⟶ℝ)
14	13	ffvelrnda 6828	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐷‘𝑗) ∈ ℝ)
15	1	adantr 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝐵 ∈ ℝ)
16	14, 15	ifcld 4470	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵) ∈ ℝ)
17		volicore 43220	. . . . . . . . . . 11 ⊢ (((𝐶‘𝑗) ∈ ℝ ∧ if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵) ∈ ℝ) → (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵))) ∈ ℝ)
18	12, 16, 17	syl2anc 587	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵))) ∈ ℝ)
19	18	rexrd 10680	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵))) ∈ ℝ*)
20	16	rexrd 10680	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵) ∈ ℝ*)
21		icombl 24168	. . . . . . . . . . 11 ⊢ (((𝐶‘𝑗) ∈ ℝ ∧ if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵) ∈ ℝ*) → ((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵)) ∈ dom vol)
22	12, 20, 21	syl2anc 587	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵)) ∈ dom vol)
23		volge0 42603	. . . . . . . . . 10 ⊢ (((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵)) ∈ dom vol → 0 ≤ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵))))
24	22, 23	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 0 ≤ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵))))
25	18	ltpnfd 12504	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵))) < +∞)
26	8, 10, 19, 24, 25	elicod 12775	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵))) ∈ (0[,)+∞))
27	6, 26	sseldi 3913	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵))) ∈ (0[,]+∞))
28		eqid 2798	. . . . . . 7 ⊢ (𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵)))) = (𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵))))
29	27, 28	fmptd 6855	. . . . . 6 ⊢ (𝜑 → (𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵)))):ℕ⟶(0[,]+∞))
30	5, 29	sge0xrcl 43024	. . . . 5 ⊢ (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵))))) ∈ ℝ*)
31	9	a1i 11	. . . . 5 ⊢ (𝜑 → +∞ ∈ ℝ*)
32		hoidmv1lelem3.r	. . . . . . 7 ⊢ (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)(𝐷‘𝑗))))) ∈ ℝ)
33	32	rexrd 10680	. . . . . 6 ⊢ (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)(𝐷‘𝑗))))) ∈ ℝ*)
34		nfv 1915	. . . . . . 7 ⊢ Ⅎ𝑗𝜑
35		volf 24133	. . . . . . . . 9 ⊢ vol:dom vol⟶(0[,]+∞)
36	35	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → vol:dom vol⟶(0[,]+∞))
37	14	rexrd 10680	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐷‘𝑗) ∈ ℝ*)
38		icombl 24168	. . . . . . . . 9 ⊢ (((𝐶‘𝑗) ∈ ℝ ∧ (𝐷‘𝑗) ∈ ℝ*) → ((𝐶‘𝑗)[,)(𝐷‘𝑗)) ∈ dom vol)
39	12, 37, 38	syl2anc 587	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)[,)(𝐷‘𝑗)) ∈ dom vol)
40	36, 39	ffvelrnd 6829	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (vol‘((𝐶‘𝑗)[,)(𝐷‘𝑗))) ∈ (0[,]+∞))
41	12	rexrd 10680	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐶‘𝑗) ∈ ℝ*)
42	12	leidd 11195	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐶‘𝑗) ≤ (𝐶‘𝑗))
43		min1 12570	. . . . . . . . . 10 ⊢ (((𝐷‘𝑗) ∈ ℝ ∧ 𝐵 ∈ ℝ) → if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵) ≤ (𝐷‘𝑗))
44	14, 15, 43	syl2anc 587	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵) ≤ (𝐷‘𝑗))
45		icossico 12795	. . . . . . . . 9 ⊢ ((((𝐶‘𝑗) ∈ ℝ* ∧ (𝐷‘𝑗) ∈ ℝ*) ∧ ((𝐶‘𝑗) ≤ (𝐶‘𝑗) ∧ if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵) ≤ (𝐷‘𝑗))) → ((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵)) ⊆ ((𝐶‘𝑗)[,)(𝐷‘𝑗)))
46	41, 37, 42, 44, 45	syl22anc 837	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵)) ⊆ ((𝐶‘𝑗)[,)(𝐷‘𝑗)))
47		volss 24137	. . . . . . . 8 ⊢ ((((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵)) ∈ dom vol ∧ ((𝐶‘𝑗)[,)(𝐷‘𝑗)) ∈ dom vol ∧ ((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵)) ⊆ ((𝐶‘𝑗)[,)(𝐷‘𝑗))) → (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵))) ≤ (vol‘((𝐶‘𝑗)[,)(𝐷‘𝑗))))
48	22, 39, 46, 47	syl3anc 1368	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵))) ≤ (vol‘((𝐶‘𝑗)[,)(𝐷‘𝑗))))
49	34, 5, 27, 40, 48	sge0lempt 43049	. . . . . 6 ⊢ (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵))))) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)(𝐷‘𝑗))))))
50	32	ltpnfd 12504	. . . . . 6 ⊢ (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)(𝐷‘𝑗))))) < +∞)
51	30, 33, 31, 49, 50	xrlelttrd 12541	. . . . 5 ⊢ (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵))))) < +∞)
52	30, 31, 51	xrltned 41989	. . . 4 ⊢ (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵))))) ≠ +∞)
53	52	neneqd 2992	. . 3 ⊢ (𝜑 → ¬ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵))))) = +∞)
54	5, 29	sge0repnf 43025	. . 3 ⊢ (𝜑 → ((Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵))))) ∈ ℝ ↔ ¬ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵))))) = +∞))
55	53, 54	mpbird 260	. 2 ⊢ (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵))))) ∈ ℝ)
56	1	rexrd 10680	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℝ*)
57	2, 1	iccssred 12812	. . . . . . . . 9 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
58		hoidmv1lelem3.u	. . . . . . . . . . 11 ⊢ 𝑈 = {𝑧 ∈ (𝐴[,]𝐵) ∣ (𝑧 − 𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧)))))}
59		ssrab2 4007	. . . . . . . . . . 11 ⊢ {𝑧 ∈ (𝐴[,]𝐵) ∣ (𝑧 − 𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧)))))} ⊆ (𝐴[,]𝐵)
60	58, 59	eqsstri 3949	. . . . . . . . . 10 ⊢ 𝑈 ⊆ (𝐴[,]𝐵)
61		hoidmv1lelem3.l	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 < 𝐵)
62		hoidmv1lelem3.s	. . . . . . . . . . . 12 ⊢ 𝑆 = sup(𝑈, ℝ, < )
63	2, 1, 61, 11, 13, 32, 58, 62	hoidmv1lelem1 43230	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑆 ∈ 𝑈 ∧ 𝐴 ∈ 𝑈 ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝑈 𝑦 ≤ 𝑥))
64	63	simp1d 1139	. . . . . . . . . 10 ⊢ (𝜑 → 𝑆 ∈ 𝑈)
65	60, 64	sseldi 3913	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ∈ (𝐴[,]𝐵))
66	57, 65	sseldd 3916	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ ℝ)
67	66	rexrd 10680	. . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ ℝ*)
68		simpl 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐵 ≤ 𝑆) → 𝜑)
69		simpr 488	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝐵 ≤ 𝑆) → ¬ 𝐵 ≤ 𝑆)
70	68, 66	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 𝐵 ≤ 𝑆) → 𝑆 ∈ ℝ)
71	68, 1	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 𝐵 ≤ 𝑆) → 𝐵 ∈ ℝ)
72	70, 71	ltnled 10776	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝐵 ≤ 𝑆) → (𝑆 < 𝐵 ↔ ¬ 𝐵 ≤ 𝑆))
73	69, 72	mpbird 260	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐵 ≤ 𝑆) → 𝑆 < 𝐵)
74		hoidmv1lelem3.x	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐴[,)𝐵) ⊆ ∪ 𝑗 ∈ ℕ ((𝐶‘𝑗)[,)(𝐷‘𝑗)))
75	74	adantr 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑆 < 𝐵) → (𝐴[,)𝐵) ⊆ ∪ 𝑗 ∈ ℕ ((𝐶‘𝑗)[,)(𝐷‘𝑗)))
76	2	rexrd 10680	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐴 ∈ ℝ*)
77	76	adantr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑆 < 𝐵) → 𝐴 ∈ ℝ*)
78	56	adantr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑆 < 𝐵) → 𝐵 ∈ ℝ*)
79	67	adantr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑆 < 𝐵) → 𝑆 ∈ ℝ*)
80	60, 57	sstrid 3926	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑈 ⊆ ℝ)
81	64	ne0d 4251	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑈 ≠ ∅)
82	63	simp3d 1141	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝑈 𝑦 ≤ 𝑥)
83	63	simp2d 1140	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐴 ∈ 𝑈)
84		suprub 11589	. . . . . . . . . . . . . . . 16 ⊢ (((𝑈 ⊆ ℝ ∧ 𝑈 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝑈 𝑦 ≤ 𝑥) ∧ 𝐴 ∈ 𝑈) → 𝐴 ≤ sup(𝑈, ℝ, < ))
85	80, 81, 82, 83, 84	syl31anc 1370	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐴 ≤ sup(𝑈, ℝ, < ))
86	85, 62	breqtrrdi 5072	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐴 ≤ 𝑆)
87	86	adantr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑆 < 𝐵) → 𝐴 ≤ 𝑆)
88		simpr 488	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑆 < 𝐵) → 𝑆 < 𝐵)
89	77, 78, 79, 87, 88	elicod 12775	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑆 < 𝐵) → 𝑆 ∈ (𝐴[,)𝐵))
90	75, 89	sseldd 3916	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑆 < 𝐵) → 𝑆 ∈ ∪ 𝑗 ∈ ℕ ((𝐶‘𝑗)[,)(𝐷‘𝑗)))
91		eliun 4885	. . . . . . . . . . 11 ⊢ (𝑆 ∈ ∪ 𝑗 ∈ ℕ ((𝐶‘𝑗)[,)(𝐷‘𝑗)) ↔ ∃𝑗 ∈ ℕ 𝑆 ∈ ((𝐶‘𝑗)[,)(𝐷‘𝑗)))
92	90, 91	sylib 221	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑆 < 𝐵) → ∃𝑗 ∈ ℕ 𝑆 ∈ ((𝐶‘𝑗)[,)(𝐷‘𝑗)))
93	2	adantr 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑆 < 𝐵) → 𝐴 ∈ ℝ)
94	93	3ad2ant1 1130	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑆 < 𝐵) ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ ((𝐶‘𝑗)[,)(𝐷‘𝑗))) → 𝐴 ∈ ℝ)
95	1	adantr 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑆 < 𝐵) → 𝐵 ∈ ℝ)
96	95	3ad2ant1 1130	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑆 < 𝐵) ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ ((𝐶‘𝑗)[,)(𝐷‘𝑗))) → 𝐵 ∈ ℝ)
97	11	adantr 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑆 < 𝐵) → 𝐶:ℕ⟶ℝ)
98	97	3ad2ant1 1130	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑆 < 𝐵) ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ ((𝐶‘𝑗)[,)(𝐷‘𝑗))) → 𝐶:ℕ⟶ℝ)
99	13	adantr 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑆 < 𝐵) → 𝐷:ℕ⟶ℝ)
100	99	3ad2ant1 1130	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑆 < 𝐵) ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ ((𝐶‘𝑗)[,)(𝐷‘𝑗))) → 𝐷:ℕ⟶ℝ)
101		fveq2 6645	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 = 𝑗 → (𝐶‘𝑖) = (𝐶‘𝑗))
102		fveq2 6645	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 = 𝑗 → (𝐷‘𝑖) = (𝐷‘𝑗))
103	101, 102	oveq12d 7153	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 = 𝑗 → ((𝐶‘𝑖)[,)(𝐷‘𝑖)) = ((𝐶‘𝑗)[,)(𝐷‘𝑗)))
104	103	fveq2d 6649	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 = 𝑗 → (vol‘((𝐶‘𝑖)[,)(𝐷‘𝑖))) = (vol‘((𝐶‘𝑗)[,)(𝐷‘𝑗))))
105	104	cbvmptv 5133	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ ℕ ↦ (vol‘((𝐶‘𝑖)[,)(𝐷‘𝑖)))) = (𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)(𝐷‘𝑗))))
106	105	fveq2i 6648	. . . . . . . . . . . . . . . 16 ⊢ (Σ^‘(𝑖 ∈ ℕ ↦ (vol‘((𝐶‘𝑖)[,)(𝐷‘𝑖))))) = (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)(𝐷‘𝑗)))))
107	106, 32	eqeltrid 2894	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (Σ^‘(𝑖 ∈ ℕ ↦ (vol‘((𝐶‘𝑖)[,)(𝐷‘𝑖))))) ∈ ℝ)
108	107	adantr 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑆 < 𝐵) → (Σ^‘(𝑖 ∈ ℕ ↦ (vol‘((𝐶‘𝑖)[,)(𝐷‘𝑖))))) ∈ ℝ)
109	108	3ad2ant1 1130	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑆 < 𝐵) ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ ((𝐶‘𝑗)[,)(𝐷‘𝑗))) → (Σ^‘(𝑖 ∈ ℕ ↦ (vol‘((𝐶‘𝑖)[,)(𝐷‘𝑖))))) ∈ ℝ)
110	102	breq1d 5040	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑖 = 𝑗 → ((𝐷‘𝑖) ≤ 𝑧 ↔ (𝐷‘𝑗) ≤ 𝑧))
111	110, 102	ifbieq1d 4448	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑖 = 𝑗 → if((𝐷‘𝑖) ≤ 𝑧, (𝐷‘𝑖), 𝑧) = if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧))
112	101, 111	oveq12d 7153	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 = 𝑗 → ((𝐶‘𝑖)[,)if((𝐷‘𝑖) ≤ 𝑧, (𝐷‘𝑖), 𝑧)) = ((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧)))
113	112	fveq2d 6649	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 = 𝑗 → (vol‘((𝐶‘𝑖)[,)if((𝐷‘𝑖) ≤ 𝑧, (𝐷‘𝑖), 𝑧))) = (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧))))
114	113	cbvmptv 5133	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 ∈ ℕ ↦ (vol‘((𝐶‘𝑖)[,)if((𝐷‘𝑖) ≤ 𝑧, (𝐷‘𝑖), 𝑧)))) = (𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧))))
115	114	eqcomi 2807	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧)))) = (𝑖 ∈ ℕ ↦ (vol‘((𝐶‘𝑖)[,)if((𝐷‘𝑖) ≤ 𝑧, (𝐷‘𝑖), 𝑧))))
116	115	fveq2i 6648	. . . . . . . . . . . . . . . 16 ⊢ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧))))) = (Σ^‘(𝑖 ∈ ℕ ↦ (vol‘((𝐶‘𝑖)[,)if((𝐷‘𝑖) ≤ 𝑧, (𝐷‘𝑖), 𝑧)))))
117	116	breq2i 5038	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 − 𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧))))) ↔ (𝑧 − 𝐴) ≤ (Σ^‘(𝑖 ∈ ℕ ↦ (vol‘((𝐶‘𝑖)[,)if((𝐷‘𝑖) ≤ 𝑧, (𝐷‘𝑖), 𝑧))))))
118	117	rabbii 3420	. . . . . . . . . . . . . 14 ⊢ {𝑧 ∈ (𝐴[,]𝐵) ∣ (𝑧 − 𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧)))))} = {𝑧 ∈ (𝐴[,]𝐵) ∣ (𝑧 − 𝐴) ≤ (Σ^‘(𝑖 ∈ ℕ ↦ (vol‘((𝐶‘𝑖)[,)if((𝐷‘𝑖) ≤ 𝑧, (𝐷‘𝑖), 𝑧)))))}
119	58, 118	eqtri 2821	. . . . . . . . . . . . 13 ⊢ 𝑈 = {𝑧 ∈ (𝐴[,]𝐵) ∣ (𝑧 − 𝐴) ≤ (Σ^‘(𝑖 ∈ ℕ ↦ (vol‘((𝐶‘𝑖)[,)if((𝐷‘𝑖) ≤ 𝑧, (𝐷‘𝑖), 𝑧)))))}
120	64	adantr 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑆 < 𝐵) → 𝑆 ∈ 𝑈)
121	120	3ad2ant1 1130	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑆 < 𝐵) ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ ((𝐶‘𝑗)[,)(𝐷‘𝑗))) → 𝑆 ∈ 𝑈)
122	87	3ad2ant1 1130	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑆 < 𝐵) ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ ((𝐶‘𝑗)[,)(𝐷‘𝑗))) → 𝐴 ≤ 𝑆)
123	88	3ad2ant1 1130	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑆 < 𝐵) ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ ((𝐶‘𝑗)[,)(𝐷‘𝑗))) → 𝑆 < 𝐵)
124		simp2 1134	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑆 < 𝐵) ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ ((𝐶‘𝑗)[,)(𝐷‘𝑗))) → 𝑗 ∈ ℕ)
125		simp3 1135	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑆 < 𝐵) ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ ((𝐶‘𝑗)[,)(𝐷‘𝑗))) → 𝑆 ∈ ((𝐶‘𝑗)[,)(𝐷‘𝑗)))
126		eqid 2798	. . . . . . . . . . . . 13 ⊢ if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵) = if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵)
127	94, 96, 98, 100, 109, 119, 121, 122, 123, 124, 125, 126	hoidmv1lelem2 43231	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑆 < 𝐵) ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ ((𝐶‘𝑗)[,)(𝐷‘𝑗))) → ∃𝑢 ∈ 𝑈 𝑆 < 𝑢)
128	127	3exp 1116	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑆 < 𝐵) → (𝑗 ∈ ℕ → (𝑆 ∈ ((𝐶‘𝑗)[,)(𝐷‘𝑗)) → ∃𝑢 ∈ 𝑈 𝑆 < 𝑢)))
129	128	rexlimdv 3242	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑆 < 𝐵) → (∃𝑗 ∈ ℕ 𝑆 ∈ ((𝐶‘𝑗)[,)(𝐷‘𝑗)) → ∃𝑢 ∈ 𝑈 𝑆 < 𝑢))
130	92, 129	mpd 15	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑆 < 𝐵) → ∃𝑢 ∈ 𝑈 𝑆 < 𝑢)
131	68, 73, 130	syl2anc 587	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐵 ≤ 𝑆) → ∃𝑢 ∈ 𝑈 𝑆 < 𝑢)
132	57	adantr 484	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (𝐴[,]𝐵) ⊆ ℝ)
133	60, 132	sstrid 3926	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 𝑈 ⊆ ℝ)
134	81	adantr 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 𝑈 ≠ ∅)
135	2, 1	jca 515	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
136	135	adantr 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
137	60	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 𝑈 ⊆ (𝐴[,]𝐵))
138	64	adantr 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 𝑆 ∈ 𝑈)
139		iccsupr 12820	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝑈 ⊆ (𝐴[,]𝐵) ∧ 𝑆 ∈ 𝑈) → (𝑈 ⊆ ℝ ∧ 𝑈 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝑈 𝑦 ≤ 𝑥))
140	136, 137, 138, 139	syl3anc 1368	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (𝑈 ⊆ ℝ ∧ 𝑈 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝑈 𝑦 ≤ 𝑥))
141	140	simp3d 1141	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝑈 𝑦 ≤ 𝑥)
142		simpr 488	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 𝑢 ∈ 𝑈)
143		suprub 11589	. . . . . . . . . . . . . 14 ⊢ (((𝑈 ⊆ ℝ ∧ 𝑈 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝑈 𝑦 ≤ 𝑥) ∧ 𝑢 ∈ 𝑈) → 𝑢 ≤ sup(𝑈, ℝ, < ))
144	133, 134, 141, 142, 143	syl31anc 1370	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 𝑢 ≤ sup(𝑈, ℝ, < ))
145	144, 62	breqtrrdi 5072	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 𝑢 ≤ 𝑆)
146	145	ralrimiva 3149	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑢 ∈ 𝑈 𝑢 ≤ 𝑆)
147	60	sseli 3911	. . . . . . . . . . . . . . 15 ⊢ (𝑢 ∈ 𝑈 → 𝑢 ∈ (𝐴[,]𝐵))
148	147	adantl 485	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 𝑢 ∈ (𝐴[,]𝐵))
149	132, 148	sseldd 3916	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 𝑢 ∈ ℝ)
150	66	adantr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 𝑆 ∈ ℝ)
151	149, 150	lenltd 10775	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (𝑢 ≤ 𝑆 ↔ ¬ 𝑆 < 𝑢))
152	151	ralbidva 3161	. . . . . . . . . . 11 ⊢ (𝜑 → (∀𝑢 ∈ 𝑈 𝑢 ≤ 𝑆 ↔ ∀𝑢 ∈ 𝑈 ¬ 𝑆 < 𝑢))
153	146, 152	mpbid 235	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑢 ∈ 𝑈 ¬ 𝑆 < 𝑢)
154		ralnex 3199	. . . . . . . . . 10 ⊢ (∀𝑢 ∈ 𝑈 ¬ 𝑆 < 𝑢 ↔ ¬ ∃𝑢 ∈ 𝑈 𝑆 < 𝑢)
155	153, 154	sylib 221	. . . . . . . . 9 ⊢ (𝜑 → ¬ ∃𝑢 ∈ 𝑈 𝑆 < 𝑢)
156	155	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐵 ≤ 𝑆) → ¬ ∃𝑢 ∈ 𝑈 𝑆 < 𝑢)
157	131, 156	condan 817	. . . . . . 7 ⊢ (𝜑 → 𝐵 ≤ 𝑆)
158		iccleub 12780	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝑆 ∈ (𝐴[,]𝐵)) → 𝑆 ≤ 𝐵)
159	76, 56, 65, 158	syl3anc 1368	. . . . . . 7 ⊢ (𝜑 → 𝑆 ≤ 𝐵)
160	56, 67, 157, 159	xrletrid 12536	. . . . . 6 ⊢ (𝜑 → 𝐵 = 𝑆)
161	160, 64	eqeltrd 2890	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑈)
162	161, 58	eleqtrdi 2900	. . . 4 ⊢ (𝜑 → 𝐵 ∈ {𝑧 ∈ (𝐴[,]𝐵) ∣ (𝑧 − 𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧)))))})
163		oveq1 7142	. . . . . 6 ⊢ (𝑧 = 𝐵 → (𝑧 − 𝐴) = (𝐵 − 𝐴))
164		breq2 5034	. . . . . . . . . . 11 ⊢ (𝑧 = 𝐵 → ((𝐷‘𝑗) ≤ 𝑧 ↔ (𝐷‘𝑗) ≤ 𝐵))
165		id 22	. . . . . . . . . . 11 ⊢ (𝑧 = 𝐵 → 𝑧 = 𝐵)
166	164, 165	ifbieq2d 4450	. . . . . . . . . 10 ⊢ (𝑧 = 𝐵 → if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧) = if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵))
167	166	oveq2d 7151	. . . . . . . . 9 ⊢ (𝑧 = 𝐵 → ((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧)) = ((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵)))
168	167	fveq2d 6649	. . . . . . . 8 ⊢ (𝑧 = 𝐵 → (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧))) = (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵))))
169	168	mpteq2dv 5126	. . . . . . 7 ⊢ (𝑧 = 𝐵 → (𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧)))) = (𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵)))))
170	169	fveq2d 6649	. . . . . 6 ⊢ (𝑧 = 𝐵 → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧))))) = (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵))))))
171	163, 170	breq12d 5043	. . . . 5 ⊢ (𝑧 = 𝐵 → ((𝑧 − 𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧))))) ↔ (𝐵 − 𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵)))))))
172	171	elrab 3628	. . . 4 ⊢ (𝐵 ∈ {𝑧 ∈ (𝐴[,]𝐵) ∣ (𝑧 − 𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧)))))} ↔ (𝐵 ∈ (𝐴[,]𝐵) ∧ (𝐵 − 𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵)))))))
173	162, 172	sylib 221	. . 3 ⊢ (𝜑 → (𝐵 ∈ (𝐴[,]𝐵) ∧ (𝐵 − 𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵)))))))
174	173	simprd 499	. 2 ⊢ (𝜑 → (𝐵 − 𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵))))))
175	3, 55, 32, 174, 49	letrd 10786	1 ⊢ (𝜑 → (𝐵 − 𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)(𝐷‘𝑗))))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  ∀wral 3106  ∃wrex 3107  {crab 3110  Vcvv 3441   ⊆ wss 3881  ∅c0 4243  ifcif 4425  ∪ ciun 4881   class class class wbr 5030   ↦ cmpt 5110  dom cdm 5519  ⟶wf 6320  ‘cfv 6324  (class class class)co 7135  supcsup 8888  ℝcr 10525  0cc0 10526  +∞cpnf 10661  ℝ^*cxr 10663   < clt 10664   ≤ cle 10665   − cmin 10859  ℕcn 11625  [,)cico 12728  [,]cicc 12729  volcvol 24067  Σ^{^}csumge0 43001

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-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-inf2 9088  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-fal 1551  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-int 4839  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-se 5479  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-isom 6333  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-of 7389  df-om 7561  df-1st 7671  df-2nd 7672  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-2o 8086  df-oadd 8089  df-er 8272  df-map 8391  df-pm 8392  df-en 8493  df-dom 8494  df-sdom 8495  df-fin 8496  df-fi 8859  df-sup 8890  df-inf 8891  df-oi 8958  df-dju 9314  df-card 9352  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-q 12337  df-rp 12378  df-xneg 12495  df-xadd 12496  df-xmul 12497  df-ioo 12730  df-ico 12732  df-icc 12733  df-fz 12886  df-fzo 13029  df-fl 13157  df-seq 13365  df-exp 13426  df-hash 13687  df-cj 14450  df-re 14451  df-im 14452  df-sqrt 14586  df-abs 14587  df-clim 14837  df-rlim 14838  df-sum 15035  df-rest 16688  df-topgen 16709  df-psmet 20083  df-xmet 20084  df-met 20085  df-bl 20086  df-mopn 20087  df-top 21499  df-topon 21516  df-bases 21551  df-cmp 21992  df-ovol 24068  df-vol 24069  df-sumge0 43002

This theorem is referenced by:  hoidmv1le  43233