| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | ovolicc2.7 | . . . 4
⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ∪ 𝑈) | 
| 2 |  | ovolicc.1 | . . . . . 6
⊢ (𝜑 → 𝐴 ∈ ℝ) | 
| 3 | 2 | rexrd 11311 | . . . . 5
⊢ (𝜑 → 𝐴 ∈
ℝ*) | 
| 4 |  | ovolicc.2 | . . . . . 6
⊢ (𝜑 → 𝐵 ∈ ℝ) | 
| 5 | 4 | rexrd 11311 | . . . . 5
⊢ (𝜑 → 𝐵 ∈
ℝ*) | 
| 6 |  | ovolicc.3 | . . . . 5
⊢ (𝜑 → 𝐴 ≤ 𝐵) | 
| 7 |  | lbicc2 13504 | . . . . 5
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ* ∧ 𝐴
≤ 𝐵) → 𝐴 ∈ (𝐴[,]𝐵)) | 
| 8 | 3, 5, 6, 7 | syl3anc 1373 | . . . 4
⊢ (𝜑 → 𝐴 ∈ (𝐴[,]𝐵)) | 
| 9 | 1, 8 | sseldd 3984 | . . 3
⊢ (𝜑 → 𝐴 ∈ ∪ 𝑈) | 
| 10 |  | eluni2 4911 | . . 3
⊢ (𝐴 ∈ ∪ 𝑈
↔ ∃𝑧 ∈
𝑈 𝐴 ∈ 𝑧) | 
| 11 | 9, 10 | sylib 218 | . 2
⊢ (𝜑 → ∃𝑧 ∈ 𝑈 𝐴 ∈ 𝑧) | 
| 12 |  | ovolicc2.6 | . . . . . . 7
⊢ (𝜑 → 𝑈 ∈ (𝒫 ran ((,) ∘ 𝐹) ∩ Fin)) | 
| 13 | 12 | elin2d 4205 | . . . . . 6
⊢ (𝜑 → 𝑈 ∈ Fin) | 
| 14 |  | ovolicc2.10 | . . . . . . 7
⊢ 𝑇 = {𝑢 ∈ 𝑈 ∣ (𝑢 ∩ (𝐴[,]𝐵)) ≠ ∅} | 
| 15 | 14 | ssrab3 4082 | . . . . . 6
⊢ 𝑇 ⊆ 𝑈 | 
| 16 |  | ssfi 9213 | . . . . . 6
⊢ ((𝑈 ∈ Fin ∧ 𝑇 ⊆ 𝑈) → 𝑇 ∈ Fin) | 
| 17 | 13, 15, 16 | sylancl 586 | . . . . 5
⊢ (𝜑 → 𝑇 ∈ Fin) | 
| 18 | 1 | adantr 480 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐴[,]𝐵) ⊆ ∪ 𝑈) | 
| 19 |  | ovolicc2.8 | . . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝐺:𝑈⟶ℕ) | 
| 20 |  | ineq1 4213 | . . . . . . . . . . . . . . . . . 18
⊢ (𝑢 = 𝑡 → (𝑢 ∩ (𝐴[,]𝐵)) = (𝑡 ∩ (𝐴[,]𝐵))) | 
| 21 | 20 | neeq1d 3000 | . . . . . . . . . . . . . . . . 17
⊢ (𝑢 = 𝑡 → ((𝑢 ∩ (𝐴[,]𝐵)) ≠ ∅ ↔ (𝑡 ∩ (𝐴[,]𝐵)) ≠ ∅)) | 
| 22 | 21, 14 | elrab2 3695 | . . . . . . . . . . . . . . . 16
⊢ (𝑡 ∈ 𝑇 ↔ (𝑡 ∈ 𝑈 ∧ (𝑡 ∩ (𝐴[,]𝐵)) ≠ ∅)) | 
| 23 | 22 | simplbi 497 | . . . . . . . . . . . . . . 15
⊢ (𝑡 ∈ 𝑇 → 𝑡 ∈ 𝑈) | 
| 24 |  | ffvelcdm 7101 | . . . . . . . . . . . . . . 15
⊢ ((𝐺:𝑈⟶ℕ ∧ 𝑡 ∈ 𝑈) → (𝐺‘𝑡) ∈ ℕ) | 
| 25 | 19, 23, 24 | syl2an 596 | . . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐺‘𝑡) ∈ ℕ) | 
| 26 |  | ovolicc2.5 | . . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝐹:ℕ⟶( ≤ ∩ (ℝ
× ℝ))) | 
| 27 | 26 | ffvelcdmda 7104 | . . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝐺‘𝑡) ∈ ℕ) → (𝐹‘(𝐺‘𝑡)) ∈ ( ≤ ∩ (ℝ ×
ℝ))) | 
| 28 | 25, 27 | syldan 591 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐹‘(𝐺‘𝑡)) ∈ ( ≤ ∩ (ℝ ×
ℝ))) | 
| 29 | 28 | elin2d 4205 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐹‘(𝐺‘𝑡)) ∈ (ℝ ×
ℝ)) | 
| 30 |  | xp2nd 8047 | . . . . . . . . . . . 12
⊢ ((𝐹‘(𝐺‘𝑡)) ∈ (ℝ × ℝ) →
(2nd ‘(𝐹‘(𝐺‘𝑡))) ∈ ℝ) | 
| 31 | 29, 30 | syl 17 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (2nd ‘(𝐹‘(𝐺‘𝑡))) ∈ ℝ) | 
| 32 | 4 | adantr 480 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝐵 ∈ ℝ) | 
| 33 | 31, 32 | ifcld 4572 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ ℝ) | 
| 34 | 22 | simprbi 496 | . . . . . . . . . . . . . 14
⊢ (𝑡 ∈ 𝑇 → (𝑡 ∩ (𝐴[,]𝐵)) ≠ ∅) | 
| 35 | 34 | adantl 481 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝑡 ∩ (𝐴[,]𝐵)) ≠ ∅) | 
| 36 |  | n0 4353 | . . . . . . . . . . . . 13
⊢ ((𝑡 ∩ (𝐴[,]𝐵)) ≠ ∅ ↔ ∃𝑦 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵))) | 
| 37 | 35, 36 | sylib 218 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ∃𝑦 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵))) | 
| 38 | 2 | adantr 480 | . . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → 𝐴 ∈ ℝ) | 
| 39 |  | simprr 773 | . . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵))) | 
| 40 | 39 | elin2d 4205 | . . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → 𝑦 ∈ (𝐴[,]𝐵)) | 
| 41 | 4 | adantr 480 | . . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → 𝐵 ∈ ℝ) | 
| 42 |  | elicc2 13452 | . . . . . . . . . . . . . . . . . 18
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝑦 ∈ (𝐴[,]𝐵) ↔ (𝑦 ∈ ℝ ∧ 𝐴 ≤ 𝑦 ∧ 𝑦 ≤ 𝐵))) | 
| 43 | 2, 41, 42 | syl2an2r 685 | . . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → (𝑦 ∈ (𝐴[,]𝐵) ↔ (𝑦 ∈ ℝ ∧ 𝐴 ≤ 𝑦 ∧ 𝑦 ≤ 𝐵))) | 
| 44 | 40, 43 | mpbid 232 | . . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → (𝑦 ∈ ℝ ∧ 𝐴 ≤ 𝑦 ∧ 𝑦 ≤ 𝐵)) | 
| 45 | 44 | simp1d 1143 | . . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → 𝑦 ∈ ℝ) | 
| 46 | 29 | adantrr 717 | . . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → (𝐹‘(𝐺‘𝑡)) ∈ (ℝ ×
ℝ)) | 
| 47 | 46, 30 | syl 17 | . . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → (2nd ‘(𝐹‘(𝐺‘𝑡))) ∈ ℝ) | 
| 48 | 44 | simp2d 1144 | . . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → 𝐴 ≤ 𝑦) | 
| 49 | 39 | elin1d 4204 | . . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → 𝑦 ∈ 𝑡) | 
| 50 | 25 | adantrr 717 | . . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → (𝐺‘𝑡) ∈ ℕ) | 
| 51 |  | fvco3 7008 | . . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝐹:ℕ⟶( ≤ ∩
(ℝ × ℝ)) ∧ (𝐺‘𝑡) ∈ ℕ) → (((,) ∘ 𝐹)‘(𝐺‘𝑡)) = ((,)‘(𝐹‘(𝐺‘𝑡)))) | 
| 52 | 26, 50, 51 | syl2an2r 685 | . . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → (((,) ∘ 𝐹)‘(𝐺‘𝑡)) = ((,)‘(𝐹‘(𝐺‘𝑡)))) | 
| 53 |  | ovolicc2.9 | . . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑈) → (((,) ∘ 𝐹)‘(𝐺‘𝑡)) = 𝑡) | 
| 54 | 23, 53 | sylan2 593 | . . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (((,) ∘ 𝐹)‘(𝐺‘𝑡)) = 𝑡) | 
| 55 | 54 | adantrr 717 | . . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → (((,) ∘ 𝐹)‘(𝐺‘𝑡)) = 𝑡) | 
| 56 |  | 1st2nd2 8053 | . . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝐹‘(𝐺‘𝑡)) ∈ (ℝ × ℝ) →
(𝐹‘(𝐺‘𝑡)) = 〈(1st ‘(𝐹‘(𝐺‘𝑡))), (2nd ‘(𝐹‘(𝐺‘𝑡)))〉) | 
| 57 | 46, 56 | syl 17 | . . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → (𝐹‘(𝐺‘𝑡)) = 〈(1st ‘(𝐹‘(𝐺‘𝑡))), (2nd ‘(𝐹‘(𝐺‘𝑡)))〉) | 
| 58 | 57 | fveq2d 6910 | . . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → ((,)‘(𝐹‘(𝐺‘𝑡))) = ((,)‘〈(1st
‘(𝐹‘(𝐺‘𝑡))), (2nd ‘(𝐹‘(𝐺‘𝑡)))〉)) | 
| 59 |  | df-ov 7434 | . . . . . . . . . . . . . . . . . . . . 21
⊢
((1st ‘(𝐹‘(𝐺‘𝑡)))(,)(2nd ‘(𝐹‘(𝐺‘𝑡)))) = ((,)‘〈(1st
‘(𝐹‘(𝐺‘𝑡))), (2nd ‘(𝐹‘(𝐺‘𝑡)))〉) | 
| 60 | 58, 59 | eqtr4di 2795 | . . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → ((,)‘(𝐹‘(𝐺‘𝑡))) = ((1st ‘(𝐹‘(𝐺‘𝑡)))(,)(2nd ‘(𝐹‘(𝐺‘𝑡))))) | 
| 61 | 52, 55, 60 | 3eqtr3d 2785 | . . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → 𝑡 = ((1st ‘(𝐹‘(𝐺‘𝑡)))(,)(2nd ‘(𝐹‘(𝐺‘𝑡))))) | 
| 62 | 49, 61 | eleqtrd 2843 | . . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → 𝑦 ∈ ((1st ‘(𝐹‘(𝐺‘𝑡)))(,)(2nd ‘(𝐹‘(𝐺‘𝑡))))) | 
| 63 |  | xp1st 8046 | . . . . . . . . . . . . . . . . . . . 20
⊢ ((𝐹‘(𝐺‘𝑡)) ∈ (ℝ × ℝ) →
(1st ‘(𝐹‘(𝐺‘𝑡))) ∈ ℝ) | 
| 64 | 46, 63 | syl 17 | . . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → (1st ‘(𝐹‘(𝐺‘𝑡))) ∈ ℝ) | 
| 65 |  | rexr 11307 | . . . . . . . . . . . . . . . . . . . 20
⊢
((1st ‘(𝐹‘(𝐺‘𝑡))) ∈ ℝ → (1st
‘(𝐹‘(𝐺‘𝑡))) ∈
ℝ*) | 
| 66 |  | rexr 11307 | . . . . . . . . . . . . . . . . . . . 20
⊢
((2nd ‘(𝐹‘(𝐺‘𝑡))) ∈ ℝ → (2nd
‘(𝐹‘(𝐺‘𝑡))) ∈
ℝ*) | 
| 67 |  | elioo2 13428 | . . . . . . . . . . . . . . . . . . . 20
⊢
(((1st ‘(𝐹‘(𝐺‘𝑡))) ∈ ℝ* ∧
(2nd ‘(𝐹‘(𝐺‘𝑡))) ∈ ℝ*) → (𝑦 ∈ ((1st
‘(𝐹‘(𝐺‘𝑡)))(,)(2nd ‘(𝐹‘(𝐺‘𝑡)))) ↔ (𝑦 ∈ ℝ ∧ (1st
‘(𝐹‘(𝐺‘𝑡))) < 𝑦 ∧ 𝑦 < (2nd ‘(𝐹‘(𝐺‘𝑡)))))) | 
| 68 | 65, 66, 67 | syl2an 596 | . . . . . . . . . . . . . . . . . . 19
⊢
(((1st ‘(𝐹‘(𝐺‘𝑡))) ∈ ℝ ∧ (2nd
‘(𝐹‘(𝐺‘𝑡))) ∈ ℝ) → (𝑦 ∈ ((1st
‘(𝐹‘(𝐺‘𝑡)))(,)(2nd ‘(𝐹‘(𝐺‘𝑡)))) ↔ (𝑦 ∈ ℝ ∧ (1st
‘(𝐹‘(𝐺‘𝑡))) < 𝑦 ∧ 𝑦 < (2nd ‘(𝐹‘(𝐺‘𝑡)))))) | 
| 69 | 64, 47, 68 | syl2anc 584 | . . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → (𝑦 ∈ ((1st ‘(𝐹‘(𝐺‘𝑡)))(,)(2nd ‘(𝐹‘(𝐺‘𝑡)))) ↔ (𝑦 ∈ ℝ ∧ (1st
‘(𝐹‘(𝐺‘𝑡))) < 𝑦 ∧ 𝑦 < (2nd ‘(𝐹‘(𝐺‘𝑡)))))) | 
| 70 | 62, 69 | mpbid 232 | . . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → (𝑦 ∈ ℝ ∧ (1st
‘(𝐹‘(𝐺‘𝑡))) < 𝑦 ∧ 𝑦 < (2nd ‘(𝐹‘(𝐺‘𝑡))))) | 
| 71 | 70 | simp3d 1145 | . . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → 𝑦 < (2nd ‘(𝐹‘(𝐺‘𝑡)))) | 
| 72 | 45, 47, 71 | ltled 11409 | . . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → 𝑦 ≤ (2nd ‘(𝐹‘(𝐺‘𝑡)))) | 
| 73 | 38, 45, 47, 48, 72 | letrd 11418 | . . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)))) → 𝐴 ≤ (2nd ‘(𝐹‘(𝐺‘𝑡)))) | 
| 74 | 73 | expr 456 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)) → 𝐴 ≤ (2nd ‘(𝐹‘(𝐺‘𝑡))))) | 
| 75 | 74 | exlimdv 1933 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (∃𝑦 𝑦 ∈ (𝑡 ∩ (𝐴[,]𝐵)) → 𝐴 ≤ (2nd ‘(𝐹‘(𝐺‘𝑡))))) | 
| 76 | 37, 75 | mpd 15 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝐴 ≤ (2nd ‘(𝐹‘(𝐺‘𝑡)))) | 
| 77 | 6 | adantr 480 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝐴 ≤ 𝐵) | 
| 78 |  | breq2 5147 | . . . . . . . . . . . 12
⊢
((2nd ‘(𝐹‘(𝐺‘𝑡))) = if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) → (𝐴 ≤ (2nd ‘(𝐹‘(𝐺‘𝑡))) ↔ 𝐴 ≤ if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵))) | 
| 79 |  | breq2 5147 | . . . . . . . . . . . 12
⊢ (𝐵 = if((2nd
‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) → (𝐴 ≤ 𝐵 ↔ 𝐴 ≤ if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵))) | 
| 80 | 78, 79 | ifboth 4565 | . . . . . . . . . . 11
⊢ ((𝐴 ≤ (2nd
‘(𝐹‘(𝐺‘𝑡))) ∧ 𝐴 ≤ 𝐵) → 𝐴 ≤ if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵)) | 
| 81 | 76, 77, 80 | syl2anc 584 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝐴 ≤ if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵)) | 
| 82 |  | min2 13232 | . . . . . . . . . . 11
⊢
(((2nd ‘(𝐹‘(𝐺‘𝑡))) ∈ ℝ ∧ 𝐵 ∈ ℝ) → if((2nd
‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ≤ 𝐵) | 
| 83 | 31, 32, 82 | syl2anc 584 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ≤ 𝐵) | 
| 84 |  | elicc2 13452 | . . . . . . . . . . . 12
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) →
(if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ (𝐴[,]𝐵) ↔ (if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ ℝ ∧ 𝐴 ≤ if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∧ if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ≤ 𝐵))) | 
| 85 | 2, 4, 84 | syl2anc 584 | . . . . . . . . . . 11
⊢ (𝜑 → (if((2nd
‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ (𝐴[,]𝐵) ↔ (if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ ℝ ∧ 𝐴 ≤ if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∧ if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ≤ 𝐵))) | 
| 86 | 85 | adantr 480 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ (𝐴[,]𝐵) ↔ (if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ ℝ ∧ 𝐴 ≤ if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∧ if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ≤ 𝐵))) | 
| 87 | 33, 81, 83, 86 | mpbir3and 1343 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ (𝐴[,]𝐵)) | 
| 88 | 18, 87 | sseldd 3984 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ ∪ 𝑈) | 
| 89 |  | eluni2 4911 | . . . . . . . 8
⊢
(if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ ∪ 𝑈 ↔ ∃𝑥 ∈ 𝑈 if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ 𝑥) | 
| 90 | 88, 89 | sylib 218 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ∃𝑥 ∈ 𝑈 if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ 𝑥) | 
| 91 |  | simprl 771 | . . . . . . . 8
⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ (𝑥 ∈ 𝑈 ∧ if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ 𝑥)) → 𝑥 ∈ 𝑈) | 
| 92 |  | simprr 773 | . . . . . . . . 9
⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ (𝑥 ∈ 𝑈 ∧ if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ 𝑥)) → if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ 𝑥) | 
| 93 | 87 | adantr 480 | . . . . . . . . 9
⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ (𝑥 ∈ 𝑈 ∧ if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ 𝑥)) → if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ (𝐴[,]𝐵)) | 
| 94 |  | inelcm 4465 | . . . . . . . . 9
⊢
((if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ 𝑥 ∧ if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ (𝐴[,]𝐵)) → (𝑥 ∩ (𝐴[,]𝐵)) ≠ ∅) | 
| 95 | 92, 93, 94 | syl2anc 584 | . . . . . . . 8
⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ (𝑥 ∈ 𝑈 ∧ if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ 𝑥)) → (𝑥 ∩ (𝐴[,]𝐵)) ≠ ∅) | 
| 96 |  | ineq1 4213 | . . . . . . . . . 10
⊢ (𝑢 = 𝑥 → (𝑢 ∩ (𝐴[,]𝐵)) = (𝑥 ∩ (𝐴[,]𝐵))) | 
| 97 | 96 | neeq1d 3000 | . . . . . . . . 9
⊢ (𝑢 = 𝑥 → ((𝑢 ∩ (𝐴[,]𝐵)) ≠ ∅ ↔ (𝑥 ∩ (𝐴[,]𝐵)) ≠ ∅)) | 
| 98 | 97, 14 | elrab2 3695 | . . . . . . . 8
⊢ (𝑥 ∈ 𝑇 ↔ (𝑥 ∈ 𝑈 ∧ (𝑥 ∩ (𝐴[,]𝐵)) ≠ ∅)) | 
| 99 | 91, 95, 98 | sylanbrc 583 | . . . . . . 7
⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ (𝑥 ∈ 𝑈 ∧ if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ 𝑥)) → 𝑥 ∈ 𝑇) | 
| 100 | 90, 99, 92 | reximssdv 3173 | . . . . . 6
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ∃𝑥 ∈ 𝑇 if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ 𝑥) | 
| 101 | 100 | ralrimiva 3146 | . . . . 5
⊢ (𝜑 → ∀𝑡 ∈ 𝑇 ∃𝑥 ∈ 𝑇 if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ 𝑥) | 
| 102 |  | eleq2 2830 | . . . . . 6
⊢ (𝑥 = (ℎ‘𝑡) → (if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ 𝑥 ↔ if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ (ℎ‘𝑡))) | 
| 103 | 102 | ac6sfi 9320 | . . . . 5
⊢ ((𝑇 ∈ Fin ∧ ∀𝑡 ∈ 𝑇 ∃𝑥 ∈ 𝑇 if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ 𝑥) → ∃ℎ(ℎ:𝑇⟶𝑇 ∧ ∀𝑡 ∈ 𝑇 if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ (ℎ‘𝑡))) | 
| 104 | 17, 101, 103 | syl2anc 584 | . . . 4
⊢ (𝜑 → ∃ℎ(ℎ:𝑇⟶𝑇 ∧ ∀𝑡 ∈ 𝑇 if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ (ℎ‘𝑡))) | 
| 105 | 104 | adantr 480 | . . 3
⊢ ((𝜑 ∧ (𝑧 ∈ 𝑈 ∧ 𝐴 ∈ 𝑧)) → ∃ℎ(ℎ:𝑇⟶𝑇 ∧ ∀𝑡 ∈ 𝑇 if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ (ℎ‘𝑡))) | 
| 106 |  | 2fveq3 6911 | . . . . . . . . . . 11
⊢ (𝑥 = 𝑡 → (𝐹‘(𝐺‘𝑥)) = (𝐹‘(𝐺‘𝑡))) | 
| 107 | 106 | fveq2d 6910 | . . . . . . . . . 10
⊢ (𝑥 = 𝑡 → (2nd ‘(𝐹‘(𝐺‘𝑥))) = (2nd ‘(𝐹‘(𝐺‘𝑡)))) | 
| 108 | 107 | breq1d 5153 | . . . . . . . . 9
⊢ (𝑥 = 𝑡 → ((2nd ‘(𝐹‘(𝐺‘𝑥))) ≤ 𝐵 ↔ (2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵)) | 
| 109 | 108, 107 | ifbieq1d 4550 | . . . . . . . 8
⊢ (𝑥 = 𝑡 → if((2nd ‘(𝐹‘(𝐺‘𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑥))), 𝐵) = if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵)) | 
| 110 |  | fveq2 6906 | . . . . . . . 8
⊢ (𝑥 = 𝑡 → (ℎ‘𝑥) = (ℎ‘𝑡)) | 
| 111 | 109, 110 | eleq12d 2835 | . . . . . . 7
⊢ (𝑥 = 𝑡 → (if((2nd ‘(𝐹‘(𝐺‘𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑥))), 𝐵) ∈ (ℎ‘𝑥) ↔ if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ (ℎ‘𝑡))) | 
| 112 | 111 | cbvralvw 3237 | . . . . . 6
⊢
(∀𝑥 ∈
𝑇 if((2nd
‘(𝐹‘(𝐺‘𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑥))), 𝐵) ∈ (ℎ‘𝑥) ↔ ∀𝑡 ∈ 𝑇 if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ (ℎ‘𝑡)) | 
| 113 | 2 | adantr 480 | . . . . . . . . 9
⊢ ((𝜑 ∧ ((𝑧 ∈ 𝑈 ∧ 𝐴 ∈ 𝑧) ∧ (ℎ:𝑇⟶𝑇 ∧ ∀𝑥 ∈ 𝑇 if((2nd ‘(𝐹‘(𝐺‘𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑥))), 𝐵) ∈ (ℎ‘𝑥)))) → 𝐴 ∈ ℝ) | 
| 114 | 4 | adantr 480 | . . . . . . . . 9
⊢ ((𝜑 ∧ ((𝑧 ∈ 𝑈 ∧ 𝐴 ∈ 𝑧) ∧ (ℎ:𝑇⟶𝑇 ∧ ∀𝑥 ∈ 𝑇 if((2nd ‘(𝐹‘(𝐺‘𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑥))), 𝐵) ∈ (ℎ‘𝑥)))) → 𝐵 ∈ ℝ) | 
| 115 | 6 | adantr 480 | . . . . . . . . 9
⊢ ((𝜑 ∧ ((𝑧 ∈ 𝑈 ∧ 𝐴 ∈ 𝑧) ∧ (ℎ:𝑇⟶𝑇 ∧ ∀𝑥 ∈ 𝑇 if((2nd ‘(𝐹‘(𝐺‘𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑥))), 𝐵) ∈ (ℎ‘𝑥)))) → 𝐴 ≤ 𝐵) | 
| 116 |  | ovolicc2.4 | . . . . . . . . 9
⊢ 𝑆 = seq1( + , ((abs ∘
− ) ∘ 𝐹)) | 
| 117 | 26 | adantr 480 | . . . . . . . . 9
⊢ ((𝜑 ∧ ((𝑧 ∈ 𝑈 ∧ 𝐴 ∈ 𝑧) ∧ (ℎ:𝑇⟶𝑇 ∧ ∀𝑥 ∈ 𝑇 if((2nd ‘(𝐹‘(𝐺‘𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑥))), 𝐵) ∈ (ℎ‘𝑥)))) → 𝐹:ℕ⟶( ≤ ∩ (ℝ
× ℝ))) | 
| 118 | 12 | adantr 480 | . . . . . . . . 9
⊢ ((𝜑 ∧ ((𝑧 ∈ 𝑈 ∧ 𝐴 ∈ 𝑧) ∧ (ℎ:𝑇⟶𝑇 ∧ ∀𝑥 ∈ 𝑇 if((2nd ‘(𝐹‘(𝐺‘𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑥))), 𝐵) ∈ (ℎ‘𝑥)))) → 𝑈 ∈ (𝒫 ran ((,) ∘ 𝐹) ∩ Fin)) | 
| 119 | 1 | adantr 480 | . . . . . . . . 9
⊢ ((𝜑 ∧ ((𝑧 ∈ 𝑈 ∧ 𝐴 ∈ 𝑧) ∧ (ℎ:𝑇⟶𝑇 ∧ ∀𝑥 ∈ 𝑇 if((2nd ‘(𝐹‘(𝐺‘𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑥))), 𝐵) ∈ (ℎ‘𝑥)))) → (𝐴[,]𝐵) ⊆ ∪ 𝑈) | 
| 120 | 19 | adantr 480 | . . . . . . . . 9
⊢ ((𝜑 ∧ ((𝑧 ∈ 𝑈 ∧ 𝐴 ∈ 𝑧) ∧ (ℎ:𝑇⟶𝑇 ∧ ∀𝑥 ∈ 𝑇 if((2nd ‘(𝐹‘(𝐺‘𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑥))), 𝐵) ∈ (ℎ‘𝑥)))) → 𝐺:𝑈⟶ℕ) | 
| 121 | 53 | adantlr 715 | . . . . . . . . 9
⊢ (((𝜑 ∧ ((𝑧 ∈ 𝑈 ∧ 𝐴 ∈ 𝑧) ∧ (ℎ:𝑇⟶𝑇 ∧ ∀𝑥 ∈ 𝑇 if((2nd ‘(𝐹‘(𝐺‘𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑥))), 𝐵) ∈ (ℎ‘𝑥)))) ∧ 𝑡 ∈ 𝑈) → (((,) ∘ 𝐹)‘(𝐺‘𝑡)) = 𝑡) | 
| 122 |  | simprrl 781 | . . . . . . . . 9
⊢ ((𝜑 ∧ ((𝑧 ∈ 𝑈 ∧ 𝐴 ∈ 𝑧) ∧ (ℎ:𝑇⟶𝑇 ∧ ∀𝑥 ∈ 𝑇 if((2nd ‘(𝐹‘(𝐺‘𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑥))), 𝐵) ∈ (ℎ‘𝑥)))) → ℎ:𝑇⟶𝑇) | 
| 123 |  | simprrr 782 | . . . . . . . . . 10
⊢ ((𝜑 ∧ ((𝑧 ∈ 𝑈 ∧ 𝐴 ∈ 𝑧) ∧ (ℎ:𝑇⟶𝑇 ∧ ∀𝑥 ∈ 𝑇 if((2nd ‘(𝐹‘(𝐺‘𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑥))), 𝐵) ∈ (ℎ‘𝑥)))) → ∀𝑥 ∈ 𝑇 if((2nd ‘(𝐹‘(𝐺‘𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑥))), 𝐵) ∈ (ℎ‘𝑥)) | 
| 124 | 111 | rspccva 3621 | . . . . . . . . . 10
⊢
((∀𝑥 ∈
𝑇 if((2nd
‘(𝐹‘(𝐺‘𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑥))), 𝐵) ∈ (ℎ‘𝑥) ∧ 𝑡 ∈ 𝑇) → if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ (ℎ‘𝑡)) | 
| 125 | 123, 124 | sylan 580 | . . . . . . . . 9
⊢ (((𝜑 ∧ ((𝑧 ∈ 𝑈 ∧ 𝐴 ∈ 𝑧) ∧ (ℎ:𝑇⟶𝑇 ∧ ∀𝑥 ∈ 𝑇 if((2nd ‘(𝐹‘(𝐺‘𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑥))), 𝐵) ∈ (ℎ‘𝑥)))) ∧ 𝑡 ∈ 𝑇) → if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ (ℎ‘𝑡)) | 
| 126 |  | simprlr 780 | . . . . . . . . 9
⊢ ((𝜑 ∧ ((𝑧 ∈ 𝑈 ∧ 𝐴 ∈ 𝑧) ∧ (ℎ:𝑇⟶𝑇 ∧ ∀𝑥 ∈ 𝑇 if((2nd ‘(𝐹‘(𝐺‘𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑥))), 𝐵) ∈ (ℎ‘𝑥)))) → 𝐴 ∈ 𝑧) | 
| 127 |  | simprll 779 | . . . . . . . . . 10
⊢ ((𝜑 ∧ ((𝑧 ∈ 𝑈 ∧ 𝐴 ∈ 𝑧) ∧ (ℎ:𝑇⟶𝑇 ∧ ∀𝑥 ∈ 𝑇 if((2nd ‘(𝐹‘(𝐺‘𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑥))), 𝐵) ∈ (ℎ‘𝑥)))) → 𝑧 ∈ 𝑈) | 
| 128 | 8 | adantr 480 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ ((𝑧 ∈ 𝑈 ∧ 𝐴 ∈ 𝑧) ∧ (ℎ:𝑇⟶𝑇 ∧ ∀𝑥 ∈ 𝑇 if((2nd ‘(𝐹‘(𝐺‘𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑥))), 𝐵) ∈ (ℎ‘𝑥)))) → 𝐴 ∈ (𝐴[,]𝐵)) | 
| 129 |  | inelcm 4465 | . . . . . . . . . . 11
⊢ ((𝐴 ∈ 𝑧 ∧ 𝐴 ∈ (𝐴[,]𝐵)) → (𝑧 ∩ (𝐴[,]𝐵)) ≠ ∅) | 
| 130 | 126, 128,
129 | syl2anc 584 | . . . . . . . . . 10
⊢ ((𝜑 ∧ ((𝑧 ∈ 𝑈 ∧ 𝐴 ∈ 𝑧) ∧ (ℎ:𝑇⟶𝑇 ∧ ∀𝑥 ∈ 𝑇 if((2nd ‘(𝐹‘(𝐺‘𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑥))), 𝐵) ∈ (ℎ‘𝑥)))) → (𝑧 ∩ (𝐴[,]𝐵)) ≠ ∅) | 
| 131 |  | ineq1 4213 | . . . . . . . . . . . 12
⊢ (𝑢 = 𝑧 → (𝑢 ∩ (𝐴[,]𝐵)) = (𝑧 ∩ (𝐴[,]𝐵))) | 
| 132 | 131 | neeq1d 3000 | . . . . . . . . . . 11
⊢ (𝑢 = 𝑧 → ((𝑢 ∩ (𝐴[,]𝐵)) ≠ ∅ ↔ (𝑧 ∩ (𝐴[,]𝐵)) ≠ ∅)) | 
| 133 | 132, 14 | elrab2 3695 | . . . . . . . . . 10
⊢ (𝑧 ∈ 𝑇 ↔ (𝑧 ∈ 𝑈 ∧ (𝑧 ∩ (𝐴[,]𝐵)) ≠ ∅)) | 
| 134 | 127, 130,
133 | sylanbrc 583 | . . . . . . . . 9
⊢ ((𝜑 ∧ ((𝑧 ∈ 𝑈 ∧ 𝐴 ∈ 𝑧) ∧ (ℎ:𝑇⟶𝑇 ∧ ∀𝑥 ∈ 𝑇 if((2nd ‘(𝐹‘(𝐺‘𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑥))), 𝐵) ∈ (ℎ‘𝑥)))) → 𝑧 ∈ 𝑇) | 
| 135 |  | eqid 2737 | . . . . . . . . 9
⊢
seq1((ℎ ∘
1st ), (ℕ × {𝑧})) = seq1((ℎ ∘ 1st ), (ℕ ×
{𝑧})) | 
| 136 |  | fveq2 6906 | . . . . . . . . . . 11
⊢ (𝑚 = 𝑛 → (seq1((ℎ ∘ 1st ), (ℕ ×
{𝑧}))‘𝑚) = (seq1((ℎ ∘ 1st ), (ℕ ×
{𝑧}))‘𝑛)) | 
| 137 | 136 | eleq2d 2827 | . . . . . . . . . 10
⊢ (𝑚 = 𝑛 → (𝐵 ∈ (seq1((ℎ ∘ 1st ), (ℕ ×
{𝑧}))‘𝑚) ↔ 𝐵 ∈ (seq1((ℎ ∘ 1st ), (ℕ ×
{𝑧}))‘𝑛))) | 
| 138 | 137 | cbvrabv 3447 | . . . . . . . . 9
⊢ {𝑚 ∈ ℕ ∣ 𝐵 ∈ (seq1((ℎ ∘ 1st ),
(ℕ × {𝑧}))‘𝑚)} = {𝑛 ∈ ℕ ∣ 𝐵 ∈ (seq1((ℎ ∘ 1st ), (ℕ ×
{𝑧}))‘𝑛)} | 
| 139 |  | eqid 2737 | . . . . . . . . 9
⊢
inf({𝑚 ∈
ℕ ∣ 𝐵 ∈
(seq1((ℎ ∘
1st ), (ℕ × {𝑧}))‘𝑚)}, ℝ, < ) = inf({𝑚 ∈ ℕ ∣ 𝐵 ∈ (seq1((ℎ ∘ 1st ), (ℕ ×
{𝑧}))‘𝑚)}, ℝ, <
) | 
| 140 | 113, 114,
115, 116, 117, 118, 119, 120, 121, 14, 122, 125, 126, 134, 135, 138, 139 | ovolicc2lem4 25555 | . . . . . . . 8
⊢ ((𝜑 ∧ ((𝑧 ∈ 𝑈 ∧ 𝐴 ∈ 𝑧) ∧ (ℎ:𝑇⟶𝑇 ∧ ∀𝑥 ∈ 𝑇 if((2nd ‘(𝐹‘(𝐺‘𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑥))), 𝐵) ∈ (ℎ‘𝑥)))) → (𝐵 − 𝐴) ≤ sup(ran 𝑆, ℝ*, <
)) | 
| 141 | 140 | anassrs 467 | . . . . . . 7
⊢ (((𝜑 ∧ (𝑧 ∈ 𝑈 ∧ 𝐴 ∈ 𝑧)) ∧ (ℎ:𝑇⟶𝑇 ∧ ∀𝑥 ∈ 𝑇 if((2nd ‘(𝐹‘(𝐺‘𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑥))), 𝐵) ∈ (ℎ‘𝑥))) → (𝐵 − 𝐴) ≤ sup(ran 𝑆, ℝ*, <
)) | 
| 142 | 141 | expr 456 | . . . . . 6
⊢ (((𝜑 ∧ (𝑧 ∈ 𝑈 ∧ 𝐴 ∈ 𝑧)) ∧ ℎ:𝑇⟶𝑇) → (∀𝑥 ∈ 𝑇 if((2nd ‘(𝐹‘(𝐺‘𝑥))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑥))), 𝐵) ∈ (ℎ‘𝑥) → (𝐵 − 𝐴) ≤ sup(ran 𝑆, ℝ*, <
))) | 
| 143 | 112, 142 | biimtrrid 243 | . . . . 5
⊢ (((𝜑 ∧ (𝑧 ∈ 𝑈 ∧ 𝐴 ∈ 𝑧)) ∧ ℎ:𝑇⟶𝑇) → (∀𝑡 ∈ 𝑇 if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ (ℎ‘𝑡) → (𝐵 − 𝐴) ≤ sup(ran 𝑆, ℝ*, <
))) | 
| 144 | 143 | expimpd 453 | . . . 4
⊢ ((𝜑 ∧ (𝑧 ∈ 𝑈 ∧ 𝐴 ∈ 𝑧)) → ((ℎ:𝑇⟶𝑇 ∧ ∀𝑡 ∈ 𝑇 if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ (ℎ‘𝑡)) → (𝐵 − 𝐴) ≤ sup(ran 𝑆, ℝ*, <
))) | 
| 145 | 144 | exlimdv 1933 | . . 3
⊢ ((𝜑 ∧ (𝑧 ∈ 𝑈 ∧ 𝐴 ∈ 𝑧)) → (∃ℎ(ℎ:𝑇⟶𝑇 ∧ ∀𝑡 ∈ 𝑇 if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ (ℎ‘𝑡)) → (𝐵 − 𝐴) ≤ sup(ran 𝑆, ℝ*, <
))) | 
| 146 | 105, 145 | mpd 15 | . 2
⊢ ((𝜑 ∧ (𝑧 ∈ 𝑈 ∧ 𝐴 ∈ 𝑧)) → (𝐵 − 𝐴) ≤ sup(ran 𝑆, ℝ*, <
)) | 
| 147 | 11, 146 | rexlimddv 3161 | 1
⊢ (𝜑 → (𝐵 − 𝐴) ≤ sup(ran 𝑆, ℝ*, <
)) |