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Theorem hoidmv1lelem3 42860
 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 (𝜑𝐴 ∈ ℝ)
hoidmv1lelem3.b (𝜑𝐵 ∈ ℝ)
hoidmv1lelem3.l (𝜑𝐴 < 𝐵)
hoidmv1lelem3.c (𝜑𝐶:ℕ⟶ℝ)
hoidmv1lelem3.d (𝜑𝐷:ℕ⟶ℝ)
hoidmv1lelem3.x (𝜑 → (𝐴[,)𝐵) ⊆ 𝑗 ∈ ℕ ((𝐶𝑗)[,)(𝐷𝑗)))
hoidmv1lelem3.r (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)(𝐷𝑗))))) ∈ ℝ)
hoidmv1lelem3.u 𝑈 = {𝑧 ∈ (𝐴[,]𝐵) ∣ (𝑧𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)))))}
hoidmv1lelem3.s 𝑆 = sup(𝑈, ℝ, < )
Assertion
Ref Expression
hoidmv1lelem3 (𝜑 → (𝐵𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)(𝐷𝑗))))))
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.
StepHypRef Expression
1 hoidmv1lelem3.b . . 3 (𝜑𝐵 ∈ ℝ)
2 hoidmv1lelem3.a . . 3 (𝜑𝐴 ∈ ℝ)
31, 2resubcld 11060 . 2 (𝜑 → (𝐵𝐴) ∈ ℝ)
4 nnex 11636 . . . . . . 7 ℕ ∈ V
54a1i 11 . . . . . 6 (𝜑 → ℕ ∈ V)
6 icossicc 12816 . . . . . . . 8 (0[,)+∞) ⊆ (0[,]+∞)
7 0xr 10680 . . . . . . . . . 10 0 ∈ ℝ*
87a1i 11 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → 0 ∈ ℝ*)
9 pnfxr 10687 . . . . . . . . . 10 +∞ ∈ ℝ*
109a1i 11 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → +∞ ∈ ℝ*)
11 hoidmv1lelem3.c . . . . . . . . . . . 12 (𝜑𝐶:ℕ⟶ℝ)
1211ffvelrnda 6844 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ℕ) → (𝐶𝑗) ∈ ℝ)
13 hoidmv1lelem3.d . . . . . . . . . . . . 13 (𝜑𝐷:ℕ⟶ℝ)
1413ffvelrnda 6844 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ℕ) → (𝐷𝑗) ∈ ℝ)
151adantr 483 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ℕ) → 𝐵 ∈ ℝ)
1614, 15ifcld 4510 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ℕ) → if((𝐷𝑗) ≤ 𝐵, (𝐷𝑗), 𝐵) ∈ ℝ)
17 volicore 42848 . . . . . . . . . . 11 (((𝐶𝑗) ∈ ℝ ∧ if((𝐷𝑗) ≤ 𝐵, (𝐷𝑗), 𝐵) ∈ ℝ) → (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐵, (𝐷𝑗), 𝐵))) ∈ ℝ)
1812, 16, 17syl2anc 586 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐵, (𝐷𝑗), 𝐵))) ∈ ℝ)
1918rexrd 10683 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐵, (𝐷𝑗), 𝐵))) ∈ ℝ*)
2016rexrd 10683 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ℕ) → if((𝐷𝑗) ≤ 𝐵, (𝐷𝑗), 𝐵) ∈ ℝ*)
21 icombl 24157 . . . . . . . . . . 11 (((𝐶𝑗) ∈ ℝ ∧ if((𝐷𝑗) ≤ 𝐵, (𝐷𝑗), 𝐵) ∈ ℝ*) → ((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐵, (𝐷𝑗), 𝐵)) ∈ dom vol)
2212, 20, 21syl2anc 586 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → ((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐵, (𝐷𝑗), 𝐵)) ∈ dom vol)
23 volge0 42230 . . . . . . . . . 10 (((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐵, (𝐷𝑗), 𝐵)) ∈ dom vol → 0 ≤ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐵, (𝐷𝑗), 𝐵))))
2422, 23syl 17 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → 0 ≤ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐵, (𝐷𝑗), 𝐵))))
2518ltpnfd 12508 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐵, (𝐷𝑗), 𝐵))) < +∞)
268, 10, 19, 24, 25elicod 12779 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐵, (𝐷𝑗), 𝐵))) ∈ (0[,)+∞))
276, 26sseldi 3963 . . . . . . 7 ((𝜑𝑗 ∈ ℕ) → (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐵, (𝐷𝑗), 𝐵))) ∈ (0[,]+∞))
28 eqid 2819 . . . . . . 7 (𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐵, (𝐷𝑗), 𝐵)))) = (𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐵, (𝐷𝑗), 𝐵))))
2927, 28fmptd 6871 . . . . . 6 (𝜑 → (𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐵, (𝐷𝑗), 𝐵)))):ℕ⟶(0[,]+∞))
305, 29sge0xrcl 42652 . . . . 5 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐵, (𝐷𝑗), 𝐵))))) ∈ ℝ*)
319a1i 11 . . . . 5 (𝜑 → +∞ ∈ ℝ*)
32 hoidmv1lelem3.r . . . . . . 7 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)(𝐷𝑗))))) ∈ ℝ)
3332rexrd 10683 . . . . . 6 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)(𝐷𝑗))))) ∈ ℝ*)
34 nfv 1908 . . . . . . 7 𝑗𝜑
35 volf 24122 . . . . . . . . 9 vol:dom vol⟶(0[,]+∞)
3635a1i 11 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → vol:dom vol⟶(0[,]+∞))
3714rexrd 10683 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → (𝐷𝑗) ∈ ℝ*)
38 icombl 24157 . . . . . . . . 9 (((𝐶𝑗) ∈ ℝ ∧ (𝐷𝑗) ∈ ℝ*) → ((𝐶𝑗)[,)(𝐷𝑗)) ∈ dom vol)
3912, 37, 38syl2anc 586 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → ((𝐶𝑗)[,)(𝐷𝑗)) ∈ dom vol)
4036, 39ffvelrnd 6845 . . . . . . 7 ((𝜑𝑗 ∈ ℕ) → (vol‘((𝐶𝑗)[,)(𝐷𝑗))) ∈ (0[,]+∞))
4112rexrd 10683 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → (𝐶𝑗) ∈ ℝ*)
4212leidd 11198 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → (𝐶𝑗) ≤ (𝐶𝑗))
43 min1 12574 . . . . . . . . . 10 (((𝐷𝑗) ∈ ℝ ∧ 𝐵 ∈ ℝ) → if((𝐷𝑗) ≤ 𝐵, (𝐷𝑗), 𝐵) ≤ (𝐷𝑗))
4414, 15, 43syl2anc 586 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → if((𝐷𝑗) ≤ 𝐵, (𝐷𝑗), 𝐵) ≤ (𝐷𝑗))
45 icossico 12798 . . . . . . . . 9 ((((𝐶𝑗) ∈ ℝ* ∧ (𝐷𝑗) ∈ ℝ*) ∧ ((𝐶𝑗) ≤ (𝐶𝑗) ∧ if((𝐷𝑗) ≤ 𝐵, (𝐷𝑗), 𝐵) ≤ (𝐷𝑗))) → ((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐵, (𝐷𝑗), 𝐵)) ⊆ ((𝐶𝑗)[,)(𝐷𝑗)))
4641, 37, 42, 44, 45syl22anc 836 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → ((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐵, (𝐷𝑗), 𝐵)) ⊆ ((𝐶𝑗)[,)(𝐷𝑗)))
47 volss 24126 . . . . . . . 8 ((((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐵, (𝐷𝑗), 𝐵)) ∈ dom vol ∧ ((𝐶𝑗)[,)(𝐷𝑗)) ∈ dom vol ∧ ((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐵, (𝐷𝑗), 𝐵)) ⊆ ((𝐶𝑗)[,)(𝐷𝑗))) → (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐵, (𝐷𝑗), 𝐵))) ≤ (vol‘((𝐶𝑗)[,)(𝐷𝑗))))
4822, 39, 46, 47syl3anc 1365 . . . . . . 7 ((𝜑𝑗 ∈ ℕ) → (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐵, (𝐷𝑗), 𝐵))) ≤ (vol‘((𝐶𝑗)[,)(𝐷𝑗))))
4934, 5, 27, 40, 48sge0lempt 42677 . . . . . 6 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐵, (𝐷𝑗), 𝐵))))) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)(𝐷𝑗))))))
5032ltpnfd 12508 . . . . . 6 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)(𝐷𝑗))))) < +∞)
5130, 33, 31, 49, 50xrlelttrd 12545 . . . . 5 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐵, (𝐷𝑗), 𝐵))))) < +∞)
5230, 31, 51xrltned 41609 . . . 4 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐵, (𝐷𝑗), 𝐵))))) ≠ +∞)
5352neneqd 3019 . . 3 (𝜑 → ¬ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐵, (𝐷𝑗), 𝐵))))) = +∞)
545, 29sge0repnf 42653 . . 3 (𝜑 → ((Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐵, (𝐷𝑗), 𝐵))))) ∈ ℝ ↔ ¬ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐵, (𝐷𝑗), 𝐵))))) = +∞))
5553, 54mpbird 259 . 2 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐵, (𝐷𝑗), 𝐵))))) ∈ ℝ)
561rexrd 10683 . . . . . . 7 (𝜑𝐵 ∈ ℝ*)
572, 1iccssred 41764 . . . . . . . . 9 (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
58 hoidmv1lelem3.u . . . . . . . . . . 11 𝑈 = {𝑧 ∈ (𝐴[,]𝐵) ∣ (𝑧𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)))))}
59 ssrab2 4054 . . . . . . . . . . 11 {𝑧 ∈ (𝐴[,]𝐵) ∣ (𝑧𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)))))} ⊆ (𝐴[,]𝐵)
6058, 59eqsstri 3999 . . . . . . . . . 10 𝑈 ⊆ (𝐴[,]𝐵)
61 hoidmv1lelem3.l . . . . . . . . . . . 12 (𝜑𝐴 < 𝐵)
62 hoidmv1lelem3.s . . . . . . . . . . . 12 𝑆 = sup(𝑈, ℝ, < )
632, 1, 61, 11, 13, 32, 58, 62hoidmv1lelem1 42858 . . . . . . . . . . 11 (𝜑 → (𝑆𝑈𝐴𝑈 ∧ ∃𝑥 ∈ ℝ ∀𝑦𝑈 𝑦𝑥))
6463simp1d 1136 . . . . . . . . . 10 (𝜑𝑆𝑈)
6560, 64sseldi 3963 . . . . . . . . 9 (𝜑𝑆 ∈ (𝐴[,]𝐵))
6657, 65sseldd 3966 . . . . . . . 8 (𝜑𝑆 ∈ ℝ)
6766rexrd 10683 . . . . . . 7 (𝜑𝑆 ∈ ℝ*)
68 simpl 485 . . . . . . . . 9 ((𝜑 ∧ ¬ 𝐵𝑆) → 𝜑)
69 simpr 487 . . . . . . . . . 10 ((𝜑 ∧ ¬ 𝐵𝑆) → ¬ 𝐵𝑆)
7068, 66syl 17 . . . . . . . . . . 11 ((𝜑 ∧ ¬ 𝐵𝑆) → 𝑆 ∈ ℝ)
7168, 1syl 17 . . . . . . . . . . 11 ((𝜑 ∧ ¬ 𝐵𝑆) → 𝐵 ∈ ℝ)
7270, 71ltnled 10779 . . . . . . . . . 10 ((𝜑 ∧ ¬ 𝐵𝑆) → (𝑆 < 𝐵 ↔ ¬ 𝐵𝑆))
7369, 72mpbird 259 . . . . . . . . 9 ((𝜑 ∧ ¬ 𝐵𝑆) → 𝑆 < 𝐵)
74 hoidmv1lelem3.x . . . . . . . . . . . . 13 (𝜑 → (𝐴[,)𝐵) ⊆ 𝑗 ∈ ℕ ((𝐶𝑗)[,)(𝐷𝑗)))
7574adantr 483 . . . . . . . . . . . 12 ((𝜑𝑆 < 𝐵) → (𝐴[,)𝐵) ⊆ 𝑗 ∈ ℕ ((𝐶𝑗)[,)(𝐷𝑗)))
762rexrd 10683 . . . . . . . . . . . . . 14 (𝜑𝐴 ∈ ℝ*)
7776adantr 483 . . . . . . . . . . . . 13 ((𝜑𝑆 < 𝐵) → 𝐴 ∈ ℝ*)
7856adantr 483 . . . . . . . . . . . . 13 ((𝜑𝑆 < 𝐵) → 𝐵 ∈ ℝ*)
7967adantr 483 . . . . . . . . . . . . 13 ((𝜑𝑆 < 𝐵) → 𝑆 ∈ ℝ*)
8060, 57sstrid 3976 . . . . . . . . . . . . . . . 16 (𝜑𝑈 ⊆ ℝ)
8164ne0d 4299 . . . . . . . . . . . . . . . 16 (𝜑𝑈 ≠ ∅)
8263simp3d 1138 . . . . . . . . . . . . . . . 16 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦𝑈 𝑦𝑥)
8363simp2d 1137 . . . . . . . . . . . . . . . 16 (𝜑𝐴𝑈)
84 suprub 11594 . . . . . . . . . . . . . . . 16 (((𝑈 ⊆ ℝ ∧ 𝑈 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦𝑈 𝑦𝑥) ∧ 𝐴𝑈) → 𝐴 ≤ sup(𝑈, ℝ, < ))
8580, 81, 82, 83, 84syl31anc 1367 . . . . . . . . . . . . . . 15 (𝜑𝐴 ≤ sup(𝑈, ℝ, < ))
8685, 62breqtrrdi 5099 . . . . . . . . . . . . . 14 (𝜑𝐴𝑆)
8786adantr 483 . . . . . . . . . . . . 13 ((𝜑𝑆 < 𝐵) → 𝐴𝑆)
88 simpr 487 . . . . . . . . . . . . 13 ((𝜑𝑆 < 𝐵) → 𝑆 < 𝐵)
8977, 78, 79, 87, 88elicod 12779 . . . . . . . . . . . 12 ((𝜑𝑆 < 𝐵) → 𝑆 ∈ (𝐴[,)𝐵))
9075, 89sseldd 3966 . . . . . . . . . . 11 ((𝜑𝑆 < 𝐵) → 𝑆 𝑗 ∈ ℕ ((𝐶𝑗)[,)(𝐷𝑗)))
91 eliun 4914 . . . . . . . . . . 11 (𝑆 𝑗 ∈ ℕ ((𝐶𝑗)[,)(𝐷𝑗)) ↔ ∃𝑗 ∈ ℕ 𝑆 ∈ ((𝐶𝑗)[,)(𝐷𝑗)))
9290, 91sylib 220 . . . . . . . . . 10 ((𝜑𝑆 < 𝐵) → ∃𝑗 ∈ ℕ 𝑆 ∈ ((𝐶𝑗)[,)(𝐷𝑗)))
932adantr 483 . . . . . . . . . . . . . 14 ((𝜑𝑆 < 𝐵) → 𝐴 ∈ ℝ)
94933ad2ant1 1127 . . . . . . . . . . . . 13 (((𝜑𝑆 < 𝐵) ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ ((𝐶𝑗)[,)(𝐷𝑗))) → 𝐴 ∈ ℝ)
951adantr 483 . . . . . . . . . . . . . 14 ((𝜑𝑆 < 𝐵) → 𝐵 ∈ ℝ)
96953ad2ant1 1127 . . . . . . . . . . . . 13 (((𝜑𝑆 < 𝐵) ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ ((𝐶𝑗)[,)(𝐷𝑗))) → 𝐵 ∈ ℝ)
9711adantr 483 . . . . . . . . . . . . . 14 ((𝜑𝑆 < 𝐵) → 𝐶:ℕ⟶ℝ)
98973ad2ant1 1127 . . . . . . . . . . . . 13 (((𝜑𝑆 < 𝐵) ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ ((𝐶𝑗)[,)(𝐷𝑗))) → 𝐶:ℕ⟶ℝ)
9913adantr 483 . . . . . . . . . . . . . 14 ((𝜑𝑆 < 𝐵) → 𝐷:ℕ⟶ℝ)
100993ad2ant1 1127 . . . . . . . . . . . . 13 (((𝜑𝑆 < 𝐵) ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ ((𝐶𝑗)[,)(𝐷𝑗))) → 𝐷:ℕ⟶ℝ)
101 fveq2 6663 . . . . . . . . . . . . . . . . . . . 20 (𝑖 = 𝑗 → (𝐶𝑖) = (𝐶𝑗))
102 fveq2 6663 . . . . . . . . . . . . . . . . . . . 20 (𝑖 = 𝑗 → (𝐷𝑖) = (𝐷𝑗))
103101, 102oveq12d 7166 . . . . . . . . . . . . . . . . . . 19 (𝑖 = 𝑗 → ((𝐶𝑖)[,)(𝐷𝑖)) = ((𝐶𝑗)[,)(𝐷𝑗)))
104103fveq2d 6667 . . . . . . . . . . . . . . . . . 18 (𝑖 = 𝑗 → (vol‘((𝐶𝑖)[,)(𝐷𝑖))) = (vol‘((𝐶𝑗)[,)(𝐷𝑗))))
105104cbvmptv 5160 . . . . . . . . . . . . . . . . 17 (𝑖 ∈ ℕ ↦ (vol‘((𝐶𝑖)[,)(𝐷𝑖)))) = (𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)(𝐷𝑗))))
106105fveq2i 6666 . . . . . . . . . . . . . . . 16 ^‘(𝑖 ∈ ℕ ↦ (vol‘((𝐶𝑖)[,)(𝐷𝑖))))) = (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)(𝐷𝑗)))))
107106, 32eqeltrid 2915 . . . . . . . . . . . . . . 15 (𝜑 → (Σ^‘(𝑖 ∈ ℕ ↦ (vol‘((𝐶𝑖)[,)(𝐷𝑖))))) ∈ ℝ)
108107adantr 483 . . . . . . . . . . . . . 14 ((𝜑𝑆 < 𝐵) → (Σ^‘(𝑖 ∈ ℕ ↦ (vol‘((𝐶𝑖)[,)(𝐷𝑖))))) ∈ ℝ)
1091083ad2ant1 1127 . . . . . . . . . . . . 13 (((𝜑𝑆 < 𝐵) ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ ((𝐶𝑗)[,)(𝐷𝑗))) → (Σ^‘(𝑖 ∈ ℕ ↦ (vol‘((𝐶𝑖)[,)(𝐷𝑖))))) ∈ ℝ)
110102breq1d 5067 . . . . . . . . . . . . . . . . . . . . . 22 (𝑖 = 𝑗 → ((𝐷𝑖) ≤ 𝑧 ↔ (𝐷𝑗) ≤ 𝑧))
111110, 102ifbieq1d 4488 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 = 𝑗 → if((𝐷𝑖) ≤ 𝑧, (𝐷𝑖), 𝑧) = if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))
112101, 111oveq12d 7166 . . . . . . . . . . . . . . . . . . . 20 (𝑖 = 𝑗 → ((𝐶𝑖)[,)if((𝐷𝑖) ≤ 𝑧, (𝐷𝑖), 𝑧)) = ((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)))
113112fveq2d 6667 . . . . . . . . . . . . . . . . . . 19 (𝑖 = 𝑗 → (vol‘((𝐶𝑖)[,)if((𝐷𝑖) ≤ 𝑧, (𝐷𝑖), 𝑧))) = (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))))
114113cbvmptv 5160 . . . . . . . . . . . . . . . . . 18 (𝑖 ∈ ℕ ↦ (vol‘((𝐶𝑖)[,)if((𝐷𝑖) ≤ 𝑧, (𝐷𝑖), 𝑧)))) = (𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))))
115114eqcomi 2828 . . . . . . . . . . . . . . . . 17 (𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)))) = (𝑖 ∈ ℕ ↦ (vol‘((𝐶𝑖)[,)if((𝐷𝑖) ≤ 𝑧, (𝐷𝑖), 𝑧))))
116115fveq2i 6666 . . . . . . . . . . . . . . . 16 ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))))) = (Σ^‘(𝑖 ∈ ℕ ↦ (vol‘((𝐶𝑖)[,)if((𝐷𝑖) ≤ 𝑧, (𝐷𝑖), 𝑧)))))
117116breq2i 5065 . . . . . . . . . . . . . . 15 ((𝑧𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))))) ↔ (𝑧𝐴) ≤ (Σ^‘(𝑖 ∈ ℕ ↦ (vol‘((𝐶𝑖)[,)if((𝐷𝑖) ≤ 𝑧, (𝐷𝑖), 𝑧))))))
118117rabbii 3472 . . . . . . . . . . . . . 14 {𝑧 ∈ (𝐴[,]𝐵) ∣ (𝑧𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)))))} = {𝑧 ∈ (𝐴[,]𝐵) ∣ (𝑧𝐴) ≤ (Σ^‘(𝑖 ∈ ℕ ↦ (vol‘((𝐶𝑖)[,)if((𝐷𝑖) ≤ 𝑧, (𝐷𝑖), 𝑧)))))}
11958, 118eqtri 2842 . . . . . . . . . . . . 13 𝑈 = {𝑧 ∈ (𝐴[,]𝐵) ∣ (𝑧𝐴) ≤ (Σ^‘(𝑖 ∈ ℕ ↦ (vol‘((𝐶𝑖)[,)if((𝐷𝑖) ≤ 𝑧, (𝐷𝑖), 𝑧)))))}
12064adantr 483 . . . . . . . . . . . . . 14 ((𝜑𝑆 < 𝐵) → 𝑆𝑈)
1211203ad2ant1 1127 . . . . . . . . . . . . 13 (((𝜑𝑆 < 𝐵) ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ ((𝐶𝑗)[,)(𝐷𝑗))) → 𝑆𝑈)
122873ad2ant1 1127 . . . . . . . . . . . . 13 (((𝜑𝑆 < 𝐵) ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ ((𝐶𝑗)[,)(𝐷𝑗))) → 𝐴𝑆)
123883ad2ant1 1127 . . . . . . . . . . . . 13 (((𝜑𝑆 < 𝐵) ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ ((𝐶𝑗)[,)(𝐷𝑗))) → 𝑆 < 𝐵)
124 simp2 1131 . . . . . . . . . . . . 13 (((𝜑𝑆 < 𝐵) ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ ((𝐶𝑗)[,)(𝐷𝑗))) → 𝑗 ∈ ℕ)
125 simp3 1132 . . . . . . . . . . . . 13 (((𝜑𝑆 < 𝐵) ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ ((𝐶𝑗)[,)(𝐷𝑗))) → 𝑆 ∈ ((𝐶𝑗)[,)(𝐷𝑗)))
126 eqid 2819 . . . . . . . . . . . . 13 if((𝐷𝑗) ≤ 𝐵, (𝐷𝑗), 𝐵) = if((𝐷𝑗) ≤ 𝐵, (𝐷𝑗), 𝐵)
12794, 96, 98, 100, 109, 119, 121, 122, 123, 124, 125, 126hoidmv1lelem2 42859 . . . . . . . . . . . 12 (((𝜑𝑆 < 𝐵) ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ ((𝐶𝑗)[,)(𝐷𝑗))) → ∃𝑢𝑈 𝑆 < 𝑢)
1281273exp 1113 . . . . . . . . . . 11 ((𝜑𝑆 < 𝐵) → (𝑗 ∈ ℕ → (𝑆 ∈ ((𝐶𝑗)[,)(𝐷𝑗)) → ∃𝑢𝑈 𝑆 < 𝑢)))
129128rexlimdv 3281 . . . . . . . . . 10 ((𝜑𝑆 < 𝐵) → (∃𝑗 ∈ ℕ 𝑆 ∈ ((𝐶𝑗)[,)(𝐷𝑗)) → ∃𝑢𝑈 𝑆 < 𝑢))
13092, 129mpd 15 . . . . . . . . 9 ((𝜑𝑆 < 𝐵) → ∃𝑢𝑈 𝑆 < 𝑢)
13168, 73, 130syl2anc 586 . . . . . . . 8 ((𝜑 ∧ ¬ 𝐵𝑆) → ∃𝑢𝑈 𝑆 < 𝑢)
13257adantr 483 . . . . . . . . . . . . . . 15 ((𝜑𝑢𝑈) → (𝐴[,]𝐵) ⊆ ℝ)
13360, 132sstrid 3976 . . . . . . . . . . . . . 14 ((𝜑𝑢𝑈) → 𝑈 ⊆ ℝ)
13481adantr 483 . . . . . . . . . . . . . 14 ((𝜑𝑢𝑈) → 𝑈 ≠ ∅)
1352, 1jca 514 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
136135adantr 483 . . . . . . . . . . . . . . . 16 ((𝜑𝑢𝑈) → (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
13760a1i 11 . . . . . . . . . . . . . . . 16 ((𝜑𝑢𝑈) → 𝑈 ⊆ (𝐴[,]𝐵))
13864adantr 483 . . . . . . . . . . . . . . . 16 ((𝜑𝑢𝑈) → 𝑆𝑈)
139 iccsupr 12822 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝑈 ⊆ (𝐴[,]𝐵) ∧ 𝑆𝑈) → (𝑈 ⊆ ℝ ∧ 𝑈 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦𝑈 𝑦𝑥))
140136, 137, 138, 139syl3anc 1365 . . . . . . . . . . . . . . 15 ((𝜑𝑢𝑈) → (𝑈 ⊆ ℝ ∧ 𝑈 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦𝑈 𝑦𝑥))
141140simp3d 1138 . . . . . . . . . . . . . 14 ((𝜑𝑢𝑈) → ∃𝑥 ∈ ℝ ∀𝑦𝑈 𝑦𝑥)
142 simpr 487 . . . . . . . . . . . . . 14 ((𝜑𝑢𝑈) → 𝑢𝑈)
143 suprub 11594 . . . . . . . . . . . . . 14 (((𝑈 ⊆ ℝ ∧ 𝑈 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦𝑈 𝑦𝑥) ∧ 𝑢𝑈) → 𝑢 ≤ sup(𝑈, ℝ, < ))
144133, 134, 141, 142, 143syl31anc 1367 . . . . . . . . . . . . 13 ((𝜑𝑢𝑈) → 𝑢 ≤ sup(𝑈, ℝ, < ))
145144, 62breqtrrdi 5099 . . . . . . . . . . . 12 ((𝜑𝑢𝑈) → 𝑢𝑆)
146145ralrimiva 3180 . . . . . . . . . . 11 (𝜑 → ∀𝑢𝑈 𝑢𝑆)
14760sseli 3961 . . . . . . . . . . . . . . 15 (𝑢𝑈𝑢 ∈ (𝐴[,]𝐵))
148147adantl 484 . . . . . . . . . . . . . 14 ((𝜑𝑢𝑈) → 𝑢 ∈ (𝐴[,]𝐵))
149132, 148sseldd 3966 . . . . . . . . . . . . 13 ((𝜑𝑢𝑈) → 𝑢 ∈ ℝ)
15066adantr 483 . . . . . . . . . . . . 13 ((𝜑𝑢𝑈) → 𝑆 ∈ ℝ)
151149, 150lenltd 10778 . . . . . . . . . . . 12 ((𝜑𝑢𝑈) → (𝑢𝑆 ↔ ¬ 𝑆 < 𝑢))
152151ralbidva 3194 . . . . . . . . . . 11 (𝜑 → (∀𝑢𝑈 𝑢𝑆 ↔ ∀𝑢𝑈 ¬ 𝑆 < 𝑢))
153146, 152mpbid 234 . . . . . . . . . 10 (𝜑 → ∀𝑢𝑈 ¬ 𝑆 < 𝑢)
154 ralnex 3234 . . . . . . . . . 10 (∀𝑢𝑈 ¬ 𝑆 < 𝑢 ↔ ¬ ∃𝑢𝑈 𝑆 < 𝑢)
155153, 154sylib 220 . . . . . . . . 9 (𝜑 → ¬ ∃𝑢𝑈 𝑆 < 𝑢)
156155adantr 483 . . . . . . . 8 ((𝜑 ∧ ¬ 𝐵𝑆) → ¬ ∃𝑢𝑈 𝑆 < 𝑢)
157131, 156condan 816 . . . . . . 7 (𝜑𝐵𝑆)
158 iccleub 12784 . . . . . . . 8 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝑆 ∈ (𝐴[,]𝐵)) → 𝑆𝐵)
15976, 56, 65, 158syl3anc 1365 . . . . . . 7 (𝜑𝑆𝐵)
16056, 67, 157, 159xrletrid 12540 . . . . . 6 (𝜑𝐵 = 𝑆)
161160, 64eqeltrd 2911 . . . . 5 (𝜑𝐵𝑈)
162161, 58eleqtrdi 2921 . . . 4 (𝜑𝐵 ∈ {𝑧 ∈ (𝐴[,]𝐵) ∣ (𝑧𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)))))})
163 oveq1 7155 . . . . . 6 (𝑧 = 𝐵 → (𝑧𝐴) = (𝐵𝐴))
164 breq2 5061 . . . . . . . . . . 11 (𝑧 = 𝐵 → ((𝐷𝑗) ≤ 𝑧 ↔ (𝐷𝑗) ≤ 𝐵))
165 id 22 . . . . . . . . . . 11 (𝑧 = 𝐵𝑧 = 𝐵)
166164, 165ifbieq2d 4490 . . . . . . . . . 10 (𝑧 = 𝐵 → if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧) = if((𝐷𝑗) ≤ 𝐵, (𝐷𝑗), 𝐵))
167166oveq2d 7164 . . . . . . . . 9 (𝑧 = 𝐵 → ((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)) = ((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐵, (𝐷𝑗), 𝐵)))
168167fveq2d 6667 . . . . . . . 8 (𝑧 = 𝐵 → (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))) = (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐵, (𝐷𝑗), 𝐵))))
169168mpteq2dv 5153 . . . . . . 7 (𝑧 = 𝐵 → (𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)))) = (𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐵, (𝐷𝑗), 𝐵)))))
170169fveq2d 6667 . . . . . 6 (𝑧 = 𝐵 → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))))) = (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐵, (𝐷𝑗), 𝐵))))))
171163, 170breq12d 5070 . . . . 5 (𝑧 = 𝐵 → ((𝑧𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))))) ↔ (𝐵𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐵, (𝐷𝑗), 𝐵)))))))
172171elrab 3678 . . . 4 (𝐵 ∈ {𝑧 ∈ (𝐴[,]𝐵) ∣ (𝑧𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)))))} ↔ (𝐵 ∈ (𝐴[,]𝐵) ∧ (𝐵𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐵, (𝐷𝑗), 𝐵)))))))
173162, 172sylib 220 . . 3 (𝜑 → (𝐵 ∈ (𝐴[,]𝐵) ∧ (𝐵𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐵, (𝐷𝑗), 𝐵)))))))
174173simprd 498 . 2 (𝜑 → (𝐵𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐵, (𝐷𝑗), 𝐵))))))
1753, 55, 32, 174, 49letrd 10789 1 (𝜑 → (𝐵𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)(𝐷𝑗))))))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398   ∧ w3a 1081   = wceq 1530   ∈ wcel 2107   ≠ wne 3014  ∀wral 3136  ∃wrex 3137  {crab 3140  Vcvv 3493   ⊆ wss 3934  ∅c0 4289  ifcif 4465  ∪ ciun 4910   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  +∞cpnf 10664  ℝ*cxr 10666   < clt 10667   ≤ cle 10668   − cmin 10862  ℕcn 11630  [,)cico 12732  [,]cicc 12733  volcvol 24056  Σ^csumge0 42629 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  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 1082  df-3an 1083  df-tru 1533  df-fal 1543  df-ex 1774  df-nf 1778  df-sb 2063  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-fi 8867  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-xneg 12499  df-xadd 12500  df-xmul 12501  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-rest 16688  df-topgen 16709  df-psmet 20529  df-xmet 20530  df-met 20531  df-bl 20532  df-mopn 20533  df-top 21494  df-topon 21511  df-bases 21546  df-cmp 21987  df-ovol 24057  df-vol 24058  df-sumge0 42630 This theorem is referenced by:  hoidmv1le  42861
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