Theorem hoidmv1lelem3 47876

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 11783	. 2 ⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℝ)
4		nnex 12381	. . . . . . 7 ⊢ ℕ ∈ V
5	4	a1i 11	. . . . . 6 ⊢ (𝜑 → ℕ ∈ V)
6		icossicc 13607	. . . . . . . 8 ⊢ (0[,)+∞) ⊆ (0[,]+∞)
7		0xr 11395	. . . . . . . . . 10 ⊢ 0 ∈ ℝ*
8	7	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 0 ∈ ℝ*)
9		pnfxr 11402	. . . . . . . . . 10 ⊢ +∞ ∈ ℝ*
10	9	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → +∞ ∈ ℝ*)
11		hoidmv1lelem3.c	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐶:ℕ⟶ℝ)
12	11	ffvelcdmda 7116	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐶‘𝑗) ∈ ℝ)
13		hoidmv1lelem3.d	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐷:ℕ⟶ℝ)
14	13	ffvelcdmda 7116	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐷‘𝑗) ∈ ℝ)
15	1	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝐵 ∈ ℝ)
16	14, 15	ifcld 4532	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵) ∈ ℝ)
17		volicore 47864	. . . . . . . . . . 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 596	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵))) ∈ ℝ)
19	18	rexrd 11398	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵))) ∈ ℝ*)
20	16	rexrd 11398	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵) ∈ ℝ*)
21		icombl 25940	. . . . . . . . . . 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 596	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵)) ∈ dom vol)
23		volge0 47244	. . . . . . . . . 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 18	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 0 ≤ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵))))
25	18	ltpnfd 13290	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵))) < +∞)
26	8, 10, 19, 24, 25	elicod 13566	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵))) ∈ (0[,)+∞))
27	6, 26	sselid 3911	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵))) ∈ (0[,]+∞))
28		eqid 2763	. . . . . . 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 7146	. . . . . 6 ⊢ (𝜑 → (𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵)))):ℕ⟶(0[,]+∞))
30	5, 29	sge0xrcl 47668	. . . . 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 11398	. . . . . 6 ⊢ (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)(𝐷‘𝑗))))) ∈ ℝ*)
34		nfv 1949	. . . . . . 7 ⊢ Ⅎ𝑗𝜑
35		volf 25905	. . . . . . . . 9 ⊢ vol:dom vol⟶(0[,]+∞)
36	35	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → vol:dom vol⟶(0[,]+∞))
37	14	rexrd 11398	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐷‘𝑗) ∈ ℝ*)
38		icombl 25940	. . . . . . . . 9 ⊢ (((𝐶‘𝑗) ∈ ℝ ∧ (𝐷‘𝑗) ∈ ℝ*) → ((𝐶‘𝑗)[,)(𝐷‘𝑗)) ∈ dom vol)
39	12, 37, 38	syl2anc 596	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)[,)(𝐷‘𝑗)) ∈ dom vol)
40	36, 39	ffvelcdmd 7117	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (vol‘((𝐶‘𝑗)[,)(𝐷‘𝑗))) ∈ (0[,]+∞))
41	12	rexrd 11398	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐶‘𝑗) ∈ ℝ*)
42	12	leidd 11922	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐶‘𝑗) ≤ (𝐶‘𝑗))
43		min1 13359	. . . . . . . . . 10 ⊢ (((𝐷‘𝑗) ∈ ℝ ∧ 𝐵 ∈ ℝ) → if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵) ≤ (𝐷‘𝑗))
44	14, 15, 43	syl2anc 596	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵) ≤ (𝐷‘𝑗))
45		icossico 13587	. . . . . . . . 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 852	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵)) ⊆ ((𝐶‘𝑗)[,)(𝐷‘𝑗)))
47		volss 25909	. . . . . . . 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 1399	. . . . . . 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 47693	. . . . . 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 13290	. . . . . 6 ⊢ (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)(𝐷‘𝑗))))) < +∞)
51	30, 33, 31, 49, 50	xrlelttrd 13329	. . . . 5 ⊢ (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵))))) < +∞)
52	30, 31, 51	xrltned 46642	. . . 4 ⊢ (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵))))) ≠ +∞)
53	52	neneqd 2964	. . 3 ⊢ (𝜑 → ¬ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵))))) = +∞)
54	5, 29	sge0repnf 47669	. . 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 11398	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℝ*)
57	2, 1	iccssred 13605	. . . . . . . . 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 4008	. . . . . . . . . . 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 3959	. . . . . . . . . 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 47874	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑆 ∈ 𝑈 ∧ 𝐴 ∈ 𝑈 ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝑈 𝑦 ≤ 𝑥))
64	63	simp1d 1160	. . . . . . . . . 10 ⊢ (𝜑 → 𝑆 ∈ 𝑈)
65	60, 64	sselid 3911	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ∈ (𝐴[,]𝐵))
66	57, 65	sseldd 3914	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ ℝ)
67	66	rexrd 11398	. . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ ℝ*)
68		simpl 488	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐵 ≤ 𝑆) → 𝜑)
69		simpr 490	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝐵 ≤ 𝑆) → ¬ 𝐵 ≤ 𝑆)
70	68, 66	syl 18	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 𝐵 ≤ 𝑆) → 𝑆 ∈ ℝ)
71	68, 1	syl 18	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 𝐵 ≤ 𝑆) → 𝐵 ∈ ℝ)
72	70, 71	ltnled 11496	. . . . . . . . . 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 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑆 < 𝐵) → (𝐴[,)𝐵) ⊆ ∪ 𝑗 ∈ ℕ ((𝐶‘𝑗)[,)(𝐷‘𝑗)))
76	2	rexrd 11398	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐴 ∈ ℝ*)
77	76	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑆 < 𝐵) → 𝐴 ∈ ℝ*)
78	56	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑆 < 𝐵) → 𝐵 ∈ ℝ*)
79	67	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑆 < 𝐵) → 𝑆 ∈ ℝ*)
80	60, 57	sstrid 3924	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑈 ⊆ ℝ)
81	64	ne0d 4291	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑈 ≠ ∅)
82	63	simp3d 1162	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝑈 𝑦 ≤ 𝑥)
83	63	simp2d 1161	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐴 ∈ 𝑈)
84		suprub 12318	. . . . . . . . . . . . . . . 16 ⊢ (((𝑈 ⊆ ℝ ∧ 𝑈 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝑈 𝑦 ≤ 𝑥) ∧ 𝐴 ∈ 𝑈) → 𝐴 ≤ sup(𝑈, ℝ, < ))
85	80, 81, 82, 83, 84	syl31anc 1401	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐴 ≤ sup(𝑈, ℝ, < ))
86	85, 62	breqtrrdi 5153	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐴 ≤ 𝑆)
87	86	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑆 < 𝐵) → 𝐴 ≤ 𝑆)
88		simpr 490	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑆 < 𝐵) → 𝑆 < 𝐵)
89	77, 78, 79, 87, 88	elicod 13566	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑆 < 𝐵) → 𝑆 ∈ (𝐴[,)𝐵))
90	75, 89	sseldd 3914	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑆 < 𝐵) → 𝑆 ∈ ∪ 𝑗 ∈ ℕ ((𝐶‘𝑗)[,)(𝐷‘𝑗)))
91		eliun 4960	. . . . . . . . . . 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 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑆 < 𝐵) → 𝐴 ∈ ℝ)
94	93	3ad2ant1 1151	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑆 < 𝐵) ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ ((𝐶‘𝑗)[,)(𝐷‘𝑗))) → 𝐴 ∈ ℝ)
95	1	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑆 < 𝐵) → 𝐵 ∈ ℝ)
96	95	3ad2ant1 1151	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑆 < 𝐵) ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ ((𝐶‘𝑗)[,)(𝐷‘𝑗))) → 𝐵 ∈ ℝ)
97	11	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑆 < 𝐵) → 𝐶:ℕ⟶ℝ)
98	97	3ad2ant1 1151	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑆 < 𝐵) ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ ((𝐶‘𝑗)[,)(𝐷‘𝑗))) → 𝐶:ℕ⟶ℝ)
99	13	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑆 < 𝐵) → 𝐷:ℕ⟶ℝ)
100	99	3ad2ant1 1151	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑆 < 𝐵) ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ ((𝐶‘𝑗)[,)(𝐷‘𝑗))) → 𝐷:ℕ⟶ℝ)
101		fveq2 6916	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 = 𝑗 → (𝐶‘𝑖) = (𝐶‘𝑗))
102		fveq2 6916	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 = 𝑗 → (𝐷‘𝑖) = (𝐷‘𝑗))
103	101, 102	oveq12d 7469	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 = 𝑗 → ((𝐶‘𝑖)[,)(𝐷‘𝑖)) = ((𝐶‘𝑗)[,)(𝐷‘𝑗)))
104	103	fveq2d 6920	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 = 𝑗 → (vol‘((𝐶‘𝑖)[,)(𝐷‘𝑖))) = (vol‘((𝐶‘𝑗)[,)(𝐷‘𝑗))))
105	104	cbvmptv 5215	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ ℕ ↦ (vol‘((𝐶‘𝑖)[,)(𝐷‘𝑖)))) = (𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)(𝐷‘𝑗))))
106	105	fveq2i 6919	. . . . . . . . . . . . . . . 16 ⊢ (Σ^‘(𝑖 ∈ ℕ ↦ (vol‘((𝐶‘𝑖)[,)(𝐷‘𝑖))))) = (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)(𝐷‘𝑗)))))
107	106, 32	eqeltrid 2868	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (Σ^‘(𝑖 ∈ ℕ ↦ (vol‘((𝐶‘𝑖)[,)(𝐷‘𝑖))))) ∈ ℝ)
108	107	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑆 < 𝐵) → (Σ^‘(𝑖 ∈ ℕ ↦ (vol‘((𝐶‘𝑖)[,)(𝐷‘𝑖))))) ∈ ℝ)
109	108	3ad2ant1 1151	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑆 < 𝐵) ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ ((𝐶‘𝑗)[,)(𝐷‘𝑗))) → (Σ^‘(𝑖 ∈ ℕ ↦ (vol‘((𝐶‘𝑖)[,)(𝐷‘𝑖))))) ∈ ℝ)
110	102	breq1d 5118	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑖 = 𝑗 → ((𝐷‘𝑖) ≤ 𝑧 ↔ (𝐷‘𝑗) ≤ 𝑧))
111	110, 102	ifbieq1d 4510	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑖 = 𝑗 → if((𝐷‘𝑖) ≤ 𝑧, (𝐷‘𝑖), 𝑧) = if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧))
112	101, 111	oveq12d 7469	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 = 𝑗 → ((𝐶‘𝑖)[,)if((𝐷‘𝑖) ≤ 𝑧, (𝐷‘𝑖), 𝑧)) = ((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧)))
113	112	fveq2d 6920	. . . . . . . . . . . . . . . . . . 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 5215	. . . . . . . . . . . . . . . . . 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 2772	. . . . . . . . . . . . . . . . 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 6919	. . . . . . . . . . . . . . . 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 5116	. . . . . . . . . . . . . . 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 3421	. . . . . . . . . . . . . 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 2786	. . . . . . . . . . . . 13 ⊢ 𝑈 = {𝑧 ∈ (𝐴[,]𝐵) ∣ (𝑧 − 𝐴) ≤ (Σ^‘(𝑖 ∈ ℕ ↦ (vol‘((𝐶‘𝑖)[,)if((𝐷‘𝑖) ≤ 𝑧, (𝐷‘𝑖), 𝑧)))))}
120	64	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑆 < 𝐵) → 𝑆 ∈ 𝑈)
121	120	3ad2ant1 1151	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑆 < 𝐵) ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ ((𝐶‘𝑗)[,)(𝐷‘𝑗))) → 𝑆 ∈ 𝑈)
122	87	3ad2ant1 1151	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑆 < 𝐵) ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ ((𝐶‘𝑗)[,)(𝐷‘𝑗))) → 𝐴 ≤ 𝑆)
123	88	3ad2ant1 1151	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑆 < 𝐵) ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ ((𝐶‘𝑗)[,)(𝐷‘𝑗))) → 𝑆 < 𝐵)
124		simp2 1155	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑆 < 𝐵) ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ ((𝐶‘𝑗)[,)(𝐷‘𝑗))) → 𝑗 ∈ ℕ)
125		simp3 1156	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑆 < 𝐵) ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ ((𝐶‘𝑗)[,)(𝐷‘𝑗))) → 𝑆 ∈ ((𝐶‘𝑗)[,)(𝐷‘𝑗)))
126		eqid 2763	. . . . . . . . . . . . 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 47875	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑆 < 𝐵) ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ ((𝐶‘𝑗)[,)(𝐷‘𝑗))) → ∃𝑢 ∈ 𝑈 𝑆 < 𝑢)
128	127	3exp 1137	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑆 < 𝐵) → (𝑗 ∈ ℕ → (𝑆 ∈ ((𝐶‘𝑗)[,)(𝐷‘𝑗)) → ∃𝑢 ∈ 𝑈 𝑆 < 𝑢)))
129	128	rexlimdv 3165	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑆 < 𝐵) → (∃𝑗 ∈ ℕ 𝑆 ∈ ((𝐶‘𝑗)[,)(𝐷‘𝑗)) → ∃𝑢 ∈ 𝑈 𝑆 < 𝑢))
130	92, 129	mpd 16	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑆 < 𝐵) → ∃𝑢 ∈ 𝑈 𝑆 < 𝑢)
131	68, 73, 130	syl2anc 596	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐵 ≤ 𝑆) → ∃𝑢 ∈ 𝑈 𝑆 < 𝑢)
132	57	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (𝐴[,]𝐵) ⊆ ℝ)
133	60, 132	sstrid 3924	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 𝑈 ⊆ ℝ)
134	81	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 𝑈 ≠ ∅)
135	2, 1	jca 521	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
136	135	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
137	60	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 𝑈 ⊆ (𝐴[,]𝐵))
138	64	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 𝑆 ∈ 𝑈)
139		iccsupr 13613	. . . . . . . . . . . . . . . 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 1399	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (𝑈 ⊆ ℝ ∧ 𝑈 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝑈 𝑦 ≤ 𝑥))
141	140	simp3d 1162	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝑈 𝑦 ≤ 𝑥)
142		simpr 490	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 𝑢 ∈ 𝑈)
143		suprub 12318	. . . . . . . . . . . . . 14 ⊢ (((𝑈 ⊆ ℝ ∧ 𝑈 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝑈 𝑦 ≤ 𝑥) ∧ 𝑢 ∈ 𝑈) → 𝑢 ≤ sup(𝑈, ℝ, < ))
144	133, 134, 141, 142, 143	syl31anc 1401	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 𝑢 ≤ sup(𝑈, ℝ, < ))
145	144, 62	breqtrrdi 5153	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 𝑢 ≤ 𝑆)
146	145	ralrimiva 3158	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑢 ∈ 𝑈 𝑢 ≤ 𝑆)
147	60	sseli 3909	. . . . . . . . . . . . . . 15 ⊢ (𝑢 ∈ 𝑈 → 𝑢 ∈ (𝐴[,]𝐵))
148	147	adantl 487	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 𝑢 ∈ (𝐴[,]𝐵))
149	132, 148	sseldd 3914	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 𝑢 ∈ ℝ)
150	66	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 𝑆 ∈ ℝ)
151	149, 150	lenltd 11495	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (𝑢 ≤ 𝑆 ↔ ¬ 𝑆 < 𝑢))
152	151	ralbidva 3187	. . . . . . . . . . 11 ⊢ (𝜑 → (∀𝑢 ∈ 𝑈 𝑢 ≤ 𝑆 ↔ ∀𝑢 ∈ 𝑈 ¬ 𝑆 < 𝑢))
153	146, 152	mpbid 235	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑢 ∈ 𝑈 ¬ 𝑆 < 𝑢)
154		ralnex 3092	. . . . . . . . . 10 ⊢ (∀𝑢 ∈ 𝑈 ¬ 𝑆 < 𝑢 ↔ ¬ ∃𝑢 ∈ 𝑈 𝑆 < 𝑢)
155	153, 154	sylib 221	. . . . . . . . 9 ⊢ (𝜑 → ¬ ∃𝑢 ∈ 𝑈 𝑆 < 𝑢)
156	155	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐵 ≤ 𝑆) → ¬ ∃𝑢 ∈ 𝑈 𝑆 < 𝑢)
157	131, 156	condan 830	. . . . . . 7 ⊢ (𝜑 → 𝐵 ≤ 𝑆)
158		iccleub 13572	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝑆 ∈ (𝐴[,]𝐵)) → 𝑆 ≤ 𝐵)
159	76, 56, 65, 158	syl3anc 1399	. . . . . . 7 ⊢ (𝜑 → 𝑆 ≤ 𝐵)
160	56, 67, 157, 159	xrletrid 13324	. . . . . 6 ⊢ (𝜑 → 𝐵 = 𝑆)
161	160, 64	eqeltrd 2864	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑈)
162	161, 58	eleqtrdi 2874	. . . 4 ⊢ (𝜑 → 𝐵 ∈ {𝑧 ∈ (𝐴[,]𝐵) ∣ (𝑧 − 𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧)))))})
163		oveq1 7458	. . . . . 6 ⊢ (𝑧 = 𝐵 → (𝑧 − 𝐴) = (𝐵 − 𝐴))
164		breq2 5112	. . . . . . . . . . 11 ⊢ (𝑧 = 𝐵 → ((𝐷‘𝑗) ≤ 𝑧 ↔ (𝐷‘𝑗) ≤ 𝐵))
165		id 23	. . . . . . . . . . 11 ⊢ (𝑧 = 𝐵 → 𝑧 = 𝐵)
166	164, 165	ifbieq2d 4512	. . . . . . . . . 10 ⊢ (𝑧 = 𝐵 → if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧) = if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵))
167	166	oveq2d 7467	. . . . . . . . 9 ⊢ (𝑧 = 𝐵 → ((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧)) = ((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵)))
168	167	fveq2d 6920	. . . . . . . 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 5205	. . . . . . 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 6920	. . . . . 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 5121	. . . . 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 3648	. . . 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 501	. 2 ⊢ (𝜑 → (𝐵 − 𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵))))))
175	3, 55, 32, 174, 49	letrd 11506	1 ⊢ (𝜑 → (𝐵 − 𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)(𝐷‘𝑗))))))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147   ≠ wne 2959  ∀wral 3080  ∃wrex 3090  {crab 3416  Vcvv 3454   ⊆ wss 3900  ∅c0 4281  ifcif 4485  ∪ ciun 4956   class class class wbr 5108   ↦ cmpt 5192  dom cdm 5658  ⟶wf 6562  ‘cfv 6566  (class class class)co 7451  supcsup 9469  ℝcr 11238  0cc0 11239  +∞cpnf 11379  ℝ^*cxr 11381   < clt 11382   ≤ cle 11383   − cmin 11580  ℕcn 12375  [,)cico 13518  [,]cicc 13519  volcvol 25839  Σ^{^}csumge0 47645

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783  ax-inf2 9679  ax-cnex 11295  ax-resscn 11296  ax-1cn 11297  ax-icn 11298  ax-addcl 11299  ax-addrcl 11300  ax-mulcl 11301  ax-mulrcl 11302  ax-mulcom 11303  ax-addass 11304  ax-mulass 11305  ax-distr 11306  ax-i2m1 11307  ax-1ne0 11308  ax-1rid 11309  ax-rnegex 11310  ax-rrecex 11311  ax-cnre 11312  ax-pre-lttri 11313  ax-pre-lttrn 11314  ax-pre-ltadd 11315  ax-pre-mulgt0 11316  ax-pre-sup 11317

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-int 4913  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-se 5612  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-isom 6575  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-of 7725  df-om 7912  df-1st 8035  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456  df-1o 8514  df-2o 8515  df-er 8755  df-map 8887  df-pm 8888  df-en 9012  df-dom 9013  df-sdom 9014  df-fin 9015  df-fi 9440  df-sup 9471  df-inf 9472  df-oi 9541  df-dju 10019  df-card 10057  df-pnf 11384  df-mnf 11385  df-xr 11386  df-ltxr 11387  df-le 11388  df-sub 11582  df-neg 11583  df-div 12014  df-nn 12376  df-2 12445  df-3 12446  df-n0 12647  df-z 12734  df-uz 13006  df-q 13116  df-rp 13161  df-xneg 13281  df-xadd 13282  df-xmul 13283  df-ioo 13520  df-ico 13522  df-icc 13523  df-fz 13680  df-fzo 13829  df-fl 13972  df-seq 14185  df-exp 14245  df-hash 14516  df-cj 15309  df-re 15310  df-im 15311  df-sqrt 15445  df-abs 15446  df-clim 15699  df-rlim 15700  df-sum 15898  df-rest 17643  df-topgen 17664  df-psmet 21721  df-xmet 21722  df-met 21723  df-bl 21724  df-mopn 21725  df-top 23267  df-topon 23284  df-bases 23319  df-cmp 23760  df-ovol 25840  df-vol 25841  df-sumge0 47646

This theorem is used by:  hoidmv1le  47877