Proof of Theorem cdleme31sn1
| Step | Hyp | Ref
| Expression |
| 1 | | cdleme31sn1.n |
. . . 4
⊢ 𝑁 = if(𝑠 ≤ (𝑃 ∨ 𝑄), 𝐼, 𝐷) |
| 2 | | eqid 2737 |
. . . 4
⊢ if(𝑅 ≤ (𝑃 ∨ 𝑄), ⦋𝑅 / 𝑠⦌𝐼, ⦋𝑅 / 𝑠⦌𝐷) = if(𝑅 ≤ (𝑃 ∨ 𝑄), ⦋𝑅 / 𝑠⦌𝐼, ⦋𝑅 / 𝑠⦌𝐷) |
| 3 | 1, 2 | cdleme31sn 40382 |
. . 3
⊢ (𝑅 ∈ 𝐴 → ⦋𝑅 / 𝑠⦌𝑁 = if(𝑅 ≤ (𝑃 ∨ 𝑄), ⦋𝑅 / 𝑠⦌𝐼, ⦋𝑅 / 𝑠⦌𝐷)) |
| 4 | 3 | adantr 480 |
. 2
⊢ ((𝑅 ∈ 𝐴 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) → ⦋𝑅 / 𝑠⦌𝑁 = if(𝑅 ≤ (𝑃 ∨ 𝑄), ⦋𝑅 / 𝑠⦌𝐼, ⦋𝑅 / 𝑠⦌𝐷)) |
| 5 | | iftrue 4531 |
. . . . 5
⊢ (𝑅 ≤ (𝑃 ∨ 𝑄) → if(𝑅 ≤ (𝑃 ∨ 𝑄), ⦋𝑅 / 𝑠⦌𝐼, ⦋𝑅 / 𝑠⦌𝐷) = ⦋𝑅 / 𝑠⦌𝐼) |
| 6 | | cdleme31sn1.i |
. . . . . 6
⊢ 𝐼 = (℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = 𝐺)) |
| 7 | 6 | csbeq2i 3907 |
. . . . 5
⊢
⦋𝑅 /
𝑠⦌𝐼 = ⦋𝑅 / 𝑠⦌(℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = 𝐺)) |
| 8 | 5, 7 | eqtrdi 2793 |
. . . 4
⊢ (𝑅 ≤ (𝑃 ∨ 𝑄) → if(𝑅 ≤ (𝑃 ∨ 𝑄), ⦋𝑅 / 𝑠⦌𝐼, ⦋𝑅 / 𝑠⦌𝐷) = ⦋𝑅 / 𝑠⦌(℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = 𝐺))) |
| 9 | | nfcv 2905 |
. . . . . . . 8
⊢
Ⅎ𝑠𝐴 |
| 10 | | nfv 1914 |
. . . . . . . . 9
⊢
Ⅎ𝑠(¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) |
| 11 | | nfcsb1v 3923 |
. . . . . . . . . 10
⊢
Ⅎ𝑠⦋𝑅 / 𝑠⦌𝐺 |
| 12 | 11 | nfeq2 2923 |
. . . . . . . . 9
⊢
Ⅎ𝑠 𝑦 = ⦋𝑅 / 𝑠⦌𝐺 |
| 13 | 10, 12 | nfim 1896 |
. . . . . . . 8
⊢
Ⅎ𝑠((¬
𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = ⦋𝑅 / 𝑠⦌𝐺) |
| 14 | 9, 13 | nfralw 3311 |
. . . . . . 7
⊢
Ⅎ𝑠∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = ⦋𝑅 / 𝑠⦌𝐺) |
| 15 | | nfcv 2905 |
. . . . . . 7
⊢
Ⅎ𝑠𝐵 |
| 16 | 14, 15 | nfriota 7400 |
. . . . . 6
⊢
Ⅎ𝑠(℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = ⦋𝑅 / 𝑠⦌𝐺)) |
| 17 | 16 | a1i 11 |
. . . . 5
⊢ (𝑅 ∈ 𝐴 → Ⅎ𝑠(℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = ⦋𝑅 / 𝑠⦌𝐺))) |
| 18 | | csbeq1a 3913 |
. . . . . . . . 9
⊢ (𝑠 = 𝑅 → 𝐺 = ⦋𝑅 / 𝑠⦌𝐺) |
| 19 | 18 | eqeq2d 2748 |
. . . . . . . 8
⊢ (𝑠 = 𝑅 → (𝑦 = 𝐺 ↔ 𝑦 = ⦋𝑅 / 𝑠⦌𝐺)) |
| 20 | 19 | imbi2d 340 |
. . . . . . 7
⊢ (𝑠 = 𝑅 → (((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = 𝐺) ↔ ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = ⦋𝑅 / 𝑠⦌𝐺))) |
| 21 | 20 | ralbidv 3178 |
. . . . . 6
⊢ (𝑠 = 𝑅 → (∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = 𝐺) ↔ ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = ⦋𝑅 / 𝑠⦌𝐺))) |
| 22 | 21 | riotabidv 7390 |
. . . . 5
⊢ (𝑠 = 𝑅 → (℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = 𝐺)) = (℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = ⦋𝑅 / 𝑠⦌𝐺))) |
| 23 | 17, 22 | csbiegf 3932 |
. . . 4
⊢ (𝑅 ∈ 𝐴 → ⦋𝑅 / 𝑠⦌(℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = 𝐺)) = (℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = ⦋𝑅 / 𝑠⦌𝐺))) |
| 24 | 8, 23 | sylan9eqr 2799 |
. . 3
⊢ ((𝑅 ∈ 𝐴 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) → if(𝑅 ≤ (𝑃 ∨ 𝑄), ⦋𝑅 / 𝑠⦌𝐼, ⦋𝑅 / 𝑠⦌𝐷) = (℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = ⦋𝑅 / 𝑠⦌𝐺))) |
| 25 | | cdleme31sn1.c |
. . 3
⊢ 𝐶 = (℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = ⦋𝑅 / 𝑠⦌𝐺)) |
| 26 | 24, 25 | eqtr4di 2795 |
. 2
⊢ ((𝑅 ∈ 𝐴 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) → if(𝑅 ≤ (𝑃 ∨ 𝑄), ⦋𝑅 / 𝑠⦌𝐼, ⦋𝑅 / 𝑠⦌𝐷) = 𝐶) |
| 27 | 4, 26 | eqtrd 2777 |
1
⊢ ((𝑅 ∈ 𝐴 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) → ⦋𝑅 / 𝑠⦌𝑁 = 𝐶) |