hoidmv1lelem1 - Mathbox for Glauco Siliprandi

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Theorem hoidmv1lelem1 42863

Description: The supremum of 𝑈 belongs to 𝑈. This is the last part of step (a) and the whole step (b) in the proof of Lemma 114B of [Fremlin1] p. 23. (Contributed by Glauco Siliprandi, 21-Nov-2020.)

**Hypotheses**
Ref	Expression
hoidmv1lelem1.a	⊢ (𝜑 → 𝐴 ∈ ℝ)
hoidmv1lelem1.b	⊢ (𝜑 → 𝐵 ∈ ℝ)
hoidmv1lelem1.l	⊢ (𝜑 → 𝐴 < 𝐵)
hoidmv1lelem1.c	⊢ (𝜑 → 𝐶:ℕ⟶ℝ)
hoidmv1lelem1.d	⊢ (𝜑 → 𝐷:ℕ⟶ℝ)
hoidmv1lelem1.r	⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)(𝐷‘𝑗))))) ∈ ℝ)
hoidmv1lelem1.u	⊢ 𝑈 = {𝑧 ∈ (𝐴[,]𝐵) ∣ (𝑧 − 𝐴) ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧)))))}
hoidmv1lelem1.s	⊢ 𝑆 = sup(𝑈, ℝ, < )

**Assertion**
Ref	Expression
hoidmv1lelem1	⊢ (𝜑 → (𝑆 ∈ 𝑈 ∧ 𝐴 ∈ 𝑈 ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝑈 𝑦 ≤ 𝑥))

Distinct variable groups: 𝐴,𝑗,𝑧 𝑦,𝐴 𝑥,𝐵,𝑦 𝑧,𝐵 𝑧,𝐶 𝑧,𝐷 𝑆,𝑗,𝑧 𝑈,𝑗,𝑧 𝑥,𝑈,𝑦 𝜑,𝑗,𝑧 Allowed substitution hints: 𝜑(𝑥,𝑦) 𝐴(𝑥) 𝐵(𝑗) 𝐶(𝑥,𝑦,𝑗) 𝐷(𝑥,𝑦,𝑗) 𝑆(𝑥,𝑦)

**Proof of Theorem hoidmv1lelem1**
Step	Hyp	Ref	Expression
1		hoidmv1lelem1.s	. . . . . 6 ⊢ 𝑆 = sup(𝑈, ℝ, < )
2		hoidmv1lelem1.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℝ)
3		hoidmv1lelem1.b	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℝ)
4		hoidmv1lelem1.u	. . . . . . . . 9 ⊢ 𝑈 = {𝑧 ∈ (𝐴[,]𝐵) ∣ (𝑧 − 𝐴) ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧)))))}
5		ssrab2 4054	. . . . . . . . 9 ⊢ {𝑧 ∈ (𝐴[,]𝐵) ∣ (𝑧 − 𝐴) ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧)))))} ⊆ (𝐴[,]𝐵)
6	4, 5	eqsstri 3999	. . . . . . . 8 ⊢ 𝑈 ⊆ (𝐴[,]𝐵)
7	6	a1i 11	. . . . . . 7 ⊢ (𝜑 → 𝑈 ⊆ (𝐴[,]𝐵))
8	2	rexrd 10683	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
9	3	rexrd 10683	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
10		hoidmv1lelem1.l	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴 < 𝐵)
11	2, 3, 10	ltled 10780	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
12		lbicc2 12844	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → 𝐴 ∈ (𝐴[,]𝐵))
13	8, 9, 11, 12	syl3anc 1366	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ (𝐴[,]𝐵))
14	2	recnd 10661	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴 ∈ ℂ)
15	14	subidd 10977	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴 − 𝐴) = 0)
16		nfv 1909	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗𝜑
17		nnex 11636	. . . . . . . . . . . . . 14 ⊢ ℕ ∈ V
18	17	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → ℕ ∈ V)
19		volf 24122	. . . . . . . . . . . . . . 15 ⊢ vol:dom vol⟶(0[,]+∞)
20	19	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → vol:dom vol⟶(0[,]+∞))
21		hoidmv1lelem1.c	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐶:ℕ⟶ℝ)
22	21	ffvelrnda 6844	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐶‘𝑗) ∈ ℝ)
23		hoidmv1lelem1.d	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐷:ℕ⟶ℝ)
24	23	ffvelrnda 6844	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐷‘𝑗) ∈ ℝ)
25	2	adantr 483	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝐴 ∈ ℝ)
26	24, 25	ifcld 4510	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → if((𝐷‘𝑗) ≤ 𝐴, (𝐷‘𝑗), 𝐴) ∈ ℝ)
27	26	rexrd 10683	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → if((𝐷‘𝑗) ≤ 𝐴, (𝐷‘𝑗), 𝐴) ∈ ℝ^*)
28		icombl 24157	. . . . . . . . . . . . . . 15 ⊢ (((𝐶‘𝑗) ∈ ℝ ∧ if((𝐷‘𝑗) ≤ 𝐴, (𝐷‘𝑗), 𝐴) ∈ ℝ^*) → ((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐴, (𝐷‘𝑗), 𝐴)) ∈ dom vol)
29	22, 27, 28	syl2anc 586	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐴, (𝐷‘𝑗), 𝐴)) ∈ dom vol)
30	20, 29	ffvelrnd 6845	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐴, (𝐷‘𝑗), 𝐴))) ∈ (0[,]+∞))
31	16, 18, 30	sge0ge0mpt 42710	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐴, (𝐷‘𝑗), 𝐴))))))
32	15, 31	eqbrtrd 5079	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴 − 𝐴) ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐴, (𝐷‘𝑗), 𝐴))))))
33	13, 32	jca 514	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 ∈ (𝐴[,]𝐵) ∧ (𝐴 − 𝐴) ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐴, (𝐷‘𝑗), 𝐴)))))))
34		oveq1 7155	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝐴 → (𝑧 − 𝐴) = (𝐴 − 𝐴))
35		breq2 5061	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = 𝐴 → ((𝐷‘𝑗) ≤ 𝑧 ↔ (𝐷‘𝑗) ≤ 𝐴))
36		id 22	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = 𝐴 → 𝑧 = 𝐴)
37	35, 36	ifbieq2d 4490	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝐴 → if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧) = if((𝐷‘𝑗) ≤ 𝐴, (𝐷‘𝑗), 𝐴))
38	37	oveq2d 7164	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝐴 → ((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧)) = ((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐴, (𝐷‘𝑗), 𝐴)))
39	38	fveq2d 6667	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝐴 → (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧))) = (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐴, (𝐷‘𝑗), 𝐴))))
40	39	mpteq2dv 5153	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝐴 → (𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧)))) = (𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐴, (𝐷‘𝑗), 𝐴)))))
41	40	fveq2d 6667	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝐴 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧))))) = (Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐴, (𝐷‘𝑗), 𝐴))))))
42	34, 41	breq12d 5070	. . . . . . . . . . 11 ⊢ (𝑧 = 𝐴 → ((𝑧 − 𝐴) ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧))))) ↔ (𝐴 − 𝐴) ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐴, (𝐷‘𝑗), 𝐴)))))))
43	42	elrab 3678	. . . . . . . . . 10 ⊢ (𝐴 ∈ {𝑧 ∈ (𝐴[,]𝐵) ∣ (𝑧 − 𝐴) ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧)))))} ↔ (𝐴 ∈ (𝐴[,]𝐵) ∧ (𝐴 − 𝐴) ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐴, (𝐷‘𝑗), 𝐴)))))))
44	33, 43	sylibr 236	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ {𝑧 ∈ (𝐴[,]𝐵) ∣ (𝑧 − 𝐴) ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧)))))})
45	44, 4	eleqtrrdi 2922	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ 𝑈)
46	45	ne0d 4299	. . . . . . 7 ⊢ (𝜑 → 𝑈 ≠ ∅)
47	2, 3, 7, 46	supicc 12878	. . . . . 6 ⊢ (𝜑 → sup(𝑈, ℝ, < ) ∈ (𝐴[,]𝐵))
48	1, 47	eqeltrid 2915	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ (𝐴[,]𝐵))
49	1	a1i 11	. . . . . . 7 ⊢ (𝜑 → 𝑆 = sup(𝑈, ℝ, < ))
50		nfv 1909	. . . . . . . . 9 ⊢ Ⅎ𝑧𝜑
51	2, 3	iccssred 41769	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
52	7, 51	sstrd 3975	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑈 ⊆ ℝ)
53	52	sselda 3965	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → 𝑧 ∈ ℝ)
54		nfv 1909	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑗(𝜑 ∧ 𝑧 ∈ 𝑈)
55	17	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → ℕ ∈ V)
56	19	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) → vol:dom vol⟶(0[,]+∞))
57	22	adantlr 713	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) → (𝐶‘𝑗) ∈ ℝ)
58	24	adantlr 713	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) → (𝐷‘𝑗) ∈ ℝ)
59	53	adantr 483	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) → 𝑧 ∈ ℝ)
60	58, 59	ifcld 4510	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) → if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧) ∈ ℝ)
61	60	rexrd 10683	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) → if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧) ∈ ℝ^*)
62		icombl 24157	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐶‘𝑗) ∈ ℝ ∧ if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧) ∈ ℝ^*) → ((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧)) ∈ dom vol)
63	57, 61, 62	syl2anc 586	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧)) ∈ dom vol)
64	56, 63	ffvelrnd 6845	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) → (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧))) ∈ (0[,]+∞))
65	54, 55, 64	sge0xrclmpt 42700	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → (Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧))))) ∈ ℝ^*)
66		pnfxr 10687	. . . . . . . . . . . . . . . 16 ⊢ +∞ ∈ ℝ^*
67	66	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → +∞ ∈ ℝ^*)
68		hoidmv1lelem1.r	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)(𝐷‘𝑗))))) ∈ ℝ)
69	68	rexrd 10683	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)(𝐷‘𝑗))))) ∈ ℝ^*)
70	69	adantr 483	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → (Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)(𝐷‘𝑗))))) ∈ ℝ^*)
71	24	rexrd 10683	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐷‘𝑗) ∈ ℝ^*)
72		icombl 24157	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐶‘𝑗) ∈ ℝ ∧ (𝐷‘𝑗) ∈ ℝ^*) → ((𝐶‘𝑗)[,)(𝐷‘𝑗)) ∈ dom vol)
73	22, 71, 72	syl2anc 586	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)[,)(𝐷‘𝑗)) ∈ dom vol)
74	20, 73	ffvelrnd 6845	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (vol‘((𝐶‘𝑗)[,)(𝐷‘𝑗))) ∈ (0[,]+∞))
75	74	adantlr 713	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) → (vol‘((𝐶‘𝑗)[,)(𝐷‘𝑗))) ∈ (0[,]+∞))
76	73	adantlr 713	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)[,)(𝐷‘𝑗)) ∈ dom vol)
77	22	rexrd 10683	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐶‘𝑗) ∈ ℝ^*)
78	77	adantlr 713	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) → (𝐶‘𝑗) ∈ ℝ^*)
79	71	adantlr 713	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) → (𝐷‘𝑗) ∈ ℝ^*)
80	22	leidd 11198	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐶‘𝑗) ≤ (𝐶‘𝑗))
81	80	adantlr 713	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) → (𝐶‘𝑗) ≤ (𝐶‘𝑗))
82		min1 12574	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐷‘𝑗) ∈ ℝ ∧ 𝑧 ∈ ℝ) → if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧) ≤ (𝐷‘𝑗))
83	58, 59, 82	syl2anc 586	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) → if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧) ≤ (𝐷‘𝑗))
84		icossico 12798	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐶‘𝑗) ∈ ℝ^* ∧ (𝐷‘𝑗) ∈ ℝ^*) ∧ ((𝐶‘𝑗) ≤ (𝐶‘𝑗) ∧ if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧) ≤ (𝐷‘𝑗))) → ((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧)) ⊆ ((𝐶‘𝑗)[,)(𝐷‘𝑗)))
85	78, 79, 81, 83, 84	syl22anc 836	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧)) ⊆ ((𝐶‘𝑗)[,)(𝐷‘𝑗)))
86		volss 24126	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧)) ∈ dom vol ∧ ((𝐶‘𝑗)[,)(𝐷‘𝑗)) ∈ dom vol ∧ ((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧)) ⊆ ((𝐶‘𝑗)[,)(𝐷‘𝑗))) → (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧))) ≤ (vol‘((𝐶‘𝑗)[,)(𝐷‘𝑗))))
87	63, 76, 85, 86	syl3anc 1366	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) → (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧))) ≤ (vol‘((𝐶‘𝑗)[,)(𝐷‘𝑗))))
88	54, 55, 64, 75, 87	sge0lempt 42682	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → (Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧))))) ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)(𝐷‘𝑗))))))
89	68	ltpnfd 12508	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)(𝐷‘𝑗))))) < +∞)
90	89	adantr 483	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → (Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)(𝐷‘𝑗))))) < +∞)
91	65, 70, 67, 88, 90	xrlelttrd 12545	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → (Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧))))) < +∞)
92	65, 67, 91	xrltned 41614	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → (Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧))))) ≠ +∞)
93	92	neneqd 3019	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → ¬ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧))))) = +∞)
94		eqid 2819	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧)))) = (𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧))))
95	64, 94	fmptd 6871	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → (𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧)))):ℕ⟶(0[,]+∞))
96	55, 95	sge0repnf 42658	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → ((Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧))))) ∈ ℝ ↔ ¬ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧))))) = +∞))
97	93, 96	mpbird 259	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → (Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧))))) ∈ ℝ)
98	2	adantr 483	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → 𝐴 ∈ ℝ)
99	97, 98	readdcld 10662	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → ((Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧))))) + 𝐴) ∈ ℝ)
100	51, 48	sseldd 3966	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 𝑆 ∈ ℝ)
101	100	adantr 483	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑆 ∈ ℝ)
102	24, 101	ifcld 4510	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆) ∈ ℝ)
103	102	rexrd 10683	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆) ∈ ℝ^*)
104		icombl 24157	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐶‘𝑗) ∈ ℝ ∧ if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆) ∈ ℝ^*) → ((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆)) ∈ dom vol)
105	22, 103, 104	syl2anc 586	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆)) ∈ dom vol)
106	20, 105	ffvelrnd 6845	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆))) ∈ (0[,]+∞))
107	16, 18, 106	sge0xrclmpt 42700	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆))))) ∈ ℝ^*)
108	66	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → +∞ ∈ ℝ^*)
109		min1 12574	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐷‘𝑗) ∈ ℝ ∧ 𝑆 ∈ ℝ) → if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆) ≤ (𝐷‘𝑗))
110	24, 101, 109	syl2anc 586	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆) ≤ (𝐷‘𝑗))
111		icossico 12798	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐶‘𝑗) ∈ ℝ^* ∧ (𝐷‘𝑗) ∈ ℝ^*) ∧ ((𝐶‘𝑗) ≤ (𝐶‘𝑗) ∧ if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆) ≤ (𝐷‘𝑗))) → ((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆)) ⊆ ((𝐶‘𝑗)[,)(𝐷‘𝑗)))
112	77, 71, 80, 110, 111	syl22anc 836	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆)) ⊆ ((𝐶‘𝑗)[,)(𝐷‘𝑗)))
113		volss 24126	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆)) ∈ dom vol ∧ ((𝐶‘𝑗)[,)(𝐷‘𝑗)) ∈ dom vol ∧ ((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆)) ⊆ ((𝐶‘𝑗)[,)(𝐷‘𝑗))) → (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆))) ≤ (vol‘((𝐶‘𝑗)[,)(𝐷‘𝑗))))
114	105, 73, 112, 113	syl3anc 1366	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆))) ≤ (vol‘((𝐶‘𝑗)[,)(𝐷‘𝑗))))
115	16, 18, 106, 74, 114	sge0lempt 42682	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆))))) ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)(𝐷‘𝑗))))))
116	107, 69, 108, 115, 89	xrlelttrd 12545	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆))))) < +∞)
117	107, 108, 116	xrltned 41614	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆))))) ≠ +∞)
118	117	neneqd 3019	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ¬ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆))))) = +∞)
119		eqid 2819	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆)))) = (𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆))))
120	106, 119	fmptd 6871	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆)))):ℕ⟶(0[,]+∞))
121	18, 120	sge0repnf 42658	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆))))) ∈ ℝ ↔ ¬ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆))))) = +∞))
122	118, 121	mpbird 259	. . . . . . . . . . . . 13 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆))))) ∈ ℝ)
123	122, 2	readdcld 10662	. . . . . . . . . . . 12 ⊢ (𝜑 → ((Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆))))) + 𝐴) ∈ ℝ)
124	123	adantr 483	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → ((Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆))))) + 𝐴) ∈ ℝ)
125	4	eleq2i 2902	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ 𝑈 ↔ 𝑧 ∈ {𝑧 ∈ (𝐴[,]𝐵) ∣ (𝑧 − 𝐴) ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧)))))})
126	125	biimpi 218	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ 𝑈 → 𝑧 ∈ {𝑧 ∈ (𝐴[,]𝐵) ∣ (𝑧 − 𝐴) ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧)))))})
127	126	adantl 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → 𝑧 ∈ {𝑧 ∈ (𝐴[,]𝐵) ∣ (𝑧 − 𝐴) ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧)))))})
128		rabid 3377	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ {𝑧 ∈ (𝐴[,]𝐵) ∣ (𝑧 − 𝐴) ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧)))))} ↔ (𝑧 ∈ (𝐴[,]𝐵) ∧ (𝑧 − 𝐴) ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧)))))))
129	127, 128	sylib 220	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → (𝑧 ∈ (𝐴[,]𝐵) ∧ (𝑧 − 𝐴) ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧)))))))
130	129	simprd 498	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → (𝑧 − 𝐴) ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧))))))
131	53, 98, 97	lesubaddd 11229	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → ((𝑧 − 𝐴) ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧))))) ↔ 𝑧 ≤ ((Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧))))) + 𝐴)))
132	130, 131	mpbid 234	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → 𝑧 ≤ ((Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧))))) + 𝐴))
133	122	adantr 483	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → (Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆))))) ∈ ℝ)
134	106	adantlr 713	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) → (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆))) ∈ (0[,]+∞))
135	105	adantlr 713	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆)) ∈ dom vol)
136	103	adantlr 713	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) → if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆) ∈ ℝ^*)
137	60	adantr 483	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) ∧ (𝐷‘𝑗) ≤ 𝑧) → if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧) ∈ ℝ)
138		eqidd 2820	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) ∧ (𝐷‘𝑗) ≤ 𝑧) → (𝐷‘𝑗) = (𝐷‘𝑗))
139		iftrue 4471	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐷‘𝑗) ≤ 𝑧 → if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧) = (𝐷‘𝑗))
140	139	adantl 484	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) ∧ (𝐷‘𝑗) ≤ 𝑧) → if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧) = (𝐷‘𝑗))
141	58	adantr 483	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) ∧ (𝐷‘𝑗) ≤ 𝑧) → (𝐷‘𝑗) ∈ ℝ)
142	59	adantr 483	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) ∧ (𝐷‘𝑗) ≤ 𝑧) → 𝑧 ∈ ℝ)
143	100	ad3antrrr 728	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) ∧ (𝐷‘𝑗) ≤ 𝑧) → 𝑆 ∈ ℝ)
144		simpr 487	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) ∧ (𝐷‘𝑗) ≤ 𝑧) → (𝐷‘𝑗) ≤ 𝑧)
145	52	adantr 483	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → 𝑈 ⊆ ℝ)
146	46	adantr 483	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → 𝑈 ≠ ∅)
147	2, 3	jca 514	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
148		iccsupr 12822	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝑈 ⊆ (𝐴[,]𝐵) ∧ 𝐴 ∈ 𝑈) → (𝑈 ⊆ ℝ ∧ 𝑈 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝑈 𝑦 ≤ 𝑥))
149	147, 7, 45, 148	syl3anc 1366	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → (𝑈 ⊆ ℝ ∧ 𝑈 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝑈 𝑦 ≤ 𝑥))
150	149	simp3d 1139	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝑈 𝑦 ≤ 𝑥)
151	150	adantr 483	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝑈 𝑦 ≤ 𝑥)
152	127, 125	sylibr 236	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → 𝑧 ∈ 𝑈)
153		suprub 11594	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑈 ⊆ ℝ ∧ 𝑈 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝑈 𝑦 ≤ 𝑥) ∧ 𝑧 ∈ 𝑈) → 𝑧 ≤ sup(𝑈, ℝ, < ))
154	145, 146, 151, 152, 153	syl31anc 1368	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → 𝑧 ≤ sup(𝑈, ℝ, < ))
155	154, 1	breqtrrdi 5099	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → 𝑧 ≤ 𝑆)
156	155	ad2antrr 724	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) ∧ (𝐷‘𝑗) ≤ 𝑧) → 𝑧 ≤ 𝑆)
157	141, 142, 143, 144, 156	letrd 10789	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) ∧ (𝐷‘𝑗) ≤ 𝑧) → (𝐷‘𝑗) ≤ 𝑆)
158	157	iftrued 4473	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) ∧ (𝐷‘𝑗) ≤ 𝑧) → if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆) = (𝐷‘𝑗))
159	138, 140, 158	3eqtr4d 2864	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) ∧ (𝐷‘𝑗) ≤ 𝑧) → if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧) = if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆))
160	137, 159	eqled 10735	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) ∧ (𝐷‘𝑗) ≤ 𝑧) → if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧) ≤ if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆))
161	59	adantr 483	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷‘𝑗) ≤ 𝑧) → 𝑧 ∈ ℝ)
162	58	adantr 483	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷‘𝑗) ≤ 𝑧) → (𝐷‘𝑗) ∈ ℝ)
163		simpr 487	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷‘𝑗) ≤ 𝑧) → ¬ (𝐷‘𝑗) ≤ 𝑧)
164	161, 162	ltnled 10779	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷‘𝑗) ≤ 𝑧) → (𝑧 < (𝐷‘𝑗) ↔ ¬ (𝐷‘𝑗) ≤ 𝑧))
165	163, 164	mpbird 259	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷‘𝑗) ≤ 𝑧) → 𝑧 < (𝐷‘𝑗))
166	161, 162, 165	ltled 10780	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷‘𝑗) ≤ 𝑧) → 𝑧 ≤ (𝐷‘𝑗))
167	166	adantr 483	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷‘𝑗) ≤ 𝑧) ∧ (𝐷‘𝑗) ≤ 𝑆) → 𝑧 ≤ (𝐷‘𝑗))
168		iffalse 4474	. . . . . . . . . . . . . . . . . . . 20 ⊢ (¬ (𝐷‘𝑗) ≤ 𝑧 → if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧) = 𝑧)
169	168	ad2antlr 725	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷‘𝑗) ≤ 𝑧) ∧ (𝐷‘𝑗) ≤ 𝑆) → if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧) = 𝑧)
170		iftrue 4471	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐷‘𝑗) ≤ 𝑆 → if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆) = (𝐷‘𝑗))
171	170	adantl 484	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷‘𝑗) ≤ 𝑧) ∧ (𝐷‘𝑗) ≤ 𝑆) → if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆) = (𝐷‘𝑗))
172	169, 171	breq12d 5070	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷‘𝑗) ≤ 𝑧) ∧ (𝐷‘𝑗) ≤ 𝑆) → (if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧) ≤ if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆) ↔ 𝑧 ≤ (𝐷‘𝑗)))
173	167, 172	mpbird 259	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷‘𝑗) ≤ 𝑧) ∧ (𝐷‘𝑗) ≤ 𝑆) → if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧) ≤ if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆))
174	155	ad3antrrr 728	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷‘𝑗) ≤ 𝑧) ∧ ¬ (𝐷‘𝑗) ≤ 𝑆) → 𝑧 ≤ 𝑆)
175	168	ad2antlr 725	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷‘𝑗) ≤ 𝑧) ∧ ¬ (𝐷‘𝑗) ≤ 𝑆) → if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧) = 𝑧)
176		iffalse 4474	. . . . . . . . . . . . . . . . . . . 20 ⊢ (¬ (𝐷‘𝑗) ≤ 𝑆 → if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆) = 𝑆)
177	176	adantl 484	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷‘𝑗) ≤ 𝑧) ∧ ¬ (𝐷‘𝑗) ≤ 𝑆) → if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆) = 𝑆)
178	175, 177	breq12d 5070	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷‘𝑗) ≤ 𝑧) ∧ ¬ (𝐷‘𝑗) ≤ 𝑆) → (if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧) ≤ if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆) ↔ 𝑧 ≤ 𝑆))
179	174, 178	mpbird 259	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷‘𝑗) ≤ 𝑧) ∧ ¬ (𝐷‘𝑗) ≤ 𝑆) → if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧) ≤ if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆))
180	173, 179	pm2.61dan 811	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷‘𝑗) ≤ 𝑧) → if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧) ≤ if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆))
181	160, 180	pm2.61dan 811	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) → if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧) ≤ if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆))
182		icossico 12798	. . . . . . . . . . . . . . 15 ⊢ ((((𝐶‘𝑗) ∈ ℝ^* ∧ if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆) ∈ ℝ^*) ∧ ((𝐶‘𝑗) ≤ (𝐶‘𝑗) ∧ if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧) ≤ if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆))) → ((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧)) ⊆ ((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆)))
183	78, 136, 81, 181, 182	syl22anc 836	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧)) ⊆ ((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆)))
184		volss 24126	. . . . . . . . . . . . . 14 ⊢ ((((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧)) ∈ dom vol ∧ ((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆)) ∈ dom vol ∧ ((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧)) ⊆ ((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆))) → (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧))) ≤ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆))))
185	63, 135, 183, 184	syl3anc 1366	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) → (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧))) ≤ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆))))
186	54, 55, 64, 134, 185	sge0lempt 42682	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → (Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧))))) ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆))))))
187	97, 133, 98, 186	leadd1dd 11246	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → ((Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧))))) + 𝐴) ≤ ((Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆))))) + 𝐴))
188	53, 99, 124, 132, 187	letrd 10789	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → 𝑧 ≤ ((Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆))))) + 𝐴))
189	188	ex 415	. . . . . . . . 9 ⊢ (𝜑 → (𝑧 ∈ 𝑈 → 𝑧 ≤ ((Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆))))) + 𝐴)))
190	50, 189	ralrimi 3214	. . . . . . . 8 ⊢ (𝜑 → ∀𝑧 ∈ 𝑈 𝑧 ≤ ((Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆))))) + 𝐴))
191		suprleub 11599	. . . . . . . . 9 ⊢ (((𝑈 ⊆ ℝ ∧ 𝑈 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝑈 𝑦 ≤ 𝑥) ∧ ((Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆))))) + 𝐴) ∈ ℝ) → (sup(𝑈, ℝ, < ) ≤ ((Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆))))) + 𝐴) ↔ ∀𝑧 ∈ 𝑈 𝑧 ≤ ((Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆))))) + 𝐴)))
192	52, 46, 150, 123, 191	syl31anc 1368	. . . . . . . 8 ⊢ (𝜑 → (sup(𝑈, ℝ, < ) ≤ ((Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆))))) + 𝐴) ↔ ∀𝑧 ∈ 𝑈 𝑧 ≤ ((Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆))))) + 𝐴)))
193	190, 192	mpbird 259	. . . . . . 7 ⊢ (𝜑 → sup(𝑈, ℝ, < ) ≤ ((Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆))))) + 𝐴))
194	49, 193	eqbrtrd 5079	. . . . . 6 ⊢ (𝜑 → 𝑆 ≤ ((Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆))))) + 𝐴))
195	100, 2, 122	lesubaddd 11229	. . . . . 6 ⊢ (𝜑 → ((𝑆 − 𝐴) ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆))))) ↔ 𝑆 ≤ ((Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆))))) + 𝐴)))
196	194, 195	mpbird 259	. . . . 5 ⊢ (𝜑 → (𝑆 − 𝐴) ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆))))))
197	48, 196	jca 514	. . . 4 ⊢ (𝜑 → (𝑆 ∈ (𝐴[,]𝐵) ∧ (𝑆 − 𝐴) ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆)))))))
198		oveq1 7155	. . . . . 6 ⊢ (𝑧 = 𝑆 → (𝑧 − 𝐴) = (𝑆 − 𝐴))
199		breq2 5061	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑆 → ((𝐷‘𝑗) ≤ 𝑧 ↔ (𝐷‘𝑗) ≤ 𝑆))
200		id 22	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑆 → 𝑧 = 𝑆)
201	199, 200	ifbieq2d 4490	. . . . . . . . . 10 ⊢ (𝑧 = 𝑆 → if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧) = if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆))
202	201	oveq2d 7164	. . . . . . . . 9 ⊢ (𝑧 = 𝑆 → ((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧)) = ((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆)))
203	202	fveq2d 6667	. . . . . . . 8 ⊢ (𝑧 = 𝑆 → (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧))) = (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆))))
204	203	mpteq2dv 5153	. . . . . . 7 ⊢ (𝑧 = 𝑆 → (𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧)))) = (𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆)))))
205	204	fveq2d 6667	. . . . . 6 ⊢ (𝑧 = 𝑆 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧))))) = (Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆))))))
206	198, 205	breq12d 5070	. . . . 5 ⊢ (𝑧 = 𝑆 → ((𝑧 − 𝐴) ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧))))) ↔ (𝑆 − 𝐴) ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆)))))))
207	206	elrab 3678	. . . 4 ⊢ (𝑆 ∈ {𝑧 ∈ (𝐴[,]𝐵) ∣ (𝑧 − 𝐴) ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧)))))} ↔ (𝑆 ∈ (𝐴[,]𝐵) ∧ (𝑆 − 𝐴) ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆)))))))
208	197, 207	sylibr 236	. . 3 ⊢ (𝜑 → 𝑆 ∈ {𝑧 ∈ (𝐴[,]𝐵) ∣ (𝑧 − 𝐴) ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧)))))})
209	208, 4	eleqtrrdi 2922	. 2 ⊢ (𝜑 → 𝑆 ∈ 𝑈)
210	209, 45, 150	3jca 1123	1 ⊢ (𝜑 → (𝑆 ∈ 𝑈 ∧ 𝐴 ∈ 𝑈 ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝑈 𝑦 ≤ 𝑥))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1082 = wceq 1531 ∈ wcel 2108 ≠ wne 3014 ∀wral 3136 ∃wrex 3137 {crab 3140 Vcvv 3493 ⊆ wss 3934 ∅c0 4289 ifcif 4465 class class class wbr 5057 ↦ cmpt 5137 dom cdm 5548 ⟶wf 6344 ‘cfv 6348 (class class class)co 7148 supcsup 8896 ℝcr 10528 0cc0 10529 + caddc 10532 +∞cpnf 10664 ℝ^*cxr 10666 < clt 10667 ≤ cle 10668 − cmin 10862 ℕcn 11630 [,)cico 12732 [,]cicc 12733 volcvol 24056 Σ^{^}csumge0 42634

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-inf2 9096 ax-cnex 10585 ax-resscn 10586 ax-1cn 10587 ax-icn 10588 ax-addcl 10589 ax-addrcl 10590 ax-mulcl 10591 ax-mulrcl 10592 ax-mulcom 10593 ax-addass 10594 ax-mulass 10595 ax-distr 10596 ax-i2m1 10597 ax-1ne0 10598 ax-1rid 10599 ax-rnegex 10600 ax-rrecex 10601 ax-cnre 10602 ax-pre-lttri 10603 ax-pre-lttrn 10604 ax-pre-ltadd 10605 ax-pre-mulgt0 10606 ax-pre-sup 10607

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-fal 1544 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-reu 3143 df-rmo 3144 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-pss 3952 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7106 df-ov 7151 df-oprab 7152 df-mpo 7153 df-of 7401 df-om 7573 df-1st 7681 df-2nd 7682 df-wrecs 7939 df-recs 8000 df-rdg 8038 df-1o 8094 df-2o 8095 df-oadd 8098 df-er 8281 df-map 8400 df-pm 8401 df-en 8502 df-dom 8503 df-sdom 8504 df-fin 8505 df-sup 8898 df-inf 8899 df-oi 8966 df-dju 9322 df-card 9360 df-pnf 10669 df-mnf 10670 df-xr 10671 df-ltxr 10672 df-le 10673 df-sub 10864 df-neg 10865 df-div 11290 df-nn 11631 df-2 11692 df-3 11693 df-n0 11890 df-z 11974 df-uz 12236 df-q 12341 df-rp 12382 df-xadd 12500 df-ioo 12734 df-ico 12736 df-icc 12737 df-fz 12885 df-fzo 13026 df-fl 13154 df-seq 13362 df-exp 13422 df-hash 13683 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-xmet 20530 df-met 20531 df-ovol 24057 df-vol 24058 df-sumge0 42635

This theorem is referenced by: hoidmv1lelem3 42865

W3C validator