| Step | Hyp | Ref
| Expression |
| 1 | | srgpcomp.k |
. 2
⊢ (𝜑 → 𝐾 ∈
ℕ0) |
| 2 | | oveq1 7438 |
. . . . . 6
⊢ (𝑥 = 0 → (𝑥 ↑ 𝐵) = (0 ↑ 𝐵)) |
| 3 | 2 | oveq1d 7446 |
. . . . 5
⊢ (𝑥 = 0 → ((𝑥 ↑ 𝐵) × 𝐴) = ((0 ↑ 𝐵) × 𝐴)) |
| 4 | 2 | oveq2d 7447 |
. . . . 5
⊢ (𝑥 = 0 → (𝐴 × (𝑥 ↑ 𝐵)) = (𝐴 × (0 ↑ 𝐵))) |
| 5 | 3, 4 | eqeq12d 2753 |
. . . 4
⊢ (𝑥 = 0 → (((𝑥 ↑ 𝐵) × 𝐴) = (𝐴 × (𝑥 ↑ 𝐵)) ↔ ((0 ↑ 𝐵) × 𝐴) = (𝐴 × (0 ↑ 𝐵)))) |
| 6 | 5 | imbi2d 340 |
. . 3
⊢ (𝑥 = 0 → ((𝜑 → ((𝑥 ↑ 𝐵) × 𝐴) = (𝐴 × (𝑥 ↑ 𝐵))) ↔ (𝜑 → ((0 ↑ 𝐵) × 𝐴) = (𝐴 × (0 ↑ 𝐵))))) |
| 7 | | oveq1 7438 |
. . . . . 6
⊢ (𝑥 = 𝑦 → (𝑥 ↑ 𝐵) = (𝑦 ↑ 𝐵)) |
| 8 | 7 | oveq1d 7446 |
. . . . 5
⊢ (𝑥 = 𝑦 → ((𝑥 ↑ 𝐵) × 𝐴) = ((𝑦 ↑ 𝐵) × 𝐴)) |
| 9 | 7 | oveq2d 7447 |
. . . . 5
⊢ (𝑥 = 𝑦 → (𝐴 × (𝑥 ↑ 𝐵)) = (𝐴 × (𝑦 ↑ 𝐵))) |
| 10 | 8, 9 | eqeq12d 2753 |
. . . 4
⊢ (𝑥 = 𝑦 → (((𝑥 ↑ 𝐵) × 𝐴) = (𝐴 × (𝑥 ↑ 𝐵)) ↔ ((𝑦 ↑ 𝐵) × 𝐴) = (𝐴 × (𝑦 ↑ 𝐵)))) |
| 11 | 10 | imbi2d 340 |
. . 3
⊢ (𝑥 = 𝑦 → ((𝜑 → ((𝑥 ↑ 𝐵) × 𝐴) = (𝐴 × (𝑥 ↑ 𝐵))) ↔ (𝜑 → ((𝑦 ↑ 𝐵) × 𝐴) = (𝐴 × (𝑦 ↑ 𝐵))))) |
| 12 | | oveq1 7438 |
. . . . . 6
⊢ (𝑥 = (𝑦 + 1) → (𝑥 ↑ 𝐵) = ((𝑦 + 1) ↑ 𝐵)) |
| 13 | 12 | oveq1d 7446 |
. . . . 5
⊢ (𝑥 = (𝑦 + 1) → ((𝑥 ↑ 𝐵) × 𝐴) = (((𝑦 + 1) ↑ 𝐵) × 𝐴)) |
| 14 | 12 | oveq2d 7447 |
. . . . 5
⊢ (𝑥 = (𝑦 + 1) → (𝐴 × (𝑥 ↑ 𝐵)) = (𝐴 × ((𝑦 + 1) ↑ 𝐵))) |
| 15 | 13, 14 | eqeq12d 2753 |
. . . 4
⊢ (𝑥 = (𝑦 + 1) → (((𝑥 ↑ 𝐵) × 𝐴) = (𝐴 × (𝑥 ↑ 𝐵)) ↔ (((𝑦 + 1) ↑ 𝐵) × 𝐴) = (𝐴 × ((𝑦 + 1) ↑ 𝐵)))) |
| 16 | 15 | imbi2d 340 |
. . 3
⊢ (𝑥 = (𝑦 + 1) → ((𝜑 → ((𝑥 ↑ 𝐵) × 𝐴) = (𝐴 × (𝑥 ↑ 𝐵))) ↔ (𝜑 → (((𝑦 + 1) ↑ 𝐵) × 𝐴) = (𝐴 × ((𝑦 + 1) ↑ 𝐵))))) |
| 17 | | oveq1 7438 |
. . . . . 6
⊢ (𝑥 = 𝐾 → (𝑥 ↑ 𝐵) = (𝐾 ↑ 𝐵)) |
| 18 | 17 | oveq1d 7446 |
. . . . 5
⊢ (𝑥 = 𝐾 → ((𝑥 ↑ 𝐵) × 𝐴) = ((𝐾 ↑ 𝐵) × 𝐴)) |
| 19 | 17 | oveq2d 7447 |
. . . . 5
⊢ (𝑥 = 𝐾 → (𝐴 × (𝑥 ↑ 𝐵)) = (𝐴 × (𝐾 ↑ 𝐵))) |
| 20 | 18, 19 | eqeq12d 2753 |
. . . 4
⊢ (𝑥 = 𝐾 → (((𝑥 ↑ 𝐵) × 𝐴) = (𝐴 × (𝑥 ↑ 𝐵)) ↔ ((𝐾 ↑ 𝐵) × 𝐴) = (𝐴 × (𝐾 ↑ 𝐵)))) |
| 21 | 20 | imbi2d 340 |
. . 3
⊢ (𝑥 = 𝐾 → ((𝜑 → ((𝑥 ↑ 𝐵) × 𝐴) = (𝐴 × (𝑥 ↑ 𝐵))) ↔ (𝜑 → ((𝐾 ↑ 𝐵) × 𝐴) = (𝐴 × (𝐾 ↑ 𝐵))))) |
| 22 | | srgpcomp.b |
. . . . . 6
⊢ (𝜑 → 𝐵 ∈ 𝑆) |
| 23 | | srgpcomp.g |
. . . . . . . 8
⊢ 𝐺 = (mulGrp‘𝑅) |
| 24 | | srgpcomp.s |
. . . . . . . 8
⊢ 𝑆 = (Base‘𝑅) |
| 25 | 23, 24 | mgpbas 20142 |
. . . . . . 7
⊢ 𝑆 = (Base‘𝐺) |
| 26 | | eqid 2737 |
. . . . . . . 8
⊢
(1r‘𝑅) = (1r‘𝑅) |
| 27 | 23, 26 | ringidval 20180 |
. . . . . . 7
⊢
(1r‘𝑅) = (0g‘𝐺) |
| 28 | | srgpcomp.e |
. . . . . . 7
⊢ ↑ =
(.g‘𝐺) |
| 29 | 25, 27, 28 | mulg0 19092 |
. . . . . 6
⊢ (𝐵 ∈ 𝑆 → (0 ↑ 𝐵) = (1r‘𝑅)) |
| 30 | 22, 29 | syl 17 |
. . . . 5
⊢ (𝜑 → (0 ↑ 𝐵) = (1r‘𝑅)) |
| 31 | 30 | oveq1d 7446 |
. . . 4
⊢ (𝜑 → ((0 ↑ 𝐵) × 𝐴) = ((1r‘𝑅) × 𝐴)) |
| 32 | | srgpcomp.r |
. . . . . 6
⊢ (𝜑 → 𝑅 ∈ SRing) |
| 33 | | srgpcomp.a |
. . . . . 6
⊢ (𝜑 → 𝐴 ∈ 𝑆) |
| 34 | | srgpcomp.m |
. . . . . . 7
⊢ × =
(.r‘𝑅) |
| 35 | 24, 34, 26 | srgridm 20200 |
. . . . . 6
⊢ ((𝑅 ∈ SRing ∧ 𝐴 ∈ 𝑆) → (𝐴 ×
(1r‘𝑅)) =
𝐴) |
| 36 | 32, 33, 35 | syl2anc 584 |
. . . . 5
⊢ (𝜑 → (𝐴 ×
(1r‘𝑅)) =
𝐴) |
| 37 | 30 | oveq2d 7447 |
. . . . 5
⊢ (𝜑 → (𝐴 × (0 ↑ 𝐵)) = (𝐴 ×
(1r‘𝑅))) |
| 38 | 24, 34, 26 | srglidm 20199 |
. . . . . 6
⊢ ((𝑅 ∈ SRing ∧ 𝐴 ∈ 𝑆) → ((1r‘𝑅) × 𝐴) = 𝐴) |
| 39 | 32, 33, 38 | syl2anc 584 |
. . . . 5
⊢ (𝜑 →
((1r‘𝑅)
×
𝐴) = 𝐴) |
| 40 | 36, 37, 39 | 3eqtr4rd 2788 |
. . . 4
⊢ (𝜑 →
((1r‘𝑅)
×
𝐴) = (𝐴 × (0 ↑ 𝐵))) |
| 41 | 31, 40 | eqtrd 2777 |
. . 3
⊢ (𝜑 → ((0 ↑ 𝐵) × 𝐴) = (𝐴 × (0 ↑ 𝐵))) |
| 42 | 23 | srgmgp 20188 |
. . . . . . . . . . . . 13
⊢ (𝑅 ∈ SRing → 𝐺 ∈ Mnd) |
| 43 | 32, 42 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐺 ∈ Mnd) |
| 44 | 43 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑦 ∈ ℕ0) → 𝐺 ∈ Mnd) |
| 45 | | simpr 484 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑦 ∈ ℕ0) → 𝑦 ∈
ℕ0) |
| 46 | 22 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑦 ∈ ℕ0) → 𝐵 ∈ 𝑆) |
| 47 | 23, 34 | mgpplusg 20141 |
. . . . . . . . . . . 12
⊢ × =
(+g‘𝐺) |
| 48 | 25, 28, 47 | mulgnn0p1 19103 |
. . . . . . . . . . 11
⊢ ((𝐺 ∈ Mnd ∧ 𝑦 ∈ ℕ0
∧ 𝐵 ∈ 𝑆) → ((𝑦 + 1) ↑ 𝐵) = ((𝑦 ↑ 𝐵) × 𝐵)) |
| 49 | 44, 45, 46, 48 | syl3anc 1373 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑦 ∈ ℕ0) → ((𝑦 + 1) ↑ 𝐵) = ((𝑦 ↑ 𝐵) × 𝐵)) |
| 50 | 49 | oveq1d 7446 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 ∈ ℕ0) → (((𝑦 + 1) ↑ 𝐵) × 𝐴) = (((𝑦 ↑ 𝐵) × 𝐵) × 𝐴)) |
| 51 | | srgpcomp.c |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝐴 × 𝐵) = (𝐵 × 𝐴)) |
| 52 | 51 | eqcomd 2743 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝐵 × 𝐴) = (𝐴 × 𝐵)) |
| 53 | 52 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑦 ∈ ℕ0) → (𝐵 × 𝐴) = (𝐴 × 𝐵)) |
| 54 | 53 | oveq2d 7447 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑦 ∈ ℕ0) → ((𝑦 ↑ 𝐵) × (𝐵 × 𝐴)) = ((𝑦 ↑ 𝐵) × (𝐴 × 𝐵))) |
| 55 | 32 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑦 ∈ ℕ0) → 𝑅 ∈ SRing) |
| 56 | 25, 28, 44, 45, 46 | mulgnn0cld 19113 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑦 ∈ ℕ0) → (𝑦 ↑ 𝐵) ∈ 𝑆) |
| 57 | 33 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑦 ∈ ℕ0) → 𝐴 ∈ 𝑆) |
| 58 | 24, 34 | srgass 20191 |
. . . . . . . . . . 11
⊢ ((𝑅 ∈ SRing ∧ ((𝑦 ↑ 𝐵) ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆)) → (((𝑦 ↑ 𝐵) × 𝐵) × 𝐴) = ((𝑦 ↑ 𝐵) × (𝐵 × 𝐴))) |
| 59 | 55, 56, 46, 57, 58 | syl13anc 1374 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑦 ∈ ℕ0) → (((𝑦 ↑ 𝐵) × 𝐵) × 𝐴) = ((𝑦 ↑ 𝐵) × (𝐵 × 𝐴))) |
| 60 | 24, 34 | srgass 20191 |
. . . . . . . . . . 11
⊢ ((𝑅 ∈ SRing ∧ ((𝑦 ↑ 𝐵) ∈ 𝑆 ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆)) → (((𝑦 ↑ 𝐵) × 𝐴) × 𝐵) = ((𝑦 ↑ 𝐵) × (𝐴 × 𝐵))) |
| 61 | 55, 56, 57, 46, 60 | syl13anc 1374 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑦 ∈ ℕ0) → (((𝑦 ↑ 𝐵) × 𝐴) × 𝐵) = ((𝑦 ↑ 𝐵) × (𝐴 × 𝐵))) |
| 62 | 54, 59, 61 | 3eqtr4d 2787 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 ∈ ℕ0) → (((𝑦 ↑ 𝐵) × 𝐵) × 𝐴) = (((𝑦 ↑ 𝐵) × 𝐴) × 𝐵)) |
| 63 | 50, 62 | eqtrd 2777 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑦 ∈ ℕ0) → (((𝑦 + 1) ↑ 𝐵) × 𝐴) = (((𝑦 ↑ 𝐵) × 𝐴) × 𝐵)) |
| 64 | 63 | adantr 480 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑦 ∈ ℕ0) ∧ ((𝑦 ↑ 𝐵) × 𝐴) = (𝐴 × (𝑦 ↑ 𝐵))) → (((𝑦 + 1) ↑ 𝐵) × 𝐴) = (((𝑦 ↑ 𝐵) × 𝐴) × 𝐵)) |
| 65 | | oveq1 7438 |
. . . . . . . 8
⊢ (((𝑦 ↑ 𝐵) × 𝐴) = (𝐴 × (𝑦 ↑ 𝐵)) → (((𝑦 ↑ 𝐵) × 𝐴) × 𝐵) = ((𝐴 × (𝑦 ↑ 𝐵)) × 𝐵)) |
| 66 | 24, 34 | srgass 20191 |
. . . . . . . . . 10
⊢ ((𝑅 ∈ SRing ∧ (𝐴 ∈ 𝑆 ∧ (𝑦 ↑ 𝐵) ∈ 𝑆 ∧ 𝐵 ∈ 𝑆)) → ((𝐴 × (𝑦 ↑ 𝐵)) × 𝐵) = (𝐴 × ((𝑦 ↑ 𝐵) × 𝐵))) |
| 67 | 55, 57, 56, 46, 66 | syl13anc 1374 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 ∈ ℕ0) → ((𝐴 × (𝑦 ↑ 𝐵)) × 𝐵) = (𝐴 × ((𝑦 ↑ 𝐵) × 𝐵))) |
| 68 | 49 | eqcomd 2743 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑦 ∈ ℕ0) → ((𝑦 ↑ 𝐵) × 𝐵) = ((𝑦 + 1) ↑ 𝐵)) |
| 69 | 68 | oveq2d 7447 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 ∈ ℕ0) → (𝐴 × ((𝑦 ↑ 𝐵) × 𝐵)) = (𝐴 × ((𝑦 + 1) ↑ 𝐵))) |
| 70 | 67, 69 | eqtrd 2777 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑦 ∈ ℕ0) → ((𝐴 × (𝑦 ↑ 𝐵)) × 𝐵) = (𝐴 × ((𝑦 + 1) ↑ 𝐵))) |
| 71 | 65, 70 | sylan9eqr 2799 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑦 ∈ ℕ0) ∧ ((𝑦 ↑ 𝐵) × 𝐴) = (𝐴 × (𝑦 ↑ 𝐵))) → (((𝑦 ↑ 𝐵) × 𝐴) × 𝐵) = (𝐴 × ((𝑦 + 1) ↑ 𝐵))) |
| 72 | 64, 71 | eqtrd 2777 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑦 ∈ ℕ0) ∧ ((𝑦 ↑ 𝐵) × 𝐴) = (𝐴 × (𝑦 ↑ 𝐵))) → (((𝑦 + 1) ↑ 𝐵) × 𝐴) = (𝐴 × ((𝑦 + 1) ↑ 𝐵))) |
| 73 | 72 | ex 412 |
. . . . 5
⊢ ((𝜑 ∧ 𝑦 ∈ ℕ0) → (((𝑦 ↑ 𝐵) × 𝐴) = (𝐴 × (𝑦 ↑ 𝐵)) → (((𝑦 + 1) ↑ 𝐵) × 𝐴) = (𝐴 × ((𝑦 + 1) ↑ 𝐵)))) |
| 74 | 73 | expcom 413 |
. . . 4
⊢ (𝑦 ∈ ℕ0
→ (𝜑 → (((𝑦 ↑ 𝐵) × 𝐴) = (𝐴 × (𝑦 ↑ 𝐵)) → (((𝑦 + 1) ↑ 𝐵) × 𝐴) = (𝐴 × ((𝑦 + 1) ↑ 𝐵))))) |
| 75 | 74 | a2d 29 |
. . 3
⊢ (𝑦 ∈ ℕ0
→ ((𝜑 → ((𝑦 ↑ 𝐵) × 𝐴) = (𝐴 × (𝑦 ↑ 𝐵))) → (𝜑 → (((𝑦 + 1) ↑ 𝐵) × 𝐴) = (𝐴 × ((𝑦 + 1) ↑ 𝐵))))) |
| 76 | 6, 11, 16, 21, 41, 75 | nn0ind 12713 |
. 2
⊢ (𝐾 ∈ ℕ0
→ (𝜑 → ((𝐾 ↑ 𝐵) × 𝐴) = (𝐴 × (𝐾 ↑ 𝐵)))) |
| 77 | 1, 76 | mpcom 38 |
1
⊢ (𝜑 → ((𝐾 ↑ 𝐵) × 𝐴) = (𝐴 × (𝐾 ↑ 𝐵))) |