| Step | Hyp | Ref
| Expression |
| 1 | | fourierdlem82.2 |
. . . . 5
⊢ (𝜑 → 𝐴 ∈ ℝ) |
| 2 | | fourierdlem82.3 |
. . . . 5
⊢ (𝜑 → 𝐵 ∈ ℝ) |
| 3 | | fourierdlem82.9 |
. . . . 5
⊢ (𝜑 → 𝑋 ∈ ℝ) |
| 4 | | fourierdlem82.4 |
. . . . . 6
⊢ (𝜑 → 𝐴 < 𝐵) |
| 5 | 1, 2, 4 | ltled 11409 |
. . . . 5
⊢ (𝜑 → 𝐴 ≤ 𝐵) |
| 6 | 1, 2, 3, 5 | lesub1dd 11879 |
. . . 4
⊢ (𝜑 → (𝐴 − 𝑋) ≤ (𝐵 − 𝑋)) |
| 7 | 6 | ditgpos 25891 |
. . 3
⊢ (𝜑 → ⨜[(𝐴 − 𝑋) → (𝐵 − 𝑋)](𝐺‘(𝑋 + 𝑡)) d𝑡 = ∫((𝐴 − 𝑋)(,)(𝐵 − 𝑋))(𝐺‘(𝑋 + 𝑡)) d𝑡) |
| 8 | | fourierdlem82.1 |
. . . . . . 7
⊢ 𝐺 = (𝑥 ∈ (𝐴[,]𝐵) ↦ if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, ((𝐹 ↾ (𝐴(,)𝐵))‘𝑥)))) |
| 9 | | iftrue 4531 |
. . . . . . . . . . . 12
⊢ (𝑥 = 𝐴 → if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, ((𝐹 ↾ (𝐴(,)𝐵))‘𝑥))) = 𝑅) |
| 10 | 9 | adantl 481 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 = 𝐴) → if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, ((𝐹 ↾ (𝐴(,)𝐵))‘𝑥))) = 𝑅) |
| 11 | | iftrue 4531 |
. . . . . . . . . . . 12
⊢ (𝑥 = 𝐴 → if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥))) = 𝑅) |
| 12 | 11 | adantl 481 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 = 𝐴) → if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥))) = 𝑅) |
| 13 | 10, 12 | eqtr4d 2780 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 = 𝐴) → if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, ((𝐹 ↾ (𝐴(,)𝐵))‘𝑥))) = if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥)))) |
| 14 | 13 | adantlr 715 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ 𝑥 = 𝐴) → if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, ((𝐹 ↾ (𝐴(,)𝐵))‘𝑥))) = if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥)))) |
| 15 | | iffalse 4534 |
. . . . . . . . . . . . 13
⊢ (¬
𝑥 = 𝐴 → if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, ((𝐹 ↾ (𝐴(,)𝐵))‘𝑥))) = if(𝑥 = 𝐵, 𝐿, ((𝐹 ↾ (𝐴(,)𝐵))‘𝑥))) |
| 16 | | iftrue 4531 |
. . . . . . . . . . . . 13
⊢ (𝑥 = 𝐵 → if(𝑥 = 𝐵, 𝐿, ((𝐹 ↾ (𝐴(,)𝐵))‘𝑥)) = 𝐿) |
| 17 | 15, 16 | sylan9eq 2797 |
. . . . . . . . . . . 12
⊢ ((¬
𝑥 = 𝐴 ∧ 𝑥 = 𝐵) → if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, ((𝐹 ↾ (𝐴(,)𝐵))‘𝑥))) = 𝐿) |
| 18 | 17 | adantll 714 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) ∧ 𝑥 = 𝐵) → if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, ((𝐹 ↾ (𝐴(,)𝐵))‘𝑥))) = 𝐿) |
| 19 | | iffalse 4534 |
. . . . . . . . . . . . 13
⊢ (¬
𝑥 = 𝐴 → if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥))) = if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥))) |
| 20 | | iftrue 4531 |
. . . . . . . . . . . . 13
⊢ (𝑥 = 𝐵 → if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥)) = 𝐿) |
| 21 | 19, 20 | sylan9eq 2797 |
. . . . . . . . . . . 12
⊢ ((¬
𝑥 = 𝐴 ∧ 𝑥 = 𝐵) → if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥))) = 𝐿) |
| 22 | 21 | adantll 714 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) ∧ 𝑥 = 𝐵) → if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥))) = 𝐿) |
| 23 | 18, 22 | eqtr4d 2780 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) ∧ 𝑥 = 𝐵) → if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, ((𝐹 ↾ (𝐴(,)𝐵))‘𝑥))) = if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥)))) |
| 24 | | iffalse 4534 |
. . . . . . . . . . . 12
⊢ (¬
𝑥 = 𝐵 → if(𝑥 = 𝐵, 𝐿, ((𝐹 ↾ (𝐴(,)𝐵))‘𝑥)) = ((𝐹 ↾ (𝐴(,)𝐵))‘𝑥)) |
| 25 | 24 | adantl 481 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) ∧ ¬ 𝑥 = 𝐵) → if(𝑥 = 𝐵, 𝐿, ((𝐹 ↾ (𝐴(,)𝐵))‘𝑥)) = ((𝐹 ↾ (𝐴(,)𝐵))‘𝑥)) |
| 26 | 15 | ad2antlr 727 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) ∧ ¬ 𝑥 = 𝐵) → if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, ((𝐹 ↾ (𝐴(,)𝐵))‘𝑥))) = if(𝑥 = 𝐵, 𝐿, ((𝐹 ↾ (𝐴(,)𝐵))‘𝑥))) |
| 27 | | iffalse 4534 |
. . . . . . . . . . . . 13
⊢ (¬
𝑥 = 𝐵 → if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥)) = (𝐹‘𝑥)) |
| 28 | 27 | adantl 481 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) ∧ ¬ 𝑥 = 𝐵) → if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥)) = (𝐹‘𝑥)) |
| 29 | 19 | ad2antlr 727 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) ∧ ¬ 𝑥 = 𝐵) → if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥))) = if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥))) |
| 30 | 1 | rexrd 11311 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝐴 ∈
ℝ*) |
| 31 | 30 | ad3antrrr 730 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) ∧ ¬ 𝑥 = 𝐵) → 𝐴 ∈
ℝ*) |
| 32 | 2 | rexrd 11311 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝐵 ∈
ℝ*) |
| 33 | 32 | ad3antrrr 730 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) ∧ ¬ 𝑥 = 𝐵) → 𝐵 ∈
ℝ*) |
| 34 | 1 | adantr 480 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝐴 ∈ ℝ) |
| 35 | 2 | adantr 480 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝐵 ∈ ℝ) |
| 36 | | simpr 484 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝑥 ∈ (𝐴[,]𝐵)) |
| 37 | | eliccre 45518 |
. . . . . . . . . . . . . . . 16
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝑥 ∈ ℝ) |
| 38 | 34, 35, 36, 37 | syl3anc 1373 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝑥 ∈ ℝ) |
| 39 | 38 | ad2antrr 726 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) ∧ ¬ 𝑥 = 𝐵) → 𝑥 ∈ ℝ) |
| 40 | 1 | ad2antrr 726 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) → 𝐴 ∈ ℝ) |
| 41 | 38 | adantr 480 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) → 𝑥 ∈ ℝ) |
| 42 | | elicc2 13452 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝑥 ∈ (𝐴[,]𝐵) ↔ (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥 ∧ 𝑥 ≤ 𝐵))) |
| 43 | 34, 35, 42 | syl2anc 584 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝑥 ∈ (𝐴[,]𝐵) ↔ (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥 ∧ 𝑥 ≤ 𝐵))) |
| 44 | 36, 43 | mpbid 232 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥 ∧ 𝑥 ≤ 𝐵)) |
| 45 | 44 | simp2d 1144 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝐴 ≤ 𝑥) |
| 46 | 45 | adantr 480 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) → 𝐴 ≤ 𝑥) |
| 47 | | neqne 2948 |
. . . . . . . . . . . . . . . . 17
⊢ (¬
𝑥 = 𝐴 → 𝑥 ≠ 𝐴) |
| 48 | 47 | adantl 481 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) → 𝑥 ≠ 𝐴) |
| 49 | 40, 41, 46, 48 | leneltd 11415 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) → 𝐴 < 𝑥) |
| 50 | 49 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) ∧ ¬ 𝑥 = 𝐵) → 𝐴 < 𝑥) |
| 51 | 38 | adantr 480 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐵) → 𝑥 ∈ ℝ) |
| 52 | 2 | ad2antrr 726 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐵) → 𝐵 ∈ ℝ) |
| 53 | 44 | simp3d 1145 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝑥 ≤ 𝐵) |
| 54 | 53 | adantr 480 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐵) → 𝑥 ≤ 𝐵) |
| 55 | | nesym 2997 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝐵 ≠ 𝑥 ↔ ¬ 𝑥 = 𝐵) |
| 56 | 55 | biimpri 228 |
. . . . . . . . . . . . . . . . 17
⊢ (¬
𝑥 = 𝐵 → 𝐵 ≠ 𝑥) |
| 57 | 56 | adantl 481 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐵) → 𝐵 ≠ 𝑥) |
| 58 | 51, 52, 54, 57 | leneltd 11415 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐵) → 𝑥 < 𝐵) |
| 59 | 58 | adantlr 715 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) ∧ ¬ 𝑥 = 𝐵) → 𝑥 < 𝐵) |
| 60 | 31, 33, 39, 50, 59 | eliood 45511 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) ∧ ¬ 𝑥 = 𝐵) → 𝑥 ∈ (𝐴(,)𝐵)) |
| 61 | | fvres 6925 |
. . . . . . . . . . . . 13
⊢ (𝑥 ∈ (𝐴(,)𝐵) → ((𝐹 ↾ (𝐴(,)𝐵))‘𝑥) = (𝐹‘𝑥)) |
| 62 | 60, 61 | syl 17 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) ∧ ¬ 𝑥 = 𝐵) → ((𝐹 ↾ (𝐴(,)𝐵))‘𝑥) = (𝐹‘𝑥)) |
| 63 | 28, 29, 62 | 3eqtr4d 2787 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) ∧ ¬ 𝑥 = 𝐵) → if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥))) = ((𝐹 ↾ (𝐴(,)𝐵))‘𝑥)) |
| 64 | 25, 26, 63 | 3eqtr4d 2787 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) ∧ ¬ 𝑥 = 𝐵) → if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, ((𝐹 ↾ (𝐴(,)𝐵))‘𝑥))) = if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥)))) |
| 65 | 23, 64 | pm2.61dan 813 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑥 = 𝐴) → if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, ((𝐹 ↾ (𝐴(,)𝐵))‘𝑥))) = if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥)))) |
| 66 | 14, 65 | pm2.61dan 813 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, ((𝐹 ↾ (𝐴(,)𝐵))‘𝑥))) = if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥)))) |
| 67 | 66 | mpteq2dva 5242 |
. . . . . . 7
⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↦ if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, ((𝐹 ↾ (𝐴(,)𝐵))‘𝑥)))) = (𝑥 ∈ (𝐴[,]𝐵) ↦ if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥))))) |
| 68 | 8, 67 | eqtrid 2789 |
. . . . . 6
⊢ (𝜑 → 𝐺 = (𝑥 ∈ (𝐴[,]𝐵) ↦ if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥))))) |
| 69 | 68 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → 𝐺 = (𝑥 ∈ (𝐴[,]𝐵) ↦ if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥))))) |
| 70 | | eqeq1 2741 |
. . . . . . 7
⊢ (𝑥 = (𝑋 + 𝑡) → (𝑥 = 𝐴 ↔ (𝑋 + 𝑡) = 𝐴)) |
| 71 | | eqeq1 2741 |
. . . . . . . 8
⊢ (𝑥 = (𝑋 + 𝑡) → (𝑥 = 𝐵 ↔ (𝑋 + 𝑡) = 𝐵)) |
| 72 | | fveq2 6906 |
. . . . . . . 8
⊢ (𝑥 = (𝑋 + 𝑡) → (𝐹‘𝑥) = (𝐹‘(𝑋 + 𝑡))) |
| 73 | 71, 72 | ifbieq2d 4552 |
. . . . . . 7
⊢ (𝑥 = (𝑋 + 𝑡) → if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥)) = if((𝑋 + 𝑡) = 𝐵, 𝐿, (𝐹‘(𝑋 + 𝑡)))) |
| 74 | 70, 73 | ifbieq2d 4552 |
. . . . . 6
⊢ (𝑥 = (𝑋 + 𝑡) → if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥))) = if((𝑋 + 𝑡) = 𝐴, 𝑅, if((𝑋 + 𝑡) = 𝐵, 𝐿, (𝐹‘(𝑋 + 𝑡))))) |
| 75 | 1 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → 𝐴 ∈ ℝ) |
| 76 | | simpr 484 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) |
| 77 | 1, 3 | resubcld 11691 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (𝐴 − 𝑋) ∈ ℝ) |
| 78 | 77 | rexrd 11311 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝐴 − 𝑋) ∈
ℝ*) |
| 79 | 78 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → (𝐴 − 𝑋) ∈
ℝ*) |
| 80 | 2, 3 | resubcld 11691 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (𝐵 − 𝑋) ∈ ℝ) |
| 81 | 80 | rexrd 11311 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝐵 − 𝑋) ∈
ℝ*) |
| 82 | 81 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → (𝐵 − 𝑋) ∈
ℝ*) |
| 83 | | elioo2 13428 |
. . . . . . . . . . . . . 14
⊢ (((𝐴 − 𝑋) ∈ ℝ* ∧ (𝐵 − 𝑋) ∈ ℝ*) → (𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋)) ↔ (𝑡 ∈ ℝ ∧ (𝐴 − 𝑋) < 𝑡 ∧ 𝑡 < (𝐵 − 𝑋)))) |
| 84 | 79, 82, 83 | syl2anc 584 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → (𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋)) ↔ (𝑡 ∈ ℝ ∧ (𝐴 − 𝑋) < 𝑡 ∧ 𝑡 < (𝐵 − 𝑋)))) |
| 85 | 76, 84 | mpbid 232 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → (𝑡 ∈ ℝ ∧ (𝐴 − 𝑋) < 𝑡 ∧ 𝑡 < (𝐵 − 𝑋))) |
| 86 | 85 | simp2d 1144 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → (𝐴 − 𝑋) < 𝑡) |
| 87 | 3 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → 𝑋 ∈ ℝ) |
| 88 | 85 | simp1d 1143 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → 𝑡 ∈ ℝ) |
| 89 | 75, 87, 88 | ltsubadd2d 11861 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → ((𝐴 − 𝑋) < 𝑡 ↔ 𝐴 < (𝑋 + 𝑡))) |
| 90 | 86, 89 | mpbid 232 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → 𝐴 < (𝑋 + 𝑡)) |
| 91 | 75, 90 | gtned 11396 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → (𝑋 + 𝑡) ≠ 𝐴) |
| 92 | 91 | neneqd 2945 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → ¬ (𝑋 + 𝑡) = 𝐴) |
| 93 | 92 | iffalsed 4536 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → if((𝑋 + 𝑡) = 𝐴, 𝑅, if((𝑋 + 𝑡) = 𝐵, 𝐿, (𝐹‘(𝑋 + 𝑡)))) = if((𝑋 + 𝑡) = 𝐵, 𝐿, (𝐹‘(𝑋 + 𝑡)))) |
| 94 | 87, 88 | readdcld 11290 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → (𝑋 + 𝑡) ∈ ℝ) |
| 95 | 85 | simp3d 1145 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → 𝑡 < (𝐵 − 𝑋)) |
| 96 | 2 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → 𝐵 ∈ ℝ) |
| 97 | 87, 88, 96 | ltaddsub2d 11864 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → ((𝑋 + 𝑡) < 𝐵 ↔ 𝑡 < (𝐵 − 𝑋))) |
| 98 | 95, 97 | mpbird 257 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → (𝑋 + 𝑡) < 𝐵) |
| 99 | 94, 98 | ltned 11397 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → (𝑋 + 𝑡) ≠ 𝐵) |
| 100 | 99 | neneqd 2945 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → ¬ (𝑋 + 𝑡) = 𝐵) |
| 101 | 100 | iffalsed 4536 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → if((𝑋 + 𝑡) = 𝐵, 𝐿, (𝐹‘(𝑋 + 𝑡))) = (𝐹‘(𝑋 + 𝑡))) |
| 102 | 93, 101 | eqtrd 2777 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → if((𝑋 + 𝑡) = 𝐴, 𝑅, if((𝑋 + 𝑡) = 𝐵, 𝐿, (𝐹‘(𝑋 + 𝑡)))) = (𝐹‘(𝑋 + 𝑡))) |
| 103 | 74, 102 | sylan9eqr 2799 |
. . . . 5
⊢ (((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) ∧ 𝑥 = (𝑋 + 𝑡)) → if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥))) = (𝐹‘(𝑋 + 𝑡))) |
| 104 | 75, 94, 90 | ltled 11409 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → 𝐴 ≤ (𝑋 + 𝑡)) |
| 105 | 94, 96, 98 | ltled 11409 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → (𝑋 + 𝑡) ≤ 𝐵) |
| 106 | 75, 96, 94, 104, 105 | eliccd 45517 |
. . . . 5
⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → (𝑋 + 𝑡) ∈ (𝐴[,]𝐵)) |
| 107 | | fourierdlem82.5 |
. . . . . . . 8
⊢ (𝜑 → 𝐹:(𝐴[,]𝐵)⟶ℂ) |
| 108 | 107 | ffund 6740 |
. . . . . . 7
⊢ (𝜑 → Fun 𝐹) |
| 109 | 108 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → Fun 𝐹) |
| 110 | 107 | fdmd 6746 |
. . . . . . . . 9
⊢ (𝜑 → dom 𝐹 = (𝐴[,]𝐵)) |
| 111 | 110 | eqcomd 2743 |
. . . . . . . 8
⊢ (𝜑 → (𝐴[,]𝐵) = dom 𝐹) |
| 112 | 111 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → (𝐴[,]𝐵) = dom 𝐹) |
| 113 | 106, 112 | eleqtrd 2843 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → (𝑋 + 𝑡) ∈ dom 𝐹) |
| 114 | | fvelrn 7096 |
. . . . . 6
⊢ ((Fun
𝐹 ∧ (𝑋 + 𝑡) ∈ dom 𝐹) → (𝐹‘(𝑋 + 𝑡)) ∈ ran 𝐹) |
| 115 | 109, 113,
114 | syl2anc 584 |
. . . . 5
⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → (𝐹‘(𝑋 + 𝑡)) ∈ ran 𝐹) |
| 116 | 69, 103, 106, 115 | fvmptd 7023 |
. . . 4
⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)(,)(𝐵 − 𝑋))) → (𝐺‘(𝑋 + 𝑡)) = (𝐹‘(𝑋 + 𝑡))) |
| 117 | 116 | itgeq2dv 25817 |
. . 3
⊢ (𝜑 → ∫((𝐴 − 𝑋)(,)(𝐵 − 𝑋))(𝐺‘(𝑋 + 𝑡)) d𝑡 = ∫((𝐴 − 𝑋)(,)(𝐵 − 𝑋))(𝐹‘(𝑋 + 𝑡)) d𝑡) |
| 118 | 107 | frnd 6744 |
. . . . . 6
⊢ (𝜑 → ran 𝐹 ⊆ ℂ) |
| 119 | 118 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → ran 𝐹 ⊆ ℂ) |
| 120 | 108 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → Fun 𝐹) |
| 121 | 1 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → 𝐴 ∈ ℝ) |
| 122 | 2 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → 𝐵 ∈ ℝ) |
| 123 | 3 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → 𝑋 ∈ ℝ) |
| 124 | 77 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → (𝐴 − 𝑋) ∈ ℝ) |
| 125 | 80 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → (𝐵 − 𝑋) ∈ ℝ) |
| 126 | | simpr 484 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → 𝑡 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) |
| 127 | | eliccre 45518 |
. . . . . . . . . 10
⊢ (((𝐴 − 𝑋) ∈ ℝ ∧ (𝐵 − 𝑋) ∈ ℝ ∧ 𝑡 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → 𝑡 ∈ ℝ) |
| 128 | 124, 125,
126, 127 | syl3anc 1373 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → 𝑡 ∈ ℝ) |
| 129 | 123, 128 | readdcld 11290 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → (𝑋 + 𝑡) ∈ ℝ) |
| 130 | | elicc2 13452 |
. . . . . . . . . . . 12
⊢ (((𝐴 − 𝑋) ∈ ℝ ∧ (𝐵 − 𝑋) ∈ ℝ) → (𝑡 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋)) ↔ (𝑡 ∈ ℝ ∧ (𝐴 − 𝑋) ≤ 𝑡 ∧ 𝑡 ≤ (𝐵 − 𝑋)))) |
| 131 | 124, 125,
130 | syl2anc 584 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → (𝑡 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋)) ↔ (𝑡 ∈ ℝ ∧ (𝐴 − 𝑋) ≤ 𝑡 ∧ 𝑡 ≤ (𝐵 − 𝑋)))) |
| 132 | 126, 131 | mpbid 232 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → (𝑡 ∈ ℝ ∧ (𝐴 − 𝑋) ≤ 𝑡 ∧ 𝑡 ≤ (𝐵 − 𝑋))) |
| 133 | 132 | simp2d 1144 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → (𝐴 − 𝑋) ≤ 𝑡) |
| 134 | 121, 123,
128 | lesubadd2d 11862 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → ((𝐴 − 𝑋) ≤ 𝑡 ↔ 𝐴 ≤ (𝑋 + 𝑡))) |
| 135 | 133, 134 | mpbid 232 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → 𝐴 ≤ (𝑋 + 𝑡)) |
| 136 | 132 | simp3d 1145 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → 𝑡 ≤ (𝐵 − 𝑋)) |
| 137 | 123, 128,
122 | leaddsub2d 11865 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → ((𝑋 + 𝑡) ≤ 𝐵 ↔ 𝑡 ≤ (𝐵 − 𝑋))) |
| 138 | 136, 137 | mpbird 257 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → (𝑋 + 𝑡) ≤ 𝐵) |
| 139 | 121, 122,
129, 135, 138 | eliccd 45517 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → (𝑋 + 𝑡) ∈ (𝐴[,]𝐵)) |
| 140 | 111 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → (𝐴[,]𝐵) = dom 𝐹) |
| 141 | 139, 140 | eleqtrd 2843 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → (𝑋 + 𝑡) ∈ dom 𝐹) |
| 142 | 120, 141,
114 | syl2anc 584 |
. . . . 5
⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → (𝐹‘(𝑋 + 𝑡)) ∈ ran 𝐹) |
| 143 | 119, 142 | sseldd 3984 |
. . . 4
⊢ ((𝜑 ∧ 𝑡 ∈ ((𝐴 − 𝑋)[,](𝐵 − 𝑋))) → (𝐹‘(𝑋 + 𝑡)) ∈ ℂ) |
| 144 | 77, 80, 143 | itgioo 25851 |
. . 3
⊢ (𝜑 → ∫((𝐴 − 𝑋)(,)(𝐵 − 𝑋))(𝐹‘(𝑋 + 𝑡)) d𝑡 = ∫((𝐴 − 𝑋)[,](𝐵 − 𝑋))(𝐹‘(𝑋 + 𝑡)) d𝑡) |
| 145 | 7, 117, 144 | 3eqtrrd 2782 |
. 2
⊢ (𝜑 → ∫((𝐴 − 𝑋)[,](𝐵 − 𝑋))(𝐹‘(𝑋 + 𝑡)) d𝑡 = ⨜[(𝐴 − 𝑋) → (𝐵 − 𝑋)](𝐺‘(𝑋 + 𝑡)) d𝑡) |
| 146 | | nfv 1914 |
. . . 4
⊢
Ⅎ𝑥𝜑 |
| 147 | | fourierdlem82.6 |
. . . 4
⊢ (𝜑 → (𝐹 ↾ (𝐴(,)𝐵)) ∈ ((𝐴(,)𝐵)–cn→ℂ)) |
| 148 | | fourierdlem82.7 |
. . . . 5
⊢ (𝜑 → 𝐿 ∈ (𝐹 limℂ 𝐵)) |
| 149 | 1, 2, 4, 107 | limcicciooub 45652 |
. . . . 5
⊢ (𝜑 → ((𝐹 ↾ (𝐴(,)𝐵)) limℂ 𝐵) = (𝐹 limℂ 𝐵)) |
| 150 | 148, 149 | eleqtrrd 2844 |
. . . 4
⊢ (𝜑 → 𝐿 ∈ ((𝐹 ↾ (𝐴(,)𝐵)) limℂ 𝐵)) |
| 151 | | fourierdlem82.8 |
. . . . 5
⊢ (𝜑 → 𝑅 ∈ (𝐹 limℂ 𝐴)) |
| 152 | 1, 2, 4, 107 | limciccioolb 45636 |
. . . . 5
⊢ (𝜑 → ((𝐹 ↾ (𝐴(,)𝐵)) limℂ 𝐴) = (𝐹 limℂ 𝐴)) |
| 153 | 151, 152 | eleqtrrd 2844 |
. . . 4
⊢ (𝜑 → 𝑅 ∈ ((𝐹 ↾ (𝐴(,)𝐵)) limℂ 𝐴)) |
| 154 | 146, 8, 1, 2, 147, 150, 153 | cncfiooicc 45909 |
. . 3
⊢ (𝜑 → 𝐺 ∈ ((𝐴[,]𝐵)–cn→ℂ)) |
| 155 | 1, 2, 5, 3, 154 | itgsbtaddcnst 45997 |
. 2
⊢ (𝜑 → ⨜[(𝐴 − 𝑋) → (𝐵 − 𝑋)](𝐺‘(𝑋 + 𝑡)) d𝑡 = ⨜[𝐴 → 𝐵](𝐺‘𝑠) d𝑠) |
| 156 | 5 | ditgpos 25891 |
. . 3
⊢ (𝜑 → ⨜[𝐴 → 𝐵](𝐺‘𝑠) d𝑠 = ∫(𝐴(,)𝐵)(𝐺‘𝑠) d𝑠) |
| 157 | | fveq2 6906 |
. . . . 5
⊢ (𝑠 = 𝑡 → (𝐺‘𝑠) = (𝐺‘𝑡)) |
| 158 | 157 | cbvitgv 25812 |
. . . 4
⊢
∫(𝐴(,)𝐵)(𝐺‘𝑠) d𝑠 = ∫(𝐴(,)𝐵)(𝐺‘𝑡) d𝑡 |
| 159 | 8 | a1i 11 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → 𝐺 = (𝑥 ∈ (𝐴[,]𝐵) ↦ if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, ((𝐹 ↾ (𝐴(,)𝐵))‘𝑥))))) |
| 160 | 1 | ad2antrr 726 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝑡) → 𝐴 ∈ ℝ) |
| 161 | | simplr 769 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝑡) → 𝑡 ∈ (𝐴(,)𝐵)) |
| 162 | 30 | ad2antrr 726 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝑡) → 𝐴 ∈
ℝ*) |
| 163 | 32 | ad2antrr 726 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝑡) → 𝐵 ∈
ℝ*) |
| 164 | | elioo2 13428 |
. . . . . . . . . . . . . 14
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ*) → (𝑡 ∈ (𝐴(,)𝐵) ↔ (𝑡 ∈ ℝ ∧ 𝐴 < 𝑡 ∧ 𝑡 < 𝐵))) |
| 165 | 162, 163,
164 | syl2anc 584 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝑡) → (𝑡 ∈ (𝐴(,)𝐵) ↔ (𝑡 ∈ ℝ ∧ 𝐴 < 𝑡 ∧ 𝑡 < 𝐵))) |
| 166 | 161, 165 | mpbid 232 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝑡) → (𝑡 ∈ ℝ ∧ 𝐴 < 𝑡 ∧ 𝑡 < 𝐵)) |
| 167 | 166 | simp2d 1144 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝑡) → 𝐴 < 𝑡) |
| 168 | | simpr 484 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝑡) → 𝑥 = 𝑡) |
| 169 | 167, 168 | breqtrrd 5171 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝑡) → 𝐴 < 𝑥) |
| 170 | 160, 169 | gtned 11396 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝑡) → 𝑥 ≠ 𝐴) |
| 171 | 170 | neneqd 2945 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝑡) → ¬ 𝑥 = 𝐴) |
| 172 | 171 | iffalsed 4536 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝑡) → if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, ((𝐹 ↾ (𝐴(,)𝐵))‘𝑥))) = if(𝑥 = 𝐵, 𝐿, ((𝐹 ↾ (𝐴(,)𝐵))‘𝑥))) |
| 173 | 166 | simp1d 1143 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝑡) → 𝑡 ∈ ℝ) |
| 174 | 168, 173 | eqeltrd 2841 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝑡) → 𝑥 ∈ ℝ) |
| 175 | 166 | simp3d 1145 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝑡) → 𝑡 < 𝐵) |
| 176 | 168, 175 | eqbrtrd 5165 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝑡) → 𝑥 < 𝐵) |
| 177 | 174, 176 | ltned 11397 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝑡) → 𝑥 ≠ 𝐵) |
| 178 | 177 | neneqd 2945 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝑡) → ¬ 𝑥 = 𝐵) |
| 179 | 178 | iffalsed 4536 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝑡) → if(𝑥 = 𝐵, 𝐿, ((𝐹 ↾ (𝐴(,)𝐵))‘𝑥)) = ((𝐹 ↾ (𝐴(,)𝐵))‘𝑥)) |
| 180 | 168, 161 | eqeltrd 2841 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝑡) → 𝑥 ∈ (𝐴(,)𝐵)) |
| 181 | 180, 61 | syl 17 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝑡) → ((𝐹 ↾ (𝐴(,)𝐵))‘𝑥) = (𝐹‘𝑥)) |
| 182 | | fveq2 6906 |
. . . . . . . . 9
⊢ (𝑥 = 𝑡 → (𝐹‘𝑥) = (𝐹‘𝑡)) |
| 183 | 182 | adantl 481 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝑡) → (𝐹‘𝑥) = (𝐹‘𝑡)) |
| 184 | 181, 183 | eqtrd 2777 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝑡) → ((𝐹 ↾ (𝐴(,)𝐵))‘𝑥) = (𝐹‘𝑡)) |
| 185 | 172, 179,
184 | 3eqtrd 2781 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) ∧ 𝑥 = 𝑡) → if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, ((𝐹 ↾ (𝐴(,)𝐵))‘𝑥))) = (𝐹‘𝑡)) |
| 186 | | ioossicc 13473 |
. . . . . . 7
⊢ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵) |
| 187 | | simpr 484 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → 𝑡 ∈ (𝐴(,)𝐵)) |
| 188 | 186, 187 | sselid 3981 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → 𝑡 ∈ (𝐴[,]𝐵)) |
| 189 | 108 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → Fun 𝐹) |
| 190 | 111 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → (𝐴[,]𝐵) = dom 𝐹) |
| 191 | 188, 190 | eleqtrd 2843 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → 𝑡 ∈ dom 𝐹) |
| 192 | | fvelrn 7096 |
. . . . . . 7
⊢ ((Fun
𝐹 ∧ 𝑡 ∈ dom 𝐹) → (𝐹‘𝑡) ∈ ran 𝐹) |
| 193 | 189, 191,
192 | syl2anc 584 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → (𝐹‘𝑡) ∈ ran 𝐹) |
| 194 | 159, 185,
188, 193 | fvmptd 7023 |
. . . . 5
⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → (𝐺‘𝑡) = (𝐹‘𝑡)) |
| 195 | 194 | itgeq2dv 25817 |
. . . 4
⊢ (𝜑 → ∫(𝐴(,)𝐵)(𝐺‘𝑡) d𝑡 = ∫(𝐴(,)𝐵)(𝐹‘𝑡) d𝑡) |
| 196 | 158, 195 | eqtrid 2789 |
. . 3
⊢ (𝜑 → ∫(𝐴(,)𝐵)(𝐺‘𝑠) d𝑠 = ∫(𝐴(,)𝐵)(𝐹‘𝑡) d𝑡) |
| 197 | 107 | ffvelcdmda 7104 |
. . . 4
⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑡) ∈ ℂ) |
| 198 | 1, 2, 197 | itgioo 25851 |
. . 3
⊢ (𝜑 → ∫(𝐴(,)𝐵)(𝐹‘𝑡) d𝑡 = ∫(𝐴[,]𝐵)(𝐹‘𝑡) d𝑡) |
| 199 | 156, 196,
198 | 3eqtrd 2781 |
. 2
⊢ (𝜑 → ⨜[𝐴 → 𝐵](𝐺‘𝑠) d𝑠 = ∫(𝐴[,]𝐵)(𝐹‘𝑡) d𝑡) |
| 200 | 145, 155,
199 | 3eqtrrd 2782 |
1
⊢ (𝜑 → ∫(𝐴[,]𝐵)(𝐹‘𝑡) d𝑡 = ∫((𝐴 − 𝑋)[,](𝐵 − 𝑋))(𝐹‘(𝑋 + 𝑡)) d𝑡) |