hoidmv1lelem2 - Mathbox for Glauco Siliprandi

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Theorem hoidmv1lelem2 45767

Description: This is the contradiction proven in step (c) in the proof of Lemma 114B of [Fremlin1] p. 23. (Contributed by Glauco Siliprandi, 21-Nov-2020.)

**Hypotheses**
Ref	Expression
hoidmv1lelem2.a	⊢ (𝜑 → 𝐴 ∈ ℝ)
hoidmv1lelem2.b	⊢ (𝜑 → 𝐵 ∈ ℝ)
hoidmv1lelem2.c	⊢ (𝜑 → 𝐶:ℕ⟶ℝ)
hoidmv1lelem2.d	⊢ (𝜑 → 𝐷:ℕ⟶ℝ)
hoidmv1lelem2.r	⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)(𝐷‘𝑗))))) ∈ ℝ)
hoidmv1lelem2.u	⊢ 𝑈 = {𝑧 ∈ (𝐴[,]𝐵) ∣ (𝑧 − 𝐴) ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧)))))}
hoidmv1lelem2.e	⊢ (𝜑 → 𝑆 ∈ 𝑈)
hoidmv1lelem2.g	⊢ (𝜑 → 𝐴 ≤ 𝑆)
hoidmv1lelem2.l	⊢ (𝜑 → 𝑆 < 𝐵)
hoidmv1lelem2.k	⊢ (𝜑 → 𝐾 ∈ ℕ)
hoidmv1lelem2.s	⊢ (𝜑 → 𝑆 ∈ ((𝐶‘𝐾)[,)(𝐷‘𝐾)))
hoidmv1lelem2.m	⊢ 𝑀 = if((𝐷‘𝐾) ≤ 𝐵, (𝐷‘𝐾), 𝐵)

**Assertion**
Ref	Expression
hoidmv1lelem2	⊢ (𝜑 → ∃𝑢 ∈ 𝑈 𝑆 < 𝑢)

Distinct variable groups: 𝑧,𝐴 𝑧,𝐵 𝐶,𝑗,𝑧 𝐷,𝑗,𝑧 𝑗,𝐾 𝑗,𝑀,𝑧 𝑢,𝑀 𝑆,𝑗,𝑧 𝑢,𝑆 𝑢,𝑈 𝜑,𝑗 Allowed substitution hints: 𝜑(𝑧,𝑢) 𝐴(𝑢,𝑗) 𝐵(𝑢,𝑗) 𝐶(𝑢) 𝐷(𝑢) 𝑈(𝑧,𝑗) 𝐾(𝑧,𝑢)

**Proof of Theorem hoidmv1lelem2**
Step	Hyp	Ref	Expression
1		hoidmv1lelem2.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℝ)
2		hoidmv1lelem2.b	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℝ)
3		hoidmv1lelem2.m	. . . . . . . 8 ⊢ 𝑀 = if((𝐷‘𝐾) ≤ 𝐵, (𝐷‘𝐾), 𝐵)
4	3	a1i 11	. . . . . . 7 ⊢ (𝜑 → 𝑀 = if((𝐷‘𝐾) ≤ 𝐵, (𝐷‘𝐾), 𝐵))
5		hoidmv1lelem2.d	. . . . . . . . 9 ⊢ (𝜑 → 𝐷:ℕ⟶ℝ)
6		hoidmv1lelem2.k	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ ℕ)
7	5, 6	ffvelcdmd 7087	. . . . . . . 8 ⊢ (𝜑 → (𝐷‘𝐾) ∈ ℝ)
8	7, 2	ifcld 4574	. . . . . . 7 ⊢ (𝜑 → if((𝐷‘𝐾) ≤ 𝐵, (𝐷‘𝐾), 𝐵) ∈ ℝ)
9	4, 8	eqeltrd 2832	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℝ)
10		hoidmv1lelem2.c	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐶:ℕ⟶ℝ)
11	10, 6	ffvelcdmd 7087	. . . . . . . . . 10 ⊢ (𝜑 → (𝐶‘𝐾) ∈ ℝ)
12	7	rexrd 11271	. . . . . . . . . 10 ⊢ (𝜑 → (𝐷‘𝐾) ∈ ℝ^*)
13		icossre 13412	. . . . . . . . . 10 ⊢ (((𝐶‘𝐾) ∈ ℝ ∧ (𝐷‘𝐾) ∈ ℝ^*) → ((𝐶‘𝐾)[,)(𝐷‘𝐾)) ⊆ ℝ)
14	11, 12, 13	syl2anc 583	. . . . . . . . 9 ⊢ (𝜑 → ((𝐶‘𝐾)[,)(𝐷‘𝐾)) ⊆ ℝ)
15		hoidmv1lelem2.s	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ∈ ((𝐶‘𝐾)[,)(𝐷‘𝐾)))
16	14, 15	sseldd 3983	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ ℝ)
17		hoidmv1lelem2.g	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ≤ 𝑆)
18	11	rexrd 11271	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶‘𝐾) ∈ ℝ^*)
19		icoltub 44680	. . . . . . . . . . . 12 ⊢ (((𝐶‘𝐾) ∈ ℝ^* ∧ (𝐷‘𝐾) ∈ ℝ^* ∧ 𝑆 ∈ ((𝐶‘𝐾)[,)(𝐷‘𝐾))) → 𝑆 < (𝐷‘𝐾))
20	18, 12, 15, 19	syl3anc 1370	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑆 < (𝐷‘𝐾))
21	16, 7, 20	ltled 11369	. . . . . . . . . 10 ⊢ (𝜑 → 𝑆 ≤ (𝐷‘𝐾))
22		hoidmv1lelem2.l	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑆 < 𝐵)
23	16, 2, 22	ltled 11369	. . . . . . . . . 10 ⊢ (𝜑 → 𝑆 ≤ 𝐵)
24	21, 23	jca 511	. . . . . . . . 9 ⊢ (𝜑 → (𝑆 ≤ (𝐷‘𝐾) ∧ 𝑆 ≤ 𝐵))
25		lemin 13178	. . . . . . . . . 10 ⊢ ((𝑆 ∈ ℝ ∧ (𝐷‘𝐾) ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝑆 ≤ if((𝐷‘𝐾) ≤ 𝐵, (𝐷‘𝐾), 𝐵) ↔ (𝑆 ≤ (𝐷‘𝐾) ∧ 𝑆 ≤ 𝐵)))
26	16, 7, 2, 25	syl3anc 1370	. . . . . . . . 9 ⊢ (𝜑 → (𝑆 ≤ if((𝐷‘𝐾) ≤ 𝐵, (𝐷‘𝐾), 𝐵) ↔ (𝑆 ≤ (𝐷‘𝐾) ∧ 𝑆 ≤ 𝐵)))
27	24, 26	mpbird 257	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ≤ if((𝐷‘𝐾) ≤ 𝐵, (𝐷‘𝐾), 𝐵))
28	1, 16, 8, 17, 27	letrd 11378	. . . . . . 7 ⊢ (𝜑 → 𝐴 ≤ if((𝐷‘𝐾) ≤ 𝐵, (𝐷‘𝐾), 𝐵))
29	4	eqcomd 2737	. . . . . . 7 ⊢ (𝜑 → if((𝐷‘𝐾) ≤ 𝐵, (𝐷‘𝐾), 𝐵) = 𝑀)
30	28, 29	breqtrd 5174	. . . . . 6 ⊢ (𝜑 → 𝐴 ≤ 𝑀)
31		min2 13176	. . . . . . . 8 ⊢ (((𝐷‘𝐾) ∈ ℝ ∧ 𝐵 ∈ ℝ) → if((𝐷‘𝐾) ≤ 𝐵, (𝐷‘𝐾), 𝐵) ≤ 𝐵)
32	7, 2, 31	syl2anc 583	. . . . . . 7 ⊢ (𝜑 → if((𝐷‘𝐾) ≤ 𝐵, (𝐷‘𝐾), 𝐵) ≤ 𝐵)
33	4, 32	eqbrtrd 5170	. . . . . 6 ⊢ (𝜑 → 𝑀 ≤ 𝐵)
34	1, 2, 9, 30, 33	eliccd 44676	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ (𝐴[,]𝐵))
35	9	recnd 11249	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℂ)
36	16	recnd 11249	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ ℂ)
37	1	recnd 11249	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℂ)
38	35, 36, 37	npncand 11602	. . . . . . 7 ⊢ (𝜑 → ((𝑀 − 𝑆) + (𝑆 − 𝐴)) = (𝑀 − 𝐴))
39	38	eqcomd 2737	. . . . . 6 ⊢ (𝜑 → (𝑀 − 𝐴) = ((𝑀 − 𝑆) + (𝑆 − 𝐴)))
40	9, 16	resubcld 11649	. . . . . . . 8 ⊢ (𝜑 → (𝑀 − 𝑆) ∈ ℝ)
41	16, 1	resubcld 11649	. . . . . . . 8 ⊢ (𝜑 → (𝑆 − 𝐴) ∈ ℝ)
42	40, 41	readdcld 11250	. . . . . . 7 ⊢ (𝜑 → ((𝑀 − 𝑆) + (𝑆 − 𝐴)) ∈ ℝ)
43		nnex 12225	. . . . . . . . . . . . 13 ⊢ ℕ ∈ V
44	43	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → ℕ ∈ V)
45		volf 25378	. . . . . . . . . . . . . . 15 ⊢ vol:dom vol⟶(0[,]+∞)
46	45	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → vol:dom vol⟶(0[,]+∞))
47	10	ffvelcdmda 7086	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐶‘𝑗) ∈ ℝ)
48	5	ffvelcdmda 7086	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐷‘𝑗) ∈ ℝ)
49	16	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑆 ∈ ℝ)
50	48, 49	ifcld 4574	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆) ∈ ℝ)
51	50	rexrd 11271	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆) ∈ ℝ^*)
52		icombl 25413	. . . . . . . . . . . . . . 15 ⊢ (((𝐶‘𝑗) ∈ ℝ ∧ if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆) ∈ ℝ^*) → ((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆)) ∈ dom vol)
53	47, 51, 52	syl2anc 583	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆)) ∈ dom vol)
54	46, 53	ffvelcdmd 7087	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆))) ∈ (0[,]+∞))
55		eqid 2731	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆)))) = (𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆))))
56	54, 55	fmptd 7115	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆)))):ℕ⟶(0[,]+∞))
57	44, 56	sge0xrcl 45560	. . . . . . . . . . 11 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆))))) ∈ ℝ^*)
58		pnfxr 11275	. . . . . . . . . . . 12 ⊢ +∞ ∈ ℝ^*
59	58	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → +∞ ∈ ℝ^*)
60		hoidmv1lelem2.r	. . . . . . . . . . . . 13 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)(𝐷‘𝑗))))) ∈ ℝ)
61	60	rexrd 11271	. . . . . . . . . . . 12 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)(𝐷‘𝑗))))) ∈ ℝ^*)
62		nfv 1916	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗𝜑
63	48	rexrd 11271	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐷‘𝑗) ∈ ℝ^*)
64		icombl 25413	. . . . . . . . . . . . . . 15 ⊢ (((𝐶‘𝑗) ∈ ℝ ∧ (𝐷‘𝑗) ∈ ℝ^*) → ((𝐶‘𝑗)[,)(𝐷‘𝑗)) ∈ dom vol)
65	47, 63, 64	syl2anc 583	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)[,)(𝐷‘𝑗)) ∈ dom vol)
66	46, 65	ffvelcdmd 7087	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (vol‘((𝐶‘𝑗)[,)(𝐷‘𝑗))) ∈ (0[,]+∞))
67	47	rexrd 11271	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐶‘𝑗) ∈ ℝ^*)
68	47	leidd 11787	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐶‘𝑗) ≤ (𝐶‘𝑗))
69		min1 13175	. . . . . . . . . . . . . . . 16 ⊢ (((𝐷‘𝑗) ∈ ℝ ∧ 𝑆 ∈ ℝ) → if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆) ≤ (𝐷‘𝑗))
70	48, 49, 69	syl2anc 583	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆) ≤ (𝐷‘𝑗))
71		icossico 13401	. . . . . . . . . . . . . . 15 ⊢ ((((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)))
72	67, 63, 68, 70, 71	syl22anc 836	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆)) ⊆ ((𝐶‘𝑗)[,)(𝐷‘𝑗)))
73		volss 25382	. . . . . . . . . . . . . 14 ⊢ ((((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))))
74	53, 65, 72, 73	syl3anc 1370	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆))) ≤ (vol‘((𝐶‘𝑗)[,)(𝐷‘𝑗))))
75	62, 44, 54, 66, 74	sge0lempt 45585	. . . . . . . . . . . 12 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆))))) ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)(𝐷‘𝑗))))))
76	60	ltpnfd 13108	. . . . . . . . . . . 12 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)(𝐷‘𝑗))))) < +∞)
77	57, 61, 59, 75, 76	xrlelttrd 13146	. . . . . . . . . . 11 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆))))) < +∞)
78	57, 59, 77	xrltned 44526	. . . . . . . . . 10 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆))))) ≠ +∞)
79	78	neneqd 2944	. . . . . . . . 9 ⊢ (𝜑 → ¬ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆))))) = +∞)
80	44, 56	sge0repnf 45561	. . . . . . . . 9 ⊢ (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))))) = +∞))
81	79, 80	mpbird 257	. . . . . . . 8 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆))))) ∈ ℝ)
82	40, 81	readdcld 11250	. . . . . . 7 ⊢ (𝜑 → ((𝑀 − 𝑆) + (Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆)))))) ∈ ℝ)
83	9	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑀 ∈ ℝ)
84	48, 83	ifcld 4574	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → if((𝐷‘𝑗) ≤ 𝑀, (𝐷‘𝑗), 𝑀) ∈ ℝ)
85	84	rexrd 11271	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → if((𝐷‘𝑗) ≤ 𝑀, (𝐷‘𝑗), 𝑀) ∈ ℝ^*)
86		icombl 25413	. . . . . . . . . . . . . 14 ⊢ (((𝐶‘𝑗) ∈ ℝ ∧ if((𝐷‘𝑗) ≤ 𝑀, (𝐷‘𝑗), 𝑀) ∈ ℝ^*) → ((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑀, (𝐷‘𝑗), 𝑀)) ∈ dom vol)
87	47, 85, 86	syl2anc 583	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑀, (𝐷‘𝑗), 𝑀)) ∈ dom vol)
88	46, 87	ffvelcdmd 7087	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑀, (𝐷‘𝑗), 𝑀))) ∈ (0[,]+∞))
89		eqid 2731	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑀, (𝐷‘𝑗), 𝑀)))) = (𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑀, (𝐷‘𝑗), 𝑀))))
90	88, 89	fmptd 7115	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑀, (𝐷‘𝑗), 𝑀)))):ℕ⟶(0[,]+∞))
91	44, 90	sge0xrcl 45560	. . . . . . . . . 10 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑀, (𝐷‘𝑗), 𝑀))))) ∈ ℝ^*)
92		min1 13175	. . . . . . . . . . . . . . 15 ⊢ (((𝐷‘𝑗) ∈ ℝ ∧ 𝑀 ∈ ℝ) → if((𝐷‘𝑗) ≤ 𝑀, (𝐷‘𝑗), 𝑀) ≤ (𝐷‘𝑗))
93	48, 83, 92	syl2anc 583	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → if((𝐷‘𝑗) ≤ 𝑀, (𝐷‘𝑗), 𝑀) ≤ (𝐷‘𝑗))
94		icossico 13401	. . . . . . . . . . . . . 14 ⊢ ((((𝐶‘𝑗) ∈ ℝ^* ∧ (𝐷‘𝑗) ∈ ℝ^*) ∧ ((𝐶‘𝑗) ≤ (𝐶‘𝑗) ∧ if((𝐷‘𝑗) ≤ 𝑀, (𝐷‘𝑗), 𝑀) ≤ (𝐷‘𝑗))) → ((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑀, (𝐷‘𝑗), 𝑀)) ⊆ ((𝐶‘𝑗)[,)(𝐷‘𝑗)))
95	67, 63, 68, 93, 94	syl22anc 836	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑀, (𝐷‘𝑗), 𝑀)) ⊆ ((𝐶‘𝑗)[,)(𝐷‘𝑗)))
96		volss 25382	. . . . . . . . . . . . 13 ⊢ ((((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑀, (𝐷‘𝑗), 𝑀)) ∈ dom vol ∧ ((𝐶‘𝑗)[,)(𝐷‘𝑗)) ∈ dom vol ∧ ((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑀, (𝐷‘𝑗), 𝑀)) ⊆ ((𝐶‘𝑗)[,)(𝐷‘𝑗))) → (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑀, (𝐷‘𝑗), 𝑀))) ≤ (vol‘((𝐶‘𝑗)[,)(𝐷‘𝑗))))
97	87, 65, 95, 96	syl3anc 1370	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑀, (𝐷‘𝑗), 𝑀))) ≤ (vol‘((𝐶‘𝑗)[,)(𝐷‘𝑗))))
98	62, 44, 88, 66, 97	sge0lempt 45585	. . . . . . . . . . 11 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑀, (𝐷‘𝑗), 𝑀))))) ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)(𝐷‘𝑗))))))
99	91, 61, 59, 98, 76	xrlelttrd 13146	. . . . . . . . . 10 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑀, (𝐷‘𝑗), 𝑀))))) < +∞)
100	91, 59, 99	xrltned 44526	. . . . . . . . 9 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑀, (𝐷‘𝑗), 𝑀))))) ≠ +∞)
101	100	neneqd 2944	. . . . . . . 8 ⊢ (𝜑 → ¬ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑀, (𝐷‘𝑗), 𝑀))))) = +∞)
102	44, 90	sge0repnf 45561	. . . . . . . 8 ⊢ (𝜑 → ((Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑀, (𝐷‘𝑗), 𝑀))))) ∈ ℝ ↔ ¬ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑀, (𝐷‘𝑗), 𝑀))))) = +∞))
103	101, 102	mpbird 257	. . . . . . 7 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑀, (𝐷‘𝑗), 𝑀))))) ∈ ℝ)
104		hoidmv1lelem2.e	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑆 ∈ 𝑈)
105		hoidmv1lelem2.u	. . . . . . . . . . 11 ⊢ 𝑈 = {𝑧 ∈ (𝐴[,]𝐵) ∣ (𝑧 − 𝐴) ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧)))))}
106	104, 105	eleqtrdi 2842	. . . . . . . . . 10 ⊢ (𝜑 → 𝑆 ∈ {𝑧 ∈ (𝐴[,]𝐵) ∣ (𝑧 − 𝐴) ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧)))))})
107		oveq1 7419	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑆 → (𝑧 − 𝐴) = (𝑆 − 𝐴))
108		simpl 482	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑧 = 𝑆 ∧ 𝑗 ∈ ℕ) → 𝑧 = 𝑆)
109	108	breq2d 5160	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 = 𝑆 ∧ 𝑗 ∈ ℕ) → ((𝐷‘𝑗) ≤ 𝑧 ↔ (𝐷‘𝑗) ≤ 𝑆))
110	109, 108	ifbieq2d 4554	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 = 𝑆 ∧ 𝑗 ∈ ℕ) → if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧) = if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆))
111	110	oveq2d 7428	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 = 𝑆 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧)) = ((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆)))
112	111	fveq2d 6895	. . . . . . . . . . . . . 14 ⊢ ((𝑧 = 𝑆 ∧ 𝑗 ∈ ℕ) → (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧))) = (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆))))
113	112	mpteq2dva 5248	. . . . . . . . . . . . 13 ⊢ (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)))))
114	113	fveq2d 6895	. . . . . . . . . . . 12 ⊢ (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))))))
115	107, 114	breq12d 5161	. . . . . . . . . . 11 ⊢ (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)))))))
116	115	elrab 3683	. . . . . . . . . 10 ⊢ (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)))))))
117	106, 116	sylib 217	. . . . . . . . 9 ⊢ (𝜑 → (𝑆 ∈ (𝐴[,]𝐵) ∧ (𝑆 − 𝐴) ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆)))))))
118	117	simprd 495	. . . . . . . 8 ⊢ (𝜑 → (𝑆 − 𝐴) ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆))))))
119	41, 81, 40, 118	leadd2dd 11836	. . . . . . 7 ⊢ (𝜑 → ((𝑀 − 𝑆) + (𝑆 − 𝐴)) ≤ ((𝑀 − 𝑆) + (Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆)))))))
120		difssd 4132	. . . . . . . . . 10 ⊢ (𝜑 → (ℕ ∖ {𝐾}) ⊆ ℕ)
121	62, 44, 54, 81, 120	sge0ssrempt 45580	. . . . . . . . 9 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ (ℕ ∖ {𝐾}) ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆))))) ∈ ℝ)
122		difexg 5327	. . . . . . . . . . . . . . 15 ⊢ (ℕ ∈ V → (ℕ ∖ {𝐾}) ∈ V)
123	43, 122	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ (ℕ ∖ {𝐾}) ∈ V
124	123	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → (ℕ ∖ {𝐾}) ∈ V)
125	45	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℕ ∖ {𝐾})) → vol:dom vol⟶(0[,]+∞))
126		simpl 482	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℕ ∖ {𝐾})) → 𝜑)
127		eldifi 4126	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 ∈ (ℕ ∖ {𝐾}) → 𝑗 ∈ ℕ)
128	127	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℕ ∖ {𝐾})) → 𝑗 ∈ ℕ)
129	126, 128, 47	syl2anc 583	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℕ ∖ {𝐾})) → (𝐶‘𝑗) ∈ ℝ)
130	128, 85	syldan 590	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℕ ∖ {𝐾})) → if((𝐷‘𝑗) ≤ 𝑀, (𝐷‘𝑗), 𝑀) ∈ ℝ^*)
131	129, 130, 86	syl2anc 583	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℕ ∖ {𝐾})) → ((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑀, (𝐷‘𝑗), 𝑀)) ∈ dom vol)
132	125, 131	ffvelcdmd 7087	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℕ ∖ {𝐾})) → (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑀, (𝐷‘𝑗), 𝑀))) ∈ (0[,]+∞))
133	62, 124, 132	sge0xrclmpt 45603	. . . . . . . . . . . 12 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ (ℕ ∖ {𝐾}) ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑀, (𝐷‘𝑗), 𝑀))))) ∈ ℝ^*)
134	44, 88, 120	sge0lessmpt 45574	. . . . . . . . . . . . 13 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ (ℕ ∖ {𝐾}) ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑀, (𝐷‘𝑗), 𝑀))))) ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑀, (𝐷‘𝑗), 𝑀))))))
135	133, 91, 59, 134, 99	xrlelttrd 13146	. . . . . . . . . . . 12 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ (ℕ ∖ {𝐾}) ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑀, (𝐷‘𝑗), 𝑀))))) < +∞)
136	133, 59, 135	xrltned 44526	. . . . . . . . . . 11 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ (ℕ ∖ {𝐾}) ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑀, (𝐷‘𝑗), 𝑀))))) ≠ +∞)
137	136	neneqd 2944	. . . . . . . . . 10 ⊢ (𝜑 → ¬ (Σ^{^}‘(𝑗 ∈ (ℕ ∖ {𝐾}) ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑀, (𝐷‘𝑗), 𝑀))))) = +∞)
138	62, 124, 132	sge0repnfmpt 45614	. . . . . . . . . 10 ⊢ (𝜑 → ((Σ^{^}‘(𝑗 ∈ (ℕ ∖ {𝐾}) ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑀, (𝐷‘𝑗), 𝑀))))) ∈ ℝ ↔ ¬ (Σ^{^}‘(𝑗 ∈ (ℕ ∖ {𝐾}) ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑀, (𝐷‘𝑗), 𝑀))))) = +∞))
139	137, 138	mpbird 257	. . . . . . . . 9 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ (ℕ ∖ {𝐾}) ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑀, (𝐷‘𝑗), 𝑀))))) ∈ ℝ)
140	9, 11	resubcld 11649	. . . . . . . . 9 ⊢ (𝜑 → (𝑀 − (𝐶‘𝐾)) ∈ ℝ)
141	128, 54	syldan 590	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℕ ∖ {𝐾})) → (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆))) ∈ (0[,]+∞))
142	128, 53	syldan 590	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℕ ∖ {𝐾})) → ((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆)) ∈ dom vol)
143	128, 67	syldan 590	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℕ ∖ {𝐾})) → (𝐶‘𝑗) ∈ ℝ^*)
144	128, 68	syldan 590	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℕ ∖ {𝐾})) → (𝐶‘𝑗) ≤ (𝐶‘𝑗))
145		iftrue 4534	. . . . . . . . . . . . . . . 16 ⊢ ((𝐷‘𝑗) ≤ 𝑆 → if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆) = (𝐷‘𝑗))
146	145	adantl 481	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝐷‘𝑗) ≤ 𝑆) → if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆) = (𝐷‘𝑗))
147	48	leidd 11787	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐷‘𝑗) ≤ (𝐷‘𝑗))
148	147	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝐷‘𝑗) ≤ 𝑆) → (𝐷‘𝑗) ≤ (𝐷‘𝑗))
149	48	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝐷‘𝑗) ≤ 𝑆) → (𝐷‘𝑗) ∈ ℝ)
150	83	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝐷‘𝑗) ≤ 𝑆) → 𝑀 ∈ ℝ)
151	49	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝐷‘𝑗) ≤ 𝑆) → 𝑆 ∈ ℝ)
152		simpr 484	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝐷‘𝑗) ≤ 𝑆) → (𝐷‘𝑗) ≤ 𝑆)
153	20, 22	jca 511	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (𝑆 < (𝐷‘𝐾) ∧ 𝑆 < 𝐵))
154		ltmin 13180	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑆 ∈ ℝ ∧ (𝐷‘𝐾) ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝑆 < if((𝐷‘𝐾) ≤ 𝐵, (𝐷‘𝐾), 𝐵) ↔ (𝑆 < (𝐷‘𝐾) ∧ 𝑆 < 𝐵)))
155	16, 7, 2, 154	syl3anc 1370	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (𝑆 < if((𝐷‘𝐾) ≤ 𝐵, (𝐷‘𝐾), 𝐵) ↔ (𝑆 < (𝐷‘𝐾) ∧ 𝑆 < 𝐵)))
156	153, 155	mpbird 257	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝑆 < if((𝐷‘𝐾) ≤ 𝐵, (𝐷‘𝐾), 𝐵))
157	156, 29	breqtrd 5174	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑆 < 𝑀)
158	157	ad2antrr 723	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝐷‘𝑗) ≤ 𝑆) → 𝑆 < 𝑀)
159	149, 151, 150, 152, 158	lelttrd 11379	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝐷‘𝑗) ≤ 𝑆) → (𝐷‘𝑗) < 𝑀)
160	149, 150, 159	ltled 11369	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝐷‘𝑗) ≤ 𝑆) → (𝐷‘𝑗) ≤ 𝑀)
161	148, 160	jca 511	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝐷‘𝑗) ≤ 𝑆) → ((𝐷‘𝑗) ≤ (𝐷‘𝑗) ∧ (𝐷‘𝑗) ≤ 𝑀))
162		lemin 13178	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐷‘𝑗) ∈ ℝ ∧ (𝐷‘𝑗) ∈ ℝ ∧ 𝑀 ∈ ℝ) → ((𝐷‘𝑗) ≤ if((𝐷‘𝑗) ≤ 𝑀, (𝐷‘𝑗), 𝑀) ↔ ((𝐷‘𝑗) ≤ (𝐷‘𝑗) ∧ (𝐷‘𝑗) ≤ 𝑀)))
163	149, 149, 150, 162	syl3anc 1370	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝐷‘𝑗) ≤ 𝑆) → ((𝐷‘𝑗) ≤ if((𝐷‘𝑗) ≤ 𝑀, (𝐷‘𝑗), 𝑀) ↔ ((𝐷‘𝑗) ≤ (𝐷‘𝑗) ∧ (𝐷‘𝑗) ≤ 𝑀)))
164	161, 163	mpbird 257	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝐷‘𝑗) ≤ 𝑆) → (𝐷‘𝑗) ≤ if((𝐷‘𝑗) ≤ 𝑀, (𝐷‘𝑗), 𝑀))
165	146, 164	eqbrtrd 5170	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝐷‘𝑗) ≤ 𝑆) → if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆) ≤ if((𝐷‘𝑗) ≤ 𝑀, (𝐷‘𝑗), 𝑀))
166		iffalse 4537	. . . . . . . . . . . . . . . 16 ⊢ (¬ (𝐷‘𝑗) ≤ 𝑆 → if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆) = 𝑆)
167	166	adantl 481	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷‘𝑗) ≤ 𝑆) → if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆) = 𝑆)
168	49	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷‘𝑗) ≤ 𝑆) → 𝑆 ∈ ℝ)
169	84	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷‘𝑗) ≤ 𝑆) → if((𝐷‘𝑗) ≤ 𝑀, (𝐷‘𝑗), 𝑀) ∈ ℝ)
170		simpr 484	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷‘𝑗) ≤ 𝑆) → ¬ (𝐷‘𝑗) ≤ 𝑆)
171	48	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷‘𝑗) ≤ 𝑆) → (𝐷‘𝑗) ∈ ℝ)
172	168, 171	ltnled 11368	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷‘𝑗) ≤ 𝑆) → (𝑆 < (𝐷‘𝑗) ↔ ¬ (𝐷‘𝑗) ≤ 𝑆))
173	170, 172	mpbird 257	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷‘𝑗) ≤ 𝑆) → 𝑆 < (𝐷‘𝑗))
174	157	ad2antrr 723	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷‘𝑗) ≤ 𝑆) → 𝑆 < 𝑀)
175	173, 174	jca 511	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷‘𝑗) ≤ 𝑆) → (𝑆 < (𝐷‘𝑗) ∧ 𝑆 < 𝑀))
176	83	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷‘𝑗) ≤ 𝑆) → 𝑀 ∈ ℝ)
177		ltmin 13180	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑆 ∈ ℝ ∧ (𝐷‘𝑗) ∈ ℝ ∧ 𝑀 ∈ ℝ) → (𝑆 < if((𝐷‘𝑗) ≤ 𝑀, (𝐷‘𝑗), 𝑀) ↔ (𝑆 < (𝐷‘𝑗) ∧ 𝑆 < 𝑀)))
178	168, 171, 176, 177	syl3anc 1370	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷‘𝑗) ≤ 𝑆) → (𝑆 < if((𝐷‘𝑗) ≤ 𝑀, (𝐷‘𝑗), 𝑀) ↔ (𝑆 < (𝐷‘𝑗) ∧ 𝑆 < 𝑀)))
179	175, 178	mpbird 257	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷‘𝑗) ≤ 𝑆) → 𝑆 < if((𝐷‘𝑗) ≤ 𝑀, (𝐷‘𝑗), 𝑀))
180	168, 169, 179	ltled 11369	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷‘𝑗) ≤ 𝑆) → 𝑆 ≤ if((𝐷‘𝑗) ≤ 𝑀, (𝐷‘𝑗), 𝑀))
181	167, 180	eqbrtrd 5170	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷‘𝑗) ≤ 𝑆) → if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆) ≤ if((𝐷‘𝑗) ≤ 𝑀, (𝐷‘𝑗), 𝑀))
182	165, 181	pm2.61dan 810	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆) ≤ if((𝐷‘𝑗) ≤ 𝑀, (𝐷‘𝑗), 𝑀))
183	128, 182	syldan 590	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℕ ∖ {𝐾})) → if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆) ≤ if((𝐷‘𝑗) ≤ 𝑀, (𝐷‘𝑗), 𝑀))
184		icossico 13401	. . . . . . . . . . . 12 ⊢ ((((𝐶‘𝑗) ∈ ℝ^* ∧ if((𝐷‘𝑗) ≤ 𝑀, (𝐷‘𝑗), 𝑀) ∈ ℝ^*) ∧ ((𝐶‘𝑗) ≤ (𝐶‘𝑗) ∧ if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆) ≤ if((𝐷‘𝑗) ≤ 𝑀, (𝐷‘𝑗), 𝑀))) → ((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆)) ⊆ ((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑀, (𝐷‘𝑗), 𝑀)))
185	143, 130, 144, 183, 184	syl22anc 836	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℕ ∖ {𝐾})) → ((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆)) ⊆ ((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑀, (𝐷‘𝑗), 𝑀)))
186		volss 25382	. . . . . . . . . . 11 ⊢ ((((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆)) ∈ dom vol ∧ ((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑀, (𝐷‘𝑗), 𝑀)) ∈ dom vol ∧ ((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆)) ⊆ ((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑀, (𝐷‘𝑗), 𝑀))) → (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆))) ≤ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑀, (𝐷‘𝑗), 𝑀))))
187	142, 131, 185, 186	syl3anc 1370	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℕ ∖ {𝐾})) → (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆))) ≤ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑀, (𝐷‘𝑗), 𝑀))))
188	62, 124, 141, 132, 187	sge0lempt 45585	. . . . . . . . 9 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ (ℕ ∖ {𝐾}) ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆))))) ≤ (Σ^{^}‘(𝑗 ∈ (ℕ ∖ {𝐾}) ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑀, (𝐷‘𝑗), 𝑀))))))
189	121, 139, 140, 188	leadd2dd 11836	. . . . . . . 8 ⊢ (𝜑 → ((𝑀 − (𝐶‘𝐾)) + (Σ^{^}‘(𝑗 ∈ (ℕ ∖ {𝐾}) ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆)))))) ≤ ((𝑀 − (𝐶‘𝐾)) + (Σ^{^}‘(𝑗 ∈ (ℕ ∖ {𝐾}) ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑀, (𝐷‘𝑗), 𝑀)))))))
190		difsnid 4813	. . . . . . . . . . . . . . . 16 ⊢ (𝐾 ∈ ℕ → ((ℕ ∖ {𝐾}) ∪ {𝐾}) = ℕ)
191	6, 190	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((ℕ ∖ {𝐾}) ∪ {𝐾}) = ℕ)
192	191	eqcomd 2737	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ℕ = ((ℕ ∖ {𝐾}) ∪ {𝐾}))
193	192	mpteq1d 5243	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆)))) = (𝑗 ∈ ((ℕ ∖ {𝐾}) ∪ {𝐾}) ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆)))))
194	193	fveq2d 6895	. . . . . . . . . . . 12 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆))))) = (Σ^{^}‘(𝑗 ∈ ((ℕ ∖ {𝐾}) ∪ {𝐾}) ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆))))))
195		neldifsnd 4796	. . . . . . . . . . . . 13 ⊢ (𝜑 → ¬ 𝐾 ∈ (ℕ ∖ {𝐾}))
196		fveq2 6891	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝐾 → (𝐶‘𝑗) = (𝐶‘𝐾))
197		fveq2 6891	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = 𝐾 → (𝐷‘𝑗) = (𝐷‘𝐾))
198	197	breq1d 5158	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 𝐾 → ((𝐷‘𝑗) ≤ 𝑆 ↔ (𝐷‘𝐾) ≤ 𝑆))
199	198, 197	ifbieq1d 4552	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝐾 → if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆) = if((𝐷‘𝐾) ≤ 𝑆, (𝐷‘𝐾), 𝑆))
200	196, 199	oveq12d 7430	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝐾 → ((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆)) = ((𝐶‘𝐾)[,)if((𝐷‘𝐾) ≤ 𝑆, (𝐷‘𝐾), 𝑆)))
201	200	fveq2d 6895	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝐾 → (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆))) = (vol‘((𝐶‘𝐾)[,)if((𝐷‘𝐾) ≤ 𝑆, (𝐷‘𝐾), 𝑆))))
202	45	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → vol:dom vol⟶(0[,]+∞))
203	7, 16	ifcld 4574	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → if((𝐷‘𝐾) ≤ 𝑆, (𝐷‘𝐾), 𝑆) ∈ ℝ)
204	203	rexrd 11271	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → if((𝐷‘𝐾) ≤ 𝑆, (𝐷‘𝐾), 𝑆) ∈ ℝ^*)
205		icombl 25413	. . . . . . . . . . . . . . 15 ⊢ (((𝐶‘𝐾) ∈ ℝ ∧ if((𝐷‘𝐾) ≤ 𝑆, (𝐷‘𝐾), 𝑆) ∈ ℝ^*) → ((𝐶‘𝐾)[,)if((𝐷‘𝐾) ≤ 𝑆, (𝐷‘𝐾), 𝑆)) ∈ dom vol)
206	11, 204, 205	syl2anc 583	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝐶‘𝐾)[,)if((𝐷‘𝐾) ≤ 𝑆, (𝐷‘𝐾), 𝑆)) ∈ dom vol)
207	202, 206	ffvelcdmd 7087	. . . . . . . . . . . . 13 ⊢ (𝜑 → (vol‘((𝐶‘𝐾)[,)if((𝐷‘𝐾) ≤ 𝑆, (𝐷‘𝐾), 𝑆))) ∈ (0[,]+∞))
208	62, 124, 6, 195, 141, 201, 207	sge0splitsn 45616	. . . . . . . . . . . 12 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ((ℕ ∖ {𝐾}) ∪ {𝐾}) ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆))))) = ((Σ^{^}‘(𝑗 ∈ (ℕ ∖ {𝐾}) ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆))))) +_𝑒 (vol‘((𝐶‘𝐾)[,)if((𝐷‘𝐾) ≤ 𝑆, (𝐷‘𝐾), 𝑆)))))
209		volicore 45756	. . . . . . . . . . . . . . 15 ⊢ (((𝐶‘𝐾) ∈ ℝ ∧ if((𝐷‘𝐾) ≤ 𝑆, (𝐷‘𝐾), 𝑆) ∈ ℝ) → (vol‘((𝐶‘𝐾)[,)if((𝐷‘𝐾) ≤ 𝑆, (𝐷‘𝐾), 𝑆))) ∈ ℝ)
210	11, 203, 209	syl2anc 583	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (vol‘((𝐶‘𝐾)[,)if((𝐷‘𝐾) ≤ 𝑆, (𝐷‘𝐾), 𝑆))) ∈ ℝ)
211		rexadd 13218	. . . . . . . . . . . . . 14 ⊢ (((Σ^{^}‘(𝑗 ∈ (ℕ ∖ {𝐾}) ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆))))) ∈ ℝ ∧ (vol‘((𝐶‘𝐾)[,)if((𝐷‘𝐾) ≤ 𝑆, (𝐷‘𝐾), 𝑆))) ∈ ℝ) → ((Σ^{^}‘(𝑗 ∈ (ℕ ∖ {𝐾}) ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆))))) +_𝑒 (vol‘((𝐶‘𝐾)[,)if((𝐷‘𝐾) ≤ 𝑆, (𝐷‘𝐾), 𝑆)))) = ((Σ^{^}‘(𝑗 ∈ (ℕ ∖ {𝐾}) ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆))))) + (vol‘((𝐶‘𝐾)[,)if((𝐷‘𝐾) ≤ 𝑆, (𝐷‘𝐾), 𝑆)))))
212	121, 210, 211	syl2anc 583	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((Σ^{^}‘(𝑗 ∈ (ℕ ∖ {𝐾}) ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆))))) +_𝑒 (vol‘((𝐶‘𝐾)[,)if((𝐷‘𝐾) ≤ 𝑆, (𝐷‘𝐾), 𝑆)))) = ((Σ^{^}‘(𝑗 ∈ (ℕ ∖ {𝐾}) ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆))))) + (vol‘((𝐶‘𝐾)[,)if((𝐷‘𝐾) ≤ 𝑆, (𝐷‘𝐾), 𝑆)))))
213		volico 45158	. . . . . . . . . . . . . . . 16 ⊢ (((𝐶‘𝐾) ∈ ℝ ∧ if((𝐷‘𝐾) ≤ 𝑆, (𝐷‘𝐾), 𝑆) ∈ ℝ) → (vol‘((𝐶‘𝐾)[,)if((𝐷‘𝐾) ≤ 𝑆, (𝐷‘𝐾), 𝑆))) = if((𝐶‘𝐾) < if((𝐷‘𝐾) ≤ 𝑆, (𝐷‘𝐾), 𝑆), (if((𝐷‘𝐾) ≤ 𝑆, (𝐷‘𝐾), 𝑆) − (𝐶‘𝐾)), 0))
214	11, 203, 213	syl2anc 583	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (vol‘((𝐶‘𝐾)[,)if((𝐷‘𝐾) ≤ 𝑆, (𝐷‘𝐾), 𝑆))) = if((𝐶‘𝐾) < if((𝐷‘𝐾) ≤ 𝑆, (𝐷‘𝐾), 𝑆), (if((𝐷‘𝐾) ≤ 𝑆, (𝐷‘𝐾), 𝑆) − (𝐶‘𝐾)), 0))
215	16, 7	ltnled 11368	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑆 < (𝐷‘𝐾) ↔ ¬ (𝐷‘𝐾) ≤ 𝑆))
216	20, 215	mpbid 231	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ¬ (𝐷‘𝐾) ≤ 𝑆)
217	216	iffalsed 4539	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → if((𝐷‘𝐾) ≤ 𝑆, (𝐷‘𝐾), 𝑆) = 𝑆)
218	217	breq2d 5160	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((𝐶‘𝐾) < if((𝐷‘𝐾) ≤ 𝑆, (𝐷‘𝐾), 𝑆) ↔ (𝐶‘𝐾) < 𝑆))
219	218	ifbid 4551	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → if((𝐶‘𝐾) < if((𝐷‘𝐾) ≤ 𝑆, (𝐷‘𝐾), 𝑆), (if((𝐷‘𝐾) ≤ 𝑆, (𝐷‘𝐾), 𝑆) − (𝐶‘𝐾)), 0) = if((𝐶‘𝐾) < 𝑆, (if((𝐷‘𝐾) ≤ 𝑆, (𝐷‘𝐾), 𝑆) − (𝐶‘𝐾)), 0))
220	217	oveq1d 7427	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (if((𝐷‘𝐾) ≤ 𝑆, (𝐷‘𝐾), 𝑆) − (𝐶‘𝐾)) = (𝑆 − (𝐶‘𝐾)))
221	220	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝐶‘𝐾) < 𝑆) → (if((𝐷‘𝐾) ≤ 𝑆, (𝐷‘𝐾), 𝑆) − (𝐶‘𝐾)) = (𝑆 − (𝐶‘𝐾)))
222	217, 204	eqeltrrd 2833	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑆 ∈ ℝ^*)
223	222	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ ¬ (𝐶‘𝐾) < 𝑆) → 𝑆 ∈ ℝ^*)
224	18	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ ¬ (𝐶‘𝐾) < 𝑆) → (𝐶‘𝐾) ∈ ℝ^*)
225		simpr 484	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ ¬ (𝐶‘𝐾) < 𝑆) → ¬ (𝐶‘𝐾) < 𝑆)
226	16	adantr 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ ¬ (𝐶‘𝐾) < 𝑆) → 𝑆 ∈ ℝ)
227	11	adantr 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ ¬ (𝐶‘𝐾) < 𝑆) → (𝐶‘𝐾) ∈ ℝ)
228	226, 227	lenltd 11367	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ ¬ (𝐶‘𝐾) < 𝑆) → (𝑆 ≤ (𝐶‘𝐾) ↔ ¬ (𝐶‘𝐾) < 𝑆))
229	225, 228	mpbird 257	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ ¬ (𝐶‘𝐾) < 𝑆) → 𝑆 ≤ (𝐶‘𝐾))
230		icogelb 13382	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐶‘𝐾) ∈ ℝ^* ∧ (𝐷‘𝐾) ∈ ℝ^* ∧ 𝑆 ∈ ((𝐶‘𝐾)[,)(𝐷‘𝐾))) → (𝐶‘𝐾) ≤ 𝑆)
231	18, 12, 15, 230	syl3anc 1370	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝐶‘𝐾) ≤ 𝑆)
232	231	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ ¬ (𝐶‘𝐾) < 𝑆) → (𝐶‘𝐾) ≤ 𝑆)
233	223, 224, 229, 232	xrletrid 13141	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ ¬ (𝐶‘𝐾) < 𝑆) → 𝑆 = (𝐶‘𝐾))
234	233	oveq1d 7427	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ¬ (𝐶‘𝐾) < 𝑆) → (𝑆 − (𝐶‘𝐾)) = ((𝐶‘𝐾) − (𝐶‘𝐾)))
235	227	recnd 11249	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ ¬ (𝐶‘𝐾) < 𝑆) → (𝐶‘𝐾) ∈ ℂ)
236	235	subidd 11566	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ¬ (𝐶‘𝐾) < 𝑆) → ((𝐶‘𝐾) − (𝐶‘𝐾)) = 0)
237	234, 236	eqtr2d 2772	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ¬ (𝐶‘𝐾) < 𝑆) → 0 = (𝑆 − (𝐶‘𝐾)))
238	221, 237	ifeqda 4564	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → if((𝐶‘𝐾) < 𝑆, (if((𝐷‘𝐾) ≤ 𝑆, (𝐷‘𝐾), 𝑆) − (𝐶‘𝐾)), 0) = (𝑆 − (𝐶‘𝐾)))
239	214, 219, 238	3eqtrd 2775	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (vol‘((𝐶‘𝐾)[,)if((𝐷‘𝐾) ≤ 𝑆, (𝐷‘𝐾), 𝑆))) = (𝑆 − (𝐶‘𝐾)))
240	239	oveq2d 7428	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((Σ^{^}‘(𝑗 ∈ (ℕ ∖ {𝐾}) ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆))))) + (vol‘((𝐶‘𝐾)[,)if((𝐷‘𝐾) ≤ 𝑆, (𝐷‘𝐾), 𝑆)))) = ((Σ^{^}‘(𝑗 ∈ (ℕ ∖ {𝐾}) ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆))))) + (𝑆 − (𝐶‘𝐾))))
241	121	recnd 11249	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ (ℕ ∖ {𝐾}) ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆))))) ∈ ℂ)
242	11	recnd 11249	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐶‘𝐾) ∈ ℂ)
243	36, 242	subcld 11578	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑆 − (𝐶‘𝐾)) ∈ ℂ)
244	241, 243	addcomd 11423	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((Σ^{^}‘(𝑗 ∈ (ℕ ∖ {𝐾}) ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆))))) + (𝑆 − (𝐶‘𝐾))) = ((𝑆 − (𝐶‘𝐾)) + (Σ^{^}‘(𝑗 ∈ (ℕ ∖ {𝐾}) ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆)))))))
245	212, 240, 244	3eqtrd 2775	. . . . . . . . . . . 12 ⊢ (𝜑 → ((Σ^{^}‘(𝑗 ∈ (ℕ ∖ {𝐾}) ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆))))) +_𝑒 (vol‘((𝐶‘𝐾)[,)if((𝐷‘𝐾) ≤ 𝑆, (𝐷‘𝐾), 𝑆)))) = ((𝑆 − (𝐶‘𝐾)) + (Σ^{^}‘(𝑗 ∈ (ℕ ∖ {𝐾}) ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆)))))))
246	194, 208, 245	3eqtrd 2775	. . . . . . . . . . 11 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆))))) = ((𝑆 − (𝐶‘𝐾)) + (Σ^{^}‘(𝑗 ∈ (ℕ ∖ {𝐾}) ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆)))))))
247	246	oveq2d 7428	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑀 − 𝑆) + (Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆)))))) = ((𝑀 − 𝑆) + ((𝑆 − (𝐶‘𝐾)) + (Σ^{^}‘(𝑗 ∈ (ℕ ∖ {𝐾}) ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆))))))))
248	40	recnd 11249	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑀 − 𝑆) ∈ ℂ)
249	248, 243, 241	addassd 11243	. . . . . . . . . . 11 ⊢ (𝜑 → (((𝑀 − 𝑆) + (𝑆 − (𝐶‘𝐾))) + (Σ^{^}‘(𝑗 ∈ (ℕ ∖ {𝐾}) ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆)))))) = ((𝑀 − 𝑆) + ((𝑆 − (𝐶‘𝐾)) + (Σ^{^}‘(𝑗 ∈ (ℕ ∖ {𝐾}) ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆))))))))
250	249	eqcomd 2737	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑀 − 𝑆) + ((𝑆 − (𝐶‘𝐾)) + (Σ^{^}‘(𝑗 ∈ (ℕ ∖ {𝐾}) ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆))))))) = (((𝑀 − 𝑆) + (𝑆 − (𝐶‘𝐾))) + (Σ^{^}‘(𝑗 ∈ (ℕ ∖ {𝐾}) ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆)))))))
251	35, 36, 242	npncand 11602	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑀 − 𝑆) + (𝑆 − (𝐶‘𝐾))) = (𝑀 − (𝐶‘𝐾)))
252	251	oveq1d 7427	. . . . . . . . . 10 ⊢ (𝜑 → (((𝑀 − 𝑆) + (𝑆 − (𝐶‘𝐾))) + (Σ^{^}‘(𝑗 ∈ (ℕ ∖ {𝐾}) ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆)))))) = ((𝑀 − (𝐶‘𝐾)) + (Σ^{^}‘(𝑗 ∈ (ℕ ∖ {𝐾}) ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆)))))))
253	247, 250, 252	3eqtrd 2775	. . . . . . . . 9 ⊢ (𝜑 → ((𝑀 − 𝑆) + (Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆)))))) = ((𝑀 − (𝐶‘𝐾)) + (Σ^{^}‘(𝑗 ∈ (ℕ ∖ {𝐾}) ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆)))))))
254	192	mpteq1d 5243	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑀, (𝐷‘𝑗), 𝑀)))) = (𝑗 ∈ ((ℕ ∖ {𝐾}) ∪ {𝐾}) ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑀, (𝐷‘𝑗), 𝑀)))))
255	254	fveq2d 6895	. . . . . . . . . 10 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑀, (𝐷‘𝑗), 𝑀))))) = (Σ^{^}‘(𝑗 ∈ ((ℕ ∖ {𝐾}) ∪ {𝐾}) ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑀, (𝐷‘𝑗), 𝑀))))))
256	197	breq1d 5158	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝐾 → ((𝐷‘𝑗) ≤ 𝑀 ↔ (𝐷‘𝐾) ≤ 𝑀))
257	256, 197	ifbieq1d 4552	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝐾 → if((𝐷‘𝑗) ≤ 𝑀, (𝐷‘𝑗), 𝑀) = if((𝐷‘𝐾) ≤ 𝑀, (𝐷‘𝐾), 𝑀))
258	196, 257	oveq12d 7430	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝐾 → ((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑀, (𝐷‘𝑗), 𝑀)) = ((𝐶‘𝐾)[,)if((𝐷‘𝐾) ≤ 𝑀, (𝐷‘𝐾), 𝑀)))
259	258	fveq2d 6895	. . . . . . . . . . 11 ⊢ (𝑗 = 𝐾 → (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑀, (𝐷‘𝑗), 𝑀))) = (vol‘((𝐶‘𝐾)[,)if((𝐷‘𝐾) ≤ 𝑀, (𝐷‘𝐾), 𝑀))))
260	7, 9	ifcld 4574	. . . . . . . . . . . . . 14 ⊢ (𝜑 → if((𝐷‘𝐾) ≤ 𝑀, (𝐷‘𝐾), 𝑀) ∈ ℝ)
261	260	rexrd 11271	. . . . . . . . . . . . 13 ⊢ (𝜑 → if((𝐷‘𝐾) ≤ 𝑀, (𝐷‘𝐾), 𝑀) ∈ ℝ^*)
262		icombl 25413	. . . . . . . . . . . . 13 ⊢ (((𝐶‘𝐾) ∈ ℝ ∧ if((𝐷‘𝐾) ≤ 𝑀, (𝐷‘𝐾), 𝑀) ∈ ℝ^*) → ((𝐶‘𝐾)[,)if((𝐷‘𝐾) ≤ 𝑀, (𝐷‘𝐾), 𝑀)) ∈ dom vol)
263	11, 261, 262	syl2anc 583	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐶‘𝐾)[,)if((𝐷‘𝐾) ≤ 𝑀, (𝐷‘𝐾), 𝑀)) ∈ dom vol)
264	202, 263	ffvelcdmd 7087	. . . . . . . . . . 11 ⊢ (𝜑 → (vol‘((𝐶‘𝐾)[,)if((𝐷‘𝐾) ≤ 𝑀, (𝐷‘𝐾), 𝑀))) ∈ (0[,]+∞))
265	62, 124, 6, 195, 132, 259, 264	sge0splitsn 45616	. . . . . . . . . 10 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ((ℕ ∖ {𝐾}) ∪ {𝐾}) ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑀, (𝐷‘𝑗), 𝑀))))) = ((Σ^{^}‘(𝑗 ∈ (ℕ ∖ {𝐾}) ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑀, (𝐷‘𝑗), 𝑀))))) +_𝑒 (vol‘((𝐶‘𝐾)[,)if((𝐷‘𝐾) ≤ 𝑀, (𝐷‘𝐾), 𝑀)))))
266		volicore 45756	. . . . . . . . . . . . 13 ⊢ (((𝐶‘𝐾) ∈ ℝ ∧ if((𝐷‘𝐾) ≤ 𝑀, (𝐷‘𝐾), 𝑀) ∈ ℝ) → (vol‘((𝐶‘𝐾)[,)if((𝐷‘𝐾) ≤ 𝑀, (𝐷‘𝐾), 𝑀))) ∈ ℝ)
267	11, 260, 266	syl2anc 583	. . . . . . . . . . . 12 ⊢ (𝜑 → (vol‘((𝐶‘𝐾)[,)if((𝐷‘𝐾) ≤ 𝑀, (𝐷‘𝐾), 𝑀))) ∈ ℝ)
268		rexadd 13218	. . . . . . . . . . . 12 ⊢ (((Σ^{^}‘(𝑗 ∈ (ℕ ∖ {𝐾}) ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑀, (𝐷‘𝑗), 𝑀))))) ∈ ℝ ∧ (vol‘((𝐶‘𝐾)[,)if((𝐷‘𝐾) ≤ 𝑀, (𝐷‘𝐾), 𝑀))) ∈ ℝ) → ((Σ^{^}‘(𝑗 ∈ (ℕ ∖ {𝐾}) ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑀, (𝐷‘𝑗), 𝑀))))) +_𝑒 (vol‘((𝐶‘𝐾)[,)if((𝐷‘𝐾) ≤ 𝑀, (𝐷‘𝐾), 𝑀)))) = ((Σ^{^}‘(𝑗 ∈ (ℕ ∖ {𝐾}) ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑀, (𝐷‘𝑗), 𝑀))))) + (vol‘((𝐶‘𝐾)[,)if((𝐷‘𝐾) ≤ 𝑀, (𝐷‘𝐾), 𝑀)))))
269	139, 267, 268	syl2anc 583	. . . . . . . . . . 11 ⊢ (𝜑 → ((Σ^{^}‘(𝑗 ∈ (ℕ ∖ {𝐾}) ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑀, (𝐷‘𝑗), 𝑀))))) +_𝑒 (vol‘((𝐶‘𝐾)[,)if((𝐷‘𝐾) ≤ 𝑀, (𝐷‘𝐾), 𝑀)))) = ((Σ^{^}‘(𝑗 ∈ (ℕ ∖ {𝐾}) ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑀, (𝐷‘𝑗), 𝑀))))) + (vol‘((𝐶‘𝐾)[,)if((𝐷‘𝐾) ≤ 𝑀, (𝐷‘𝐾), 𝑀)))))
270		volico 45158	. . . . . . . . . . . . . 14 ⊢ (((𝐶‘𝐾) ∈ ℝ ∧ if((𝐷‘𝐾) ≤ 𝑀, (𝐷‘𝐾), 𝑀) ∈ ℝ) → (vol‘((𝐶‘𝐾)[,)if((𝐷‘𝐾) ≤ 𝑀, (𝐷‘𝐾), 𝑀))) = if((𝐶‘𝐾) < if((𝐷‘𝐾) ≤ 𝑀, (𝐷‘𝐾), 𝑀), (if((𝐷‘𝐾) ≤ 𝑀, (𝐷‘𝐾), 𝑀) − (𝐶‘𝐾)), 0))
271	11, 260, 270	syl2anc 583	. . . . . . . . . . . . 13 ⊢ (𝜑 → (vol‘((𝐶‘𝐾)[,)if((𝐷‘𝐾) ≤ 𝑀, (𝐷‘𝐾), 𝑀))) = if((𝐶‘𝐾) < if((𝐷‘𝐾) ≤ 𝑀, (𝐷‘𝐾), 𝑀), (if((𝐷‘𝐾) ≤ 𝑀, (𝐷‘𝐾), 𝑀) − (𝐶‘𝐾)), 0))
272	20, 157	jca 511	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑆 < (𝐷‘𝐾) ∧ 𝑆 < 𝑀))
273		ltmin 13180	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑆 ∈ ℝ ∧ (𝐷‘𝐾) ∈ ℝ ∧ 𝑀 ∈ ℝ) → (𝑆 < if((𝐷‘𝐾) ≤ 𝑀, (𝐷‘𝐾), 𝑀) ↔ (𝑆 < (𝐷‘𝐾) ∧ 𝑆 < 𝑀)))
274	16, 7, 9, 273	syl3anc 1370	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑆 < if((𝐷‘𝐾) ≤ 𝑀, (𝐷‘𝐾), 𝑀) ↔ (𝑆 < (𝐷‘𝐾) ∧ 𝑆 < 𝑀)))
275	272, 274	mpbird 257	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑆 < if((𝐷‘𝐾) ≤ 𝑀, (𝐷‘𝐾), 𝑀))
276	11, 16, 260, 231, 275	lelttrd 11379	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐶‘𝐾) < if((𝐷‘𝐾) ≤ 𝑀, (𝐷‘𝐾), 𝑀))
277	276	iftrued 4536	. . . . . . . . . . . . 13 ⊢ (𝜑 → if((𝐶‘𝐾) < if((𝐷‘𝐾) ≤ 𝑀, (𝐷‘𝐾), 𝑀), (if((𝐷‘𝐾) ≤ 𝑀, (𝐷‘𝐾), 𝑀) − (𝐶‘𝐾)), 0) = (if((𝐷‘𝐾) ≤ 𝑀, (𝐷‘𝐾), 𝑀) − (𝐶‘𝐾)))
278		iftrue 4534	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐷‘𝐾) ≤ 𝑀 → if((𝐷‘𝐾) ≤ 𝑀, (𝐷‘𝐾), 𝑀) = (𝐷‘𝐾))
279	278	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝐷‘𝐾) ≤ 𝑀) → if((𝐷‘𝐾) ≤ 𝑀, (𝐷‘𝐾), 𝑀) = (𝐷‘𝐾))
280	12	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝐷‘𝐾) ≤ 𝑀) → (𝐷‘𝐾) ∈ ℝ^*)
281	9	rexrd 11271	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑀 ∈ ℝ^*)
282	281	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝐷‘𝐾) ≤ 𝑀) → 𝑀 ∈ ℝ^*)
283		simpr 484	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝐷‘𝐾) ≤ 𝑀) → (𝐷‘𝐾) ≤ 𝑀)
284		min1 13175	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐷‘𝐾) ∈ ℝ ∧ 𝐵 ∈ ℝ) → if((𝐷‘𝐾) ≤ 𝐵, (𝐷‘𝐾), 𝐵) ≤ (𝐷‘𝐾))
285	7, 2, 284	syl2anc 583	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → if((𝐷‘𝐾) ≤ 𝐵, (𝐷‘𝐾), 𝐵) ≤ (𝐷‘𝐾))
286	4, 285	eqbrtrd 5170	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑀 ≤ (𝐷‘𝐾))
287	286	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝐷‘𝐾) ≤ 𝑀) → 𝑀 ≤ (𝐷‘𝐾))
288	280, 282, 283, 287	xrletrid 13141	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝐷‘𝐾) ≤ 𝑀) → (𝐷‘𝐾) = 𝑀)
289	279, 288	eqtrd 2771	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝐷‘𝐾) ≤ 𝑀) → if((𝐷‘𝐾) ≤ 𝑀, (𝐷‘𝐾), 𝑀) = 𝑀)
290		simpr 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ¬ (𝐷‘𝐾) ≤ 𝑀) → ¬ (𝐷‘𝐾) ≤ 𝑀)
291	290	iffalsed 4539	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ¬ (𝐷‘𝐾) ≤ 𝑀) → if((𝐷‘𝐾) ≤ 𝑀, (𝐷‘𝐾), 𝑀) = 𝑀)
292	289, 291	pm2.61dan 810	. . . . . . . . . . . . . 14 ⊢ (𝜑 → if((𝐷‘𝐾) ≤ 𝑀, (𝐷‘𝐾), 𝑀) = 𝑀)
293	292	oveq1d 7427	. . . . . . . . . . . . 13 ⊢ (𝜑 → (if((𝐷‘𝐾) ≤ 𝑀, (𝐷‘𝐾), 𝑀) − (𝐶‘𝐾)) = (𝑀 − (𝐶‘𝐾)))
294	271, 277, 293	3eqtrd 2775	. . . . . . . . . . . 12 ⊢ (𝜑 → (vol‘((𝐶‘𝐾)[,)if((𝐷‘𝐾) ≤ 𝑀, (𝐷‘𝐾), 𝑀))) = (𝑀 − (𝐶‘𝐾)))
295	294	oveq2d 7428	. . . . . . . . . . 11 ⊢ (𝜑 → ((Σ^{^}‘(𝑗 ∈ (ℕ ∖ {𝐾}) ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑀, (𝐷‘𝑗), 𝑀))))) + (vol‘((𝐶‘𝐾)[,)if((𝐷‘𝐾) ≤ 𝑀, (𝐷‘𝐾), 𝑀)))) = ((Σ^{^}‘(𝑗 ∈ (ℕ ∖ {𝐾}) ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑀, (𝐷‘𝑗), 𝑀))))) + (𝑀 − (𝐶‘𝐾))))
296	139	recnd 11249	. . . . . . . . . . . 12 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ (ℕ ∖ {𝐾}) ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑀, (𝐷‘𝑗), 𝑀))))) ∈ ℂ)
297	35, 242	subcld 11578	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑀 − (𝐶‘𝐾)) ∈ ℂ)
298	296, 297	addcomd 11423	. . . . . . . . . . 11 ⊢ (𝜑 → ((Σ^{^}‘(𝑗 ∈ (ℕ ∖ {𝐾}) ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑀, (𝐷‘𝑗), 𝑀))))) + (𝑀 − (𝐶‘𝐾))) = ((𝑀 − (𝐶‘𝐾)) + (Σ^{^}‘(𝑗 ∈ (ℕ ∖ {𝐾}) ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑀, (𝐷‘𝑗), 𝑀)))))))
299	269, 295, 298	3eqtrd 2775	. . . . . . . . . 10 ⊢ (𝜑 → ((Σ^{^}‘(𝑗 ∈ (ℕ ∖ {𝐾}) ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑀, (𝐷‘𝑗), 𝑀))))) +_𝑒 (vol‘((𝐶‘𝐾)[,)if((𝐷‘𝐾) ≤ 𝑀, (𝐷‘𝐾), 𝑀)))) = ((𝑀 − (𝐶‘𝐾)) + (Σ^{^}‘(𝑗 ∈ (ℕ ∖ {𝐾}) ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑀, (𝐷‘𝑗), 𝑀)))))))
300	255, 265, 299	3eqtrd 2775	. . . . . . . . 9 ⊢ (𝜑 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑀, (𝐷‘𝑗), 𝑀))))) = ((𝑀 − (𝐶‘𝐾)) + (Σ^{^}‘(𝑗 ∈ (ℕ ∖ {𝐾}) ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑀, (𝐷‘𝑗), 𝑀)))))))
301	253, 300	breq12d 5161	. . . . . . . 8 ⊢ (𝜑 → (((𝑀 − 𝑆) + (Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆)))))) ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑀, (𝐷‘𝑗), 𝑀))))) ↔ ((𝑀 − (𝐶‘𝐾)) + (Σ^{^}‘(𝑗 ∈ (ℕ ∖ {𝐾}) ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆)))))) ≤ ((𝑀 − (𝐶‘𝐾)) + (Σ^{^}‘(𝑗 ∈ (ℕ ∖ {𝐾}) ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑀, (𝐷‘𝑗), 𝑀))))))))
302	189, 301	mpbird 257	. . . . . . 7 ⊢ (𝜑 → ((𝑀 − 𝑆) + (Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆)))))) ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑀, (𝐷‘𝑗), 𝑀))))))
303	42, 82, 103, 119, 302	letrd 11378	. . . . . 6 ⊢ (𝜑 → ((𝑀 − 𝑆) + (𝑆 − 𝐴)) ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑀, (𝐷‘𝑗), 𝑀))))))
304	39, 303	eqbrtrd 5170	. . . . 5 ⊢ (𝜑 → (𝑀 − 𝐴) ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑀, (𝐷‘𝑗), 𝑀))))))
305	34, 304	jca 511	. . . 4 ⊢ (𝜑 → (𝑀 ∈ (𝐴[,]𝐵) ∧ (𝑀 − 𝐴) ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑀, (𝐷‘𝑗), 𝑀)))))))
306		oveq1 7419	. . . . . 6 ⊢ (𝑧 = 𝑀 → (𝑧 − 𝐴) = (𝑀 − 𝐴))
307		breq2 5152	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑀 → ((𝐷‘𝑗) ≤ 𝑧 ↔ (𝐷‘𝑗) ≤ 𝑀))
308		id 22	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑀 → 𝑧 = 𝑀)
309	307, 308	ifbieq2d 4554	. . . . . . . . . 10 ⊢ (𝑧 = 𝑀 → if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧) = if((𝐷‘𝑗) ≤ 𝑀, (𝐷‘𝑗), 𝑀))
310	309	oveq2d 7428	. . . . . . . . 9 ⊢ (𝑧 = 𝑀 → ((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧)) = ((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑀, (𝐷‘𝑗), 𝑀)))
311	310	fveq2d 6895	. . . . . . . 8 ⊢ (𝑧 = 𝑀 → (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧))) = (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑀, (𝐷‘𝑗), 𝑀))))
312	311	mpteq2dv 5250	. . . . . . 7 ⊢ (𝑧 = 𝑀 → (𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧)))) = (𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑀, (𝐷‘𝑗), 𝑀)))))
313	312	fveq2d 6895	. . . . . 6 ⊢ (𝑧 = 𝑀 → (Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧))))) = (Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑀, (𝐷‘𝑗), 𝑀))))))
314	306, 313	breq12d 5161	. . . . 5 ⊢ (𝑧 = 𝑀 → ((𝑧 − 𝐴) ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧))))) ↔ (𝑀 − 𝐴) ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑀, (𝐷‘𝑗), 𝑀)))))))
315	314	elrab 3683	. . . 4 ⊢ (𝑀 ∈ {𝑧 ∈ (𝐴[,]𝐵) ∣ (𝑧 − 𝐴) ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧)))))} ↔ (𝑀 ∈ (𝐴[,]𝐵) ∧ (𝑀 − 𝐴) ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑀, (𝐷‘𝑗), 𝑀)))))))
316	305, 315	sylibr 233	. . 3 ⊢ (𝜑 → 𝑀 ∈ {𝑧 ∈ (𝐴[,]𝐵) ∣ (𝑧 − 𝐴) ≤ (Σ^{^}‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧)))))})
317	316, 105	eleqtrrdi 2843	. 2 ⊢ (𝜑 → 𝑀 ∈ 𝑈)
318	272	simprd 495	. 2 ⊢ (𝜑 → 𝑆 < 𝑀)
319		breq2 5152	. . 3 ⊢ (𝑢 = 𝑀 → (𝑆 < 𝑢 ↔ 𝑆 < 𝑀))
320	319	rspcev 3612	. 2 ⊢ ((𝑀 ∈ 𝑈 ∧ 𝑆 < 𝑀) → ∃𝑢 ∈ 𝑈 𝑆 < 𝑢)
321	317, 318, 320	syl2anc 583	1 ⊢ (𝜑 → ∃𝑢 ∈ 𝑈 𝑆 < 𝑢)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1540 ∈ wcel 2105 ∃wrex 3069 {crab 3431 Vcvv 3473 ∖ cdif 3945 ∪ cun 3946 ⊆ wss 3948 ifcif 4528 {csn 4628 class class class wbr 5148 ↦ cmpt 5231 dom cdm 5676 ⟶wf 6539 ‘cfv 6543 (class class class)co 7412 ℝcr 11115 0cc0 11116 + caddc 11119 +∞cpnf 11252 ℝ^*cxr 11254 < clt 11255 ≤ cle 11256 − cmin 11451 ℕcn 12219 +_𝑒 cxad 13097 [,)cico 13333 [,]cicc 13334 volcvol 25312 Σ^{^}csumge0 45537

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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-inf2 9642 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 ax-pre-sup 11194

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7674 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-2o 8473 df-er 8709 df-map 8828 df-pm 8829 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-fi 9412 df-sup 9443 df-inf 9444 df-oi 9511 df-dju 9902 df-card 9940 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-div 11879 df-nn 12220 df-2 12282 df-3 12283 df-n0 12480 df-z 12566 df-uz 12830 df-q 12940 df-rp 12982 df-xneg 13099 df-xadd 13100 df-xmul 13101 df-ioo 13335 df-ico 13337 df-icc 13338 df-fz 13492 df-fzo 13635 df-fl 13764 df-seq 13974 df-exp 14035 df-hash 14298 df-cj 15053 df-re 15054 df-im 15055 df-sqrt 15189 df-abs 15190 df-clim 15439 df-rlim 15440 df-sum 15640 df-rest 17375 df-topgen 17396 df-psmet 21225 df-xmet 21226 df-met 21227 df-bl 21228 df-mopn 21229 df-top 22716 df-topon 22733 df-bases 22769 df-cmp 23211 df-ovol 25313 df-vol 25314 df-sumge0 45538

This theorem is referenced by: hoidmv1lelem3 45768

W3C validator