Proof of Theorem metakunt27
| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | metakunt27.5 | . . . . 5
⊢ 𝐴 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1)))) | 
| 2 | 1 | a1i 11 | . . . 4
⊢ (𝜑 → 𝐴 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1))))) | 
| 3 |  | metakunt27.7 | . . . . . . . 8
⊢ (𝜑 → ¬ 𝑋 = 𝐼) | 
| 4 | 3 | adantr 480 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ¬ 𝑋 = 𝐼) | 
| 5 |  | simpr 484 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 = 𝑋) → 𝑥 = 𝑋) | 
| 6 | 5 | eqeq1d 2739 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝑥 = 𝐼 ↔ 𝑋 = 𝐼)) | 
| 7 | 6 | notbid 318 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (¬ 𝑥 = 𝐼 ↔ ¬ 𝑋 = 𝐼)) | 
| 8 | 4, 7 | mpbird 257 | . . . . . 6
⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ¬ 𝑥 = 𝐼) | 
| 9 | 8 | iffalsed 4536 | . . . . 5
⊢ ((𝜑 ∧ 𝑥 = 𝑋) → if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1))) = if(𝑥 < 𝐼, 𝑥, (𝑥 − 1))) | 
| 10 |  | metakunt27.8 | . . . . . . . . 9
⊢ (𝜑 → 𝑋 < 𝐼) | 
| 11 | 10 | adantr 480 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 = 𝑋) → 𝑋 < 𝐼) | 
| 12 | 5 | breq1d 5153 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝑥 < 𝐼 ↔ 𝑋 < 𝐼)) | 
| 13 | 11, 12 | mpbird 257 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑥 = 𝑋) → 𝑥 < 𝐼) | 
| 14 | 13 | iftrued 4533 | . . . . . 6
⊢ ((𝜑 ∧ 𝑥 = 𝑋) → if(𝑥 < 𝐼, 𝑥, (𝑥 − 1)) = 𝑥) | 
| 15 | 14, 5 | eqtrd 2777 | . . . . 5
⊢ ((𝜑 ∧ 𝑥 = 𝑋) → if(𝑥 < 𝐼, 𝑥, (𝑥 − 1)) = 𝑋) | 
| 16 | 9, 15 | eqtrd 2777 | . . . 4
⊢ ((𝜑 ∧ 𝑥 = 𝑋) → if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1))) = 𝑋) | 
| 17 |  | metakunt27.4 | . . . 4
⊢ (𝜑 → 𝑋 ∈ (1...𝑀)) | 
| 18 | 2, 16, 17, 17 | fvmptd 7023 | . . 3
⊢ (𝜑 → (𝐴‘𝑋) = 𝑋) | 
| 19 | 18 | fveq2d 6910 | . 2
⊢ (𝜑 → (𝐵‘(𝐴‘𝑋)) = (𝐵‘𝑋)) | 
| 20 |  | metakunt27.6 | . . . 4
⊢ 𝐵 = (𝑧 ∈ (1...𝑀) ↦ if(𝑧 = 𝑀, 𝑀, if(𝑧 < 𝐼, (𝑧 + (𝑀 − 𝐼)), (𝑧 + (1 − 𝐼))))) | 
| 21 | 20 | a1i 11 | . . 3
⊢ (𝜑 → 𝐵 = (𝑧 ∈ (1...𝑀) ↦ if(𝑧 = 𝑀, 𝑀, if(𝑧 < 𝐼, (𝑧 + (𝑀 − 𝐼)), (𝑧 + (1 − 𝐼)))))) | 
| 22 |  | elfznn 13593 | . . . . . . . . . . 11
⊢ (𝑋 ∈ (1...𝑀) → 𝑋 ∈ ℕ) | 
| 23 | 17, 22 | syl 17 | . . . . . . . . . 10
⊢ (𝜑 → 𝑋 ∈ ℕ) | 
| 24 | 23 | nnred 12281 | . . . . . . . . 9
⊢ (𝜑 → 𝑋 ∈ ℝ) | 
| 25 |  | metakunt27.2 | . . . . . . . . . . 11
⊢ (𝜑 → 𝐼 ∈ ℕ) | 
| 26 | 25 | nnred 12281 | . . . . . . . . . 10
⊢ (𝜑 → 𝐼 ∈ ℝ) | 
| 27 |  | metakunt27.1 | . . . . . . . . . . 11
⊢ (𝜑 → 𝑀 ∈ ℕ) | 
| 28 | 27 | nnred 12281 | . . . . . . . . . 10
⊢ (𝜑 → 𝑀 ∈ ℝ) | 
| 29 |  | metakunt27.3 | . . . . . . . . . 10
⊢ (𝜑 → 𝐼 ≤ 𝑀) | 
| 30 | 24, 26, 28, 10, 29 | ltletrd 11421 | . . . . . . . . 9
⊢ (𝜑 → 𝑋 < 𝑀) | 
| 31 | 24, 30 | ltned 11397 | . . . . . . . 8
⊢ (𝜑 → 𝑋 ≠ 𝑀) | 
| 32 | 31 | neneqd 2945 | . . . . . . 7
⊢ (𝜑 → ¬ 𝑋 = 𝑀) | 
| 33 | 32 | adantr 480 | . . . . . 6
⊢ ((𝜑 ∧ 𝑧 = 𝑋) → ¬ 𝑋 = 𝑀) | 
| 34 |  | simpr 484 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑧 = 𝑋) → 𝑧 = 𝑋) | 
| 35 | 34 | eqeq1d 2739 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑧 = 𝑋) → (𝑧 = 𝑀 ↔ 𝑋 = 𝑀)) | 
| 36 | 35 | notbid 318 | . . . . . 6
⊢ ((𝜑 ∧ 𝑧 = 𝑋) → (¬ 𝑧 = 𝑀 ↔ ¬ 𝑋 = 𝑀)) | 
| 37 | 33, 36 | mpbird 257 | . . . . 5
⊢ ((𝜑 ∧ 𝑧 = 𝑋) → ¬ 𝑧 = 𝑀) | 
| 38 | 37 | iffalsed 4536 | . . . 4
⊢ ((𝜑 ∧ 𝑧 = 𝑋) → if(𝑧 = 𝑀, 𝑀, if(𝑧 < 𝐼, (𝑧 + (𝑀 − 𝐼)), (𝑧 + (1 − 𝐼)))) = if(𝑧 < 𝐼, (𝑧 + (𝑀 − 𝐼)), (𝑧 + (1 − 𝐼)))) | 
| 39 | 10 | adantr 480 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑧 = 𝑋) → 𝑋 < 𝐼) | 
| 40 | 34 | breq1d 5153 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑧 = 𝑋) → (𝑧 < 𝐼 ↔ 𝑋 < 𝐼)) | 
| 41 | 39, 40 | mpbird 257 | . . . . . 6
⊢ ((𝜑 ∧ 𝑧 = 𝑋) → 𝑧 < 𝐼) | 
| 42 | 41 | iftrued 4533 | . . . . 5
⊢ ((𝜑 ∧ 𝑧 = 𝑋) → if(𝑧 < 𝐼, (𝑧 + (𝑀 − 𝐼)), (𝑧 + (1 − 𝐼))) = (𝑧 + (𝑀 − 𝐼))) | 
| 43 | 34 | oveq1d 7446 | . . . . 5
⊢ ((𝜑 ∧ 𝑧 = 𝑋) → (𝑧 + (𝑀 − 𝐼)) = (𝑋 + (𝑀 − 𝐼))) | 
| 44 | 42, 43 | eqtrd 2777 | . . . 4
⊢ ((𝜑 ∧ 𝑧 = 𝑋) → if(𝑧 < 𝐼, (𝑧 + (𝑀 − 𝐼)), (𝑧 + (1 − 𝐼))) = (𝑋 + (𝑀 − 𝐼))) | 
| 45 | 38, 44 | eqtrd 2777 | . . 3
⊢ ((𝜑 ∧ 𝑧 = 𝑋) → if(𝑧 = 𝑀, 𝑀, if(𝑧 < 𝐼, (𝑧 + (𝑀 − 𝐼)), (𝑧 + (1 − 𝐼)))) = (𝑋 + (𝑀 − 𝐼))) | 
| 46 | 17 | elfzelzd 13565 | . . . 4
⊢ (𝜑 → 𝑋 ∈ ℤ) | 
| 47 | 27 | nnzd 12640 | . . . . 5
⊢ (𝜑 → 𝑀 ∈ ℤ) | 
| 48 | 25 | nnzd 12640 | . . . . 5
⊢ (𝜑 → 𝐼 ∈ ℤ) | 
| 49 | 47, 48 | zsubcld 12727 | . . . 4
⊢ (𝜑 → (𝑀 − 𝐼) ∈ ℤ) | 
| 50 | 46, 49 | zaddcld 12726 | . . 3
⊢ (𝜑 → (𝑋 + (𝑀 − 𝐼)) ∈ ℤ) | 
| 51 | 21, 45, 17, 50 | fvmptd 7023 | . 2
⊢ (𝜑 → (𝐵‘𝑋) = (𝑋 + (𝑀 − 𝐼))) | 
| 52 | 19, 51 | eqtrd 2777 | 1
⊢ (𝜑 → (𝐵‘(𝐴‘𝑋)) = (𝑋 + (𝑀 − 𝐼))) |