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Theorem hoidmv1lelem1 41126
Description: The supremum of 𝑈 belongs to 𝑈. This is the last part of step (a) and the whole step (b) in the proof of Lemma 114B of [Fremlin1] p. 23. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
Hypotheses
Ref Expression
hoidmv1lelem1.a (𝜑𝐴 ∈ ℝ)
hoidmv1lelem1.b (𝜑𝐵 ∈ ℝ)
hoidmv1lelem1.l (𝜑𝐴 < 𝐵)
hoidmv1lelem1.c (𝜑𝐶:ℕ⟶ℝ)
hoidmv1lelem1.d (𝜑𝐷:ℕ⟶ℝ)
hoidmv1lelem1.r (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)(𝐷𝑗))))) ∈ ℝ)
hoidmv1lelem1.u 𝑈 = {𝑧 ∈ (𝐴[,]𝐵) ∣ (𝑧𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)))))}
hoidmv1lelem1.s 𝑆 = sup(𝑈, ℝ, < )
Assertion
Ref Expression
hoidmv1lelem1 (𝜑 → (𝑆𝑈𝐴𝑈 ∧ ∃𝑥 ∈ ℝ ∀𝑦𝑈 𝑦𝑥))
Distinct variable groups:   𝐴,𝑗,𝑧   𝑦,𝐴   𝑥,𝐵,𝑦   𝑧,𝐵   𝑧,𝐶   𝑧,𝐷   𝑆,𝑗,𝑧   𝑈,𝑗,𝑧   𝑥,𝑈,𝑦   𝜑,𝑗,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥)   𝐵(𝑗)   𝐶(𝑥,𝑦,𝑗)   𝐷(𝑥,𝑦,𝑗)   𝑆(𝑥,𝑦)

Proof of Theorem hoidmv1lelem1
StepHypRef Expression
1 hoidmv1lelem1.s . . . . . 6 𝑆 = sup(𝑈, ℝ, < )
2 hoidmv1lelem1.a . . . . . . 7 (𝜑𝐴 ∈ ℝ)
3 hoidmv1lelem1.b . . . . . . 7 (𝜑𝐵 ∈ ℝ)
4 hoidmv1lelem1.u . . . . . . . . 9 𝑈 = {𝑧 ∈ (𝐴[,]𝐵) ∣ (𝑧𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)))))}
5 ssrab2 3720 . . . . . . . . 9 {𝑧 ∈ (𝐴[,]𝐵) ∣ (𝑧𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)))))} ⊆ (𝐴[,]𝐵)
64, 5eqsstri 3668 . . . . . . . 8 𝑈 ⊆ (𝐴[,]𝐵)
76a1i 11 . . . . . . 7 (𝜑𝑈 ⊆ (𝐴[,]𝐵))
82rexrd 10127 . . . . . . . . . . . 12 (𝜑𝐴 ∈ ℝ*)
93rexrd 10127 . . . . . . . . . . . 12 (𝜑𝐵 ∈ ℝ*)
10 hoidmv1lelem1.l . . . . . . . . . . . . 13 (𝜑𝐴 < 𝐵)
112, 3, 10ltled 10223 . . . . . . . . . . . 12 (𝜑𝐴𝐵)
12 lbicc2 12326 . . . . . . . . . . . 12 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐴𝐵) → 𝐴 ∈ (𝐴[,]𝐵))
138, 9, 11, 12syl3anc 1366 . . . . . . . . . . 11 (𝜑𝐴 ∈ (𝐴[,]𝐵))
142recnd 10106 . . . . . . . . . . . . 13 (𝜑𝐴 ∈ ℂ)
1514subidd 10418 . . . . . . . . . . . 12 (𝜑 → (𝐴𝐴) = 0)
16 nfv 1883 . . . . . . . . . . . . 13 𝑗𝜑
17 nnex 11064 . . . . . . . . . . . . . 14 ℕ ∈ V
1817a1i 11 . . . . . . . . . . . . 13 (𝜑 → ℕ ∈ V)
19 volf 23343 . . . . . . . . . . . . . . 15 vol:dom vol⟶(0[,]+∞)
2019a1i 11 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ ℕ) → vol:dom vol⟶(0[,]+∞))
21 hoidmv1lelem1.c . . . . . . . . . . . . . . . 16 (𝜑𝐶:ℕ⟶ℝ)
2221ffvelrnda 6399 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ ℕ) → (𝐶𝑗) ∈ ℝ)
23 hoidmv1lelem1.d . . . . . . . . . . . . . . . . . 18 (𝜑𝐷:ℕ⟶ℝ)
2423ffvelrnda 6399 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 ∈ ℕ) → (𝐷𝑗) ∈ ℝ)
252adantr 480 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 ∈ ℕ) → 𝐴 ∈ ℝ)
2624, 25ifcld 4164 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ ℕ) → if((𝐷𝑗) ≤ 𝐴, (𝐷𝑗), 𝐴) ∈ ℝ)
2726rexrd 10127 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ ℕ) → if((𝐷𝑗) ≤ 𝐴, (𝐷𝑗), 𝐴) ∈ ℝ*)
28 icombl 23378 . . . . . . . . . . . . . . 15 (((𝐶𝑗) ∈ ℝ ∧ if((𝐷𝑗) ≤ 𝐴, (𝐷𝑗), 𝐴) ∈ ℝ*) → ((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐴, (𝐷𝑗), 𝐴)) ∈ dom vol)
2922, 27, 28syl2anc 694 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ ℕ) → ((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐴, (𝐷𝑗), 𝐴)) ∈ dom vol)
3020, 29ffvelrnd 6400 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ ℕ) → (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐴, (𝐷𝑗), 𝐴))) ∈ (0[,]+∞))
3116, 18, 30sge0ge0mpt 40973 . . . . . . . . . . . 12 (𝜑 → 0 ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐴, (𝐷𝑗), 𝐴))))))
3215, 31eqbrtrd 4707 . . . . . . . . . . 11 (𝜑 → (𝐴𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐴, (𝐷𝑗), 𝐴))))))
3313, 32jca 553 . . . . . . . . . 10 (𝜑 → (𝐴 ∈ (𝐴[,]𝐵) ∧ (𝐴𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐴, (𝐷𝑗), 𝐴)))))))
34 oveq1 6697 . . . . . . . . . . . 12 (𝑧 = 𝐴 → (𝑧𝐴) = (𝐴𝐴))
35 breq2 4689 . . . . . . . . . . . . . . . . 17 (𝑧 = 𝐴 → ((𝐷𝑗) ≤ 𝑧 ↔ (𝐷𝑗) ≤ 𝐴))
36 id 22 . . . . . . . . . . . . . . . . 17 (𝑧 = 𝐴𝑧 = 𝐴)
3735, 36ifbieq2d 4144 . . . . . . . . . . . . . . . 16 (𝑧 = 𝐴 → if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧) = if((𝐷𝑗) ≤ 𝐴, (𝐷𝑗), 𝐴))
3837oveq2d 6706 . . . . . . . . . . . . . . 15 (𝑧 = 𝐴 → ((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)) = ((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐴, (𝐷𝑗), 𝐴)))
3938fveq2d 6233 . . . . . . . . . . . . . 14 (𝑧 = 𝐴 → (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))) = (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐴, (𝐷𝑗), 𝐴))))
4039mpteq2dv 4778 . . . . . . . . . . . . 13 (𝑧 = 𝐴 → (𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)))) = (𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐴, (𝐷𝑗), 𝐴)))))
4140fveq2d 6233 . . . . . . . . . . . 12 (𝑧 = 𝐴 → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))))) = (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐴, (𝐷𝑗), 𝐴))))))
4234, 41breq12d 4698 . . . . . . . . . . 11 (𝑧 = 𝐴 → ((𝑧𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))))) ↔ (𝐴𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐴, (𝐷𝑗), 𝐴)))))))
4342elrab 3396 . . . . . . . . . 10 (𝐴 ∈ {𝑧 ∈ (𝐴[,]𝐵) ∣ (𝑧𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)))))} ↔ (𝐴 ∈ (𝐴[,]𝐵) ∧ (𝐴𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝐴, (𝐷𝑗), 𝐴)))))))
4433, 43sylibr 224 . . . . . . . . 9 (𝜑𝐴 ∈ {𝑧 ∈ (𝐴[,]𝐵) ∣ (𝑧𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)))))})
4544, 4syl6eleqr 2741 . . . . . . . 8 (𝜑𝐴𝑈)
46 ne0i 3954 . . . . . . . 8 (𝐴𝑈𝑈 ≠ ∅)
4745, 46syl 17 . . . . . . 7 (𝜑𝑈 ≠ ∅)
482, 3, 7, 47supicc 12358 . . . . . 6 (𝜑 → sup(𝑈, ℝ, < ) ∈ (𝐴[,]𝐵))
491, 48syl5eqel 2734 . . . . 5 (𝜑𝑆 ∈ (𝐴[,]𝐵))
501a1i 11 . . . . . . 7 (𝜑𝑆 = sup(𝑈, ℝ, < ))
51 nfv 1883 . . . . . . . . 9 𝑧𝜑
522, 3iccssred 40045 . . . . . . . . . . . . 13 (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
537, 52sstrd 3646 . . . . . . . . . . . 12 (𝜑𝑈 ⊆ ℝ)
5453sselda 3636 . . . . . . . . . . 11 ((𝜑𝑧𝑈) → 𝑧 ∈ ℝ)
55 nfv 1883 . . . . . . . . . . . . . . . 16 𝑗(𝜑𝑧𝑈)
5617a1i 11 . . . . . . . . . . . . . . . 16 ((𝜑𝑧𝑈) → ℕ ∈ V)
5719a1i 11 . . . . . . . . . . . . . . . . 17 (((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) → vol:dom vol⟶(0[,]+∞))
5822adantlr 751 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) → (𝐶𝑗) ∈ ℝ)
5924adantlr 751 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) → (𝐷𝑗) ∈ ℝ)
6054adantr 480 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) → 𝑧 ∈ ℝ)
6159, 60ifcld 4164 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) → if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧) ∈ ℝ)
6261rexrd 10127 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) → if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧) ∈ ℝ*)
63 icombl 23378 . . . . . . . . . . . . . . . . . 18 (((𝐶𝑗) ∈ ℝ ∧ if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧) ∈ ℝ*) → ((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)) ∈ dom vol)
6458, 62, 63syl2anc 694 . . . . . . . . . . . . . . . . 17 (((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) → ((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)) ∈ dom vol)
6557, 64ffvelrnd 6400 . . . . . . . . . . . . . . . 16 (((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) → (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))) ∈ (0[,]+∞))
6655, 56, 65sge0xrclmpt 40963 . . . . . . . . . . . . . . 15 ((𝜑𝑧𝑈) → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))))) ∈ ℝ*)
67 pnfxr 10130 . . . . . . . . . . . . . . . 16 +∞ ∈ ℝ*
6867a1i 11 . . . . . . . . . . . . . . 15 ((𝜑𝑧𝑈) → +∞ ∈ ℝ*)
69 hoidmv1lelem1.r . . . . . . . . . . . . . . . . . 18 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)(𝐷𝑗))))) ∈ ℝ)
7069rexrd 10127 . . . . . . . . . . . . . . . . 17 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)(𝐷𝑗))))) ∈ ℝ*)
7170adantr 480 . . . . . . . . . . . . . . . 16 ((𝜑𝑧𝑈) → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)(𝐷𝑗))))) ∈ ℝ*)
7224rexrd 10127 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ ℕ) → (𝐷𝑗) ∈ ℝ*)
73 icombl 23378 . . . . . . . . . . . . . . . . . . . 20 (((𝐶𝑗) ∈ ℝ ∧ (𝐷𝑗) ∈ ℝ*) → ((𝐶𝑗)[,)(𝐷𝑗)) ∈ dom vol)
7422, 72, 73syl2anc 694 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 ∈ ℕ) → ((𝐶𝑗)[,)(𝐷𝑗)) ∈ dom vol)
7520, 74ffvelrnd 6400 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗 ∈ ℕ) → (vol‘((𝐶𝑗)[,)(𝐷𝑗))) ∈ (0[,]+∞))
7675adantlr 751 . . . . . . . . . . . . . . . . 17 (((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) → (vol‘((𝐶𝑗)[,)(𝐷𝑗))) ∈ (0[,]+∞))
7774adantlr 751 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) → ((𝐶𝑗)[,)(𝐷𝑗)) ∈ dom vol)
7822rexrd 10127 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ ℕ) → (𝐶𝑗) ∈ ℝ*)
7978adantlr 751 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) → (𝐶𝑗) ∈ ℝ*)
8072adantlr 751 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) → (𝐷𝑗) ∈ ℝ*)
8122leidd 10632 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ ℕ) → (𝐶𝑗) ≤ (𝐶𝑗))
8281adantlr 751 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) → (𝐶𝑗) ≤ (𝐶𝑗))
83 min1 12058 . . . . . . . . . . . . . . . . . . . 20 (((𝐷𝑗) ∈ ℝ ∧ 𝑧 ∈ ℝ) → if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧) ≤ (𝐷𝑗))
8459, 60, 83syl2anc 694 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) → if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧) ≤ (𝐷𝑗))
85 icossico 12281 . . . . . . . . . . . . . . . . . . 19 ((((𝐶𝑗) ∈ ℝ* ∧ (𝐷𝑗) ∈ ℝ*) ∧ ((𝐶𝑗) ≤ (𝐶𝑗) ∧ if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧) ≤ (𝐷𝑗))) → ((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)) ⊆ ((𝐶𝑗)[,)(𝐷𝑗)))
8679, 80, 82, 84, 85syl22anc 1367 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) → ((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)) ⊆ ((𝐶𝑗)[,)(𝐷𝑗)))
87 volss 23347 . . . . . . . . . . . . . . . . . 18 ((((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)) ∈ dom vol ∧ ((𝐶𝑗)[,)(𝐷𝑗)) ∈ dom vol ∧ ((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)) ⊆ ((𝐶𝑗)[,)(𝐷𝑗))) → (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))) ≤ (vol‘((𝐶𝑗)[,)(𝐷𝑗))))
8864, 77, 86, 87syl3anc 1366 . . . . . . . . . . . . . . . . 17 (((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) → (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))) ≤ (vol‘((𝐶𝑗)[,)(𝐷𝑗))))
8955, 56, 65, 76, 88sge0lempt 40945 . . . . . . . . . . . . . . . 16 ((𝜑𝑧𝑈) → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))))) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)(𝐷𝑗))))))
9069ltpnfd 11993 . . . . . . . . . . . . . . . . 17 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)(𝐷𝑗))))) < +∞)
9190adantr 480 . . . . . . . . . . . . . . . 16 ((𝜑𝑧𝑈) → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)(𝐷𝑗))))) < +∞)
9266, 71, 68, 89, 91xrlelttrd 12029 . . . . . . . . . . . . . . 15 ((𝜑𝑧𝑈) → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))))) < +∞)
9366, 68, 92xrltned 39886 . . . . . . . . . . . . . 14 ((𝜑𝑧𝑈) → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))))) ≠ +∞)
9493neneqd 2828 . . . . . . . . . . . . 13 ((𝜑𝑧𝑈) → ¬ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))))) = +∞)
95 eqid 2651 . . . . . . . . . . . . . . 15 (𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)))) = (𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))))
9665, 95fmptd 6425 . . . . . . . . . . . . . 14 ((𝜑𝑧𝑈) → (𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)))):ℕ⟶(0[,]+∞))
9756, 96sge0repnf 40921 . . . . . . . . . . . . 13 ((𝜑𝑧𝑈) → ((Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))))) ∈ ℝ ↔ ¬ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))))) = +∞))
9894, 97mpbird 247 . . . . . . . . . . . 12 ((𝜑𝑧𝑈) → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))))) ∈ ℝ)
992adantr 480 . . . . . . . . . . . 12 ((𝜑𝑧𝑈) → 𝐴 ∈ ℝ)
10098, 99readdcld 10107 . . . . . . . . . . 11 ((𝜑𝑧𝑈) → ((Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))))) + 𝐴) ∈ ℝ)
10152, 49sseldd 3637 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑𝑆 ∈ ℝ)
102101adantr 480 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑗 ∈ ℕ) → 𝑆 ∈ ℝ)
10324, 102ifcld 4164 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ ℕ) → if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆) ∈ ℝ)
104103rexrd 10127 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 ∈ ℕ) → if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆) ∈ ℝ*)
105 icombl 23378 . . . . . . . . . . . . . . . . . . 19 (((𝐶𝑗) ∈ ℝ ∧ if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆) ∈ ℝ*) → ((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆)) ∈ dom vol)
10622, 104, 105syl2anc 694 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗 ∈ ℕ) → ((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆)) ∈ dom vol)
10720, 106ffvelrnd 6400 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 ∈ ℕ) → (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))) ∈ (0[,]+∞))
10816, 18, 107sge0xrclmpt 40963 . . . . . . . . . . . . . . . 16 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))) ∈ ℝ*)
10967a1i 11 . . . . . . . . . . . . . . . 16 (𝜑 → +∞ ∈ ℝ*)
110 min1 12058 . . . . . . . . . . . . . . . . . . . . 21 (((𝐷𝑗) ∈ ℝ ∧ 𝑆 ∈ ℝ) → if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆) ≤ (𝐷𝑗))
11124, 102, 110syl2anc 694 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ ℕ) → if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆) ≤ (𝐷𝑗))
112 icossico 12281 . . . . . . . . . . . . . . . . . . . 20 ((((𝐶𝑗) ∈ ℝ* ∧ (𝐷𝑗) ∈ ℝ*) ∧ ((𝐶𝑗) ≤ (𝐶𝑗) ∧ if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆) ≤ (𝐷𝑗))) → ((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆)) ⊆ ((𝐶𝑗)[,)(𝐷𝑗)))
11378, 72, 81, 111, 112syl22anc 1367 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 ∈ ℕ) → ((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆)) ⊆ ((𝐶𝑗)[,)(𝐷𝑗)))
114 volss 23347 . . . . . . . . . . . . . . . . . . 19 ((((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆)) ∈ dom vol ∧ ((𝐶𝑗)[,)(𝐷𝑗)) ∈ dom vol ∧ ((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆)) ⊆ ((𝐶𝑗)[,)(𝐷𝑗))) → (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))) ≤ (vol‘((𝐶𝑗)[,)(𝐷𝑗))))
115106, 74, 113, 114syl3anc 1366 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗 ∈ ℕ) → (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))) ≤ (vol‘((𝐶𝑗)[,)(𝐷𝑗))))
11616, 18, 107, 75, 115sge0lempt 40945 . . . . . . . . . . . . . . . . 17 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)(𝐷𝑗))))))
117108, 70, 109, 116, 90xrlelttrd 12029 . . . . . . . . . . . . . . . 16 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))) < +∞)
118108, 109, 117xrltned 39886 . . . . . . . . . . . . . . 15 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))) ≠ +∞)
119118neneqd 2828 . . . . . . . . . . . . . 14 (𝜑 → ¬ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))) = +∞)
120 eqid 2651 . . . . . . . . . . . . . . . 16 (𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆)))) = (𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))
121107, 120fmptd 6425 . . . . . . . . . . . . . . 15 (𝜑 → (𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆)))):ℕ⟶(0[,]+∞))
12218, 121sge0repnf 40921 . . . . . . . . . . . . . 14 (𝜑 → ((Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))) ∈ ℝ ↔ ¬ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))) = +∞))
123119, 122mpbird 247 . . . . . . . . . . . . 13 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))) ∈ ℝ)
124123, 2readdcld 10107 . . . . . . . . . . . 12 (𝜑 → ((Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))) + 𝐴) ∈ ℝ)
125124adantr 480 . . . . . . . . . . 11 ((𝜑𝑧𝑈) → ((Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))) + 𝐴) ∈ ℝ)
1264eleq2i 2722 . . . . . . . . . . . . . . . 16 (𝑧𝑈𝑧 ∈ {𝑧 ∈ (𝐴[,]𝐵) ∣ (𝑧𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)))))})
127126biimpi 206 . . . . . . . . . . . . . . 15 (𝑧𝑈𝑧 ∈ {𝑧 ∈ (𝐴[,]𝐵) ∣ (𝑧𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)))))})
128127adantl 481 . . . . . . . . . . . . . 14 ((𝜑𝑧𝑈) → 𝑧 ∈ {𝑧 ∈ (𝐴[,]𝐵) ∣ (𝑧𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)))))})
129 rabid 3145 . . . . . . . . . . . . . 14 (𝑧 ∈ {𝑧 ∈ (𝐴[,]𝐵) ∣ (𝑧𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)))))} ↔ (𝑧 ∈ (𝐴[,]𝐵) ∧ (𝑧𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)))))))
130128, 129sylib 208 . . . . . . . . . . . . 13 ((𝜑𝑧𝑈) → (𝑧 ∈ (𝐴[,]𝐵) ∧ (𝑧𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)))))))
131130simprd 478 . . . . . . . . . . . 12 ((𝜑𝑧𝑈) → (𝑧𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))))))
13254, 99, 98lesubaddd 10662 . . . . . . . . . . . 12 ((𝜑𝑧𝑈) → ((𝑧𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))))) ↔ 𝑧 ≤ ((Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))))) + 𝐴)))
133131, 132mpbid 222 . . . . . . . . . . 11 ((𝜑𝑧𝑈) → 𝑧 ≤ ((Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))))) + 𝐴))
134123adantr 480 . . . . . . . . . . . 12 ((𝜑𝑧𝑈) → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))) ∈ ℝ)
135107adantlr 751 . . . . . . . . . . . . 13 (((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) → (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))) ∈ (0[,]+∞))
136106adantlr 751 . . . . . . . . . . . . . 14 (((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) → ((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆)) ∈ dom vol)
137104adantlr 751 . . . . . . . . . . . . . . 15 (((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) → if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆) ∈ ℝ*)
13861adantr 480 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ (𝐷𝑗) ≤ 𝑧) → if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧) ∈ ℝ)
139 eqidd 2652 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ (𝐷𝑗) ≤ 𝑧) → (𝐷𝑗) = (𝐷𝑗))
140 iftrue 4125 . . . . . . . . . . . . . . . . . . 19 ((𝐷𝑗) ≤ 𝑧 → if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧) = (𝐷𝑗))
141140adantl 481 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ (𝐷𝑗) ≤ 𝑧) → if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧) = (𝐷𝑗))
14259adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ (𝐷𝑗) ≤ 𝑧) → (𝐷𝑗) ∈ ℝ)
14360adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ (𝐷𝑗) ≤ 𝑧) → 𝑧 ∈ ℝ)
144101ad3antrrr 766 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ (𝐷𝑗) ≤ 𝑧) → 𝑆 ∈ ℝ)
145 simpr 476 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ (𝐷𝑗) ≤ 𝑧) → (𝐷𝑗) ≤ 𝑧)
14653adantr 480 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑧𝑈) → 𝑈 ⊆ ℝ)
14747adantr 480 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑧𝑈) → 𝑈 ≠ ∅)
1482, 3jca 553 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑 → (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
149 iccsupr 12304 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝑈 ⊆ (𝐴[,]𝐵) ∧ 𝐴𝑈) → (𝑈 ⊆ ℝ ∧ 𝑈 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦𝑈 𝑦𝑥))
150148, 7, 45, 149syl3anc 1366 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑 → (𝑈 ⊆ ℝ ∧ 𝑈 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦𝑈 𝑦𝑥))
151150simp3d 1095 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦𝑈 𝑦𝑥)
152151adantr 480 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑧𝑈) → ∃𝑥 ∈ ℝ ∀𝑦𝑈 𝑦𝑥)
153128, 126sylibr 224 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑧𝑈) → 𝑧𝑈)
154 suprub 11022 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑈 ⊆ ℝ ∧ 𝑈 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦𝑈 𝑦𝑥) ∧ 𝑧𝑈) → 𝑧 ≤ sup(𝑈, ℝ, < ))
155146, 147, 152, 153, 154syl31anc 1369 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑧𝑈) → 𝑧 ≤ sup(𝑈, ℝ, < ))
156155, 1syl6breqr 4727 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑧𝑈) → 𝑧𝑆)
157156ad2antrr 762 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ (𝐷𝑗) ≤ 𝑧) → 𝑧𝑆)
158142, 143, 144, 145, 157letrd 10232 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ (𝐷𝑗) ≤ 𝑧) → (𝐷𝑗) ≤ 𝑆)
159158iftrued 4127 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ (𝐷𝑗) ≤ 𝑧) → if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆) = (𝐷𝑗))
160139, 141, 1593eqtr4d 2695 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ (𝐷𝑗) ≤ 𝑧) → if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧) = if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))
161138, 160eqled 10178 . . . . . . . . . . . . . . . 16 ((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ (𝐷𝑗) ≤ 𝑧) → if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧) ≤ if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))
16260adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷𝑗) ≤ 𝑧) → 𝑧 ∈ ℝ)
16359adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷𝑗) ≤ 𝑧) → (𝐷𝑗) ∈ ℝ)
164 simpr 476 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷𝑗) ≤ 𝑧) → ¬ (𝐷𝑗) ≤ 𝑧)
165162, 163ltnled 10222 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷𝑗) ≤ 𝑧) → (𝑧 < (𝐷𝑗) ↔ ¬ (𝐷𝑗) ≤ 𝑧))
166164, 165mpbird 247 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷𝑗) ≤ 𝑧) → 𝑧 < (𝐷𝑗))
167162, 163, 166ltled 10223 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷𝑗) ≤ 𝑧) → 𝑧 ≤ (𝐷𝑗))
168167adantr 480 . . . . . . . . . . . . . . . . . 18 (((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷𝑗) ≤ 𝑧) ∧ (𝐷𝑗) ≤ 𝑆) → 𝑧 ≤ (𝐷𝑗))
169 iffalse 4128 . . . . . . . . . . . . . . . . . . . 20 (¬ (𝐷𝑗) ≤ 𝑧 → if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧) = 𝑧)
170169ad2antlr 763 . . . . . . . . . . . . . . . . . . 19 (((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷𝑗) ≤ 𝑧) ∧ (𝐷𝑗) ≤ 𝑆) → if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧) = 𝑧)
171 iftrue 4125 . . . . . . . . . . . . . . . . . . . 20 ((𝐷𝑗) ≤ 𝑆 → if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆) = (𝐷𝑗))
172171adantl 481 . . . . . . . . . . . . . . . . . . 19 (((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷𝑗) ≤ 𝑧) ∧ (𝐷𝑗) ≤ 𝑆) → if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆) = (𝐷𝑗))
173170, 172breq12d 4698 . . . . . . . . . . . . . . . . . 18 (((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷𝑗) ≤ 𝑧) ∧ (𝐷𝑗) ≤ 𝑆) → (if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧) ≤ if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆) ↔ 𝑧 ≤ (𝐷𝑗)))
174168, 173mpbird 247 . . . . . . . . . . . . . . . . 17 (((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷𝑗) ≤ 𝑧) ∧ (𝐷𝑗) ≤ 𝑆) → if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧) ≤ if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))
175156ad3antrrr 766 . . . . . . . . . . . . . . . . . 18 (((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷𝑗) ≤ 𝑧) ∧ ¬ (𝐷𝑗) ≤ 𝑆) → 𝑧𝑆)
176169ad2antlr 763 . . . . . . . . . . . . . . . . . . 19 (((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷𝑗) ≤ 𝑧) ∧ ¬ (𝐷𝑗) ≤ 𝑆) → if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧) = 𝑧)
177 iffalse 4128 . . . . . . . . . . . . . . . . . . . 20 (¬ (𝐷𝑗) ≤ 𝑆 → if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆) = 𝑆)
178177adantl 481 . . . . . . . . . . . . . . . . . . 19 (((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷𝑗) ≤ 𝑧) ∧ ¬ (𝐷𝑗) ≤ 𝑆) → if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆) = 𝑆)
179176, 178breq12d 4698 . . . . . . . . . . . . . . . . . 18 (((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷𝑗) ≤ 𝑧) ∧ ¬ (𝐷𝑗) ≤ 𝑆) → (if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧) ≤ if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆) ↔ 𝑧𝑆))
180175, 179mpbird 247 . . . . . . . . . . . . . . . . 17 (((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷𝑗) ≤ 𝑧) ∧ ¬ (𝐷𝑗) ≤ 𝑆) → if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧) ≤ if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))
181174, 180pm2.61dan 849 . . . . . . . . . . . . . . . 16 ((((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷𝑗) ≤ 𝑧) → if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧) ≤ if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))
182161, 181pm2.61dan 849 . . . . . . . . . . . . . . 15 (((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) → if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧) ≤ if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))
183 icossico 12281 . . . . . . . . . . . . . . 15 ((((𝐶𝑗) ∈ ℝ* ∧ if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆) ∈ ℝ*) ∧ ((𝐶𝑗) ≤ (𝐶𝑗) ∧ if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧) ≤ if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))) → ((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)) ⊆ ((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆)))
18479, 137, 82, 182, 183syl22anc 1367 . . . . . . . . . . . . . 14 (((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) → ((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)) ⊆ ((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆)))
185 volss 23347 . . . . . . . . . . . . . 14 ((((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)) ∈ dom vol ∧ ((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆)) ∈ dom vol ∧ ((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)) ⊆ ((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))) → (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))) ≤ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))
18664, 136, 184, 185syl3anc 1366 . . . . . . . . . . . . 13 (((𝜑𝑧𝑈) ∧ 𝑗 ∈ ℕ) → (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))) ≤ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))
18755, 56, 65, 135, 186sge0lempt 40945 . . . . . . . . . . . 12 ((𝜑𝑧𝑈) → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))))) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))))
18898, 134, 99, 187leadd1dd 10679 . . . . . . . . . . 11 ((𝜑𝑧𝑈) → ((Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))))) + 𝐴) ≤ ((Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))) + 𝐴))
18954, 100, 125, 133, 188letrd 10232 . . . . . . . . . 10 ((𝜑𝑧𝑈) → 𝑧 ≤ ((Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))) + 𝐴))
190189ex 449 . . . . . . . . 9 (𝜑 → (𝑧𝑈𝑧 ≤ ((Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))) + 𝐴)))
19151, 190ralrimi 2986 . . . . . . . 8 (𝜑 → ∀𝑧𝑈 𝑧 ≤ ((Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))) + 𝐴))
192 suprleub 11027 . . . . . . . . 9 (((𝑈 ⊆ ℝ ∧ 𝑈 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦𝑈 𝑦𝑥) ∧ ((Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))) + 𝐴) ∈ ℝ) → (sup(𝑈, ℝ, < ) ≤ ((Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))) + 𝐴) ↔ ∀𝑧𝑈 𝑧 ≤ ((Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))) + 𝐴)))
19353, 47, 151, 124, 192syl31anc 1369 . . . . . . . 8 (𝜑 → (sup(𝑈, ℝ, < ) ≤ ((Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))) + 𝐴) ↔ ∀𝑧𝑈 𝑧 ≤ ((Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))) + 𝐴)))
194191, 193mpbird 247 . . . . . . 7 (𝜑 → sup(𝑈, ℝ, < ) ≤ ((Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))) + 𝐴))
19550, 194eqbrtrd 4707 . . . . . 6 (𝜑𝑆 ≤ ((Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))) + 𝐴))
196101, 2, 123lesubaddd 10662 . . . . . 6 (𝜑 → ((𝑆𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))) ↔ 𝑆 ≤ ((Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))) + 𝐴)))
197195, 196mpbird 247 . . . . 5 (𝜑 → (𝑆𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))))
19849, 197jca 553 . . . 4 (𝜑 → (𝑆 ∈ (𝐴[,]𝐵) ∧ (𝑆𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆)))))))
199 oveq1 6697 . . . . . 6 (𝑧 = 𝑆 → (𝑧𝐴) = (𝑆𝐴))
200 breq2 4689 . . . . . . . . . . 11 (𝑧 = 𝑆 → ((𝐷𝑗) ≤ 𝑧 ↔ (𝐷𝑗) ≤ 𝑆))
201 id 22 . . . . . . . . . . 11 (𝑧 = 𝑆𝑧 = 𝑆)
202200, 201ifbieq2d 4144 . . . . . . . . . 10 (𝑧 = 𝑆 → if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧) = if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))
203202oveq2d 6706 . . . . . . . . 9 (𝑧 = 𝑆 → ((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)) = ((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆)))
204203fveq2d 6233 . . . . . . . 8 (𝑧 = 𝑆 → (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))) = (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))
205204mpteq2dv 4778 . . . . . . 7 (𝑧 = 𝑆 → (𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)))) = (𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆)))))
206205fveq2d 6233 . . . . . 6 (𝑧 = 𝑆 → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))))) = (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆))))))
207199, 206breq12d 4698 . . . . 5 (𝑧 = 𝑆 → ((𝑧𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧))))) ↔ (𝑆𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆)))))))
208207elrab 3396 . . . 4 (𝑆 ∈ {𝑧 ∈ (𝐴[,]𝐵) ∣ (𝑧𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)))))} ↔ (𝑆 ∈ (𝐴[,]𝐵) ∧ (𝑆𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑆, (𝐷𝑗), 𝑆)))))))
209198, 208sylibr 224 . . 3 (𝜑𝑆 ∈ {𝑧 ∈ (𝐴[,]𝐵) ∣ (𝑧𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶𝑗)[,)if((𝐷𝑗) ≤ 𝑧, (𝐷𝑗), 𝑧)))))})
210209, 4syl6eleqr 2741 . 2 (𝜑𝑆𝑈)
211210, 45, 1513jca 1261 1 (𝜑 → (𝑆𝑈𝐴𝑈 ∧ ∃𝑥 ∈ ℝ ∀𝑦𝑈 𝑦𝑥))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wne 2823  wral 2941  wrex 2942  {crab 2945  Vcvv 3231  wss 3607  c0 3948  ifcif 4119   class class class wbr 4685  cmpt 4762  dom cdm 5143  wf 5922  cfv 5926  (class class class)co 6690  supcsup 8387  cr 9973  0cc0 9974   + caddc 9977  +∞cpnf 10109  *cxr 10111   < clt 10112  cle 10113  cmin 10304  cn 11058  [,)cico 12215  [,]cicc 12216  volcvol 23278  Σ^csumge0 40897
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-of 6939  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-map 7901  df-pm 7902  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-sup 8389  df-inf 8390  df-oi 8456  df-card 8803  df-cda 9028  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-n0 11331  df-z 11416  df-uz 11726  df-q 11827  df-rp 11871  df-xadd 11985  df-ioo 12217  df-ico 12219  df-icc 12220  df-fz 12365  df-fzo 12505  df-fl 12633  df-seq 12842  df-exp 12901  df-hash 13158  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-clim 14263  df-rlim 14264  df-sum 14461  df-xmet 19787  df-met 19788  df-ovol 23279  df-vol 23280  df-sumge0 40898
This theorem is referenced by:  hoidmv1lelem3  41128
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