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Theorem hoidmv1lelem3 39284
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 10306 . 2 (𝜑 → (𝐵𝐴) ∈ ℝ)
4 nnex 10870 . . . . . . 7 ℕ ∈ V
54a1i 11 . . . . . 6 (𝜑 → ℕ ∈ V)
6 icossicc 12084 . . . . . . . 8 (0[,)+∞) ⊆ (0[,]+∞)
7 0xr 9939 . . . . . . . . . 10 0 ∈ ℝ*
87a1i 11 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → 0 ∈ ℝ*)
9 pnfxr 11778 . . . . . . . . . 10 +∞ ∈ ℝ*
109a1i 11 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → +∞ ∈ ℝ*)
11 hoidmv1lelem3.c . . . . . . . . . . . 12 (𝜑𝐶:ℕ⟶ℝ)
1211ffvelrnda 6249 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ℕ) → (𝐶𝑗) ∈ ℝ)
13 hoidmv1lelem3.d . . . . . . . . . . . . 13 (𝜑𝐷:ℕ⟶ℝ)
1413ffvelrnda 6249 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ℕ) → (𝐷𝑗) ∈ ℝ)
151adantr 479 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ℕ) → 𝐵 ∈ ℝ)
1614, 15ifcld 4077 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ℕ) → if((𝐷𝑗) ≤ 𝐵, (𝐷𝑗), 𝐵) ∈ ℝ)
17 volicore 39272 . . . . . . . . . . 11 (((𝐶𝑗) ∈ ℝ ∧ if((𝐷𝑗) ≤ 𝐵, (𝐷𝑗), 𝐵) ∈ ℝ) → (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐵, (𝐷𝑗), 𝐵))) ∈ ℝ)
1812, 16, 17syl2anc 690 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐵, (𝐷𝑗), 𝐵))) ∈ ℝ)
1918rexrd 9942 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐵, (𝐷𝑗), 𝐵))) ∈ ℝ*)
2016rexrd 9942 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ℕ) → if((𝐷𝑗) ≤ 𝐵, (𝐷𝑗), 𝐵) ∈ ℝ*)
21 icombl 23053 . . . . . . . . . . 11 (((𝐶𝑗) ∈ ℝ ∧ if((𝐷𝑗) ≤ 𝐵, (𝐷𝑗), 𝐵) ∈ ℝ*) → ((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐵, (𝐷𝑗), 𝐵)) ∈ dom vol)
2212, 20, 21syl2anc 690 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → ((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐵, (𝐷𝑗), 𝐵)) ∈ dom vol)
23 volge0 38654 . . . . . . . . . 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 11789 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐵, (𝐷𝑗), 𝐵))) < +∞)
268, 10, 19, 24, 25elicod 12048 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐵, (𝐷𝑗), 𝐵))) ∈ (0[,)+∞))
276, 26sseldi 3562 . . . . . . 7 ((𝜑𝑗 ∈ ℕ) → (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐵, (𝐷𝑗), 𝐵))) ∈ (0[,]+∞))
28 eqid 2606 . . . . . . 7 (𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐵, (𝐷𝑗), 𝐵)))) = (𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐵, (𝐷𝑗), 𝐵))))
2927, 28fmptd 6274 . . . . . 6 (𝜑 → (𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐵, (𝐷𝑗), 𝐵)))):ℕ⟶(0[,]+∞))
305, 29sge0xrcl 39079 . . . . 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 9942 . . . . . 6 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)(𝐷𝑗))))) ∈ ℝ*)
34 nfv 1829 . . . . . . 7 𝑗𝜑
35 volf 23018 . . . . . . . . 9 vol:dom vol⟶(0[,]+∞)
3635a1i 11 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → vol:dom vol⟶(0[,]+∞))
3714rexrd 9942 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → (𝐷𝑗) ∈ ℝ*)
38 icombl 23053 . . . . . . . . 9 (((𝐶𝑗) ∈ ℝ ∧ (𝐷𝑗) ∈ ℝ*) → ((𝐶𝑗)[,)(𝐷𝑗)) ∈ dom vol)
3912, 37, 38syl2anc 690 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → ((𝐶𝑗)[,)(𝐷𝑗)) ∈ dom vol)
4036, 39ffvelrnd 6250 . . . . . . 7 ((𝜑𝑗 ∈ ℕ) → (vol‘((𝐶𝑗)[,)(𝐷𝑗))) ∈ (0[,]+∞))
4112rexrd 9942 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → (𝐶𝑗) ∈ ℝ*)
4212leidd 10440 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → (𝐶𝑗) ≤ (𝐶𝑗))
43 min1 11850 . . . . . . . . . 10 (((𝐷𝑗) ∈ ℝ ∧ 𝐵 ∈ ℝ) → if((𝐷𝑗) ≤ 𝐵, (𝐷𝑗), 𝐵) ≤ (𝐷𝑗))
4414, 15, 43syl2anc 690 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → if((𝐷𝑗) ≤ 𝐵, (𝐷𝑗), 𝐵) ≤ (𝐷𝑗))
45 icossico 12067 . . . . . . . . 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 1318 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → ((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐵, (𝐷𝑗), 𝐵)) ⊆ ((𝐶𝑗)[,)(𝐷𝑗)))
47 volss 23022 . . . . . . . 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 1317 . . . . . . 7 ((𝜑𝑗 ∈ ℕ) → (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐵, (𝐷𝑗), 𝐵))) ≤ (vol‘((𝐶𝑗)[,)(𝐷𝑗))))
4934, 5, 27, 40, 48sge0lempt 39104 . . . . . 6 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐵, (𝐷𝑗), 𝐵))))) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)(𝐷𝑗))))))
5032ltpnfd 11789 . . . . . 6 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)(𝐷𝑗))))) < +∞)
5130, 33, 31, 49, 50xrlelttrd 11823 . . . . 5 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐵, (𝐷𝑗), 𝐵))))) < +∞)
5230, 31, 51xrltned 38315 . . . 4 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐵, (𝐷𝑗), 𝐵))))) ≠ +∞)
5352neneqd 2783 . . 3 (𝜑 → ¬ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐵, (𝐷𝑗), 𝐵))))) = +∞)
545, 29sge0repnf 39080 . . 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 245 . 2 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐵, (𝐷𝑗), 𝐵))))) ∈ ℝ)
561rexrd 9942 . . . . . . 7 (𝜑𝐵 ∈ ℝ*)
572, 1iccssred 38375 . . . . . . . . 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 3646 . . . . . . . . . . 11 {𝑧 ∈ (𝐴[,]𝐵) ∣ (𝑧𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)))))} ⊆ (𝐴[,]𝐵)
6058, 59eqsstri 3594 . . . . . . . . . 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 39282 . . . . . . . . . . 11 (𝜑 → (𝑆𝑈𝐴𝑈 ∧ ∃𝑥 ∈ ℝ ∀𝑦𝑈 𝑦𝑥))
6463simp1d 1065 . . . . . . . . . 10 (𝜑𝑆𝑈)
6560, 64sseldi 3562 . . . . . . . . 9 (𝜑𝑆 ∈ (𝐴[,]𝐵))
6657, 65sseldd 3565 . . . . . . . 8 (𝜑𝑆 ∈ ℝ)
6766rexrd 9942 . . . . . . 7 (𝜑𝑆 ∈ ℝ*)
68 simpl 471 . . . . . . . . 9 ((𝜑 ∧ ¬ 𝐵𝑆) → 𝜑)
69 simpr 475 . . . . . . . . . 10 ((𝜑 ∧ ¬ 𝐵𝑆) → ¬ 𝐵𝑆)
7068, 66syl 17 . . . . . . . . . . 11 ((𝜑 ∧ ¬ 𝐵𝑆) → 𝑆 ∈ ℝ)
7168, 1syl 17 . . . . . . . . . . 11 ((𝜑 ∧ ¬ 𝐵𝑆) → 𝐵 ∈ ℝ)
7270, 71ltnled 10032 . . . . . . . . . 10 ((𝜑 ∧ ¬ 𝐵𝑆) → (𝑆 < 𝐵 ↔ ¬ 𝐵𝑆))
7369, 72mpbird 245 . . . . . . . . 9 ((𝜑 ∧ ¬ 𝐵𝑆) → 𝑆 < 𝐵)
74 hoidmv1lelem3.x . . . . . . . . . . . . 13 (𝜑 → (𝐴[,)𝐵) ⊆ 𝑗 ∈ ℕ ((𝐶𝑗)[,)(𝐷𝑗)))
7574adantr 479 . . . . . . . . . . . 12 ((𝜑𝑆 < 𝐵) → (𝐴[,)𝐵) ⊆ 𝑗 ∈ ℕ ((𝐶𝑗)[,)(𝐷𝑗)))
762rexrd 9942 . . . . . . . . . . . . . 14 (𝜑𝐴 ∈ ℝ*)
7776adantr 479 . . . . . . . . . . . . 13 ((𝜑𝑆 < 𝐵) → 𝐴 ∈ ℝ*)
7856adantr 479 . . . . . . . . . . . . 13 ((𝜑𝑆 < 𝐵) → 𝐵 ∈ ℝ*)
7967adantr 479 . . . . . . . . . . . . 13 ((𝜑𝑆 < 𝐵) → 𝑆 ∈ ℝ*)
8060, 57syl5ss 3575 . . . . . . . . . . . . . . . 16 (𝜑𝑈 ⊆ ℝ)
81 ne0i 3876 . . . . . . . . . . . . . . . . 17 (𝑆𝑈𝑈 ≠ ∅)
8264, 81syl 17 . . . . . . . . . . . . . . . 16 (𝜑𝑈 ≠ ∅)
8363simp3d 1067 . . . . . . . . . . . . . . . 16 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦𝑈 𝑦𝑥)
8463simp2d 1066 . . . . . . . . . . . . . . . 16 (𝜑𝐴𝑈)
85 suprub 10830 . . . . . . . . . . . . . . . 16 (((𝑈 ⊆ ℝ ∧ 𝑈 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦𝑈 𝑦𝑥) ∧ 𝐴𝑈) → 𝐴 ≤ sup(𝑈, ℝ, < ))
8680, 82, 83, 84, 85syl31anc 1320 . . . . . . . . . . . . . . 15 (𝜑𝐴 ≤ sup(𝑈, ℝ, < ))
8786, 62syl6breqr 4616 . . . . . . . . . . . . . 14 (𝜑𝐴𝑆)
8887adantr 479 . . . . . . . . . . . . 13 ((𝜑𝑆 < 𝐵) → 𝐴𝑆)
89 simpr 475 . . . . . . . . . . . . 13 ((𝜑𝑆 < 𝐵) → 𝑆 < 𝐵)
9077, 78, 79, 88, 89elicod 12048 . . . . . . . . . . . 12 ((𝜑𝑆 < 𝐵) → 𝑆 ∈ (𝐴[,)𝐵))
9175, 90sseldd 3565 . . . . . . . . . . 11 ((𝜑𝑆 < 𝐵) → 𝑆 𝑗 ∈ ℕ ((𝐶𝑗)[,)(𝐷𝑗)))
92 eliun 4451 . . . . . . . . . . 11 (𝑆 𝑗 ∈ ℕ ((𝐶𝑗)[,)(𝐷𝑗)) ↔ ∃𝑗 ∈ ℕ 𝑆 ∈ ((𝐶𝑗)[,)(𝐷𝑗)))
9391, 92sylib 206 . . . . . . . . . 10 ((𝜑𝑆 < 𝐵) → ∃𝑗 ∈ ℕ 𝑆 ∈ ((𝐶𝑗)[,)(𝐷𝑗)))
942adantr 479 . . . . . . . . . . . . . 14 ((𝜑𝑆 < 𝐵) → 𝐴 ∈ ℝ)
95943ad2ant1 1074 . . . . . . . . . . . . 13 (((𝜑𝑆 < 𝐵) ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ ((𝐶𝑗)[,)(𝐷𝑗))) → 𝐴 ∈ ℝ)
961adantr 479 . . . . . . . . . . . . . 14 ((𝜑𝑆 < 𝐵) → 𝐵 ∈ ℝ)
97963ad2ant1 1074 . . . . . . . . . . . . 13 (((𝜑𝑆 < 𝐵) ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ ((𝐶𝑗)[,)(𝐷𝑗))) → 𝐵 ∈ ℝ)
9811adantr 479 . . . . . . . . . . . . . 14 ((𝜑𝑆 < 𝐵) → 𝐶:ℕ⟶ℝ)
99983ad2ant1 1074 . . . . . . . . . . . . 13 (((𝜑𝑆 < 𝐵) ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ ((𝐶𝑗)[,)(𝐷𝑗))) → 𝐶:ℕ⟶ℝ)
10013adantr 479 . . . . . . . . . . . . . 14 ((𝜑𝑆 < 𝐵) → 𝐷:ℕ⟶ℝ)
1011003ad2ant1 1074 . . . . . . . . . . . . 13 (((𝜑𝑆 < 𝐵) ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ ((𝐶𝑗)[,)(𝐷𝑗))) → 𝐷:ℕ⟶ℝ)
102 fveq2 6085 . . . . . . . . . . . . . . . . . . . 20 (𝑖 = 𝑗 → (𝐶𝑖) = (𝐶𝑗))
103 fveq2 6085 . . . . . . . . . . . . . . . . . . . 20 (𝑖 = 𝑗 → (𝐷𝑖) = (𝐷𝑗))
104102, 103oveq12d 6542 . . . . . . . . . . . . . . . . . . 19 (𝑖 = 𝑗 → ((𝐶𝑖)[,)(𝐷𝑖)) = ((𝐶𝑗)[,)(𝐷𝑗)))
105104fveq2d 6089 . . . . . . . . . . . . . . . . . 18 (𝑖 = 𝑗 → (vol‘((𝐶𝑖)[,)(𝐷𝑖))) = (vol‘((𝐶𝑗)[,)(𝐷𝑗))))
106105cbvmptv 4669 . . . . . . . . . . . . . . . . 17 (𝑖 ∈ ℕ ↦ (vol‘((𝐶𝑖)[,)(𝐷𝑖)))) = (𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)(𝐷𝑗))))
107106fveq2i 6088 . . . . . . . . . . . . . . . 16 ^‘(𝑖 ∈ ℕ ↦ (vol‘((𝐶𝑖)[,)(𝐷𝑖))))) = (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)(𝐷𝑗)))))
108107, 32syl5eqel 2688 . . . . . . . . . . . . . . 15 (𝜑 → (Σ^‘(𝑖 ∈ ℕ ↦ (vol‘((𝐶𝑖)[,)(𝐷𝑖))))) ∈ ℝ)
109108adantr 479 . . . . . . . . . . . . . 14 ((𝜑𝑆 < 𝐵) → (Σ^‘(𝑖 ∈ ℕ ↦ (vol‘((𝐶𝑖)[,)(𝐷𝑖))))) ∈ ℝ)
1101093ad2ant1 1074 . . . . . . . . . . . . 13 (((𝜑𝑆 < 𝐵) ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ ((𝐶𝑗)[,)(𝐷𝑗))) → (Σ^‘(𝑖 ∈ ℕ ↦ (vol‘((𝐶𝑖)[,)(𝐷𝑖))))) ∈ ℝ)
111103breq1d 4584 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑖 = 𝑗 → ((𝐷𝑖) ≤ 𝑧 ↔ (𝐷𝑗) ≤ 𝑧))
112111, 103ifbieq1d 4055 . . . . . . . . . . . . . . . . . . . . . 22 (𝑖 = 𝑗 → if((𝐷𝑖) ≤ 𝑧, (𝐷𝑖), 𝑧) = if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))
113102, 112oveq12d 6542 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 = 𝑗 → ((𝐶𝑖)[,)if((𝐷𝑖) ≤ 𝑧, (𝐷𝑖), 𝑧)) = ((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)))
114113fveq2d 6089 . . . . . . . . . . . . . . . . . . . 20 (𝑖 = 𝑗 → (vol‘((𝐶𝑖)[,)if((𝐷𝑖) ≤ 𝑧, (𝐷𝑖), 𝑧))) = (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))))
115114cbvmptv 4669 . . . . . . . . . . . . . . . . . . 19 (𝑖 ∈ ℕ ↦ (vol‘((𝐶𝑖)[,)if((𝐷𝑖) ≤ 𝑧, (𝐷𝑖), 𝑧)))) = (𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))))
116115eqcomi 2615 . . . . . . . . . . . . . . . . . 18 (𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)))) = (𝑖 ∈ ℕ ↦ (vol‘((𝐶𝑖)[,)if((𝐷𝑖) ≤ 𝑧, (𝐷𝑖), 𝑧))))
117116fveq2i 6088 . . . . . . . . . . . . . . . . 17 ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))))) = (Σ^‘(𝑖 ∈ ℕ ↦ (vol‘((𝐶𝑖)[,)if((𝐷𝑖) ≤ 𝑧, (𝐷𝑖), 𝑧)))))
118117breq2i 4582 . . . . . . . . . . . . . . . 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 3157 . . . . . . . . . . . . . 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 2628 . . . . . . . . . . . . 13 𝑈 = {𝑧 ∈ (𝐴[,]𝐵) ∣ (𝑧𝐴) ≤ (Σ^‘(𝑖 ∈ ℕ ↦ (vol‘((𝐶𝑖)[,)if((𝐷𝑖) ≤ 𝑧, (𝐷𝑖), 𝑧)))))}
12264adantr 479 . . . . . . . . . . . . . 14 ((𝜑𝑆 < 𝐵) → 𝑆𝑈)
1231223ad2ant1 1074 . . . . . . . . . . . . 13 (((𝜑𝑆 < 𝐵) ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ ((𝐶𝑗)[,)(𝐷𝑗))) → 𝑆𝑈)
124883ad2ant1 1074 . . . . . . . . . . . . 13 (((𝜑𝑆 < 𝐵) ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ ((𝐶𝑗)[,)(𝐷𝑗))) → 𝐴𝑆)
125893ad2ant1 1074 . . . . . . . . . . . . 13 (((𝜑𝑆 < 𝐵) ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ ((𝐶𝑗)[,)(𝐷𝑗))) → 𝑆 < 𝐵)
126 simp2 1054 . . . . . . . . . . . . 13 (((𝜑𝑆 < 𝐵) ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ ((𝐶𝑗)[,)(𝐷𝑗))) → 𝑗 ∈ ℕ)
127 simp3 1055 . . . . . . . . . . . . 13 (((𝜑𝑆 < 𝐵) ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ ((𝐶𝑗)[,)(𝐷𝑗))) → 𝑆 ∈ ((𝐶𝑗)[,)(𝐷𝑗)))
128 eqid 2606 . . . . . . . . . . . . 13 if((𝐷𝑗) ≤ 𝐵, (𝐷𝑗), 𝐵) = if((𝐷𝑗) ≤ 𝐵, (𝐷𝑗), 𝐵)
12995, 97, 99, 101, 110, 121, 123, 124, 125, 126, 127, 128hoidmv1lelem2 39283 . . . . . . . . . . . 12 (((𝜑𝑆 < 𝐵) ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ ((𝐶𝑗)[,)(𝐷𝑗))) → ∃𝑢𝑈 𝑆 < 𝑢)
1301293exp 1255 . . . . . . . . . . 11 ((𝜑𝑆 < 𝐵) → (𝑗 ∈ ℕ → (𝑆 ∈ ((𝐶𝑗)[,)(𝐷𝑗)) → ∃𝑢𝑈 𝑆 < 𝑢)))
131130rexlimdv 3008 . . . . . . . . . 10 ((𝜑𝑆 < 𝐵) → (∃𝑗 ∈ ℕ 𝑆 ∈ ((𝐶𝑗)[,)(𝐷𝑗)) → ∃𝑢𝑈 𝑆 < 𝑢))
13293, 131mpd 15 . . . . . . . . 9 ((𝜑𝑆 < 𝐵) → ∃𝑢𝑈 𝑆 < 𝑢)
13368, 73, 132syl2anc 690 . . . . . . . 8 ((𝜑 ∧ ¬ 𝐵𝑆) → ∃𝑢𝑈 𝑆 < 𝑢)
13457adantr 479 . . . . . . . . . . . . . . 15 ((𝜑𝑢𝑈) → (𝐴[,]𝐵) ⊆ ℝ)
13560, 134syl5ss 3575 . . . . . . . . . . . . . 14 ((𝜑𝑢𝑈) → 𝑈 ⊆ ℝ)
13682adantr 479 . . . . . . . . . . . . . 14 ((𝜑𝑢𝑈) → 𝑈 ≠ ∅)
1372, 1jca 552 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
138137adantr 479 . . . . . . . . . . . . . . . 16 ((𝜑𝑢𝑈) → (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
13960a1i 11 . . . . . . . . . . . . . . . 16 ((𝜑𝑢𝑈) → 𝑈 ⊆ (𝐴[,]𝐵))
14064adantr 479 . . . . . . . . . . . . . . . 16 ((𝜑𝑢𝑈) → 𝑆𝑈)
141 iccsupr 12090 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝑈 ⊆ (𝐴[,]𝐵) ∧ 𝑆𝑈) → (𝑈 ⊆ ℝ ∧ 𝑈 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦𝑈 𝑦𝑥))
142138, 139, 140, 141syl3anc 1317 . . . . . . . . . . . . . . 15 ((𝜑𝑢𝑈) → (𝑈 ⊆ ℝ ∧ 𝑈 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦𝑈 𝑦𝑥))
143142simp3d 1067 . . . . . . . . . . . . . 14 ((𝜑𝑢𝑈) → ∃𝑥 ∈ ℝ ∀𝑦𝑈 𝑦𝑥)
144 simpr 475 . . . . . . . . . . . . . 14 ((𝜑𝑢𝑈) → 𝑢𝑈)
145 suprub 10830 . . . . . . . . . . . . . 14 (((𝑈 ⊆ ℝ ∧ 𝑈 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦𝑈 𝑦𝑥) ∧ 𝑢𝑈) → 𝑢 ≤ sup(𝑈, ℝ, < ))
146135, 136, 143, 144, 145syl31anc 1320 . . . . . . . . . . . . 13 ((𝜑𝑢𝑈) → 𝑢 ≤ sup(𝑈, ℝ, < ))
147146, 62syl6breqr 4616 . . . . . . . . . . . 12 ((𝜑𝑢𝑈) → 𝑢𝑆)
148147ralrimiva 2945 . . . . . . . . . . 11 (𝜑 → ∀𝑢𝑈 𝑢𝑆)
14960sseli 3560 . . . . . . . . . . . . . . 15 (𝑢𝑈𝑢 ∈ (𝐴[,]𝐵))
150149adantl 480 . . . . . . . . . . . . . 14 ((𝜑𝑢𝑈) → 𝑢 ∈ (𝐴[,]𝐵))
151134, 150sseldd 3565 . . . . . . . . . . . . 13 ((𝜑𝑢𝑈) → 𝑢 ∈ ℝ)
15266adantr 479 . . . . . . . . . . . . 13 ((𝜑𝑢𝑈) → 𝑆 ∈ ℝ)
153151, 152lenltd 10031 . . . . . . . . . . . 12 ((𝜑𝑢𝑈) → (𝑢𝑆 ↔ ¬ 𝑆 < 𝑢))
154153ralbidva 2964 . . . . . . . . . . 11 (𝜑 → (∀𝑢𝑈 𝑢𝑆 ↔ ∀𝑢𝑈 ¬ 𝑆 < 𝑢))
155148, 154mpbid 220 . . . . . . . . . 10 (𝜑 → ∀𝑢𝑈 ¬ 𝑆 < 𝑢)
156 ralnex 2971 . . . . . . . . . 10 (∀𝑢𝑈 ¬ 𝑆 < 𝑢 ↔ ¬ ∃𝑢𝑈 𝑆 < 𝑢)
157155, 156sylib 206 . . . . . . . . 9 (𝜑 → ¬ ∃𝑢𝑈 𝑆 < 𝑢)
158157adantr 479 . . . . . . . 8 ((𝜑 ∧ ¬ 𝐵𝑆) → ¬ ∃𝑢𝑈 𝑆 < 𝑢)
159133, 158condan 830 . . . . . . 7 (𝜑𝐵𝑆)
160 iccleub 12053 . . . . . . . 8 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝑆 ∈ (𝐴[,]𝐵)) → 𝑆𝐵)
16176, 56, 65, 160syl3anc 1317 . . . . . . 7 (𝜑𝑆𝐵)
16256, 67, 159, 161xrletrid 11818 . . . . . 6 (𝜑𝐵 = 𝑆)
163162, 64eqeltrd 2684 . . . . 5 (𝜑𝐵𝑈)
164163, 58syl6eleq 2694 . . . 4 (𝜑𝐵 ∈ {𝑧 ∈ (𝐴[,]𝐵) ∣ (𝑧𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)))))})
165 oveq1 6531 . . . . . 6 (𝑧 = 𝐵 → (𝑧𝐴) = (𝐵𝐴))
166 breq2 4578 . . . . . . . . . . 11 (𝑧 = 𝐵 → ((𝐷𝑗) ≤ 𝑧 ↔ (𝐷𝑗) ≤ 𝐵))
167 id 22 . . . . . . . . . . 11 (𝑧 = 𝐵𝑧 = 𝐵)
168166, 167ifbieq2d 4057 . . . . . . . . . 10 (𝑧 = 𝐵 → if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧) = if((𝐷𝑗) ≤ 𝐵, (𝐷𝑗), 𝐵))
169168oveq2d 6540 . . . . . . . . 9 (𝑧 = 𝐵 → ((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)) = ((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐵, (𝐷𝑗), 𝐵)))
170169fveq2d 6089 . . . . . . . 8 (𝑧 = 𝐵 → (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))) = (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐵, (𝐷𝑗), 𝐵))))
171170mpteq2dv 4664 . . . . . . 7 (𝑧 = 𝐵 → (𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)))) = (𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐵, (𝐷𝑗), 𝐵)))))
172171fveq2d 6089 . . . . . 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 4587 . . . . 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 3327 . . . 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 206 . . 3 (𝜑 → (𝐵 ∈ (𝐴[,]𝐵) ∧ (𝐵𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐵, (𝐷𝑗), 𝐵)))))))
176175simprd 477 . 2 (𝜑 → (𝐵𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐵, (𝐷𝑗), 𝐵))))))
1773, 55, 32, 176, 49letrd 10042 1 (𝜑 → (𝐵𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)(𝐷𝑗))))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wa 382  w3a 1030   = wceq 1474  wcel 1976  wne 2776  wral 2892  wrex 2893  {crab 2896  Vcvv 3169  wss 3536  c0 3870  ifcif 4032   ciun 4446   class class class wbr 4574  cmpt 4634  dom cdm 5025  wf 5783  cfv 5787  (class class class)co 6524  supcsup 8203  cr 9788  0cc0 9789  +∞cpnf 9924  *cxr 9926   < clt 9927  cle 9928  cmin 10114  cn 10864  [,)cico 12001  [,]cicc 12002  volcvol 22953  Σ^csumge0 39056
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2229  ax-ext 2586  ax-rep 4690  ax-sep 4700  ax-nul 4709  ax-pow 4761  ax-pr 4825  ax-un 6821  ax-inf2 8395  ax-cnex 9845  ax-resscn 9846  ax-1cn 9847  ax-icn 9848  ax-addcl 9849  ax-addrcl 9850  ax-mulcl 9851  ax-mulrcl 9852  ax-mulcom 9853  ax-addass 9854  ax-mulass 9855  ax-distr 9856  ax-i2m1 9857  ax-1ne0 9858  ax-1rid 9859  ax-rnegex 9860  ax-rrecex 9861  ax-cnre 9862  ax-pre-lttri 9863  ax-pre-lttrn 9864  ax-pre-ltadd 9865  ax-pre-mulgt0 9866  ax-pre-sup 9867
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-fal 1480  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2458  df-mo 2459  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-ne 2778  df-nel 2779  df-ral 2897  df-rex 2898  df-reu 2899  df-rmo 2900  df-rab 2901  df-v 3171  df-sbc 3399  df-csb 3496  df-dif 3539  df-un 3541  df-in 3543  df-ss 3550  df-pss 3552  df-nul 3871  df-if 4033  df-pw 4106  df-sn 4122  df-pr 4124  df-tp 4126  df-op 4128  df-uni 4364  df-int 4402  df-iun 4448  df-br 4575  df-opab 4635  df-mpt 4636  df-tr 4672  df-eprel 4936  df-id 4940  df-po 4946  df-so 4947  df-fr 4984  df-se 4985  df-we 4986  df-xp 5031  df-rel 5032  df-cnv 5033  df-co 5034  df-dm 5035  df-rn 5036  df-res 5037  df-ima 5038  df-pred 5580  df-ord 5626  df-on 5627  df-lim 5628  df-suc 5629  df-iota 5751  df-fun 5789  df-fn 5790  df-f 5791  df-f1 5792  df-fo 5793  df-f1o 5794  df-fv 5795  df-isom 5796  df-riota 6486  df-ov 6527  df-oprab 6528  df-mpt2 6529  df-of 6769  df-om 6932  df-1st 7033  df-2nd 7034  df-wrecs 7268  df-recs 7329  df-rdg 7367  df-1o 7421  df-2o 7422  df-oadd 7425  df-er 7603  df-map 7720  df-pm 7721  df-en 7816  df-dom 7817  df-sdom 7818  df-fin 7819  df-fi 8174  df-sup 8205  df-inf 8206  df-oi 8272  df-card 8622  df-cda 8847  df-pnf 9929  df-mnf 9930  df-xr 9931  df-ltxr 9932  df-le 9933  df-sub 10116  df-neg 10117  df-div 10531  df-nn 10865  df-2 10923  df-3 10924  df-n0 11137  df-z 11208  df-uz 11517  df-q 11618  df-rp 11662  df-xneg 11775  df-xadd 11776  df-xmul 11777  df-ioo 12003  df-ico 12005  df-icc 12006  df-fz 12150  df-fzo 12287  df-fl 12407  df-seq 12616  df-exp 12675  df-hash 12932  df-cj 13630  df-re 13631  df-im 13632  df-sqrt 13766  df-abs 13767  df-clim 14010  df-rlim 14011  df-sum 14208  df-rest 15849  df-topgen 15870  df-psmet 19502  df-xmet 19503  df-met 19504  df-bl 19505  df-mopn 19506  df-top 20460  df-bases 20461  df-topon 20462  df-cmp 20939  df-ovol 22954  df-vol 22955  df-sumge0 39057
This theorem is referenced by:  hoidmv1le  39285
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