| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | itg10a.1 | . . 3
⊢ (𝜑 → 𝐹 ∈ dom
∫1) | 
| 2 |  | itg1val 25719 | . . 3
⊢ (𝐹 ∈ dom ∫1
→ (∫1‘𝐹) = Σ𝑘 ∈ (ran 𝐹 ∖ {0})(𝑘 · (vol‘(◡𝐹 “ {𝑘})))) | 
| 3 | 1, 2 | syl 17 | . 2
⊢ (𝜑 →
(∫1‘𝐹)
= Σ𝑘 ∈ (ran
𝐹 ∖ {0})(𝑘 · (vol‘(◡𝐹 “ {𝑘})))) | 
| 4 |  | i1ff 25712 | . . . . . . . . . . . . . . . 16
⊢ (𝐹 ∈ dom ∫1
→ 𝐹:ℝ⟶ℝ) | 
| 5 | 1, 4 | syl 17 | . . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝐹:ℝ⟶ℝ) | 
| 6 | 5 | ffnd 6736 | . . . . . . . . . . . . . 14
⊢ (𝜑 → 𝐹 Fn ℝ) | 
| 7 | 6 | adantr 480 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → 𝐹 Fn ℝ) | 
| 8 |  | fniniseg 7079 | . . . . . . . . . . . . 13
⊢ (𝐹 Fn ℝ → (𝑥 ∈ (◡𝐹 “ {𝑘}) ↔ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) = 𝑘))) | 
| 9 | 7, 8 | syl 17 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → (𝑥 ∈ (◡𝐹 “ {𝑘}) ↔ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) = 𝑘))) | 
| 10 |  | eldifsni 4789 | . . . . . . . . . . . . . . 15
⊢ (𝑘 ∈ (ran 𝐹 ∖ {0}) → 𝑘 ≠ 0) | 
| 11 | 10 | ad2antlr 727 | . . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) = 𝑘)) → 𝑘 ≠ 0) | 
| 12 |  | simprl 770 | . . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) = 𝑘)) → 𝑥 ∈ ℝ) | 
| 13 |  | eldif 3960 | . . . . . . . . . . . . . . . . 17
⊢ (𝑥 ∈ (ℝ ∖ 𝐴) ↔ (𝑥 ∈ ℝ ∧ ¬ 𝑥 ∈ 𝐴)) | 
| 14 |  | simplrr 777 | . . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) = 𝑘)) ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) → (𝐹‘𝑥) = 𝑘) | 
| 15 |  | itg10a.4 | . . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) → (𝐹‘𝑥) = 0) | 
| 16 | 15 | ad4ant14 752 | . . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) = 𝑘)) ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) → (𝐹‘𝑥) = 0) | 
| 17 | 14, 16 | eqtr3d 2778 | . . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) = 𝑘)) ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) → 𝑘 = 0) | 
| 18 | 17 | ex 412 | . . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) = 𝑘)) → (𝑥 ∈ (ℝ ∖ 𝐴) → 𝑘 = 0)) | 
| 19 | 13, 18 | biimtrrid 243 | . . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) = 𝑘)) → ((𝑥 ∈ ℝ ∧ ¬ 𝑥 ∈ 𝐴) → 𝑘 = 0)) | 
| 20 | 12, 19 | mpand 695 | . . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) = 𝑘)) → (¬ 𝑥 ∈ 𝐴 → 𝑘 = 0)) | 
| 21 | 20 | necon1ad 2956 | . . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) = 𝑘)) → (𝑘 ≠ 0 → 𝑥 ∈ 𝐴)) | 
| 22 | 11, 21 | mpd 15 | . . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) ∧ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) = 𝑘)) → 𝑥 ∈ 𝐴) | 
| 23 | 22 | ex 412 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → ((𝑥 ∈ ℝ ∧ (𝐹‘𝑥) = 𝑘) → 𝑥 ∈ 𝐴)) | 
| 24 | 9, 23 | sylbid 240 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → (𝑥 ∈ (◡𝐹 “ {𝑘}) → 𝑥 ∈ 𝐴)) | 
| 25 | 24 | ssrdv 3988 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → (◡𝐹 “ {𝑘}) ⊆ 𝐴) | 
| 26 |  | itg10a.2 | . . . . . . . . . . 11
⊢ (𝜑 → 𝐴 ⊆ ℝ) | 
| 27 | 26 | adantr 480 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → 𝐴 ⊆ ℝ) | 
| 28 | 25, 27 | sstrd 3993 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → (◡𝐹 “ {𝑘}) ⊆ ℝ) | 
| 29 |  | itg10a.3 | . . . . . . . . . . 11
⊢ (𝜑 → (vol*‘𝐴) = 0) | 
| 30 | 29 | adantr 480 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → (vol*‘𝐴) = 0) | 
| 31 |  | ovolssnul 25523 | . . . . . . . . . 10
⊢ (((◡𝐹 “ {𝑘}) ⊆ 𝐴 ∧ 𝐴 ⊆ ℝ ∧ (vol*‘𝐴) = 0) → (vol*‘(◡𝐹 “ {𝑘})) = 0) | 
| 32 | 25, 27, 30, 31 | syl3anc 1372 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → (vol*‘(◡𝐹 “ {𝑘})) = 0) | 
| 33 |  | nulmbl 25571 | . . . . . . . . 9
⊢ (((◡𝐹 “ {𝑘}) ⊆ ℝ ∧ (vol*‘(◡𝐹 “ {𝑘})) = 0) → (◡𝐹 “ {𝑘}) ∈ dom vol) | 
| 34 | 28, 32, 33 | syl2anc 584 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → (◡𝐹 “ {𝑘}) ∈ dom vol) | 
| 35 |  | mblvol 25566 | . . . . . . . 8
⊢ ((◡𝐹 “ {𝑘}) ∈ dom vol → (vol‘(◡𝐹 “ {𝑘})) = (vol*‘(◡𝐹 “ {𝑘}))) | 
| 36 | 34, 35 | syl 17 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → (vol‘(◡𝐹 “ {𝑘})) = (vol*‘(◡𝐹 “ {𝑘}))) | 
| 37 | 36, 32 | eqtrd 2776 | . . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → (vol‘(◡𝐹 “ {𝑘})) = 0) | 
| 38 | 37 | oveq2d 7448 | . . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → (𝑘 · (vol‘(◡𝐹 “ {𝑘}))) = (𝑘 · 0)) | 
| 39 | 5 | frnd 6743 | . . . . . . . . 9
⊢ (𝜑 → ran 𝐹 ⊆ ℝ) | 
| 40 | 39 | ssdifssd 4146 | . . . . . . . 8
⊢ (𝜑 → (ran 𝐹 ∖ {0}) ⊆
ℝ) | 
| 41 | 40 | sselda 3982 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → 𝑘 ∈ ℝ) | 
| 42 | 41 | recnd 11290 | . . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → 𝑘 ∈ ℂ) | 
| 43 | 42 | mul01d 11461 | . . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → (𝑘 · 0) = 0) | 
| 44 | 38, 43 | eqtrd 2776 | . . . 4
⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐹 ∖ {0})) → (𝑘 · (vol‘(◡𝐹 “ {𝑘}))) = 0) | 
| 45 | 44 | sumeq2dv 15739 | . . 3
⊢ (𝜑 → Σ𝑘 ∈ (ran 𝐹 ∖ {0})(𝑘 · (vol‘(◡𝐹 “ {𝑘}))) = Σ𝑘 ∈ (ran 𝐹 ∖ {0})0) | 
| 46 |  | i1frn 25713 | . . . . . . 7
⊢ (𝐹 ∈ dom ∫1
→ ran 𝐹 ∈
Fin) | 
| 47 | 1, 46 | syl 17 | . . . . . 6
⊢ (𝜑 → ran 𝐹 ∈ Fin) | 
| 48 |  | difss 4135 | . . . . . 6
⊢ (ran
𝐹 ∖ {0}) ⊆ ran
𝐹 | 
| 49 |  | ssfi 9214 | . . . . . 6
⊢ ((ran
𝐹 ∈ Fin ∧ (ran
𝐹 ∖ {0}) ⊆ ran
𝐹) → (ran 𝐹 ∖ {0}) ∈
Fin) | 
| 50 | 47, 48, 49 | sylancl 586 | . . . . 5
⊢ (𝜑 → (ran 𝐹 ∖ {0}) ∈ Fin) | 
| 51 | 50 | olcd 874 | . . . 4
⊢ (𝜑 → ((ran 𝐹 ∖ {0}) ⊆
(ℤ≥‘0) ∨ (ran 𝐹 ∖ {0}) ∈ Fin)) | 
| 52 |  | sumz 15759 | . . . 4
⊢ (((ran
𝐹 ∖ {0}) ⊆
(ℤ≥‘0) ∨ (ran 𝐹 ∖ {0}) ∈ Fin) →
Σ𝑘 ∈ (ran 𝐹 ∖ {0})0 =
0) | 
| 53 | 51, 52 | syl 17 | . . 3
⊢ (𝜑 → Σ𝑘 ∈ (ran 𝐹 ∖ {0})0 = 0) | 
| 54 | 45, 53 | eqtrd 2776 | . 2
⊢ (𝜑 → Σ𝑘 ∈ (ran 𝐹 ∖ {0})(𝑘 · (vol‘(◡𝐹 “ {𝑘}))) = 0) | 
| 55 | 3, 54 | eqtrd 2776 | 1
⊢ (𝜑 →
(∫1‘𝐹)
= 0) |