Theorem hoidmv1lelem3 42895

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 11068	. 2 ⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℝ)
4		nnex 11644	. . . . . . 7 ⊢ ℕ ∈ V
5	4	a1i 11	. . . . . 6 ⊢ (𝜑 → ℕ ∈ V)
6		icossicc 12825	. . . . . . . 8 ⊢ (0[,)+∞) ⊆ (0[,]+∞)
7		0xr 10688	. . . . . . . . . 10 ⊢ 0 ∈ ℝ*
8	7	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 0 ∈ ℝ*)
9		pnfxr 10695	. . . . . . . . . 10 ⊢ +∞ ∈ ℝ*
10	9	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → +∞ ∈ ℝ*)
11		hoidmv1lelem3.c	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐶:ℕ⟶ℝ)
12	11	ffvelrnda 6851	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐶‘𝑗) ∈ ℝ)
13		hoidmv1lelem3.d	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐷:ℕ⟶ℝ)
14	13	ffvelrnda 6851	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐷‘𝑗) ∈ ℝ)
15	1	adantr 483	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝐵 ∈ ℝ)
16	14, 15	ifcld 4512	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵) ∈ ℝ)
17		volicore 42883	. . . . . . . . . . 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 586	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵))) ∈ ℝ)
19	18	rexrd 10691	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵))) ∈ ℝ*)
20	16	rexrd 10691	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵) ∈ ℝ*)
21		icombl 24165	. . . . . . . . . . 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 586	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵)) ∈ dom vol)
23		volge0 42266	. . . . . . . . . 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 12517	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵))) < +∞)
26	8, 10, 19, 24, 25	elicod 12788	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵))) ∈ (0[,)+∞))
27	6, 26	sseldi 3965	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵))) ∈ (0[,]+∞))
28		eqid 2821	. . . . . . 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 6878	. . . . . 6 ⊢ (𝜑 → (𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵)))):ℕ⟶(0[,]+∞))
30	5, 29	sge0xrcl 42687	. . . . 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 10691	. . . . . 6 ⊢ (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)(𝐷‘𝑗))))) ∈ ℝ*)
34		nfv 1915	. . . . . . 7 ⊢ Ⅎ𝑗𝜑
35		volf 24130	. . . . . . . . 9 ⊢ vol:dom vol⟶(0[,]+∞)
36	35	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → vol:dom vol⟶(0[,]+∞))
37	14	rexrd 10691	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐷‘𝑗) ∈ ℝ*)
38		icombl 24165	. . . . . . . . 9 ⊢ (((𝐶‘𝑗) ∈ ℝ ∧ (𝐷‘𝑗) ∈ ℝ*) → ((𝐶‘𝑗)[,)(𝐷‘𝑗)) ∈ dom vol)
39	12, 37, 38	syl2anc 586	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)[,)(𝐷‘𝑗)) ∈ dom vol)
40	36, 39	ffvelrnd 6852	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (vol‘((𝐶‘𝑗)[,)(𝐷‘𝑗))) ∈ (0[,]+∞))
41	12	rexrd 10691	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐶‘𝑗) ∈ ℝ*)
42	12	leidd 11206	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐶‘𝑗) ≤ (𝐶‘𝑗))
43		min1 12583	. . . . . . . . . 10 ⊢ (((𝐷‘𝑗) ∈ ℝ ∧ 𝐵 ∈ ℝ) → if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵) ≤ (𝐷‘𝑗))
44	14, 15, 43	syl2anc 586	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵) ≤ (𝐷‘𝑗))
45		icossico 12807	. . . . . . . . 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 836	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵)) ⊆ ((𝐶‘𝑗)[,)(𝐷‘𝑗)))
47		volss 24134	. . . . . . . 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 1367	. . . . . . 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 42712	. . . . . 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 12517	. . . . . 6 ⊢ (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)(𝐷‘𝑗))))) < +∞)
51	30, 33, 31, 49, 50	xrlelttrd 12554	. . . . 5 ⊢ (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵))))) < +∞)
52	30, 31, 51	xrltned 41645	. . . 4 ⊢ (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵))))) ≠ +∞)
53	52	neneqd 3021	. . 3 ⊢ (𝜑 → ¬ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵))))) = +∞)
54	5, 29	sge0repnf 42688	. . 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 259	. 2 ⊢ (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵))))) ∈ ℝ)
56	1	rexrd 10691	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℝ*)
57	2, 1	iccssred 41800	. . . . . . . . 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 4056	. . . . . . . . . . 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 4001	. . . . . . . . . 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 42893	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑆 ∈ 𝑈 ∧ 𝐴 ∈ 𝑈 ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝑈 𝑦 ≤ 𝑥))
64	63	simp1d 1138	. . . . . . . . . 10 ⊢ (𝜑 → 𝑆 ∈ 𝑈)
65	60, 64	sseldi 3965	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ∈ (𝐴[,]𝐵))
66	57, 65	sseldd 3968	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ ℝ)
67	66	rexrd 10691	. . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ ℝ*)
68		simpl 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐵 ≤ 𝑆) → 𝜑)
69		simpr 487	. . . . . . . . . 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 10787	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝐵 ≤ 𝑆) → (𝑆 < 𝐵 ↔ ¬ 𝐵 ≤ 𝑆))
73	69, 72	mpbird 259	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐵 ≤ 𝑆) → 𝑆 < 𝐵)
74		hoidmv1lelem3.x	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐴[,)𝐵) ⊆ ∪ 𝑗 ∈ ℕ ((𝐶‘𝑗)[,)(𝐷‘𝑗)))
75	74	adantr 483	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑆 < 𝐵) → (𝐴[,)𝐵) ⊆ ∪ 𝑗 ∈ ℕ ((𝐶‘𝑗)[,)(𝐷‘𝑗)))
76	2	rexrd 10691	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐴 ∈ ℝ*)
77	76	adantr 483	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑆 < 𝐵) → 𝐴 ∈ ℝ*)
78	56	adantr 483	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑆 < 𝐵) → 𝐵 ∈ ℝ*)
79	67	adantr 483	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑆 < 𝐵) → 𝑆 ∈ ℝ*)
80	60, 57	sstrid 3978	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑈 ⊆ ℝ)
81	64	ne0d 4301	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑈 ≠ ∅)
82	63	simp3d 1140	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝑈 𝑦 ≤ 𝑥)
83	63	simp2d 1139	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐴 ∈ 𝑈)
84		suprub 11602	. . . . . . . . . . . . . . . 16 ⊢ (((𝑈 ⊆ ℝ ∧ 𝑈 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝑈 𝑦 ≤ 𝑥) ∧ 𝐴 ∈ 𝑈) → 𝐴 ≤ sup(𝑈, ℝ, < ))
85	80, 81, 82, 83, 84	syl31anc 1369	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐴 ≤ sup(𝑈, ℝ, < ))
86	85, 62	breqtrrdi 5108	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐴 ≤ 𝑆)
87	86	adantr 483	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑆 < 𝐵) → 𝐴 ≤ 𝑆)
88		simpr 487	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑆 < 𝐵) → 𝑆 < 𝐵)
89	77, 78, 79, 87, 88	elicod 12788	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑆 < 𝐵) → 𝑆 ∈ (𝐴[,)𝐵))
90	75, 89	sseldd 3968	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑆 < 𝐵) → 𝑆 ∈ ∪ 𝑗 ∈ ℕ ((𝐶‘𝑗)[,)(𝐷‘𝑗)))
91		eliun 4923	. . . . . . . . . . 11 ⊢ (𝑆 ∈ ∪ 𝑗 ∈ ℕ ((𝐶‘𝑗)[,)(𝐷‘𝑗)) ↔ ∃𝑗 ∈ ℕ 𝑆 ∈ ((𝐶‘𝑗)[,)(𝐷‘𝑗)))
92	90, 91	sylib 220	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑆 < 𝐵) → ∃𝑗 ∈ ℕ 𝑆 ∈ ((𝐶‘𝑗)[,)(𝐷‘𝑗)))
93	2	adantr 483	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑆 < 𝐵) → 𝐴 ∈ ℝ)
94	93	3ad2ant1 1129	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑆 < 𝐵) ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ ((𝐶‘𝑗)[,)(𝐷‘𝑗))) → 𝐴 ∈ ℝ)
95	1	adantr 483	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑆 < 𝐵) → 𝐵 ∈ ℝ)
96	95	3ad2ant1 1129	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑆 < 𝐵) ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ ((𝐶‘𝑗)[,)(𝐷‘𝑗))) → 𝐵 ∈ ℝ)
97	11	adantr 483	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑆 < 𝐵) → 𝐶:ℕ⟶ℝ)
98	97	3ad2ant1 1129	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑆 < 𝐵) ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ ((𝐶‘𝑗)[,)(𝐷‘𝑗))) → 𝐶:ℕ⟶ℝ)
99	13	adantr 483	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑆 < 𝐵) → 𝐷:ℕ⟶ℝ)
100	99	3ad2ant1 1129	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑆 < 𝐵) ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ ((𝐶‘𝑗)[,)(𝐷‘𝑗))) → 𝐷:ℕ⟶ℝ)
101		fveq2 6670	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 = 𝑗 → (𝐶‘𝑖) = (𝐶‘𝑗))
102		fveq2 6670	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 = 𝑗 → (𝐷‘𝑖) = (𝐷‘𝑗))
103	101, 102	oveq12d 7174	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 = 𝑗 → ((𝐶‘𝑖)[,)(𝐷‘𝑖)) = ((𝐶‘𝑗)[,)(𝐷‘𝑗)))
104	103	fveq2d 6674	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 = 𝑗 → (vol‘((𝐶‘𝑖)[,)(𝐷‘𝑖))) = (vol‘((𝐶‘𝑗)[,)(𝐷‘𝑗))))
105	104	cbvmptv 5169	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ ℕ ↦ (vol‘((𝐶‘𝑖)[,)(𝐷‘𝑖)))) = (𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)(𝐷‘𝑗))))
106	105	fveq2i 6673	. . . . . . . . . . . . . . . 16 ⊢ (Σ^‘(𝑖 ∈ ℕ ↦ (vol‘((𝐶‘𝑖)[,)(𝐷‘𝑖))))) = (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)(𝐷‘𝑗)))))
107	106, 32	eqeltrid 2917	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (Σ^‘(𝑖 ∈ ℕ ↦ (vol‘((𝐶‘𝑖)[,)(𝐷‘𝑖))))) ∈ ℝ)
108	107	adantr 483	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑆 < 𝐵) → (Σ^‘(𝑖 ∈ ℕ ↦ (vol‘((𝐶‘𝑖)[,)(𝐷‘𝑖))))) ∈ ℝ)
109	108	3ad2ant1 1129	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑆 < 𝐵) ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ ((𝐶‘𝑗)[,)(𝐷‘𝑗))) → (Σ^‘(𝑖 ∈ ℕ ↦ (vol‘((𝐶‘𝑖)[,)(𝐷‘𝑖))))) ∈ ℝ)
110	102	breq1d 5076	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑖 = 𝑗 → ((𝐷‘𝑖) ≤ 𝑧 ↔ (𝐷‘𝑗) ≤ 𝑧))
111	110, 102	ifbieq1d 4490	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑖 = 𝑗 → if((𝐷‘𝑖) ≤ 𝑧, (𝐷‘𝑖), 𝑧) = if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧))
112	101, 111	oveq12d 7174	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 = 𝑗 → ((𝐶‘𝑖)[,)if((𝐷‘𝑖) ≤ 𝑧, (𝐷‘𝑖), 𝑧)) = ((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧)))
113	112	fveq2d 6674	. . . . . . . . . . . . . . . . . . 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 5169	. . . . . . . . . . . . . . . . . 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 2830	. . . . . . . . . . . . . . . . 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 6673	. . . . . . . . . . . . . . . 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 5074	. . . . . . . . . . . . . . 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 3473	. . . . . . . . . . . . . 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 2844	. . . . . . . . . . . . 13 ⊢ 𝑈 = {𝑧 ∈ (𝐴[,]𝐵) ∣ (𝑧 − 𝐴) ≤ (Σ^‘(𝑖 ∈ ℕ ↦ (vol‘((𝐶‘𝑖)[,)if((𝐷‘𝑖) ≤ 𝑧, (𝐷‘𝑖), 𝑧)))))}
120	64	adantr 483	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑆 < 𝐵) → 𝑆 ∈ 𝑈)
121	120	3ad2ant1 1129	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑆 < 𝐵) ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ ((𝐶‘𝑗)[,)(𝐷‘𝑗))) → 𝑆 ∈ 𝑈)
122	87	3ad2ant1 1129	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑆 < 𝐵) ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ ((𝐶‘𝑗)[,)(𝐷‘𝑗))) → 𝐴 ≤ 𝑆)
123	88	3ad2ant1 1129	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑆 < 𝐵) ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ ((𝐶‘𝑗)[,)(𝐷‘𝑗))) → 𝑆 < 𝐵)
124		simp2 1133	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑆 < 𝐵) ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ ((𝐶‘𝑗)[,)(𝐷‘𝑗))) → 𝑗 ∈ ℕ)
125		simp3 1134	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑆 < 𝐵) ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ ((𝐶‘𝑗)[,)(𝐷‘𝑗))) → 𝑆 ∈ ((𝐶‘𝑗)[,)(𝐷‘𝑗)))
126		eqid 2821	. . . . . . . . . . . . 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 42894	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑆 < 𝐵) ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ ((𝐶‘𝑗)[,)(𝐷‘𝑗))) → ∃𝑢 ∈ 𝑈 𝑆 < 𝑢)
128	127	3exp 1115	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑆 < 𝐵) → (𝑗 ∈ ℕ → (𝑆 ∈ ((𝐶‘𝑗)[,)(𝐷‘𝑗)) → ∃𝑢 ∈ 𝑈 𝑆 < 𝑢)))
129	128	rexlimdv 3283	. . . . . . . . . 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 586	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐵 ≤ 𝑆) → ∃𝑢 ∈ 𝑈 𝑆 < 𝑢)
132	57	adantr 483	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (𝐴[,]𝐵) ⊆ ℝ)
133	60, 132	sstrid 3978	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 𝑈 ⊆ ℝ)
134	81	adantr 483	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 𝑈 ≠ ∅)
135	2, 1	jca 514	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
136	135	adantr 483	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
137	60	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 𝑈 ⊆ (𝐴[,]𝐵))
138	64	adantr 483	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 𝑆 ∈ 𝑈)
139		iccsupr 12831	. . . . . . . . . . . . . . . 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 1367	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (𝑈 ⊆ ℝ ∧ 𝑈 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝑈 𝑦 ≤ 𝑥))
141	140	simp3d 1140	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝑈 𝑦 ≤ 𝑥)
142		simpr 487	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 𝑢 ∈ 𝑈)
143		suprub 11602	. . . . . . . . . . . . . 14 ⊢ (((𝑈 ⊆ ℝ ∧ 𝑈 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝑈 𝑦 ≤ 𝑥) ∧ 𝑢 ∈ 𝑈) → 𝑢 ≤ sup(𝑈, ℝ, < ))
144	133, 134, 141, 142, 143	syl31anc 1369	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 𝑢 ≤ sup(𝑈, ℝ, < ))
145	144, 62	breqtrrdi 5108	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 𝑢 ≤ 𝑆)
146	145	ralrimiva 3182	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑢 ∈ 𝑈 𝑢 ≤ 𝑆)
147	60	sseli 3963	. . . . . . . . . . . . . . 15 ⊢ (𝑢 ∈ 𝑈 → 𝑢 ∈ (𝐴[,]𝐵))
148	147	adantl 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 𝑢 ∈ (𝐴[,]𝐵))
149	132, 148	sseldd 3968	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 𝑢 ∈ ℝ)
150	66	adantr 483	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → 𝑆 ∈ ℝ)
151	149, 150	lenltd 10786	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑈) → (𝑢 ≤ 𝑆 ↔ ¬ 𝑆 < 𝑢))
152	151	ralbidva 3196	. . . . . . . . . . 11 ⊢ (𝜑 → (∀𝑢 ∈ 𝑈 𝑢 ≤ 𝑆 ↔ ∀𝑢 ∈ 𝑈 ¬ 𝑆 < 𝑢))
153	146, 152	mpbid 234	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑢 ∈ 𝑈 ¬ 𝑆 < 𝑢)
154		ralnex 3236	. . . . . . . . . 10 ⊢ (∀𝑢 ∈ 𝑈 ¬ 𝑆 < 𝑢 ↔ ¬ ∃𝑢 ∈ 𝑈 𝑆 < 𝑢)
155	153, 154	sylib 220	. . . . . . . . 9 ⊢ (𝜑 → ¬ ∃𝑢 ∈ 𝑈 𝑆 < 𝑢)
156	155	adantr 483	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐵 ≤ 𝑆) → ¬ ∃𝑢 ∈ 𝑈 𝑆 < 𝑢)
157	131, 156	condan 816	. . . . . . 7 ⊢ (𝜑 → 𝐵 ≤ 𝑆)
158		iccleub 12793	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝑆 ∈ (𝐴[,]𝐵)) → 𝑆 ≤ 𝐵)
159	76, 56, 65, 158	syl3anc 1367	. . . . . . 7 ⊢ (𝜑 → 𝑆 ≤ 𝐵)
160	56, 67, 157, 159	xrletrid 12549	. . . . . 6 ⊢ (𝜑 → 𝐵 = 𝑆)
161	160, 64	eqeltrd 2913	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑈)
162	161, 58	eleqtrdi 2923	. . . 4 ⊢ (𝜑 → 𝐵 ∈ {𝑧 ∈ (𝐴[,]𝐵) ∣ (𝑧 − 𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧)))))})
163		oveq1 7163	. . . . . 6 ⊢ (𝑧 = 𝐵 → (𝑧 − 𝐴) = (𝐵 − 𝐴))
164		breq2 5070	. . . . . . . . . . 11 ⊢ (𝑧 = 𝐵 → ((𝐷‘𝑗) ≤ 𝑧 ↔ (𝐷‘𝑗) ≤ 𝐵))
165		id 22	. . . . . . . . . . 11 ⊢ (𝑧 = 𝐵 → 𝑧 = 𝐵)
166	164, 165	ifbieq2d 4492	. . . . . . . . . 10 ⊢ (𝑧 = 𝐵 → if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧) = if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵))
167	166	oveq2d 7172	. . . . . . . . 9 ⊢ (𝑧 = 𝐵 → ((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧)) = ((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵)))
168	167	fveq2d 6674	. . . . . . . 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 5162	. . . . . . 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 6674	. . . . . 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 5079	. . . . 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 3680	. . . 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 220	. . 3 ⊢ (𝜑 → (𝐵 ∈ (𝐴[,]𝐵) ∧ (𝐵 − 𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵)))))))
174	173	simprd 498	. 2 ⊢ (𝜑 → (𝐵 − 𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐵, (𝐷‘𝑗), 𝐵))))))
175	3, 55, 32, 174, 49	letrd 10797	1 ⊢ (𝜑 → (𝐵 − 𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)(𝐷‘𝑗))))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114   ≠ wne 3016  ∀wral 3138  ∃wrex 3139  {crab 3142  Vcvv 3494   ⊆ wss 3936  ∅c0 4291  ifcif 4467  ∪ ciun 4919   class class class wbr 5066   ↦ cmpt 5146  dom cdm 5555  ⟶wf 6351  ‘cfv 6355  (class class class)co 7156  supcsup 8904  ℝcr 10536  0cc0 10537  +∞cpnf 10672  ℝ^*cxr 10674   < clt 10675   ≤ cle 10676   − cmin 10870  ℕcn 11638  [,)cico 12741  [,]cicc 12742  volcvol 24064  Σ^{^}csumge0 42664

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461  ax-inf2 9104  ax-cnex 10593  ax-resscn 10594  ax-1cn 10595  ax-icn 10596  ax-addcl 10597  ax-addrcl 10598  ax-mulcl 10599  ax-mulrcl 10600  ax-mulcom 10601  ax-addass 10602  ax-mulass 10603  ax-distr 10604  ax-i2m1 10605  ax-1ne0 10606  ax-1rid 10607  ax-rnegex 10608  ax-rrecex 10609  ax-cnre 10610  ax-pre-lttri 10611  ax-pre-lttrn 10612  ax-pre-ltadd 10613  ax-pre-mulgt0 10614  ax-pre-sup 10615

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-fal 1550  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-int 4877  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-se 5515  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-pred 6148  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-isom 6364  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-of 7409  df-om 7581  df-1st 7689  df-2nd 7690  df-wrecs 7947  df-recs 8008  df-rdg 8046  df-1o 8102  df-2o 8103  df-oadd 8106  df-er 8289  df-map 8408  df-pm 8409  df-en 8510  df-dom 8511  df-sdom 8512  df-fin 8513  df-fi 8875  df-sup 8906  df-inf 8907  df-oi 8974  df-dju 9330  df-card 9368  df-pnf 10677  df-mnf 10678  df-xr 10679  df-ltxr 10680  df-le 10681  df-sub 10872  df-neg 10873  df-div 11298  df-nn 11639  df-2 11701  df-3 11702  df-n0 11899  df-z 11983  df-uz 12245  df-q 12350  df-rp 12391  df-xneg 12508  df-xadd 12509  df-xmul 12510  df-ioo 12743  df-ico 12745  df-icc 12746  df-fz 12894  df-fzo 13035  df-fl 13163  df-seq 13371  df-exp 13431  df-hash 13692  df-cj 14458  df-re 14459  df-im 14460  df-sqrt 14594  df-abs 14595  df-clim 14845  df-rlim 14846  df-sum 15043  df-rest 16696  df-topgen 16717  df-psmet 20537  df-xmet 20538  df-met 20539  df-bl 20540  df-mopn 20541  df-top 21502  df-topon 21519  df-bases 21554  df-cmp 21995  df-ovol 24065  df-vol 24066  df-sumge0 42665

This theorem is referenced by:  hoidmv1le  42896