| Step | Hyp | Ref
| Expression |
| 1 | | oveq1 7438 |
. . . . . . 7
⊢ (𝑥 = 0 → (𝑥 ↑ (𝐶 · 𝑋)) = (0 ↑ (𝐶 · 𝑋))) |
| 2 | | oveq1 7438 |
. . . . . . . 8
⊢ (𝑥 = 0 → (𝑥𝐸𝐶) = (0𝐸𝐶)) |
| 3 | 2 | oveq1d 7446 |
. . . . . . 7
⊢ (𝑥 = 0 → ((𝑥𝐸𝐶) · 𝑋) = ((0𝐸𝐶) · 𝑋)) |
| 4 | 1, 3 | eqeq12d 2753 |
. . . . . 6
⊢ (𝑥 = 0 → ((𝑥 ↑ (𝐶 · 𝑋)) = ((𝑥𝐸𝐶) · 𝑋) ↔ (0 ↑ (𝐶 · 𝑋)) = ((0𝐸𝐶) · 𝑋))) |
| 5 | 4 | imbi2d 340 |
. . . . 5
⊢ (𝑥 = 0 → ((((𝐶 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) ∧ 𝑊 ∈ LMod) → (𝑥 ↑ (𝐶 · 𝑋)) = ((𝑥𝐸𝐶) · 𝑋)) ↔ (((𝐶 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) ∧ 𝑊 ∈ LMod) → (0 ↑ (𝐶 · 𝑋)) = ((0𝐸𝐶) · 𝑋)))) |
| 6 | | oveq1 7438 |
. . . . . . 7
⊢ (𝑥 = 𝑦 → (𝑥 ↑ (𝐶 · 𝑋)) = (𝑦 ↑ (𝐶 · 𝑋))) |
| 7 | | oveq1 7438 |
. . . . . . . 8
⊢ (𝑥 = 𝑦 → (𝑥𝐸𝐶) = (𝑦𝐸𝐶)) |
| 8 | 7 | oveq1d 7446 |
. . . . . . 7
⊢ (𝑥 = 𝑦 → ((𝑥𝐸𝐶) · 𝑋) = ((𝑦𝐸𝐶) · 𝑋)) |
| 9 | 6, 8 | eqeq12d 2753 |
. . . . . 6
⊢ (𝑥 = 𝑦 → ((𝑥 ↑ (𝐶 · 𝑋)) = ((𝑥𝐸𝐶) · 𝑋) ↔ (𝑦 ↑ (𝐶 · 𝑋)) = ((𝑦𝐸𝐶) · 𝑋))) |
| 10 | 9 | imbi2d 340 |
. . . . 5
⊢ (𝑥 = 𝑦 → ((((𝐶 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) ∧ 𝑊 ∈ LMod) → (𝑥 ↑ (𝐶 · 𝑋)) = ((𝑥𝐸𝐶) · 𝑋)) ↔ (((𝐶 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) ∧ 𝑊 ∈ LMod) → (𝑦 ↑ (𝐶 · 𝑋)) = ((𝑦𝐸𝐶) · 𝑋)))) |
| 11 | | oveq1 7438 |
. . . . . . 7
⊢ (𝑥 = (𝑦 + 1) → (𝑥 ↑ (𝐶 · 𝑋)) = ((𝑦 + 1) ↑ (𝐶 · 𝑋))) |
| 12 | | oveq1 7438 |
. . . . . . . 8
⊢ (𝑥 = (𝑦 + 1) → (𝑥𝐸𝐶) = ((𝑦 + 1)𝐸𝐶)) |
| 13 | 12 | oveq1d 7446 |
. . . . . . 7
⊢ (𝑥 = (𝑦 + 1) → ((𝑥𝐸𝐶) · 𝑋) = (((𝑦 + 1)𝐸𝐶) · 𝑋)) |
| 14 | 11, 13 | eqeq12d 2753 |
. . . . . 6
⊢ (𝑥 = (𝑦 + 1) → ((𝑥 ↑ (𝐶 · 𝑋)) = ((𝑥𝐸𝐶) · 𝑋) ↔ ((𝑦 + 1) ↑ (𝐶 · 𝑋)) = (((𝑦 + 1)𝐸𝐶) · 𝑋))) |
| 15 | 14 | imbi2d 340 |
. . . . 5
⊢ (𝑥 = (𝑦 + 1) → ((((𝐶 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) ∧ 𝑊 ∈ LMod) → (𝑥 ↑ (𝐶 · 𝑋)) = ((𝑥𝐸𝐶) · 𝑋)) ↔ (((𝐶 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) ∧ 𝑊 ∈ LMod) → ((𝑦 + 1) ↑ (𝐶 · 𝑋)) = (((𝑦 + 1)𝐸𝐶) · 𝑋)))) |
| 16 | | oveq1 7438 |
. . . . . . 7
⊢ (𝑥 = 𝑁 → (𝑥 ↑ (𝐶 · 𝑋)) = (𝑁 ↑ (𝐶 · 𝑋))) |
| 17 | | oveq1 7438 |
. . . . . . . 8
⊢ (𝑥 = 𝑁 → (𝑥𝐸𝐶) = (𝑁𝐸𝐶)) |
| 18 | 17 | oveq1d 7446 |
. . . . . . 7
⊢ (𝑥 = 𝑁 → ((𝑥𝐸𝐶) · 𝑋) = ((𝑁𝐸𝐶) · 𝑋)) |
| 19 | 16, 18 | eqeq12d 2753 |
. . . . . 6
⊢ (𝑥 = 𝑁 → ((𝑥 ↑ (𝐶 · 𝑋)) = ((𝑥𝐸𝐶) · 𝑋) ↔ (𝑁 ↑ (𝐶 · 𝑋)) = ((𝑁𝐸𝐶) · 𝑋))) |
| 20 | 19 | imbi2d 340 |
. . . . 5
⊢ (𝑥 = 𝑁 → ((((𝐶 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) ∧ 𝑊 ∈ LMod) → (𝑥 ↑ (𝐶 · 𝑋)) = ((𝑥𝐸𝐶) · 𝑋)) ↔ (((𝐶 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) ∧ 𝑊 ∈ LMod) → (𝑁 ↑ (𝐶 · 𝑋)) = ((𝑁𝐸𝐶) · 𝑋)))) |
| 21 | | simpr 484 |
. . . . . . 7
⊢ (((𝐶 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) ∧ 𝑊 ∈ LMod) → 𝑊 ∈ LMod) |
| 22 | | simpr 484 |
. . . . . . . 8
⊢ ((𝐶 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) → 𝑋 ∈ 𝑉) |
| 23 | 22 | adantr 480 |
. . . . . . 7
⊢ (((𝐶 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) ∧ 𝑊 ∈ LMod) → 𝑋 ∈ 𝑉) |
| 24 | | lmodvsmmulgdi.v |
. . . . . . . 8
⊢ 𝑉 = (Base‘𝑊) |
| 25 | | lmodvsmmulgdi.f |
. . . . . . . 8
⊢ 𝐹 = (Scalar‘𝑊) |
| 26 | | lmodvsmmulgdi.s |
. . . . . . . 8
⊢ · = (
·𝑠 ‘𝑊) |
| 27 | | eqid 2737 |
. . . . . . . 8
⊢
(0g‘𝐹) = (0g‘𝐹) |
| 28 | | eqid 2737 |
. . . . . . . 8
⊢
(0g‘𝑊) = (0g‘𝑊) |
| 29 | 24, 25, 26, 27, 28 | lmod0vs 20893 |
. . . . . . 7
⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → ((0g‘𝐹) · 𝑋) = (0g‘𝑊)) |
| 30 | 21, 23, 29 | syl2anc 584 |
. . . . . 6
⊢ (((𝐶 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) ∧ 𝑊 ∈ LMod) →
((0g‘𝐹)
·
𝑋) =
(0g‘𝑊)) |
| 31 | | simpl 482 |
. . . . . . . . 9
⊢ ((𝐶 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) → 𝐶 ∈ 𝐾) |
| 32 | 31 | adantr 480 |
. . . . . . . 8
⊢ (((𝐶 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) ∧ 𝑊 ∈ LMod) → 𝐶 ∈ 𝐾) |
| 33 | | lmodvsmmulgdi.k |
. . . . . . . . 9
⊢ 𝐾 = (Base‘𝐹) |
| 34 | | lmodvsmmulgdi.e |
. . . . . . . . 9
⊢ 𝐸 = (.g‘𝐹) |
| 35 | 33, 27, 34 | mulg0 19092 |
. . . . . . . 8
⊢ (𝐶 ∈ 𝐾 → (0𝐸𝐶) = (0g‘𝐹)) |
| 36 | 32, 35 | syl 17 |
. . . . . . 7
⊢ (((𝐶 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) ∧ 𝑊 ∈ LMod) → (0𝐸𝐶) = (0g‘𝐹)) |
| 37 | 36 | oveq1d 7446 |
. . . . . 6
⊢ (((𝐶 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) ∧ 𝑊 ∈ LMod) → ((0𝐸𝐶) · 𝑋) = ((0g‘𝐹) · 𝑋)) |
| 38 | 24, 25, 26, 33 | lmodvscl 20876 |
. . . . . . . 8
⊢ ((𝑊 ∈ LMod ∧ 𝐶 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) → (𝐶 · 𝑋) ∈ 𝑉) |
| 39 | 21, 32, 23, 38 | syl3anc 1373 |
. . . . . . 7
⊢ (((𝐶 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) ∧ 𝑊 ∈ LMod) → (𝐶 · 𝑋) ∈ 𝑉) |
| 40 | | lmodvsmmulgdi.p |
. . . . . . . 8
⊢ ↑ =
(.g‘𝑊) |
| 41 | 24, 28, 40 | mulg0 19092 |
. . . . . . 7
⊢ ((𝐶 · 𝑋) ∈ 𝑉 → (0 ↑ (𝐶 · 𝑋)) = (0g‘𝑊)) |
| 42 | 39, 41 | syl 17 |
. . . . . 6
⊢ (((𝐶 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) ∧ 𝑊 ∈ LMod) → (0 ↑ (𝐶 · 𝑋)) = (0g‘𝑊)) |
| 43 | 30, 37, 42 | 3eqtr4rd 2788 |
. . . . 5
⊢ (((𝐶 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) ∧ 𝑊 ∈ LMod) → (0 ↑ (𝐶 · 𝑋)) = ((0𝐸𝐶) · 𝑋)) |
| 44 | | lmodgrp 20865 |
. . . . . . . . . . . 12
⊢ (𝑊 ∈ LMod → 𝑊 ∈ Grp) |
| 45 | 44 | grpmndd 18964 |
. . . . . . . . . . 11
⊢ (𝑊 ∈ LMod → 𝑊 ∈ Mnd) |
| 46 | 45 | ad2antll 729 |
. . . . . . . . . 10
⊢ ((𝑦 ∈ ℕ0
∧ ((𝐶 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) ∧ 𝑊 ∈ LMod)) → 𝑊 ∈ Mnd) |
| 47 | | simpl 482 |
. . . . . . . . . 10
⊢ ((𝑦 ∈ ℕ0
∧ ((𝐶 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) ∧ 𝑊 ∈ LMod)) → 𝑦 ∈ ℕ0) |
| 48 | 39 | adantl 481 |
. . . . . . . . . 10
⊢ ((𝑦 ∈ ℕ0
∧ ((𝐶 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) ∧ 𝑊 ∈ LMod)) → (𝐶 · 𝑋) ∈ 𝑉) |
| 49 | | eqid 2737 |
. . . . . . . . . . 11
⊢
(+g‘𝑊) = (+g‘𝑊) |
| 50 | 24, 40, 49 | mulgnn0p1 19103 |
. . . . . . . . . 10
⊢ ((𝑊 ∈ Mnd ∧ 𝑦 ∈ ℕ0
∧ (𝐶 · 𝑋) ∈ 𝑉) → ((𝑦 + 1) ↑ (𝐶 · 𝑋)) = ((𝑦 ↑ (𝐶 · 𝑋))(+g‘𝑊)(𝐶 · 𝑋))) |
| 51 | 46, 47, 48, 50 | syl3anc 1373 |
. . . . . . . . 9
⊢ ((𝑦 ∈ ℕ0
∧ ((𝐶 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) ∧ 𝑊 ∈ LMod)) → ((𝑦 + 1) ↑ (𝐶 · 𝑋)) = ((𝑦 ↑ (𝐶 · 𝑋))(+g‘𝑊)(𝐶 · 𝑋))) |
| 52 | 51 | adantr 480 |
. . . . . . . 8
⊢ (((𝑦 ∈ ℕ0
∧ ((𝐶 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) ∧ 𝑊 ∈ LMod)) ∧ (𝑦 ↑ (𝐶 · 𝑋)) = ((𝑦𝐸𝐶) · 𝑋)) → ((𝑦 + 1) ↑ (𝐶 · 𝑋)) = ((𝑦 ↑ (𝐶 · 𝑋))(+g‘𝑊)(𝐶 · 𝑋))) |
| 53 | | oveq1 7438 |
. . . . . . . . 9
⊢ ((𝑦 ↑ (𝐶 · 𝑋)) = ((𝑦𝐸𝐶) · 𝑋) → ((𝑦 ↑ (𝐶 · 𝑋))(+g‘𝑊)(𝐶 · 𝑋)) = (((𝑦𝐸𝐶) · 𝑋)(+g‘𝑊)(𝐶 · 𝑋))) |
| 54 | 21 | adantl 481 |
. . . . . . . . . . 11
⊢ ((𝑦 ∈ ℕ0
∧ ((𝐶 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) ∧ 𝑊 ∈ LMod)) → 𝑊 ∈ LMod) |
| 55 | 25 | lmodring 20866 |
. . . . . . . . . . . . . 14
⊢ (𝑊 ∈ LMod → 𝐹 ∈ Ring) |
| 56 | | ringmnd 20240 |
. . . . . . . . . . . . . 14
⊢ (𝐹 ∈ Ring → 𝐹 ∈ Mnd) |
| 57 | 55, 56 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝑊 ∈ LMod → 𝐹 ∈ Mnd) |
| 58 | 57 | ad2antll 729 |
. . . . . . . . . . . 12
⊢ ((𝑦 ∈ ℕ0
∧ ((𝐶 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) ∧ 𝑊 ∈ LMod)) → 𝐹 ∈ Mnd) |
| 59 | | simprll 779 |
. . . . . . . . . . . 12
⊢ ((𝑦 ∈ ℕ0
∧ ((𝐶 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) ∧ 𝑊 ∈ LMod)) → 𝐶 ∈ 𝐾) |
| 60 | 33, 34, 58, 47, 59 | mulgnn0cld 19113 |
. . . . . . . . . . 11
⊢ ((𝑦 ∈ ℕ0
∧ ((𝐶 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) ∧ 𝑊 ∈ LMod)) → (𝑦𝐸𝐶) ∈ 𝐾) |
| 61 | 23 | adantl 481 |
. . . . . . . . . . 11
⊢ ((𝑦 ∈ ℕ0
∧ ((𝐶 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) ∧ 𝑊 ∈ LMod)) → 𝑋 ∈ 𝑉) |
| 62 | | eqid 2737 |
. . . . . . . . . . . 12
⊢
(+g‘𝐹) = (+g‘𝐹) |
| 63 | 24, 49, 25, 26, 33, 62 | lmodvsdir 20884 |
. . . . . . . . . . 11
⊢ ((𝑊 ∈ LMod ∧ ((𝑦𝐸𝐶) ∈ 𝐾 ∧ 𝐶 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉)) → (((𝑦𝐸𝐶)(+g‘𝐹)𝐶) · 𝑋) = (((𝑦𝐸𝐶) · 𝑋)(+g‘𝑊)(𝐶 · 𝑋))) |
| 64 | 54, 60, 59, 61, 63 | syl13anc 1374 |
. . . . . . . . . 10
⊢ ((𝑦 ∈ ℕ0
∧ ((𝐶 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) ∧ 𝑊 ∈ LMod)) → (((𝑦𝐸𝐶)(+g‘𝐹)𝐶) · 𝑋) = (((𝑦𝐸𝐶) · 𝑋)(+g‘𝑊)(𝐶 · 𝑋))) |
| 65 | 33, 34, 62 | mulgnn0p1 19103 |
. . . . . . . . . . . . 13
⊢ ((𝐹 ∈ Mnd ∧ 𝑦 ∈ ℕ0
∧ 𝐶 ∈ 𝐾) → ((𝑦 + 1)𝐸𝐶) = ((𝑦𝐸𝐶)(+g‘𝐹)𝐶)) |
| 66 | 58, 47, 59, 65 | syl3anc 1373 |
. . . . . . . . . . . 12
⊢ ((𝑦 ∈ ℕ0
∧ ((𝐶 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) ∧ 𝑊 ∈ LMod)) → ((𝑦 + 1)𝐸𝐶) = ((𝑦𝐸𝐶)(+g‘𝐹)𝐶)) |
| 67 | 66 | eqcomd 2743 |
. . . . . . . . . . 11
⊢ ((𝑦 ∈ ℕ0
∧ ((𝐶 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) ∧ 𝑊 ∈ LMod)) → ((𝑦𝐸𝐶)(+g‘𝐹)𝐶) = ((𝑦 + 1)𝐸𝐶)) |
| 68 | 67 | oveq1d 7446 |
. . . . . . . . . 10
⊢ ((𝑦 ∈ ℕ0
∧ ((𝐶 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) ∧ 𝑊 ∈ LMod)) → (((𝑦𝐸𝐶)(+g‘𝐹)𝐶) · 𝑋) = (((𝑦 + 1)𝐸𝐶) · 𝑋)) |
| 69 | 64, 68 | eqtr3d 2779 |
. . . . . . . . 9
⊢ ((𝑦 ∈ ℕ0
∧ ((𝐶 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) ∧ 𝑊 ∈ LMod)) → (((𝑦𝐸𝐶) · 𝑋)(+g‘𝑊)(𝐶 · 𝑋)) = (((𝑦 + 1)𝐸𝐶) · 𝑋)) |
| 70 | 53, 69 | sylan9eqr 2799 |
. . . . . . . 8
⊢ (((𝑦 ∈ ℕ0
∧ ((𝐶 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) ∧ 𝑊 ∈ LMod)) ∧ (𝑦 ↑ (𝐶 · 𝑋)) = ((𝑦𝐸𝐶) · 𝑋)) → ((𝑦 ↑ (𝐶 · 𝑋))(+g‘𝑊)(𝐶 · 𝑋)) = (((𝑦 + 1)𝐸𝐶) · 𝑋)) |
| 71 | 52, 70 | eqtrd 2777 |
. . . . . . 7
⊢ (((𝑦 ∈ ℕ0
∧ ((𝐶 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) ∧ 𝑊 ∈ LMod)) ∧ (𝑦 ↑ (𝐶 · 𝑋)) = ((𝑦𝐸𝐶) · 𝑋)) → ((𝑦 + 1) ↑ (𝐶 · 𝑋)) = (((𝑦 + 1)𝐸𝐶) · 𝑋)) |
| 72 | 71 | exp31 419 |
. . . . . 6
⊢ (𝑦 ∈ ℕ0
→ (((𝐶 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) ∧ 𝑊 ∈ LMod) → ((𝑦 ↑ (𝐶 · 𝑋)) = ((𝑦𝐸𝐶) · 𝑋) → ((𝑦 + 1) ↑ (𝐶 · 𝑋)) = (((𝑦 + 1)𝐸𝐶) · 𝑋)))) |
| 73 | 72 | a2d 29 |
. . . . 5
⊢ (𝑦 ∈ ℕ0
→ ((((𝐶 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) ∧ 𝑊 ∈ LMod) → (𝑦 ↑ (𝐶 · 𝑋)) = ((𝑦𝐸𝐶) · 𝑋)) → (((𝐶 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) ∧ 𝑊 ∈ LMod) → ((𝑦 + 1) ↑ (𝐶 · 𝑋)) = (((𝑦 + 1)𝐸𝐶) · 𝑋)))) |
| 74 | 5, 10, 15, 20, 43, 73 | nn0ind 12713 |
. . . 4
⊢ (𝑁 ∈ ℕ0
→ (((𝐶 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) ∧ 𝑊 ∈ LMod) → (𝑁 ↑ (𝐶 · 𝑋)) = ((𝑁𝐸𝐶) · 𝑋))) |
| 75 | 74 | exp4c 432 |
. . 3
⊢ (𝑁 ∈ ℕ0
→ (𝐶 ∈ 𝐾 → (𝑋 ∈ 𝑉 → (𝑊 ∈ LMod → (𝑁 ↑ (𝐶 · 𝑋)) = ((𝑁𝐸𝐶) · 𝑋))))) |
| 76 | 75 | 3imp21 1114 |
. 2
⊢ ((𝐶 ∈ 𝐾 ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝑉) → (𝑊 ∈ LMod → (𝑁 ↑ (𝐶 · 𝑋)) = ((𝑁𝐸𝐶) · 𝑋))) |
| 77 | 76 | impcom 407 |
1
⊢ ((𝑊 ∈ LMod ∧ (𝐶 ∈ 𝐾 ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝑉)) → (𝑁 ↑ (𝐶 · 𝑋)) = ((𝑁𝐸𝐶) · 𝑋)) |