Mathbox for Glauco Siliprandi < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  hoidmv1lelem3 Structured version   Visualization version   GIF version

Theorem hoidmv1lelem3 40144
 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 non-empty, 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 10418 . 2 (𝜑 → (𝐵𝐴) ∈ ℝ)
4 nnex 10986 . . . . . . 7 ℕ ∈ V
54a1i 11 . . . . . 6 (𝜑 → ℕ ∈ V)
6 icossicc 12218 . . . . . . . 8 (0[,)+∞) ⊆ (0[,]+∞)
7 0xr 10046 . . . . . . . . . 10 0 ∈ ℝ*
87a1i 11 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → 0 ∈ ℝ*)
9 pnfxr 10052 . . . . . . . . . 10 +∞ ∈ ℝ*
109a1i 11 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → +∞ ∈ ℝ*)
11 hoidmv1lelem3.c . . . . . . . . . . . 12 (𝜑𝐶:ℕ⟶ℝ)
1211ffvelrnda 6325 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ℕ) → (𝐶𝑗) ∈ ℝ)
13 hoidmv1lelem3.d . . . . . . . . . . . . 13 (𝜑𝐷:ℕ⟶ℝ)
1413ffvelrnda 6325 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ℕ) → (𝐷𝑗) ∈ ℝ)
151adantr 481 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ℕ) → 𝐵 ∈ ℝ)
1614, 15ifcld 4109 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ℕ) → if((𝐷𝑗) ≤ 𝐵, (𝐷𝑗), 𝐵) ∈ ℝ)
17 volicore 40132 . . . . . . . . . . 11 (((𝐶𝑗) ∈ ℝ ∧ if((𝐷𝑗) ≤ 𝐵, (𝐷𝑗), 𝐵) ∈ ℝ) → (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐵, (𝐷𝑗), 𝐵))) ∈ ℝ)
1812, 16, 17syl2anc 692 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐵, (𝐷𝑗), 𝐵))) ∈ ℝ)
1918rexrd 10049 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐵, (𝐷𝑗), 𝐵))) ∈ ℝ*)
2016rexrd 10049 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ℕ) → if((𝐷𝑗) ≤ 𝐵, (𝐷𝑗), 𝐵) ∈ ℝ*)
21 icombl 23272 . . . . . . . . . . 11 (((𝐶𝑗) ∈ ℝ ∧ if((𝐷𝑗) ≤ 𝐵, (𝐷𝑗), 𝐵) ∈ ℝ*) → ((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐵, (𝐷𝑗), 𝐵)) ∈ dom vol)
2212, 20, 21syl2anc 692 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → ((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐵, (𝐷𝑗), 𝐵)) ∈ dom vol)
23 volge0 39514 . . . . . . . . . 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 11915 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐵, (𝐷𝑗), 𝐵))) < +∞)
268, 10, 19, 24, 25elicod 12182 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐵, (𝐷𝑗), 𝐵))) ∈ (0[,)+∞))
276, 26sseldi 3586 . . . . . . 7 ((𝜑𝑗 ∈ ℕ) → (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐵, (𝐷𝑗), 𝐵))) ∈ (0[,]+∞))
28 eqid 2621 . . . . . . 7 (𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐵, (𝐷𝑗), 𝐵)))) = (𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐵, (𝐷𝑗), 𝐵))))
2927, 28fmptd 6351 . . . . . 6 (𝜑 → (𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐵, (𝐷𝑗), 𝐵)))):ℕ⟶(0[,]+∞))
305, 29sge0xrcl 39939 . . . . 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 10049 . . . . . 6 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)(𝐷𝑗))))) ∈ ℝ*)
34 nfv 1840 . . . . . . 7 𝑗𝜑
35 volf 23237 . . . . . . . . 9 vol:dom vol⟶(0[,]+∞)
3635a1i 11 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → vol:dom vol⟶(0[,]+∞))
3714rexrd 10049 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → (𝐷𝑗) ∈ ℝ*)
38 icombl 23272 . . . . . . . . 9 (((𝐶𝑗) ∈ ℝ ∧ (𝐷𝑗) ∈ ℝ*) → ((𝐶𝑗)[,)(𝐷𝑗)) ∈ dom vol)
3912, 37, 38syl2anc 692 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → ((𝐶𝑗)[,)(𝐷𝑗)) ∈ dom vol)
4036, 39ffvelrnd 6326 . . . . . . 7 ((𝜑𝑗 ∈ ℕ) → (vol‘((𝐶𝑗)[,)(𝐷𝑗))) ∈ (0[,]+∞))
4112rexrd 10049 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → (𝐶𝑗) ∈ ℝ*)
4212leidd 10554 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → (𝐶𝑗) ≤ (𝐶𝑗))
43 min1 11979 . . . . . . . . . 10 (((𝐷𝑗) ∈ ℝ ∧ 𝐵 ∈ ℝ) → if((𝐷𝑗) ≤ 𝐵, (𝐷𝑗), 𝐵) ≤ (𝐷𝑗))
4414, 15, 43syl2anc 692 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → if((𝐷𝑗) ≤ 𝐵, (𝐷𝑗), 𝐵) ≤ (𝐷𝑗))
45 icossico 12201 . . . . . . . . 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 1324 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → ((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐵, (𝐷𝑗), 𝐵)) ⊆ ((𝐶𝑗)[,)(𝐷𝑗)))
47 volss 23241 . . . . . . . 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 1323 . . . . . . 7 ((𝜑𝑗 ∈ ℕ) → (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐵, (𝐷𝑗), 𝐵))) ≤ (vol‘((𝐶𝑗)[,)(𝐷𝑗))))
4934, 5, 27, 40, 48sge0lempt 39964 . . . . . 6 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐵, (𝐷𝑗), 𝐵))))) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)(𝐷𝑗))))))
5032ltpnfd 11915 . . . . . 6 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)(𝐷𝑗))))) < +∞)
5130, 33, 31, 49, 50xrlelttrd 11951 . . . . 5 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐵, (𝐷𝑗), 𝐵))))) < +∞)
5230, 31, 51xrltned 39072 . . . 4 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐵, (𝐷𝑗), 𝐵))))) ≠ +∞)
5352neneqd 2795 . . 3 (𝜑 → ¬ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐵, (𝐷𝑗), 𝐵))))) = +∞)
545, 29sge0repnf 39940 . . 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 247 . 2 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐵, (𝐷𝑗), 𝐵))))) ∈ ℝ)
561rexrd 10049 . . . . . . 7 (𝜑𝐵 ∈ ℝ*)
572, 1iccssred 39173 . . . . . . . . 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 3672 . . . . . . . . . . 11 {𝑧 ∈ (𝐴[,]𝐵) ∣ (𝑧𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)))))} ⊆ (𝐴[,]𝐵)
6058, 59eqsstri 3620 . . . . . . . . . 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 40142 . . . . . . . . . . 11 (𝜑 → (𝑆𝑈𝐴𝑈 ∧ ∃𝑥 ∈ ℝ ∀𝑦𝑈 𝑦𝑥))
6463simp1d 1071 . . . . . . . . . 10 (𝜑𝑆𝑈)
6560, 64sseldi 3586 . . . . . . . . 9 (𝜑𝑆 ∈ (𝐴[,]𝐵))
6657, 65sseldd 3589 . . . . . . . 8 (𝜑𝑆 ∈ ℝ)
6766rexrd 10049 . . . . . . 7 (𝜑𝑆 ∈ ℝ*)
68 simpl 473 . . . . . . . . 9 ((𝜑 ∧ ¬ 𝐵𝑆) → 𝜑)
69 simpr 477 . . . . . . . . . 10 ((𝜑 ∧ ¬ 𝐵𝑆) → ¬ 𝐵𝑆)
7068, 66syl 17 . . . . . . . . . . 11 ((𝜑 ∧ ¬ 𝐵𝑆) → 𝑆 ∈ ℝ)
7168, 1syl 17 . . . . . . . . . . 11 ((𝜑 ∧ ¬ 𝐵𝑆) → 𝐵 ∈ ℝ)
7270, 71ltnled 10144 . . . . . . . . . 10 ((𝜑 ∧ ¬ 𝐵𝑆) → (𝑆 < 𝐵 ↔ ¬ 𝐵𝑆))
7369, 72mpbird 247 . . . . . . . . 9 ((𝜑 ∧ ¬ 𝐵𝑆) → 𝑆 < 𝐵)
74 hoidmv1lelem3.x . . . . . . . . . . . . 13 (𝜑 → (𝐴[,)𝐵) ⊆ 𝑗 ∈ ℕ ((𝐶𝑗)[,)(𝐷𝑗)))
7574adantr 481 . . . . . . . . . . . 12 ((𝜑𝑆 < 𝐵) → (𝐴[,)𝐵) ⊆ 𝑗 ∈ ℕ ((𝐶𝑗)[,)(𝐷𝑗)))
762rexrd 10049 . . . . . . . . . . . . . 14 (𝜑𝐴 ∈ ℝ*)
7776adantr 481 . . . . . . . . . . . . 13 ((𝜑𝑆 < 𝐵) → 𝐴 ∈ ℝ*)
7856adantr 481 . . . . . . . . . . . . 13 ((𝜑𝑆 < 𝐵) → 𝐵 ∈ ℝ*)
7967adantr 481 . . . . . . . . . . . . 13 ((𝜑𝑆 < 𝐵) → 𝑆 ∈ ℝ*)
8060, 57syl5ss 3599 . . . . . . . . . . . . . . . 16 (𝜑𝑈 ⊆ ℝ)
81 ne0i 3903 . . . . . . . . . . . . . . . . 17 (𝑆𝑈𝑈 ≠ ∅)
8264, 81syl 17 . . . . . . . . . . . . . . . 16 (𝜑𝑈 ≠ ∅)
8363simp3d 1073 . . . . . . . . . . . . . . . 16 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦𝑈 𝑦𝑥)
8463simp2d 1072 . . . . . . . . . . . . . . . 16 (𝜑𝐴𝑈)
85 suprub 10944 . . . . . . . . . . . . . . . 16 (((𝑈 ⊆ ℝ ∧ 𝑈 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦𝑈 𝑦𝑥) ∧ 𝐴𝑈) → 𝐴 ≤ sup(𝑈, ℝ, < ))
8680, 82, 83, 84, 85syl31anc 1326 . . . . . . . . . . . . . . 15 (𝜑𝐴 ≤ sup(𝑈, ℝ, < ))
8786, 62syl6breqr 4665 . . . . . . . . . . . . . 14 (𝜑𝐴𝑆)
8887adantr 481 . . . . . . . . . . . . 13 ((𝜑𝑆 < 𝐵) → 𝐴𝑆)
89 simpr 477 . . . . . . . . . . . . 13 ((𝜑𝑆 < 𝐵) → 𝑆 < 𝐵)
9077, 78, 79, 88, 89elicod 12182 . . . . . . . . . . . 12 ((𝜑𝑆 < 𝐵) → 𝑆 ∈ (𝐴[,)𝐵))
9175, 90sseldd 3589 . . . . . . . . . . 11 ((𝜑𝑆 < 𝐵) → 𝑆 𝑗 ∈ ℕ ((𝐶𝑗)[,)(𝐷𝑗)))
92 eliun 4497 . . . . . . . . . . 11 (𝑆 𝑗 ∈ ℕ ((𝐶𝑗)[,)(𝐷𝑗)) ↔ ∃𝑗 ∈ ℕ 𝑆 ∈ ((𝐶𝑗)[,)(𝐷𝑗)))
9391, 92sylib 208 . . . . . . . . . 10 ((𝜑𝑆 < 𝐵) → ∃𝑗 ∈ ℕ 𝑆 ∈ ((𝐶𝑗)[,)(𝐷𝑗)))
942adantr 481 . . . . . . . . . . . . . 14 ((𝜑𝑆 < 𝐵) → 𝐴 ∈ ℝ)
95943ad2ant1 1080 . . . . . . . . . . . . 13 (((𝜑𝑆 < 𝐵) ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ ((𝐶𝑗)[,)(𝐷𝑗))) → 𝐴 ∈ ℝ)
961adantr 481 . . . . . . . . . . . . . 14 ((𝜑𝑆 < 𝐵) → 𝐵 ∈ ℝ)
97963ad2ant1 1080 . . . . . . . . . . . . 13 (((𝜑𝑆 < 𝐵) ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ ((𝐶𝑗)[,)(𝐷𝑗))) → 𝐵 ∈ ℝ)
9811adantr 481 . . . . . . . . . . . . . 14 ((𝜑𝑆 < 𝐵) → 𝐶:ℕ⟶ℝ)
99983ad2ant1 1080 . . . . . . . . . . . . 13 (((𝜑𝑆 < 𝐵) ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ ((𝐶𝑗)[,)(𝐷𝑗))) → 𝐶:ℕ⟶ℝ)
10013adantr 481 . . . . . . . . . . . . . 14 ((𝜑𝑆 < 𝐵) → 𝐷:ℕ⟶ℝ)
1011003ad2ant1 1080 . . . . . . . . . . . . 13 (((𝜑𝑆 < 𝐵) ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ ((𝐶𝑗)[,)(𝐷𝑗))) → 𝐷:ℕ⟶ℝ)
102 fveq2 6158 . . . . . . . . . . . . . . . . . . . 20 (𝑖 = 𝑗 → (𝐶𝑖) = (𝐶𝑗))
103 fveq2 6158 . . . . . . . . . . . . . . . . . . . 20 (𝑖 = 𝑗 → (𝐷𝑖) = (𝐷𝑗))
104102, 103oveq12d 6633 . . . . . . . . . . . . . . . . . . 19 (𝑖 = 𝑗 → ((𝐶𝑖)[,)(𝐷𝑖)) = ((𝐶𝑗)[,)(𝐷𝑗)))
105104fveq2d 6162 . . . . . . . . . . . . . . . . . 18 (𝑖 = 𝑗 → (vol‘((𝐶𝑖)[,)(𝐷𝑖))) = (vol‘((𝐶𝑗)[,)(𝐷𝑗))))
106105cbvmptv 4720 . . . . . . . . . . . . . . . . 17 (𝑖 ∈ ℕ ↦ (vol‘((𝐶𝑖)[,)(𝐷𝑖)))) = (𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)(𝐷𝑗))))
107106fveq2i 6161 . . . . . . . . . . . . . . . 16 ^‘(𝑖 ∈ ℕ ↦ (vol‘((𝐶𝑖)[,)(𝐷𝑖))))) = (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)(𝐷𝑗)))))
108107, 32syl5eqel 2702 . . . . . . . . . . . . . . 15 (𝜑 → (Σ^‘(𝑖 ∈ ℕ ↦ (vol‘((𝐶𝑖)[,)(𝐷𝑖))))) ∈ ℝ)
109108adantr 481 . . . . . . . . . . . . . 14 ((𝜑𝑆 < 𝐵) → (Σ^‘(𝑖 ∈ ℕ ↦ (vol‘((𝐶𝑖)[,)(𝐷𝑖))))) ∈ ℝ)
1101093ad2ant1 1080 . . . . . . . . . . . . 13 (((𝜑𝑆 < 𝐵) ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ ((𝐶𝑗)[,)(𝐷𝑗))) → (Σ^‘(𝑖 ∈ ℕ ↦ (vol‘((𝐶𝑖)[,)(𝐷𝑖))))) ∈ ℝ)
111103breq1d 4633 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑖 = 𝑗 → ((𝐷𝑖) ≤ 𝑧 ↔ (𝐷𝑗) ≤ 𝑧))
112111, 103ifbieq1d 4087 . . . . . . . . . . . . . . . . . . . . . 22 (𝑖 = 𝑗 → if((𝐷𝑖) ≤ 𝑧, (𝐷𝑖), 𝑧) = if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))
113102, 112oveq12d 6633 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 = 𝑗 → ((𝐶𝑖)[,)if((𝐷𝑖) ≤ 𝑧, (𝐷𝑖), 𝑧)) = ((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)))
114113fveq2d 6162 . . . . . . . . . . . . . . . . . . . 20 (𝑖 = 𝑗 → (vol‘((𝐶𝑖)[,)if((𝐷𝑖) ≤ 𝑧, (𝐷𝑖), 𝑧))) = (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))))
115114cbvmptv 4720 . . . . . . . . . . . . . . . . . . 19 (𝑖 ∈ ℕ ↦ (vol‘((𝐶𝑖)[,)if((𝐷𝑖) ≤ 𝑧, (𝐷𝑖), 𝑧)))) = (𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))))
116115eqcomi 2630 . . . . . . . . . . . . . . . . . 18 (𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)))) = (𝑖 ∈ ℕ ↦ (vol‘((𝐶𝑖)[,)if((𝐷𝑖) ≤ 𝑧, (𝐷𝑖), 𝑧))))
117116fveq2i 6161 . . . . . . . . . . . . . . . . 17 ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))))) = (Σ^‘(𝑖 ∈ ℕ ↦ (vol‘((𝐶𝑖)[,)if((𝐷𝑖) ≤ 𝑧, (𝐷𝑖), 𝑧)))))
118117breq2i 4631 . . . . . . . . . . . . . . . 16 ((𝑧𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))))) ↔ (𝑧𝐴) ≤ (Σ^‘(𝑖 ∈ ℕ ↦ (vol‘((𝐶𝑖)[,)if((𝐷𝑖) ≤ 𝑧, (𝐷𝑖), 𝑧))))))
119118a1i 11 . . . . . . . . . . . . . . 15 (𝑧 ∈ (𝐴[,]𝐵) → ((𝑧𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))))) ↔ (𝑧𝐴) ≤ (Σ^‘(𝑖 ∈ ℕ ↦ (vol‘((𝐶𝑖)[,)if((𝐷𝑖) ≤ 𝑧, (𝐷𝑖), 𝑧)))))))
120119rabbiia 3177 . . . . . . . . . . . . . 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)))))}
12158, 120eqtri 2643 . . . . . . . . . . . . 13 𝑈 = {𝑧 ∈ (𝐴[,]𝐵) ∣ (𝑧𝐴) ≤ (Σ^‘(𝑖 ∈ ℕ ↦ (vol‘((𝐶𝑖)[,)if((𝐷𝑖) ≤ 𝑧, (𝐷𝑖), 𝑧)))))}
12264adantr 481 . . . . . . . . . . . . . 14 ((𝜑𝑆 < 𝐵) → 𝑆𝑈)
1231223ad2ant1 1080 . . . . . . . . . . . . 13 (((𝜑𝑆 < 𝐵) ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ ((𝐶𝑗)[,)(𝐷𝑗))) → 𝑆𝑈)
124883ad2ant1 1080 . . . . . . . . . . . . 13 (((𝜑𝑆 < 𝐵) ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ ((𝐶𝑗)[,)(𝐷𝑗))) → 𝐴𝑆)
125893ad2ant1 1080 . . . . . . . . . . . . 13 (((𝜑𝑆 < 𝐵) ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ ((𝐶𝑗)[,)(𝐷𝑗))) → 𝑆 < 𝐵)
126 simp2 1060 . . . . . . . . . . . . 13 (((𝜑𝑆 < 𝐵) ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ ((𝐶𝑗)[,)(𝐷𝑗))) → 𝑗 ∈ ℕ)
127 simp3 1061 . . . . . . . . . . . . 13 (((𝜑𝑆 < 𝐵) ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ ((𝐶𝑗)[,)(𝐷𝑗))) → 𝑆 ∈ ((𝐶𝑗)[,)(𝐷𝑗)))
128 eqid 2621 . . . . . . . . . . . . 13 if((𝐷𝑗) ≤ 𝐵, (𝐷𝑗), 𝐵) = if((𝐷𝑗) ≤ 𝐵, (𝐷𝑗), 𝐵)
12995, 97, 99, 101, 110, 121, 123, 124, 125, 126, 127, 128hoidmv1lelem2 40143 . . . . . . . . . . . 12 (((𝜑𝑆 < 𝐵) ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ ((𝐶𝑗)[,)(𝐷𝑗))) → ∃𝑢𝑈 𝑆 < 𝑢)
1301293exp 1261 . . . . . . . . . . 11 ((𝜑𝑆 < 𝐵) → (𝑗 ∈ ℕ → (𝑆 ∈ ((𝐶𝑗)[,)(𝐷𝑗)) → ∃𝑢𝑈 𝑆 < 𝑢)))
131130rexlimdv 3025 . . . . . . . . . 10 ((𝜑𝑆 < 𝐵) → (∃𝑗 ∈ ℕ 𝑆 ∈ ((𝐶𝑗)[,)(𝐷𝑗)) → ∃𝑢𝑈 𝑆 < 𝑢))
13293, 131mpd 15 . . . . . . . . 9 ((𝜑𝑆 < 𝐵) → ∃𝑢𝑈 𝑆 < 𝑢)
13368, 73, 132syl2anc 692 . . . . . . . 8 ((𝜑 ∧ ¬ 𝐵𝑆) → ∃𝑢𝑈 𝑆 < 𝑢)
13457adantr 481 . . . . . . . . . . . . . . 15 ((𝜑𝑢𝑈) → (𝐴[,]𝐵) ⊆ ℝ)
13560, 134syl5ss 3599 . . . . . . . . . . . . . 14 ((𝜑𝑢𝑈) → 𝑈 ⊆ ℝ)
13682adantr 481 . . . . . . . . . . . . . 14 ((𝜑𝑢𝑈) → 𝑈 ≠ ∅)
1372, 1jca 554 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
138137adantr 481 . . . . . . . . . . . . . . . 16 ((𝜑𝑢𝑈) → (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
13960a1i 11 . . . . . . . . . . . . . . . 16 ((𝜑𝑢𝑈) → 𝑈 ⊆ (𝐴[,]𝐵))
14064adantr 481 . . . . . . . . . . . . . . . 16 ((𝜑𝑢𝑈) → 𝑆𝑈)
141 iccsupr 12224 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝑈 ⊆ (𝐴[,]𝐵) ∧ 𝑆𝑈) → (𝑈 ⊆ ℝ ∧ 𝑈 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦𝑈 𝑦𝑥))
142138, 139, 140, 141syl3anc 1323 . . . . . . . . . . . . . . 15 ((𝜑𝑢𝑈) → (𝑈 ⊆ ℝ ∧ 𝑈 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦𝑈 𝑦𝑥))
143142simp3d 1073 . . . . . . . . . . . . . 14 ((𝜑𝑢𝑈) → ∃𝑥 ∈ ℝ ∀𝑦𝑈 𝑦𝑥)
144 simpr 477 . . . . . . . . . . . . . 14 ((𝜑𝑢𝑈) → 𝑢𝑈)
145 suprub 10944 . . . . . . . . . . . . . 14 (((𝑈 ⊆ ℝ ∧ 𝑈 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦𝑈 𝑦𝑥) ∧ 𝑢𝑈) → 𝑢 ≤ sup(𝑈, ℝ, < ))
146135, 136, 143, 144, 145syl31anc 1326 . . . . . . . . . . . . 13 ((𝜑𝑢𝑈) → 𝑢 ≤ sup(𝑈, ℝ, < ))
147146, 62syl6breqr 4665 . . . . . . . . . . . 12 ((𝜑𝑢𝑈) → 𝑢𝑆)
148147ralrimiva 2962 . . . . . . . . . . 11 (𝜑 → ∀𝑢𝑈 𝑢𝑆)
14960sseli 3584 . . . . . . . . . . . . . . 15 (𝑢𝑈𝑢 ∈ (𝐴[,]𝐵))
150149adantl 482 . . . . . . . . . . . . . 14 ((𝜑𝑢𝑈) → 𝑢 ∈ (𝐴[,]𝐵))
151134, 150sseldd 3589 . . . . . . . . . . . . 13 ((𝜑𝑢𝑈) → 𝑢 ∈ ℝ)
15266adantr 481 . . . . . . . . . . . . 13 ((𝜑𝑢𝑈) → 𝑆 ∈ ℝ)
153151, 152lenltd 10143 . . . . . . . . . . . 12 ((𝜑𝑢𝑈) → (𝑢𝑆 ↔ ¬ 𝑆 < 𝑢))
154153ralbidva 2981 . . . . . . . . . . 11 (𝜑 → (∀𝑢𝑈 𝑢𝑆 ↔ ∀𝑢𝑈 ¬ 𝑆 < 𝑢))
155148, 154mpbid 222 . . . . . . . . . 10 (𝜑 → ∀𝑢𝑈 ¬ 𝑆 < 𝑢)
156 ralnex 2988 . . . . . . . . . 10 (∀𝑢𝑈 ¬ 𝑆 < 𝑢 ↔ ¬ ∃𝑢𝑈 𝑆 < 𝑢)
157155, 156sylib 208 . . . . . . . . 9 (𝜑 → ¬ ∃𝑢𝑈 𝑆 < 𝑢)
158157adantr 481 . . . . . . . 8 ((𝜑 ∧ ¬ 𝐵𝑆) → ¬ ∃𝑢𝑈 𝑆 < 𝑢)
159133, 158condan 834 . . . . . . 7 (𝜑𝐵𝑆)
160 iccleub 12187 . . . . . . . 8 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝑆 ∈ (𝐴[,]𝐵)) → 𝑆𝐵)
16176, 56, 65, 160syl3anc 1323 . . . . . . 7 (𝜑𝑆𝐵)
16256, 67, 159, 161xrletrid 11946 . . . . . 6 (𝜑𝐵 = 𝑆)
163162, 64eqeltrd 2698 . . . . 5 (𝜑𝐵𝑈)
164163, 58syl6eleq 2708 . . . 4 (𝜑𝐵 ∈ {𝑧 ∈ (𝐴[,]𝐵) ∣ (𝑧𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)))))})
165 oveq1 6622 . . . . . 6 (𝑧 = 𝐵 → (𝑧𝐴) = (𝐵𝐴))
166 breq2 4627 . . . . . . . . . . 11 (𝑧 = 𝐵 → ((𝐷𝑗) ≤ 𝑧 ↔ (𝐷𝑗) ≤ 𝐵))
167 id 22 . . . . . . . . . . 11 (𝑧 = 𝐵𝑧 = 𝐵)
168166, 167ifbieq2d 4089 . . . . . . . . . 10 (𝑧 = 𝐵 → if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧) = if((𝐷𝑗) ≤ 𝐵, (𝐷𝑗), 𝐵))
169168oveq2d 6631 . . . . . . . . 9 (𝑧 = 𝐵 → ((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)) = ((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐵, (𝐷𝑗), 𝐵)))
170169fveq2d 6162 . . . . . . . 8 (𝑧 = 𝐵 → (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))) = (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐵, (𝐷𝑗), 𝐵))))
171170mpteq2dv 4715 . . . . . . 7 (𝑧 = 𝐵 → (𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)))) = (𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐵, (𝐷𝑗), 𝐵)))))
172171fveq2d 6162 . . . . . 6 (𝑧 = 𝐵 → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))))) = (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐵, (𝐷𝑗), 𝐵))))))
173165, 172breq12d 4636 . . . . 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)))))))
174173elrab 3351 . . . 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)))))))
175164, 174sylib 208 . . 3 (𝜑 → (𝐵 ∈ (𝐴[,]𝐵) ∧ (𝐵𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐵, (𝐷𝑗), 𝐵)))))))
176175simprd 479 . 2 (𝜑 → (𝐵𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐵, (𝐷𝑗), 𝐵))))))
1773, 55, 32, 176, 49letrd 10154 1 (𝜑 → (𝐵𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)(𝐷𝑗))))))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1036   = wceq 1480   ∈ wcel 1987   ≠ wne 2790  ∀wral 2908  ∃wrex 2909  {crab 2912  Vcvv 3190   ⊆ wss 3560  ∅c0 3897  ifcif 4064  ∪ ciun 4492   class class class wbr 4623   ↦ cmpt 4683  dom cdm 5084  ⟶wf 5853  ‘cfv 5857  (class class class)co 6615  supcsup 8306  ℝcr 9895  0cc0 9896  +∞cpnf 10031  ℝ*cxr 10033   < clt 10034   ≤ cle 10035   − cmin 10226  ℕcn 10980  [,)cico 12135  [,]cicc 12136  volcvol 23172  Σ^csumge0 39916 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-inf2 8498  ax-cnex 9952  ax-resscn 9953  ax-1cn 9954  ax-icn 9955  ax-addcl 9956  ax-addrcl 9957  ax-mulcl 9958  ax-mulrcl 9959  ax-mulcom 9960  ax-addass 9961  ax-mulass 9962  ax-distr 9963  ax-i2m1 9964  ax-1ne0 9965  ax-1rid 9966  ax-rnegex 9967  ax-rrecex 9968  ax-cnre 9969  ax-pre-lttri 9970  ax-pre-lttrn 9971  ax-pre-ltadd 9972  ax-pre-mulgt0 9973  ax-pre-sup 9974 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-int 4448  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-se 5044  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-isom 5866  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-of 6862  df-om 7028  df-1st 7128  df-2nd 7129  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-1o 7520  df-2o 7521  df-oadd 7524  df-er 7702  df-map 7819  df-pm 7820  df-en 7916  df-dom 7917  df-sdom 7918  df-fin 7919  df-fi 8277  df-sup 8308  df-inf 8309  df-oi 8375  df-card 8725  df-cda 8950  df-pnf 10036  df-mnf 10037  df-xr 10038  df-ltxr 10039  df-le 10040  df-sub 10228  df-neg 10229  df-div 10645  df-nn 10981  df-2 11039  df-3 11040  df-n0 11253  df-z 11338  df-uz 11648  df-q 11749  df-rp 11793  df-xneg 11906  df-xadd 11907  df-xmul 11908  df-ioo 12137  df-ico 12139  df-icc 12140  df-fz 12285  df-fzo 12423  df-fl 12549  df-seq 12758  df-exp 12817  df-hash 13074  df-cj 13789  df-re 13790  df-im 13791  df-sqrt 13925  df-abs 13926  df-clim 14169  df-rlim 14170  df-sum 14367  df-rest 16023  df-topgen 16044  df-psmet 19678  df-xmet 19679  df-met 19680  df-bl 19681  df-mopn 19682  df-top 20639  df-topon 20656  df-bases 20690  df-cmp 21130  df-ovol 23173  df-vol 23174  df-sumge0 39917 This theorem is referenced by:  hoidmv1le  40145
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