Proof of Theorem fmtno4nprmfac193
| Step | Hyp | Ref
| Expression |
| 1 | | 1nn0 12542 |
. . . . 5
⊢ 1 ∈
ℕ0 |
| 2 | | 9nn0 12550 |
. . . . 5
⊢ 9 ∈
ℕ0 |
| 3 | 1, 2 | deccl 12748 |
. . . 4
⊢ ;19 ∈
ℕ0 |
| 4 | | 3nn 12345 |
. . . 4
⊢ 3 ∈
ℕ |
| 5 | 3, 4 | decnncl 12753 |
. . 3
⊢ ;;193 ∈ ℕ |
| 6 | | 3nn0 12544 |
. . . . 5
⊢ 3 ∈
ℕ0 |
| 7 | 6, 6 | deccl 12748 |
. . . 4
⊢ ;33 ∈
ℕ0 |
| 8 | 7, 2 | deccl 12748 |
. . 3
⊢ ;;339 ∈ ℕ0 |
| 9 | | 1nn 12277 |
. . . . 5
⊢ 1 ∈
ℕ |
| 10 | 1, 9 | decnncl 12753 |
. . . 4
⊢ ;11 ∈ ℕ |
| 11 | 10 | decnncl2 12757 |
. . 3
⊢ ;;110 ∈ ℕ |
| 12 | | 6nn0 12547 |
. . . . . . 7
⊢ 6 ∈
ℕ0 |
| 13 | | 5nn0 12546 |
. . . . . . 7
⊢ 5 ∈
ℕ0 |
| 14 | 12, 13 | deccl 12748 |
. . . . . 6
⊢ ;65 ∈
ℕ0 |
| 15 | | 4nn0 12545 |
. . . . . 6
⊢ 4 ∈
ℕ0 |
| 16 | 14, 15 | deccl 12748 |
. . . . 5
⊢ ;;654 ∈ ℕ0 |
| 17 | | 2nn0 12543 |
. . . . 5
⊢ 2 ∈
ℕ0 |
| 18 | 16, 17 | deccl 12748 |
. . . 4
⊢ ;;;6542
∈ ℕ0 |
| 19 | | 7nn0 12548 |
. . . 4
⊢ 7 ∈
ℕ0 |
| 20 | 1, 1 | deccl 12748 |
. . . 4
⊢ ;11 ∈
ℕ0 |
| 21 | | 0nn0 12541 |
. . . 4
⊢ 0 ∈
ℕ0 |
| 22 | 3, 6 | deccl 12748 |
. . . . 5
⊢ ;;193 ∈ ℕ0 |
| 23 | | eqid 2737 |
. . . . 5
⊢ ;;339 = ;;339 |
| 24 | 1, 19 | deccl 12748 |
. . . . . 6
⊢ ;17 ∈
ℕ0 |
| 25 | 24, 6 | deccl 12748 |
. . . . 5
⊢ ;;173 ∈ ℕ0 |
| 26 | | eqid 2737 |
. . . . . 6
⊢ ;33 = ;33 |
| 27 | | eqid 2737 |
. . . . . 6
⊢ ;;173 = ;;173 |
| 28 | | 8nn0 12549 |
. . . . . . 7
⊢ 8 ∈
ℕ0 |
| 29 | 13, 28 | deccl 12748 |
. . . . . 6
⊢ ;58 ∈
ℕ0 |
| 30 | 13, 19 | deccl 12748 |
. . . . . . 7
⊢ ;57 ∈
ℕ0 |
| 31 | | eqid 2737 |
. . . . . . . 8
⊢ ;;193 = ;;193 |
| 32 | | eqid 2737 |
. . . . . . . . 9
⊢ ;19 = ;19 |
| 33 | | 3cn 12347 |
. . . . . . . . . . . 12
⊢ 3 ∈
ℂ |
| 34 | 33 | mullidi 11266 |
. . . . . . . . . . 11
⊢ (1
· 3) = 3 |
| 35 | 34 | oveq1i 7441 |
. . . . . . . . . 10
⊢ ((1
· 3) + 2) = (3 + 2) |
| 36 | | 3p2e5 12417 |
. . . . . . . . . 10
⊢ (3 + 2) =
5 |
| 37 | 35, 36 | eqtri 2765 |
. . . . . . . . 9
⊢ ((1
· 3) + 2) = 5 |
| 38 | | 9t3e27 12856 |
. . . . . . . . 9
⊢ (9
· 3) = ;27 |
| 39 | 6, 1, 2, 32, 19, 17, 37, 38 | decmul1c 12798 |
. . . . . . . 8
⊢ (;19 · 3) = ;57 |
| 40 | | 3t3e9 12433 |
. . . . . . . 8
⊢ (3
· 3) = 9 |
| 41 | 6, 3, 6, 31, 39, 40 | decmul1 12797 |
. . . . . . 7
⊢ (;;193 · 3) = ;;579 |
| 42 | | eqid 2737 |
. . . . . . . 8
⊢ ;17 = ;17 |
| 43 | | eqid 2737 |
. . . . . . . 8
⊢ ;58 = ;58 |
| 44 | | 5cn 12354 |
. . . . . . . . . . 11
⊢ 5 ∈
ℂ |
| 45 | | ax-1cn 11213 |
. . . . . . . . . . 11
⊢ 1 ∈
ℂ |
| 46 | | 5p1e6 12413 |
. . . . . . . . . . 11
⊢ (5 + 1) =
6 |
| 47 | 44, 45, 46 | addcomli 11453 |
. . . . . . . . . 10
⊢ (1 + 5) =
6 |
| 48 | 47 | oveq1i 7441 |
. . . . . . . . 9
⊢ ((1 + 5)
+ 1) = (6 + 1) |
| 49 | | 6p1e7 12414 |
. . . . . . . . 9
⊢ (6 + 1) =
7 |
| 50 | 48, 49 | eqtri 2765 |
. . . . . . . 8
⊢ ((1 + 5)
+ 1) = 7 |
| 51 | | 8cn 12363 |
. . . . . . . . 9
⊢ 8 ∈
ℂ |
| 52 | | 7cn 12360 |
. . . . . . . . 9
⊢ 7 ∈
ℂ |
| 53 | | 8p7e15 12818 |
. . . . . . . . 9
⊢ (8 + 7) =
;15 |
| 54 | 51, 52, 53 | addcomli 11453 |
. . . . . . . 8
⊢ (7 + 8) =
;15 |
| 55 | 1, 19, 13, 28, 42, 43, 50, 13, 54 | decaddc 12788 |
. . . . . . 7
⊢ (;17 + ;58) = ;75 |
| 56 | | 4p1e5 12412 |
. . . . . . . 8
⊢ (4 + 1) =
5 |
| 57 | | eqid 2737 |
. . . . . . . . 9
⊢ ;57 = ;57 |
| 58 | | 7p7e14 12812 |
. . . . . . . . 9
⊢ (7 + 7) =
;14 |
| 59 | 13, 19, 19, 57, 46, 15, 58 | decaddci 12794 |
. . . . . . . 8
⊢ (;57 + 7) = ;64 |
| 60 | 12, 15, 56, 59 | decsuc 12764 |
. . . . . . 7
⊢ ((;57 + 7) + 1) = ;65 |
| 61 | | 9p5e14 12823 |
. . . . . . 7
⊢ (9 + 5) =
;14 |
| 62 | 30, 2, 19, 13, 41, 55, 60, 15, 61 | decaddc 12788 |
. . . . . 6
⊢ ((;;193 · 3) + (;17 + ;58)) = ;;654 |
| 63 | | 7p1e8 12415 |
. . . . . . . 8
⊢ (7 + 1) =
8 |
| 64 | 13, 19, 63, 57 | decsuc 12764 |
. . . . . . 7
⊢ (;57 + 1) = ;58 |
| 65 | | 9p3e12 12821 |
. . . . . . 7
⊢ (9 + 3) =
;12 |
| 66 | 30, 2, 6, 41, 64, 17, 65 | decaddci 12794 |
. . . . . 6
⊢ ((;;193 · 3) + 3) = ;;582 |
| 67 | 6, 6, 24, 6, 26, 27, 22, 17, 29, 62, 66 | decma2c 12786 |
. . . . 5
⊢ ((;;193 · ;33) + ;;173) =
;;;6542 |
| 68 | | 9cn 12366 |
. . . . . . . . . . 11
⊢ 9 ∈
ℂ |
| 69 | 68 | mullidi 11266 |
. . . . . . . . . 10
⊢ (1
· 9) = 9 |
| 70 | 69 | oveq1i 7441 |
. . . . . . . . 9
⊢ ((1
· 9) + 8) = (9 + 8) |
| 71 | | 9p8e17 12826 |
. . . . . . . . 9
⊢ (9 + 8) =
;17 |
| 72 | 70, 71 | eqtri 2765 |
. . . . . . . 8
⊢ ((1
· 9) + 8) = ;17 |
| 73 | | 9t9e81 12862 |
. . . . . . . 8
⊢ (9
· 9) = ;81 |
| 74 | 2, 1, 2, 32, 1, 28, 72, 73 | decmul1c 12798 |
. . . . . . 7
⊢ (;19 · 9) = ;;171 |
| 75 | | 1p2e3 12409 |
. . . . . . 7
⊢ (1 + 2) =
3 |
| 76 | 24, 1, 17, 74, 75 | decaddi 12793 |
. . . . . 6
⊢ ((;19 · 9) + 2) = ;;173 |
| 77 | 68, 33, 38 | mulcomli 11270 |
. . . . . 6
⊢ (3
· 9) = ;27 |
| 78 | 2, 3, 6, 31, 19, 17, 76, 77 | decmul1c 12798 |
. . . . 5
⊢ (;;193 · 9) = ;;;1737 |
| 79 | 22, 7, 2, 23, 19, 25, 67, 78 | decmul2c 12799 |
. . . 4
⊢ (;;193 · ;;339) =
;;;;65427 |
| 80 | | eqid 2737 |
. . . 4
⊢ ;;110 = ;;110 |
| 81 | | eqid 2737 |
. . . . 5
⊢ ;;;6542 =
;;;6542 |
| 82 | | eqid 2737 |
. . . . 5
⊢ ;11 = ;11 |
| 83 | | eqid 2737 |
. . . . . 6
⊢ ;;654 = ;;654 |
| 84 | 14, 15, 56, 83 | decsuc 12764 |
. . . . 5
⊢ (;;654 + 1) = ;;655 |
| 85 | | 2p1e3 12408 |
. . . . 5
⊢ (2 + 1) =
3 |
| 86 | 16, 17, 1, 1, 81, 82, 84, 85 | decadd 12787 |
. . . 4
⊢ (;;;6542 +
;11) = ;;;6553 |
| 87 | 52 | addridi 11448 |
. . . 4
⊢ (7 + 0) =
7 |
| 88 | 18, 19, 20, 21, 79, 80, 86, 87 | decadd 12787 |
. . 3
⊢ ((;;193 · ;;339) +
;;110) = ;;;;65537 |
| 89 | | 10pos 12750 |
. . . 4
⊢ 0 <
;10 |
| 90 | | 9nn 12364 |
. . . . 5
⊢ 9 ∈
ℕ |
| 91 | | 1lt9 12472 |
. . . . 5
⊢ 1 <
9 |
| 92 | 1, 1, 90, 91 | declt 12761 |
. . . 4
⊢ ;11 < ;19 |
| 93 | 20, 3, 21, 6, 89, 92 | decltc 12762 |
. . 3
⊢ ;;110 < ;;193 |
| 94 | 5, 8, 11, 88, 93 | ndvdsi 16449 |
. 2
⊢ ¬
;;193 ∥ ;;;;65537 |
| 95 | | fmtno4 47539 |
. . 3
⊢
(FermatNo‘4) = ;;;;65537 |
| 96 | 95 | breq2i 5151 |
. 2
⊢ (;;193 ∥ (FermatNo‘4) ↔ ;;193 ∥ ;;;;65537) |
| 97 | 94, 96 | mtbir 323 |
1
⊢ ¬
;;193 ∥ (FermatNo‘4) |