Proof of Theorem hoidmv1lelem1
| Step | Hyp | Ref
| Expression |
| 1 | | hoidmv1lelem1.s |
. . . . . 6
⊢ 𝑆 = sup(𝑈, ℝ, < ) |
| 2 | | hoidmv1lelem1.a |
. . . . . . 7
⊢ (𝜑 → 𝐴 ∈ ℝ) |
| 3 | | hoidmv1lelem1.b |
. . . . . . 7
⊢ (𝜑 → 𝐵 ∈ ℝ) |
| 4 | | hoidmv1lelem1.u |
. . . . . . . . 9
⊢ 𝑈 = {𝑧 ∈ (𝐴[,]𝐵) ∣ (𝑧 − 𝐴) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧)))))} |
| 5 | | ssrab2 4080 |
. . . . . . . . 9
⊢ {𝑧 ∈ (𝐴[,]𝐵) ∣ (𝑧 − 𝐴) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧)))))} ⊆ (𝐴[,]𝐵) |
| 6 | 4, 5 | eqsstri 4030 |
. . . . . . . 8
⊢ 𝑈 ⊆ (𝐴[,]𝐵) |
| 7 | 6 | a1i 11 |
. . . . . . 7
⊢ (𝜑 → 𝑈 ⊆ (𝐴[,]𝐵)) |
| 8 | 2 | rexrd 11311 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐴 ∈
ℝ*) |
| 9 | 3 | rexrd 11311 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐵 ∈
ℝ*) |
| 10 | | hoidmv1lelem1.l |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝐴 < 𝐵) |
| 11 | 2, 3, 10 | ltled 11409 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐴 ≤ 𝐵) |
| 12 | | lbicc2 13504 |
. . . . . . . . . . . 12
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ* ∧ 𝐴
≤ 𝐵) → 𝐴 ∈ (𝐴[,]𝐵)) |
| 13 | 8, 9, 11, 12 | syl3anc 1373 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐴 ∈ (𝐴[,]𝐵)) |
| 14 | 2 | recnd 11289 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝐴 ∈ ℂ) |
| 15 | 14 | subidd 11608 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝐴 − 𝐴) = 0) |
| 16 | | nfv 1914 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑗𝜑 |
| 17 | | nnex 12272 |
. . . . . . . . . . . . . 14
⊢ ℕ
∈ V |
| 18 | 17 | a1i 11 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ℕ ∈
V) |
| 19 | | volf 25564 |
. . . . . . . . . . . . . . 15
⊢ vol:dom
vol⟶(0[,]+∞) |
| 20 | 19 | a1i 11 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → vol:dom
vol⟶(0[,]+∞)) |
| 21 | | hoidmv1lelem1.c |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 𝐶:ℕ⟶ℝ) |
| 22 | 21 | ffvelcdmda 7104 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐶‘𝑗) ∈ ℝ) |
| 23 | | hoidmv1lelem1.d |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → 𝐷:ℕ⟶ℝ) |
| 24 | 23 | ffvelcdmda 7104 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐷‘𝑗) ∈ ℝ) |
| 25 | 2 | adantr 480 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝐴 ∈ ℝ) |
| 26 | 24, 25 | ifcld 4572 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → if((𝐷‘𝑗) ≤ 𝐴, (𝐷‘𝑗), 𝐴) ∈ ℝ) |
| 27 | 26 | rexrd 11311 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → if((𝐷‘𝑗) ≤ 𝐴, (𝐷‘𝑗), 𝐴) ∈
ℝ*) |
| 28 | | icombl 25599 |
. . . . . . . . . . . . . . 15
⊢ (((𝐶‘𝑗) ∈ ℝ ∧ if((𝐷‘𝑗) ≤ 𝐴, (𝐷‘𝑗), 𝐴) ∈ ℝ*) → ((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐴, (𝐷‘𝑗), 𝐴)) ∈ dom vol) |
| 29 | 22, 27, 28 | syl2anc 584 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐴, (𝐷‘𝑗), 𝐴)) ∈ dom vol) |
| 30 | 20, 29 | ffvelcdmd 7105 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐴, (𝐷‘𝑗), 𝐴))) ∈ (0[,]+∞)) |
| 31 | 16, 18, 30 | sge0ge0mpt 46453 |
. . . . . . . . . . . 12
⊢ (𝜑 → 0 ≤
(Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐴, (𝐷‘𝑗), 𝐴)))))) |
| 32 | 15, 31 | eqbrtrd 5165 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝐴 − 𝐴) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐴, (𝐷‘𝑗), 𝐴)))))) |
| 33 | 13, 32 | jca 511 |
. . . . . . . . . 10
⊢ (𝜑 → (𝐴 ∈ (𝐴[,]𝐵) ∧ (𝐴 − 𝐴) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐴, (𝐷‘𝑗), 𝐴))))))) |
| 34 | | oveq1 7438 |
. . . . . . . . . . . 12
⊢ (𝑧 = 𝐴 → (𝑧 − 𝐴) = (𝐴 − 𝐴)) |
| 35 | | breq2 5147 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑧 = 𝐴 → ((𝐷‘𝑗) ≤ 𝑧 ↔ (𝐷‘𝑗) ≤ 𝐴)) |
| 36 | | id 22 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑧 = 𝐴 → 𝑧 = 𝐴) |
| 37 | 35, 36 | ifbieq2d 4552 |
. . . . . . . . . . . . . . . 16
⊢ (𝑧 = 𝐴 → if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧) = if((𝐷‘𝑗) ≤ 𝐴, (𝐷‘𝑗), 𝐴)) |
| 38 | 37 | oveq2d 7447 |
. . . . . . . . . . . . . . 15
⊢ (𝑧 = 𝐴 → ((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧)) = ((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐴, (𝐷‘𝑗), 𝐴))) |
| 39 | 38 | fveq2d 6910 |
. . . . . . . . . . . . . 14
⊢ (𝑧 = 𝐴 → (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧))) = (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐴, (𝐷‘𝑗), 𝐴)))) |
| 40 | 39 | mpteq2dv 5244 |
. . . . . . . . . . . . 13
⊢ (𝑧 = 𝐴 → (𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧)))) = (𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐴, (𝐷‘𝑗), 𝐴))))) |
| 41 | 40 | fveq2d 6910 |
. . . . . . . . . . . 12
⊢ (𝑧 = 𝐴 →
(Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧))))) =
(Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐴, (𝐷‘𝑗), 𝐴)))))) |
| 42 | 34, 41 | breq12d 5156 |
. . . . . . . . . . 11
⊢ (𝑧 = 𝐴 → ((𝑧 − 𝐴) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧))))) ↔ (𝐴 − 𝐴) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐴, (𝐷‘𝑗), 𝐴))))))) |
| 43 | 42 | elrab 3692 |
. . . . . . . . . 10
⊢ (𝐴 ∈ {𝑧 ∈ (𝐴[,]𝐵) ∣ (𝑧 − 𝐴) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧)))))} ↔ (𝐴 ∈ (𝐴[,]𝐵) ∧ (𝐴 − 𝐴) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝐴, (𝐷‘𝑗), 𝐴))))))) |
| 44 | 33, 43 | sylibr 234 |
. . . . . . . . 9
⊢ (𝜑 → 𝐴 ∈ {𝑧 ∈ (𝐴[,]𝐵) ∣ (𝑧 − 𝐴) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧)))))}) |
| 45 | 44, 4 | eleqtrrdi 2852 |
. . . . . . . 8
⊢ (𝜑 → 𝐴 ∈ 𝑈) |
| 46 | 45 | ne0d 4342 |
. . . . . . 7
⊢ (𝜑 → 𝑈 ≠ ∅) |
| 47 | 2, 3, 7, 46 | supicc 13541 |
. . . . . 6
⊢ (𝜑 → sup(𝑈, ℝ, < ) ∈ (𝐴[,]𝐵)) |
| 48 | 1, 47 | eqeltrid 2845 |
. . . . 5
⊢ (𝜑 → 𝑆 ∈ (𝐴[,]𝐵)) |
| 49 | 1 | a1i 11 |
. . . . . . 7
⊢ (𝜑 → 𝑆 = sup(𝑈, ℝ, < )) |
| 50 | | nfv 1914 |
. . . . . . . . 9
⊢
Ⅎ𝑧𝜑 |
| 51 | 2, 3 | iccssred 13474 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ) |
| 52 | 7, 51 | sstrd 3994 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝑈 ⊆ ℝ) |
| 53 | 52 | sselda 3983 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → 𝑧 ∈ ℝ) |
| 54 | | nfv 1914 |
. . . . . . . . . . . . . . . 16
⊢
Ⅎ𝑗(𝜑 ∧ 𝑧 ∈ 𝑈) |
| 55 | 17 | a1i 11 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → ℕ ∈ V) |
| 56 | 19 | a1i 11 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) → vol:dom
vol⟶(0[,]+∞)) |
| 57 | 22 | adantlr 715 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) → (𝐶‘𝑗) ∈ ℝ) |
| 58 | 24 | adantlr 715 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) → (𝐷‘𝑗) ∈ ℝ) |
| 59 | 53 | adantr 480 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) → 𝑧 ∈ ℝ) |
| 60 | 58, 59 | ifcld 4572 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) → if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧) ∈ ℝ) |
| 61 | 60 | rexrd 11311 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) → if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧) ∈
ℝ*) |
| 62 | | icombl 25599 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝐶‘𝑗) ∈ ℝ ∧ if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧) ∈ ℝ*) → ((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧)) ∈ dom vol) |
| 63 | 57, 61, 62 | syl2anc 584 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧)) ∈ dom vol) |
| 64 | 56, 63 | ffvelcdmd 7105 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) → (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧))) ∈ (0[,]+∞)) |
| 65 | 54, 55, 64 | sge0xrclmpt 46443 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) →
(Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧))))) ∈
ℝ*) |
| 66 | | pnfxr 11315 |
. . . . . . . . . . . . . . . 16
⊢ +∞
∈ ℝ* |
| 67 | 66 | a1i 11 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → +∞ ∈
ℝ*) |
| 68 | | hoidmv1lelem1.r |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 →
(Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)(𝐷‘𝑗))))) ∈ ℝ) |
| 69 | 68 | rexrd 11311 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 →
(Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)(𝐷‘𝑗))))) ∈
ℝ*) |
| 70 | 69 | adantr 480 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) →
(Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)(𝐷‘𝑗))))) ∈
ℝ*) |
| 71 | 24 | rexrd 11311 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐷‘𝑗) ∈
ℝ*) |
| 72 | | icombl 25599 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝐶‘𝑗) ∈ ℝ ∧ (𝐷‘𝑗) ∈ ℝ*) → ((𝐶‘𝑗)[,)(𝐷‘𝑗)) ∈ dom vol) |
| 73 | 22, 71, 72 | syl2anc 584 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)[,)(𝐷‘𝑗)) ∈ dom vol) |
| 74 | 20, 73 | ffvelcdmd 7105 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (vol‘((𝐶‘𝑗)[,)(𝐷‘𝑗))) ∈ (0[,]+∞)) |
| 75 | 74 | adantlr 715 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) → (vol‘((𝐶‘𝑗)[,)(𝐷‘𝑗))) ∈ (0[,]+∞)) |
| 76 | 73 | adantlr 715 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)[,)(𝐷‘𝑗)) ∈ dom vol) |
| 77 | 22 | rexrd 11311 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐶‘𝑗) ∈
ℝ*) |
| 78 | 77 | adantlr 715 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) → (𝐶‘𝑗) ∈
ℝ*) |
| 79 | 71 | adantlr 715 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) → (𝐷‘𝑗) ∈
ℝ*) |
| 80 | 22 | leidd 11829 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐶‘𝑗) ≤ (𝐶‘𝑗)) |
| 81 | 80 | adantlr 715 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) → (𝐶‘𝑗) ≤ (𝐶‘𝑗)) |
| 82 | | min1 13231 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝐷‘𝑗) ∈ ℝ ∧ 𝑧 ∈ ℝ) → if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧) ≤ (𝐷‘𝑗)) |
| 83 | 58, 59, 82 | syl2anc 584 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) → if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧) ≤ (𝐷‘𝑗)) |
| 84 | | icossico 13457 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝐶‘𝑗) ∈ ℝ* ∧ (𝐷‘𝑗) ∈ ℝ*) ∧ ((𝐶‘𝑗) ≤ (𝐶‘𝑗) ∧ if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧) ≤ (𝐷‘𝑗))) → ((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧)) ⊆ ((𝐶‘𝑗)[,)(𝐷‘𝑗))) |
| 85 | 78, 79, 81, 83, 84 | syl22anc 839 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧)) ⊆ ((𝐶‘𝑗)[,)(𝐷‘𝑗))) |
| 86 | | volss 25568 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧)) ∈ dom vol ∧ ((𝐶‘𝑗)[,)(𝐷‘𝑗)) ∈ dom vol ∧ ((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧)) ⊆ ((𝐶‘𝑗)[,)(𝐷‘𝑗))) → (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧))) ≤ (vol‘((𝐶‘𝑗)[,)(𝐷‘𝑗)))) |
| 87 | 63, 76, 85, 86 | syl3anc 1373 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) → (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧))) ≤ (vol‘((𝐶‘𝑗)[,)(𝐷‘𝑗)))) |
| 88 | 54, 55, 64, 75, 87 | sge0lempt 46425 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) →
(Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧))))) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)(𝐷‘𝑗)))))) |
| 89 | 68 | ltpnfd 13163 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 →
(Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)(𝐷‘𝑗))))) < +∞) |
| 90 | 89 | adantr 480 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) →
(Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)(𝐷‘𝑗))))) < +∞) |
| 91 | 65, 70, 67, 88, 90 | xrlelttrd 13202 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) →
(Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧))))) < +∞) |
| 92 | 65, 67, 91 | xrltned 45368 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) →
(Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧))))) ≠ +∞) |
| 93 | 92 | neneqd 2945 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → ¬
(Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧))))) = +∞) |
| 94 | | eqid 2737 |
. . . . . . . . . . . . . . 15
⊢ (𝑗 ∈ ℕ ↦
(vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧)))) = (𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧)))) |
| 95 | 64, 94 | fmptd 7134 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → (𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧)))):ℕ⟶(0[,]+∞)) |
| 96 | 55, 95 | sge0repnf 46401 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) →
((Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧))))) ∈ ℝ ↔ ¬
(Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧))))) = +∞)) |
| 97 | 93, 96 | mpbird 257 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) →
(Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧))))) ∈ ℝ) |
| 98 | 2 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → 𝐴 ∈ ℝ) |
| 99 | 97, 98 | readdcld 11290 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) →
((Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧))))) + 𝐴) ∈ ℝ) |
| 100 | 51, 48 | sseldd 3984 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → 𝑆 ∈ ℝ) |
| 101 | 100 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑆 ∈ ℝ) |
| 102 | 24, 101 | ifcld 4572 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆) ∈ ℝ) |
| 103 | 102 | rexrd 11311 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆) ∈
ℝ*) |
| 104 | | icombl 25599 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝐶‘𝑗) ∈ ℝ ∧ if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆) ∈ ℝ*) → ((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆)) ∈ dom vol) |
| 105 | 22, 103, 104 | syl2anc 584 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆)) ∈ dom vol) |
| 106 | 20, 105 | ffvelcdmd 7105 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆))) ∈ (0[,]+∞)) |
| 107 | 16, 18, 106 | sge0xrclmpt 46443 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 →
(Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆))))) ∈
ℝ*) |
| 108 | 66 | a1i 11 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → +∞ ∈
ℝ*) |
| 109 | | min1 13231 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝐷‘𝑗) ∈ ℝ ∧ 𝑆 ∈ ℝ) → if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆) ≤ (𝐷‘𝑗)) |
| 110 | 24, 101, 109 | syl2anc 584 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆) ≤ (𝐷‘𝑗)) |
| 111 | | icossico 13457 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((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))) |
| 112 | 77, 71, 80, 110, 111 | syl22anc 839 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆)) ⊆ ((𝐶‘𝑗)[,)(𝐷‘𝑗))) |
| 113 | | volss 25568 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((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)))) |
| 114 | 105, 73, 112, 113 | syl3anc 1373 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆))) ≤ (vol‘((𝐶‘𝑗)[,)(𝐷‘𝑗)))) |
| 115 | 16, 18, 106, 74, 114 | sge0lempt 46425 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 →
(Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆))))) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)(𝐷‘𝑗)))))) |
| 116 | 107, 69, 108, 115, 89 | xrlelttrd 13202 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 →
(Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆))))) < +∞) |
| 117 | 107, 108,
116 | xrltned 45368 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 →
(Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆))))) ≠ +∞) |
| 118 | 117 | neneqd 2945 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ¬
(Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆))))) = +∞) |
| 119 | | eqid 2737 |
. . . . . . . . . . . . . . . 16
⊢ (𝑗 ∈ ℕ ↦
(vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆)))) = (𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆)))) |
| 120 | 106, 119 | fmptd 7134 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆)))):ℕ⟶(0[,]+∞)) |
| 121 | 18, 120 | sge0repnf 46401 |
. . . . . . . . . . . . . 14
⊢ (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))))) = +∞)) |
| 122 | 118, 121 | mpbird 257 |
. . . . . . . . . . . . 13
⊢ (𝜑 →
(Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆))))) ∈ ℝ) |
| 123 | 122, 2 | readdcld 11290 |
. . . . . . . . . . . 12
⊢ (𝜑 →
((Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆))))) + 𝐴) ∈ ℝ) |
| 124 | 123 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) →
((Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆))))) + 𝐴) ∈ ℝ) |
| 125 | 4 | eleq2i 2833 |
. . . . . . . . . . . . . . . 16
⊢ (𝑧 ∈ 𝑈 ↔ 𝑧 ∈ {𝑧 ∈ (𝐴[,]𝐵) ∣ (𝑧 − 𝐴) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧)))))}) |
| 126 | 125 | biimpi 216 |
. . . . . . . . . . . . . . 15
⊢ (𝑧 ∈ 𝑈 → 𝑧 ∈ {𝑧 ∈ (𝐴[,]𝐵) ∣ (𝑧 − 𝐴) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧)))))}) |
| 127 | 126 | adantl 481 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → 𝑧 ∈ {𝑧 ∈ (𝐴[,]𝐵) ∣ (𝑧 − 𝐴) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧)))))}) |
| 128 | | rabid 3458 |
. . . . . . . . . . . . . 14
⊢ (𝑧 ∈ {𝑧 ∈ (𝐴[,]𝐵) ∣ (𝑧 − 𝐴) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧)))))} ↔ (𝑧 ∈ (𝐴[,]𝐵) ∧ (𝑧 − 𝐴) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧))))))) |
| 129 | 127, 128 | sylib 218 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → (𝑧 ∈ (𝐴[,]𝐵) ∧ (𝑧 − 𝐴) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧))))))) |
| 130 | 129 | simprd 495 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → (𝑧 − 𝐴) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧)))))) |
| 131 | 53, 98, 97 | lesubaddd 11860 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → ((𝑧 − 𝐴) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧))))) ↔ 𝑧 ≤
((Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧))))) + 𝐴))) |
| 132 | 130, 131 | mpbid 232 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → 𝑧 ≤
((Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧))))) + 𝐴)) |
| 133 | 122 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) →
(Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆))))) ∈ ℝ) |
| 134 | 106 | adantlr 715 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) → (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆))) ∈ (0[,]+∞)) |
| 135 | 105 | adantlr 715 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆)) ∈ dom vol) |
| 136 | 103 | adantlr 715 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) → if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆) ∈
ℝ*) |
| 137 | 60 | adantr 480 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) ∧ (𝐷‘𝑗) ≤ 𝑧) → if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧) ∈ ℝ) |
| 138 | | eqidd 2738 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) ∧ (𝐷‘𝑗) ≤ 𝑧) → (𝐷‘𝑗) = (𝐷‘𝑗)) |
| 139 | | iftrue 4531 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝐷‘𝑗) ≤ 𝑧 → if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧) = (𝐷‘𝑗)) |
| 140 | 139 | adantl 481 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) ∧ (𝐷‘𝑗) ≤ 𝑧) → if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧) = (𝐷‘𝑗)) |
| 141 | 58 | adantr 480 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) ∧ (𝐷‘𝑗) ≤ 𝑧) → (𝐷‘𝑗) ∈ ℝ) |
| 142 | 59 | adantr 480 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) ∧ (𝐷‘𝑗) ≤ 𝑧) → 𝑧 ∈ ℝ) |
| 143 | 100 | ad3antrrr 730 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) ∧ (𝐷‘𝑗) ≤ 𝑧) → 𝑆 ∈ ℝ) |
| 144 | | simpr 484 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) ∧ (𝐷‘𝑗) ≤ 𝑧) → (𝐷‘𝑗) ≤ 𝑧) |
| 145 | 52 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → 𝑈 ⊆ ℝ) |
| 146 | 46 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → 𝑈 ≠ ∅) |
| 147 | 2, 3 | jca 511 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝜑 → (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ)) |
| 148 | | iccsupr 13482 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝑈 ⊆ (𝐴[,]𝐵) ∧ 𝐴 ∈ 𝑈) → (𝑈 ⊆ ℝ ∧ 𝑈 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝑈 𝑦 ≤ 𝑥)) |
| 149 | 147, 7, 45, 148 | syl3anc 1373 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝜑 → (𝑈 ⊆ ℝ ∧ 𝑈 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝑈 𝑦 ≤ 𝑥)) |
| 150 | 149 | simp3d 1145 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝑈 𝑦 ≤ 𝑥) |
| 151 | 150 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝑈 𝑦 ≤ 𝑥) |
| 152 | 127, 125 | sylibr 234 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → 𝑧 ∈ 𝑈) |
| 153 | | suprub 12229 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝑈 ⊆ ℝ ∧ 𝑈 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝑈 𝑦 ≤ 𝑥) ∧ 𝑧 ∈ 𝑈) → 𝑧 ≤ sup(𝑈, ℝ, < )) |
| 154 | 145, 146,
151, 152, 153 | syl31anc 1375 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → 𝑧 ≤ sup(𝑈, ℝ, < )) |
| 155 | 154, 1 | breqtrrdi 5185 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → 𝑧 ≤ 𝑆) |
| 156 | 155 | ad2antrr 726 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) ∧ (𝐷‘𝑗) ≤ 𝑧) → 𝑧 ≤ 𝑆) |
| 157 | 141, 142,
143, 144, 156 | letrd 11418 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) ∧ (𝐷‘𝑗) ≤ 𝑧) → (𝐷‘𝑗) ≤ 𝑆) |
| 158 | 157 | iftrued 4533 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) ∧ (𝐷‘𝑗) ≤ 𝑧) → if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆) = (𝐷‘𝑗)) |
| 159 | 138, 140,
158 | 3eqtr4d 2787 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) ∧ (𝐷‘𝑗) ≤ 𝑧) → if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧) = if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆)) |
| 160 | 137, 159 | eqled 11364 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) ∧ (𝐷‘𝑗) ≤ 𝑧) → if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧) ≤ if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆)) |
| 161 | 59 | adantr 480 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷‘𝑗) ≤ 𝑧) → 𝑧 ∈ ℝ) |
| 162 | 58 | adantr 480 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷‘𝑗) ≤ 𝑧) → (𝐷‘𝑗) ∈ ℝ) |
| 163 | | simpr 484 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷‘𝑗) ≤ 𝑧) → ¬ (𝐷‘𝑗) ≤ 𝑧) |
| 164 | 161, 162 | ltnled 11408 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷‘𝑗) ≤ 𝑧) → (𝑧 < (𝐷‘𝑗) ↔ ¬ (𝐷‘𝑗) ≤ 𝑧)) |
| 165 | 163, 164 | mpbird 257 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷‘𝑗) ≤ 𝑧) → 𝑧 < (𝐷‘𝑗)) |
| 166 | 161, 162,
165 | ltled 11409 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷‘𝑗) ≤ 𝑧) → 𝑧 ≤ (𝐷‘𝑗)) |
| 167 | 166 | adantr 480 |
. . . . . . . . . . . . . . . . . 18
⊢
(((((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷‘𝑗) ≤ 𝑧) ∧ (𝐷‘𝑗) ≤ 𝑆) → 𝑧 ≤ (𝐷‘𝑗)) |
| 168 | | iffalse 4534 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (¬
(𝐷‘𝑗) ≤ 𝑧 → if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧) = 𝑧) |
| 169 | 168 | ad2antlr 727 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷‘𝑗) ≤ 𝑧) ∧ (𝐷‘𝑗) ≤ 𝑆) → if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧) = 𝑧) |
| 170 | | iftrue 4531 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝐷‘𝑗) ≤ 𝑆 → if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆) = (𝐷‘𝑗)) |
| 171 | 170 | adantl 481 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷‘𝑗) ≤ 𝑧) ∧ (𝐷‘𝑗) ≤ 𝑆) → if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆) = (𝐷‘𝑗)) |
| 172 | 169, 171 | breq12d 5156 |
. . . . . . . . . . . . . . . . . 18
⊢
(((((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷‘𝑗) ≤ 𝑧) ∧ (𝐷‘𝑗) ≤ 𝑆) → (if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧) ≤ if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆) ↔ 𝑧 ≤ (𝐷‘𝑗))) |
| 173 | 167, 172 | mpbird 257 |
. . . . . . . . . . . . . . . . 17
⊢
(((((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷‘𝑗) ≤ 𝑧) ∧ (𝐷‘𝑗) ≤ 𝑆) → if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧) ≤ if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆)) |
| 174 | 155 | ad3antrrr 730 |
. . . . . . . . . . . . . . . . . 18
⊢
(((((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷‘𝑗) ≤ 𝑧) ∧ ¬ (𝐷‘𝑗) ≤ 𝑆) → 𝑧 ≤ 𝑆) |
| 175 | 168 | ad2antlr 727 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷‘𝑗) ≤ 𝑧) ∧ ¬ (𝐷‘𝑗) ≤ 𝑆) → if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧) = 𝑧) |
| 176 | | iffalse 4534 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (¬
(𝐷‘𝑗) ≤ 𝑆 → if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆) = 𝑆) |
| 177 | 176 | adantl 481 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷‘𝑗) ≤ 𝑧) ∧ ¬ (𝐷‘𝑗) ≤ 𝑆) → if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆) = 𝑆) |
| 178 | 175, 177 | breq12d 5156 |
. . . . . . . . . . . . . . . . . 18
⊢
(((((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷‘𝑗) ≤ 𝑧) ∧ ¬ (𝐷‘𝑗) ≤ 𝑆) → (if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧) ≤ if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆) ↔ 𝑧 ≤ 𝑆)) |
| 179 | 174, 178 | mpbird 257 |
. . . . . . . . . . . . . . . . 17
⊢
(((((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷‘𝑗) ≤ 𝑧) ∧ ¬ (𝐷‘𝑗) ≤ 𝑆) → if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧) ≤ if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆)) |
| 180 | 173, 179 | pm2.61dan 813 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) ∧ ¬ (𝐷‘𝑗) ≤ 𝑧) → if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧) ≤ if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆)) |
| 181 | 160, 180 | pm2.61dan 813 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) → if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧) ≤ if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆)) |
| 182 | | icossico 13457 |
. . . . . . . . . . . . . . 15
⊢ ((((𝐶‘𝑗) ∈ ℝ* ∧ if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆) ∈ ℝ*) ∧ ((𝐶‘𝑗) ≤ (𝐶‘𝑗) ∧ if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧) ≤ if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆))) → ((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧)) ⊆ ((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆))) |
| 183 | 78, 136, 81, 181, 182 | syl22anc 839 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧)) ⊆ ((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆))) |
| 184 | | volss 25568 |
. . . . . . . . . . . . . 14
⊢ ((((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧)) ∈ dom vol ∧ ((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆)) ∈ dom vol ∧ ((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧)) ⊆ ((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆))) → (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧))) ≤ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆)))) |
| 185 | 63, 135, 183, 184 | syl3anc 1373 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑧 ∈ 𝑈) ∧ 𝑗 ∈ ℕ) → (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧))) ≤ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆)))) |
| 186 | 54, 55, 64, 134, 185 | sge0lempt 46425 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) →
(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)))))) |
| 187 | 97, 133, 98, 186 | leadd1dd 11877 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) →
((Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧))))) + 𝐴) ≤
((Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆))))) + 𝐴)) |
| 188 | 53, 99, 124, 132, 187 | letrd 11418 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑧 ∈ 𝑈) → 𝑧 ≤
((Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆))))) + 𝐴)) |
| 189 | 188 | ex 412 |
. . . . . . . . 9
⊢ (𝜑 → (𝑧 ∈ 𝑈 → 𝑧 ≤
((Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆))))) + 𝐴))) |
| 190 | 50, 189 | ralrimi 3257 |
. . . . . . . 8
⊢ (𝜑 → ∀𝑧 ∈ 𝑈 𝑧 ≤
((Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆))))) + 𝐴)) |
| 191 | | suprleub 12234 |
. . . . . . . . 9
⊢ (((𝑈 ⊆ ℝ ∧ 𝑈 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝑈 𝑦 ≤ 𝑥) ∧
((Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆))))) + 𝐴) ∈ ℝ) → (sup(𝑈, ℝ, < ) ≤
((Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆))))) + 𝐴) ↔ ∀𝑧 ∈ 𝑈 𝑧 ≤
((Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆))))) + 𝐴))) |
| 192 | 52, 46, 150, 123, 191 | syl31anc 1375 |
. . . . . . . 8
⊢ (𝜑 → (sup(𝑈, ℝ, < ) ≤
((Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆))))) + 𝐴) ↔ ∀𝑧 ∈ 𝑈 𝑧 ≤
((Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆))))) + 𝐴))) |
| 193 | 190, 192 | mpbird 257 |
. . . . . . 7
⊢ (𝜑 → sup(𝑈, ℝ, < ) ≤
((Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆))))) + 𝐴)) |
| 194 | 49, 193 | eqbrtrd 5165 |
. . . . . 6
⊢ (𝜑 → 𝑆 ≤
((Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆))))) + 𝐴)) |
| 195 | 100, 2, 122 | lesubaddd 11860 |
. . . . . 6
⊢ (𝜑 → ((𝑆 − 𝐴) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆))))) ↔ 𝑆 ≤
((Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆))))) + 𝐴))) |
| 196 | 194, 195 | mpbird 257 |
. . . . 5
⊢ (𝜑 → (𝑆 − 𝐴) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆)))))) |
| 197 | 48, 196 | jca 511 |
. . . 4
⊢ (𝜑 → (𝑆 ∈ (𝐴[,]𝐵) ∧ (𝑆 − 𝐴) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆))))))) |
| 198 | | oveq1 7438 |
. . . . . 6
⊢ (𝑧 = 𝑆 → (𝑧 − 𝐴) = (𝑆 − 𝐴)) |
| 199 | | breq2 5147 |
. . . . . . . . . . 11
⊢ (𝑧 = 𝑆 → ((𝐷‘𝑗) ≤ 𝑧 ↔ (𝐷‘𝑗) ≤ 𝑆)) |
| 200 | | id 22 |
. . . . . . . . . . 11
⊢ (𝑧 = 𝑆 → 𝑧 = 𝑆) |
| 201 | 199, 200 | ifbieq2d 4552 |
. . . . . . . . . 10
⊢ (𝑧 = 𝑆 → if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧) = if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆)) |
| 202 | 201 | oveq2d 7447 |
. . . . . . . . 9
⊢ (𝑧 = 𝑆 → ((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧)) = ((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆))) |
| 203 | 202 | fveq2d 6910 |
. . . . . . . 8
⊢ (𝑧 = 𝑆 → (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧))) = (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑆, (𝐷‘𝑗), 𝑆)))) |
| 204 | 203 | mpteq2dv 5244 |
. . . . . . 7
⊢ (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))))) |
| 205 | 204 | fveq2d 6910 |
. . . . . 6
⊢ (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)))))) |
| 206 | 198, 205 | breq12d 5156 |
. . . . 5
⊢ (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))))))) |
| 207 | 206 | elrab 3692 |
. . . 4
⊢ (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))))))) |
| 208 | 197, 207 | sylibr 234 |
. . 3
⊢ (𝜑 → 𝑆 ∈ {𝑧 ∈ (𝐴[,]𝐵) ∣ (𝑧 − 𝐴) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧)))))}) |
| 209 | 208, 4 | eleqtrrdi 2852 |
. 2
⊢ (𝜑 → 𝑆 ∈ 𝑈) |
| 210 | 209, 45, 150 | 3jca 1129 |
1
⊢ (𝜑 → (𝑆 ∈ 𝑈 ∧ 𝐴 ∈ 𝑈 ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝑈 𝑦 ≤ 𝑥)) |