Theorem hoidmv1lelem3 47421

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 11666	. 2 ⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℝ)
4		nnex 12263	. . . . . . 7 ⊢ ℕ ∈ V
5	4	a1i 11	. . . . . 6 ⊢ (𝜑 → ℕ ∈ V)
6		icossicc 13489	. . . . . . . 8 ⊢ (0[,)+∞) ⊆ (0[,]+∞)
7		0xr 11280	. . . . . . . . . 10 ⊢ 0 ∈ ℝ*
8	7	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 0 ∈ ℝ*)
9		pnfxr 11287	. . . . . . . . . 10 ⊢ +∞ ∈ ℝ*
10	9	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → +∞ ∈ ℝ*)
11		hoidmv1lelem3.c	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐶:ℕ⟶ℝ)
12	11	ffvelcdmda 7077	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐶‘𝑗) ∈ ℝ)
13		hoidmv1lelem3.d	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐷:ℕ⟶ℝ)
14	13	ffvelcdmda 7077	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐷‘𝑗) ∈ ℝ)
15	1	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝐵 ∈ ℝ)
16	14, 15	ifcld 4529	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵) ∈ ℝ)
17		volicore 47409	. . . . . . . . . . 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 11283	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵))) ∈ ℝ*)
20	16	rexrd 11283	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵) ∈ ℝ*)
21		icombl 25792	. . . . . . . . . . 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 46789	. . . . . . . . . 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 13172	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵))) < +∞)
26	8, 10, 19, 24, 25	elicod 13448	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵))) ∈ (0[,)+∞))
27	6, 26	sselid 3929	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵))) ∈ (0[,]+∞))
28		eqid 2760	. . . . . . 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 7107	. . . . . 6 ⊢ (𝜑 → (𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵)))):ℕ⟶(0[,]+∞))
30	5, 29	sge0xrcl 47213	. . . . 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 11283	. . . . . 6 ⊢ (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)(𝐷‘𝑗))))) ∈ ℝ*)
34		nfv 1947	. . . . . . 7 ⊢ Ⅎ𝑗𝜑
35		volf 25757	. . . . . . . . 9 ⊢ vol:dom vol⟶(0[,]+∞)
36	35	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → vol:dom vol⟶(0[,]+∞))
37	14	rexrd 11283	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐷‘𝑗) ∈ ℝ*)
38		icombl 25792	. . . . . . . . 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 7078	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (vol‘((𝐶‘𝑗)[,)(𝐷‘𝑗))) ∈ (0[,]+∞))
41	12	rexrd 11283	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐶‘𝑗) ∈ ℝ*)
42	12	leidd 11804	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐶‘𝑗) ≤ (𝐶‘𝑗))
43		min1 13241	. . . . . . . . . 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 13469	. . . . . . . . 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 25761	. . . . . . . 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 1398	. . . . . . 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 47238	. . . . . 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 13172	. . . . . 6 ⊢ (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)(𝐷‘𝑗))))) < +∞)
51	30, 33, 31, 49, 50	xrlelttrd 13211	. . . . 5 ⊢ (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵))))) < +∞)
52	30, 31, 51	xrltned 46187	. . . 4 ⊢ (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵))))) ≠ +∞)
53	52	neneqd 2960	. . 3 ⊢ (𝜑 → ¬ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵))))) = +∞)
54	5, 29	sge0repnf 47214	. . 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 11283	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℝ*)
57	2, 1	iccssred 13487	. . . . . . . . 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 4028	. . . . . . . . . . 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 3977	. . . . . . . . . 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 47419	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑆 ∈ 𝑈 ∧ 𝐴 ∈ 𝑈 ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝑈 𝑦 ≤ 𝑥))
64	63	simp1d 1160	. . . . . . . . . 10 ⊢ (𝜑 → 𝑆 ∈ 𝑈)
65	60, 64	sselid 3929	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ∈ (𝐴[,]𝐵))
66	57, 65	sseldd 3932	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ ℝ)
67	66	rexrd 11283	. . . . . . 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 11381	. . . . . . . . . 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 11283	. . . . . . . . . . . . . 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 3942	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑈 ⊆ ℝ)
81	64	ne0d 4288	. . . . . . . . . . . . . . . 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 12200	. . . . . . . . . . . . . . . 16 ⊢ (((𝑈 ⊆ ℝ ∧ 𝑈 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝑈 𝑦 ≤ 𝑥) ∧ 𝐴 ∈ 𝑈) → 𝐴 ≤ sup(𝑈, ℝ, < ))
85	80, 81, 82, 83, 84	syl31anc 1400	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐴 ≤ sup(𝑈, ℝ, < ))
86	85, 62	breqtrrdi 5147	. . . . . . . . . . . . . 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 13448	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑆 < 𝐵) → 𝑆 ∈ (𝐴[,)𝐵))
90	75, 89	sseldd 3932	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑆 < 𝐵) → 𝑆 ∈ ∪ 𝑗 ∈ ℕ ((𝐶‘𝑗)[,)(𝐷‘𝑗)))
91		eliun 4955	. . . . . . . . . . 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 6878	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 = 𝑗 → (𝐶‘𝑖) = (𝐶‘𝑗))
102		fveq2 6878	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 = 𝑗 → (𝐷‘𝑖) = (𝐷‘𝑗))
103	101, 102	oveq12d 7431	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 = 𝑗 → ((𝐶‘𝑖)[,)(𝐷‘𝑖)) = ((𝐶‘𝑗)[,)(𝐷‘𝑗)))
104	103	fveq2d 6882	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 = 𝑗 → (vol‘((𝐶‘𝑖)[,)(𝐷‘𝑖))) = (vol‘((𝐶‘𝑗)[,)(𝐷‘𝑗))))
105	104	cbvmptv 5209	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ ℕ ↦ (vol‘((𝐶‘𝑖)[,)(𝐷‘𝑖)))) = (𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)(𝐷‘𝑗))))
106	105	fveq2i 6881	. . . . . . . . . . . . . . . 16 ⊢ (Σ^‘(𝑖 ∈ ℕ ↦ (vol‘((𝐶‘𝑖)[,)(𝐷‘𝑖))))) = (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)(𝐷‘𝑗)))))
107	106, 32	eqeltrid 2864	. . . . . . . . . . . . . . 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 5113	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑖 = 𝑗 → ((𝐷‘𝑖) ≤ 𝑧 ↔ (𝐷‘𝑗) ≤ 𝑧))
111	110, 102	ifbieq1d 4507	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑖 = 𝑗 → if((𝐷‘𝑖) ≤ 𝑧, (𝐷‘𝑖), 𝑧) = if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧))
112	101, 111	oveq12d 7431	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 = 𝑗 → ((𝐶‘𝑖)[,)if((𝐷‘𝑖) ≤ 𝑧, (𝐷‘𝑖), 𝑧)) = ((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧)))
113	112	fveq2d 6882	. . . . . . . . . . . . . . . . . . 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 5209	. . . . . . . . . . . . . . . . . 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 2769	. . . . . . . . . . . . . . . . 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 6881	. . . . . . . . . . . . . . . 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 5111	. . . . . . . . . . . . . . 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 3417	. . . . . . . . . . . . . 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 2783	. . . . . . . . . . . . 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 2760	. . . . . . . . . . . . 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 47420	. . . . . . . . . . . 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 3161	. . . . . . . . . 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 3942	. . . . . . . . . . . . . 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 13495	. . . . . . . . . . . . . . . 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 1398	. . . . . . . . . . . . . . 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 12200	. . . . . . . . . . . . . 14 ⊢ (((𝑈 ⊆ ℝ ∧ 𝑈 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝑈 𝑦 ≤ 𝑥) ∧ 𝑢 ∈ 𝑈) → 𝑢 ≤ sup(𝑈, ℝ, < ))
144	133, 134, 141, 142, 143	syl31anc 1400	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 𝑢 ≤ sup(𝑈, ℝ, < ))
145	144, 62	breqtrrdi 5147	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 𝑢 ≤ 𝑆)
146	145	ralrimiva 3154	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑢 ∈ 𝑈 𝑢 ≤ 𝑆)
147	60	sseli 3927	. . . . . . . . . . . . . . 15 ⊢ (𝑢 ∈ 𝑈 → 𝑢 ∈ (𝐴[,]𝐵))
148	147	adantl 487	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 𝑢 ∈ (𝐴[,]𝐵))
149	132, 148	sseldd 3932	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 𝑢 ∈ ℝ)
150	66	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 𝑆 ∈ ℝ)
151	149, 150	lenltd 11380	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (𝑢 ≤ 𝑆 ↔ ¬ 𝑆 < 𝑢))
152	151	ralbidva 3183	. . . . . . . . . . 11 ⊢ (𝜑 → (∀𝑢 ∈ 𝑈 𝑢 ≤ 𝑆 ↔ ∀𝑢 ∈ 𝑈 ¬ 𝑆 < 𝑢))
153	146, 152	mpbid 235	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑢 ∈ 𝑈 ¬ 𝑆 < 𝑢)
154		ralnex 3088	. . . . . . . . . 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 13454	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝑆 ∈ (𝐴[,]𝐵)) → 𝑆 ≤ 𝐵)
159	76, 56, 65, 158	syl3anc 1398	. . . . . . 7 ⊢ (𝜑 → 𝑆 ≤ 𝐵)
160	56, 67, 157, 159	xrletrid 13206	. . . . . 6 ⊢ (𝜑 → 𝐵 = 𝑆)
161	160, 64	eqeltrd 2860	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑈)
162	161, 58	eleqtrdi 2870	. . . 4 ⊢ (𝜑 → 𝐵 ∈ {𝑧 ∈ (𝐴[,]𝐵) ∣ (𝑧 − 𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧)))))})
163		oveq1 7420	. . . . . 6 ⊢ (𝑧 = 𝐵 → (𝑧 − 𝐴) = (𝐵 − 𝐴))
164		breq2 5107	. . . . . . . . . . 11 ⊢ (𝑧 = 𝐵 → ((𝐷‘𝑗) ≤ 𝑧 ↔ (𝐷‘𝑗) ≤ 𝐵))
165		id 23	. . . . . . . . . . 11 ⊢ (𝑧 = 𝐵 → 𝑧 = 𝐵)
166	164, 165	ifbieq2d 4509	. . . . . . . . . 10 ⊢ (𝑧 = 𝐵 → if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧) = if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵))
167	166	oveq2d 7429	. . . . . . . . 9 ⊢ (𝑧 = 𝐵 → ((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧)) = ((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵)))
168	167	fveq2d 6882	. . . . . . . 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 5199	. . . . . . 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 6882	. . . . . 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 5116	. . . . 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 3645	. . . 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 11391	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 2145   ≠ wne 2955  ∀wral 3076  ∃wrex 3086  {crab 3412  Vcvv 3450   ⊆ wss 3899  ∅c0 4279  ifcif 4482  ∪ ciun 4951   class class class wbr 5103   ↦ cmpt 5186  dom cdm 5655  ⟶wf 6529  ‘cfv 6533  (class class class)co 7413  supcsup 9410  ℝcr 11123  0cc0 11124  +∞cpnf 11264  ℝ^*cxr 11266   < clt 11267   ≤ cle 11268   − cmin 11465  ℕcn 12257  [,)cico 13400  [,]cicc 13401  volcvol 25691  Σ^{^}csumge0 47190

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7736  ax-inf2 9620  ax-cnex 11180  ax-resscn 11181  ax-1cn 11182  ax-icn 11183  ax-addcl 11184  ax-addrcl 11185  ax-mulcl 11186  ax-mulrcl 11187  ax-mulcom 11188  ax-addass 11189  ax-mulass 11190  ax-distr 11191  ax-i2m1 11192  ax-1ne0 11193  ax-1rid 11194  ax-rnegex 11195  ax-rrecex 11196  ax-cnre 11197  ax-pre-lttri 11198  ax-pre-lttrn 11199  ax-pre-ltadd 11200  ax-pre-mulgt0 11201  ax-pre-sup 11202

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 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-isom 6542  df-riota 7370  df-ov 7416  df-oprab 7417  df-mpo 7418  df-of 7678  df-om 7863  df-1st 7986  df-2nd 7987  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-1o 8455  df-2o 8456  df-er 8696  df-map 8828  df-pm 8829  df-en 8953  df-dom 8954  df-sdom 8955  df-fin 8956  df-fi 9381  df-sup 9412  df-inf 9413  df-oi 9482  df-dju 9906  df-card 9944  df-pnf 11269  df-mnf 11270  df-xr 11271  df-ltxr 11272  df-le 11273  df-sub 11467  df-neg 11468  df-div 11896  df-nn 12258  df-2 12327  df-3 12328  df-n0 12529  df-z 12616  df-uz 12888  df-q 12998  df-rp 13043  df-xneg 13163  df-xadd 13164  df-xmul 13165  df-ioo 13402  df-ico 13404  df-icc 13405  df-fz 13562  df-fzo 13710  df-fl 13853  df-seq 14066  df-exp 14126  df-hash 14395  df-cj 15186  df-re 15187  df-im 15188  df-sqrt 15322  df-abs 15323  df-clim 15575  df-rlim 15576  df-sum 15774  df-rest 17507  df-topgen 17528  df-psmet 21577  df-xmet 21578  df-met 21579  df-bl 21580  df-mopn 21581  df-top 23119  df-topon 23136  df-bases 23171  df-cmp 23612  df-ovol 25692  df-vol 25693  df-sumge0 47191

This theorem is used by:  hoidmv1le  47422