Proof of Theorem 2exp16
| Step | Hyp | Ref
| Expression |
| 1 | | 2nn0 12543 |
. 2
⊢ 2 ∈
ℕ0 |
| 2 | | 8nn0 12549 |
. 2
⊢ 8 ∈
ℕ0 |
| 3 | | 8cn 12363 |
. . 3
⊢ 8 ∈
ℂ |
| 4 | | 2cn 12341 |
. . 3
⊢ 2 ∈
ℂ |
| 5 | | 8t2e16 12848 |
. . 3
⊢ (8
· 2) = ;16 |
| 6 | 3, 4, 5 | mulcomli 11270 |
. 2
⊢ (2
· 8) = ;16 |
| 7 | | 2exp8 17126 |
. 2
⊢
(2↑8) = ;;256 |
| 8 | | 5nn0 12546 |
. . . . 5
⊢ 5 ∈
ℕ0 |
| 9 | 1, 8 | deccl 12748 |
. . . 4
⊢ ;25 ∈
ℕ0 |
| 10 | | 6nn0 12547 |
. . . 4
⊢ 6 ∈
ℕ0 |
| 11 | 9, 10 | deccl 12748 |
. . 3
⊢ ;;256 ∈ ℕ0 |
| 12 | | eqid 2737 |
. . 3
⊢ ;;256 = ;;256 |
| 13 | | 1nn0 12542 |
. . . . 5
⊢ 1 ∈
ℕ0 |
| 14 | 13, 8 | deccl 12748 |
. . . 4
⊢ ;15 ∈
ℕ0 |
| 15 | | 3nn0 12544 |
. . . 4
⊢ 3 ∈
ℕ0 |
| 16 | 14, 15 | deccl 12748 |
. . 3
⊢ ;;153 ∈ ℕ0 |
| 17 | | eqid 2737 |
. . . 4
⊢ ;25 = ;25 |
| 18 | | eqid 2737 |
. . . 4
⊢ ;;153 = ;;153 |
| 19 | 13, 1 | deccl 12748 |
. . . . 5
⊢ ;12 ∈
ℕ0 |
| 20 | 19, 2 | deccl 12748 |
. . . 4
⊢ ;;128 ∈ ℕ0 |
| 21 | | 4nn0 12545 |
. . . . . 6
⊢ 4 ∈
ℕ0 |
| 22 | 13, 21 | deccl 12748 |
. . . . 5
⊢ ;14 ∈
ℕ0 |
| 23 | | eqid 2737 |
. . . . . 6
⊢ ;15 = ;15 |
| 24 | | eqid 2737 |
. . . . . 6
⊢ ;;128 = ;;128 |
| 25 | | 0nn0 12541 |
. . . . . . . 8
⊢ 0 ∈
ℕ0 |
| 26 | 13 | dec0h 12755 |
. . . . . . . 8
⊢ 1 = ;01 |
| 27 | | eqid 2737 |
. . . . . . . 8
⊢ ;12 = ;12 |
| 28 | | 0p1e1 12388 |
. . . . . . . 8
⊢ (0 + 1) =
1 |
| 29 | | 1p2e3 12409 |
. . . . . . . 8
⊢ (1 + 2) =
3 |
| 30 | 25, 13, 13, 1, 26, 27, 28, 29 | decadd 12787 |
. . . . . . 7
⊢ (1 +
;12) = ;13 |
| 31 | | 3p1e4 12411 |
. . . . . . 7
⊢ (3 + 1) =
4 |
| 32 | 13, 15, 13, 30, 31 | decaddi 12793 |
. . . . . 6
⊢ ((1 +
;12) + 1) = ;14 |
| 33 | | 5cn 12354 |
. . . . . . 7
⊢ 5 ∈
ℂ |
| 34 | | 8p5e13 12816 |
. . . . . . 7
⊢ (8 + 5) =
;13 |
| 35 | 3, 33, 34 | addcomli 11453 |
. . . . . 6
⊢ (5 + 8) =
;13 |
| 36 | 13, 8, 19, 2, 23, 24, 32, 15, 35 | decaddc 12788 |
. . . . 5
⊢ (;15 + ;;128) =
;;143 |
| 37 | | eqid 2737 |
. . . . . . 7
⊢ ;14 = ;14 |
| 38 | | 4p1e5 12412 |
. . . . . . 7
⊢ (4 + 1) =
5 |
| 39 | 13, 21, 13, 37, 38 | decaddi 12793 |
. . . . . 6
⊢ (;14 + 1) = ;15 |
| 40 | | 2t2e4 12430 |
. . . . . . . 8
⊢ (2
· 2) = 4 |
| 41 | | 1p1e2 12391 |
. . . . . . . 8
⊢ (1 + 1) =
2 |
| 42 | 40, 41 | oveq12i 7443 |
. . . . . . 7
⊢ ((2
· 2) + (1 + 1)) = (4 + 2) |
| 43 | | 4p2e6 12419 |
. . . . . . 7
⊢ (4 + 2) =
6 |
| 44 | 42, 43 | eqtri 2765 |
. . . . . 6
⊢ ((2
· 2) + (1 + 1)) = 6 |
| 45 | | 5t2e10 12833 |
. . . . . . 7
⊢ (5
· 2) = ;10 |
| 46 | 33 | addlidi 11449 |
. . . . . . 7
⊢ (0 + 5) =
5 |
| 47 | 13, 25, 8, 45, 46 | decaddi 12793 |
. . . . . 6
⊢ ((5
· 2) + 5) = ;15 |
| 48 | 1, 8, 13, 8, 17, 39, 1, 8, 13, 44, 47 | decmac 12785 |
. . . . 5
⊢ ((;25 · 2) + (;14 + 1)) = ;65 |
| 49 | | 6t2e12 12837 |
. . . . . 6
⊢ (6
· 2) = ;12 |
| 50 | | 3cn 12347 |
. . . . . . 7
⊢ 3 ∈
ℂ |
| 51 | | 3p2e5 12417 |
. . . . . . 7
⊢ (3 + 2) =
5 |
| 52 | 50, 4, 51 | addcomli 11453 |
. . . . . 6
⊢ (2 + 3) =
5 |
| 53 | 13, 1, 15, 49, 52 | decaddi 12793 |
. . . . 5
⊢ ((6
· 2) + 3) = ;15 |
| 54 | 9, 10, 22, 15, 12, 36, 1, 8, 13, 48, 53 | decmac 12785 |
. . . 4
⊢ ((;;256 · 2) + (;15 + ;;128))
= ;;655 |
| 55 | 15 | dec0h 12755 |
. . . . 5
⊢ 3 = ;03 |
| 56 | 50 | addlidi 11449 |
. . . . . . 7
⊢ (0 + 3) =
3 |
| 57 | 56, 55 | eqtri 2765 |
. . . . . 6
⊢ (0 + 3) =
;03 |
| 58 | 4 | addlidi 11449 |
. . . . . . . 8
⊢ (0 + 2) =
2 |
| 59 | 58 | oveq2i 7442 |
. . . . . . 7
⊢ ((2
· 5) + (0 + 2)) = ((2 · 5) + 2) |
| 60 | 33, 4, 45 | mulcomli 11270 |
. . . . . . . 8
⊢ (2
· 5) = ;10 |
| 61 | 13, 25, 1, 60, 58 | decaddi 12793 |
. . . . . . 7
⊢ ((2
· 5) + 2) = ;12 |
| 62 | 59, 61 | eqtri 2765 |
. . . . . 6
⊢ ((2
· 5) + (0 + 2)) = ;12 |
| 63 | | 5t5e25 12836 |
. . . . . . 7
⊢ (5
· 5) = ;25 |
| 64 | | 5p3e8 12423 |
. . . . . . 7
⊢ (5 + 3) =
8 |
| 65 | 1, 8, 15, 63, 64 | decaddi 12793 |
. . . . . 6
⊢ ((5
· 5) + 3) = ;28 |
| 66 | 1, 8, 25, 15, 17, 57, 8, 2, 1,
62, 65 | decmac 12785 |
. . . . 5
⊢ ((;25 · 5) + (0 + 3)) = ;;128 |
| 67 | | 6t5e30 12840 |
. . . . . 6
⊢ (6
· 5) = ;30 |
| 68 | 15, 25, 15, 67, 56 | decaddi 12793 |
. . . . 5
⊢ ((6
· 5) + 3) = ;33 |
| 69 | 9, 10, 25, 15, 12, 55, 8, 15, 15, 66, 68 | decmac 12785 |
. . . 4
⊢ ((;;256 · 5) + 3) = ;;;1283 |
| 70 | 1, 8, 14, 15, 17, 18, 11, 15, 20, 54, 69 | decma2c 12786 |
. . 3
⊢ ((;;256 · ;25) + ;;153) =
;;;6553 |
| 71 | | 6cn 12357 |
. . . . . . 7
⊢ 6 ∈
ℂ |
| 72 | 71, 4, 49 | mulcomli 11270 |
. . . . . 6
⊢ (2
· 6) = ;12 |
| 73 | 13, 1, 15, 72, 52 | decaddi 12793 |
. . . . 5
⊢ ((2
· 6) + 3) = ;15 |
| 74 | 71, 33, 67 | mulcomli 11270 |
. . . . . 6
⊢ (5
· 6) = ;30 |
| 75 | 15, 25, 15, 74, 56 | decaddi 12793 |
. . . . 5
⊢ ((5
· 6) + 3) = ;33 |
| 76 | 1, 8, 15, 17, 10, 15, 15, 73, 75 | decrmac 12791 |
. . . 4
⊢ ((;25 · 6) + 3) = ;;153 |
| 77 | | 6t6e36 12841 |
. . . 4
⊢ (6
· 6) = ;36 |
| 78 | 10, 9, 10, 12, 10, 15, 76, 77 | decmul1c 12798 |
. . 3
⊢ (;;256 · 6) = ;;;1536 |
| 79 | 11, 9, 10, 12, 10, 16, 70, 78 | decmul2c 12799 |
. 2
⊢ (;;256 · ;;256) =
;;;;65536 |
| 80 | 1, 2, 6, 7, 79 | numexp2x 17116 |
1
⊢
(2↑;16) = ;;;;65536 |