| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | oveq1 7439 | . . . . . . 7
⊢ (𝑥 = 0 → (𝑥 · 𝑋) = (0 · 𝑋)) | 
| 2 | 1 | oveq1d 7447 | . . . . . 6
⊢ (𝑥 = 0 → ((𝑥 · 𝑋) + 𝑋) = ((0 · 𝑋) + 𝑋)) | 
| 3 | 1 | oveq2d 7448 | . . . . . 6
⊢ (𝑥 = 0 → (𝑋 + (𝑥 · 𝑋)) = (𝑋 + (0 · 𝑋))) | 
| 4 | 2, 3 | eqeq12d 2752 | . . . . 5
⊢ (𝑥 = 0 → (((𝑥 · 𝑋) + 𝑋) = (𝑋 + (𝑥 · 𝑋)) ↔ ((0 · 𝑋) + 𝑋) = (𝑋 + (0 · 𝑋)))) | 
| 5 |  | oveq1 7439 | . . . . . . 7
⊢ (𝑥 = 𝑦 → (𝑥 · 𝑋) = (𝑦 · 𝑋)) | 
| 6 | 5 | oveq1d 7447 | . . . . . 6
⊢ (𝑥 = 𝑦 → ((𝑥 · 𝑋) + 𝑋) = ((𝑦 · 𝑋) + 𝑋)) | 
| 7 | 5 | oveq2d 7448 | . . . . . 6
⊢ (𝑥 = 𝑦 → (𝑋 + (𝑥 · 𝑋)) = (𝑋 + (𝑦 · 𝑋))) | 
| 8 | 6, 7 | eqeq12d 2752 | . . . . 5
⊢ (𝑥 = 𝑦 → (((𝑥 · 𝑋) + 𝑋) = (𝑋 + (𝑥 · 𝑋)) ↔ ((𝑦 · 𝑋) + 𝑋) = (𝑋 + (𝑦 · 𝑋)))) | 
| 9 |  | oveq1 7439 | . . . . . . 7
⊢ (𝑥 = (𝑦 + 1) → (𝑥 · 𝑋) = ((𝑦 + 1) · 𝑋)) | 
| 10 | 9 | oveq1d 7447 | . . . . . 6
⊢ (𝑥 = (𝑦 + 1) → ((𝑥 · 𝑋) + 𝑋) = (((𝑦 + 1) · 𝑋) + 𝑋)) | 
| 11 | 9 | oveq2d 7448 | . . . . . 6
⊢ (𝑥 = (𝑦 + 1) → (𝑋 + (𝑥 · 𝑋)) = (𝑋 + ((𝑦 + 1) · 𝑋))) | 
| 12 | 10, 11 | eqeq12d 2752 | . . . . 5
⊢ (𝑥 = (𝑦 + 1) → (((𝑥 · 𝑋) + 𝑋) = (𝑋 + (𝑥 · 𝑋)) ↔ (((𝑦 + 1) · 𝑋) + 𝑋) = (𝑋 + ((𝑦 + 1) · 𝑋)))) | 
| 13 |  | oveq1 7439 | . . . . . . 7
⊢ (𝑥 = -𝑦 → (𝑥 · 𝑋) = (-𝑦 · 𝑋)) | 
| 14 | 13 | oveq1d 7447 | . . . . . 6
⊢ (𝑥 = -𝑦 → ((𝑥 · 𝑋) + 𝑋) = ((-𝑦 · 𝑋) + 𝑋)) | 
| 15 | 13 | oveq2d 7448 | . . . . . 6
⊢ (𝑥 = -𝑦 → (𝑋 + (𝑥 · 𝑋)) = (𝑋 + (-𝑦 · 𝑋))) | 
| 16 | 14, 15 | eqeq12d 2752 | . . . . 5
⊢ (𝑥 = -𝑦 → (((𝑥 · 𝑋) + 𝑋) = (𝑋 + (𝑥 · 𝑋)) ↔ ((-𝑦 · 𝑋) + 𝑋) = (𝑋 + (-𝑦 · 𝑋)))) | 
| 17 |  | oveq1 7439 | . . . . . . 7
⊢ (𝑥 = 𝑁 → (𝑥 · 𝑋) = (𝑁 · 𝑋)) | 
| 18 | 17 | oveq1d 7447 | . . . . . 6
⊢ (𝑥 = 𝑁 → ((𝑥 · 𝑋) + 𝑋) = ((𝑁 · 𝑋) + 𝑋)) | 
| 19 | 17 | oveq2d 7448 | . . . . . 6
⊢ (𝑥 = 𝑁 → (𝑋 + (𝑥 · 𝑋)) = (𝑋 + (𝑁 · 𝑋))) | 
| 20 | 18, 19 | eqeq12d 2752 | . . . . 5
⊢ (𝑥 = 𝑁 → (((𝑥 · 𝑋) + 𝑋) = (𝑋 + (𝑥 · 𝑋)) ↔ ((𝑁 · 𝑋) + 𝑋) = (𝑋 + (𝑁 · 𝑋)))) | 
| 21 |  | mulgaddcom.b | . . . . . . 7
⊢ 𝐵 = (Base‘𝐺) | 
| 22 |  | mulgaddcom.p | . . . . . . 7
⊢  + =
(+g‘𝐺) | 
| 23 |  | eqid 2736 | . . . . . . 7
⊢
(0g‘𝐺) = (0g‘𝐺) | 
| 24 | 21, 22, 23 | grplid 18986 | . . . . . 6
⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ((0g‘𝐺) + 𝑋) = 𝑋) | 
| 25 |  | mulgaddcom.t | . . . . . . . . 9
⊢  · =
(.g‘𝐺) | 
| 26 | 21, 23, 25 | mulg0 19093 | . . . . . . . 8
⊢ (𝑋 ∈ 𝐵 → (0 · 𝑋) = (0g‘𝐺)) | 
| 27 | 26 | adantl 481 | . . . . . . 7
⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (0 · 𝑋) = (0g‘𝐺)) | 
| 28 | 27 | oveq1d 7447 | . . . . . 6
⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ((0 · 𝑋) + 𝑋) = ((0g‘𝐺) + 𝑋)) | 
| 29 | 27 | oveq2d 7448 | . . . . . . 7
⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑋 + (0 · 𝑋)) = (𝑋 + (0g‘𝐺))) | 
| 30 | 21, 22, 23 | grprid 18987 | . . . . . . 7
⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑋 + (0g‘𝐺)) = 𝑋) | 
| 31 | 29, 30 | eqtrd 2776 | . . . . . 6
⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑋 + (0 · 𝑋)) = 𝑋) | 
| 32 | 24, 28, 31 | 3eqtr4d 2786 | . . . . 5
⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ((0 · 𝑋) + 𝑋) = (𝑋 + (0 · 𝑋))) | 
| 33 |  | nn0z 12640 | . . . . . . . . . 10
⊢ (𝑦 ∈ ℕ0
→ 𝑦 ∈
ℤ) | 
| 34 |  | simp1 1136 | . . . . . . . . . . 11
⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑦 ∈ ℤ) → 𝐺 ∈ Grp) | 
| 35 |  | simp2 1137 | . . . . . . . . . . 11
⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑦 ∈ ℤ) → 𝑋 ∈ 𝐵) | 
| 36 | 21, 25 | mulgcl 19110 | . . . . . . . . . . . 12
⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (𝑦 · 𝑋) ∈ 𝐵) | 
| 37 | 36 | 3com23 1126 | . . . . . . . . . . 11
⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑦 ∈ ℤ) → (𝑦 · 𝑋) ∈ 𝐵) | 
| 38 | 21, 22 | grpass 18961 | . . . . . . . . . . 11
⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ (𝑦 · 𝑋) ∈ 𝐵 ∧ 𝑋 ∈ 𝐵)) → ((𝑋 + (𝑦 · 𝑋)) + 𝑋) = (𝑋 + ((𝑦 · 𝑋) + 𝑋))) | 
| 39 | 34, 35, 37, 35, 38 | syl13anc 1373 | . . . . . . . . . 10
⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑦 ∈ ℤ) → ((𝑋 + (𝑦 · 𝑋)) + 𝑋) = (𝑋 + ((𝑦 · 𝑋) + 𝑋))) | 
| 40 | 33, 39 | syl3an3 1165 | . . . . . . . . 9
⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑦 ∈ ℕ0) → ((𝑋 + (𝑦 · 𝑋)) + 𝑋) = (𝑋 + ((𝑦 · 𝑋) + 𝑋))) | 
| 41 | 40 | adantr 480 | . . . . . . . 8
⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑦 ∈ ℕ0) ∧ ((𝑦 · 𝑋) + 𝑋) = (𝑋 + (𝑦 · 𝑋))) → ((𝑋 + (𝑦 · 𝑋)) + 𝑋) = (𝑋 + ((𝑦 · 𝑋) + 𝑋))) | 
| 42 |  | grpmnd 18959 | . . . . . . . . . . . . 13
⊢ (𝐺 ∈ Grp → 𝐺 ∈ Mnd) | 
| 43 | 42 | 3ad2ant1 1133 | . . . . . . . . . . . 12
⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑦 ∈ ℕ0) → 𝐺 ∈ Mnd) | 
| 44 |  | simp3 1138 | . . . . . . . . . . . 12
⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑦 ∈ ℕ0) → 𝑦 ∈
ℕ0) | 
| 45 |  | simp2 1137 | . . . . . . . . . . . 12
⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑦 ∈ ℕ0) → 𝑋 ∈ 𝐵) | 
| 46 | 21, 25, 22 | mulgnn0p1 19104 | . . . . . . . . . . . 12
⊢ ((𝐺 ∈ Mnd ∧ 𝑦 ∈ ℕ0
∧ 𝑋 ∈ 𝐵) → ((𝑦 + 1) · 𝑋) = ((𝑦 · 𝑋) + 𝑋)) | 
| 47 | 43, 44, 45, 46 | syl3anc 1372 | . . . . . . . . . . 11
⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑦 ∈ ℕ0) → ((𝑦 + 1) · 𝑋) = ((𝑦 · 𝑋) + 𝑋)) | 
| 48 | 47 | eqeq1d 2738 | . . . . . . . . . 10
⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑦 ∈ ℕ0) → (((𝑦 + 1) · 𝑋) = (𝑋 + (𝑦 · 𝑋)) ↔ ((𝑦 · 𝑋) + 𝑋) = (𝑋 + (𝑦 · 𝑋)))) | 
| 49 | 48 | biimpar 477 | . . . . . . . . 9
⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑦 ∈ ℕ0) ∧ ((𝑦 · 𝑋) + 𝑋) = (𝑋 + (𝑦 · 𝑋))) → ((𝑦 + 1) · 𝑋) = (𝑋 + (𝑦 · 𝑋))) | 
| 50 | 49 | oveq1d 7447 | . . . . . . . 8
⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑦 ∈ ℕ0) ∧ ((𝑦 · 𝑋) + 𝑋) = (𝑋 + (𝑦 · 𝑋))) → (((𝑦 + 1) · 𝑋) + 𝑋) = ((𝑋 + (𝑦 · 𝑋)) + 𝑋)) | 
| 51 | 47 | oveq2d 7448 | . . . . . . . . 9
⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑦 ∈ ℕ0) → (𝑋 + ((𝑦 + 1) · 𝑋)) = (𝑋 + ((𝑦 · 𝑋) + 𝑋))) | 
| 52 | 51 | adantr 480 | . . . . . . . 8
⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑦 ∈ ℕ0) ∧ ((𝑦 · 𝑋) + 𝑋) = (𝑋 + (𝑦 · 𝑋))) → (𝑋 + ((𝑦 + 1) · 𝑋)) = (𝑋 + ((𝑦 · 𝑋) + 𝑋))) | 
| 53 | 41, 50, 52 | 3eqtr4d 2786 | . . . . . . 7
⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑦 ∈ ℕ0) ∧ ((𝑦 · 𝑋) + 𝑋) = (𝑋 + (𝑦 · 𝑋))) → (((𝑦 + 1) · 𝑋) + 𝑋) = (𝑋 + ((𝑦 + 1) · 𝑋))) | 
| 54 | 53 | ex 412 | . . . . . 6
⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑦 ∈ ℕ0) → (((𝑦 · 𝑋) + 𝑋) = (𝑋 + (𝑦 · 𝑋)) → (((𝑦 + 1) · 𝑋) + 𝑋) = (𝑋 + ((𝑦 + 1) · 𝑋)))) | 
| 55 | 54 | 3expia 1121 | . . . . 5
⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑦 ∈ ℕ0 → (((𝑦 · 𝑋) + 𝑋) = (𝑋 + (𝑦 · 𝑋)) → (((𝑦 + 1) · 𝑋) + 𝑋) = (𝑋 + ((𝑦 + 1) · 𝑋))))) | 
| 56 |  | nnz 12636 | . . . . . 6
⊢ (𝑦 ∈ ℕ → 𝑦 ∈
ℤ) | 
| 57 | 21, 25, 22 | mulgaddcomlem 19116 | . . . . . . . . 9
⊢ (((𝐺 ∈ Grp ∧ 𝑦 ∈ ℤ ∧ 𝑋 ∈ 𝐵) ∧ ((𝑦 · 𝑋) + 𝑋) = (𝑋 + (𝑦 · 𝑋))) → ((-𝑦 · 𝑋) + 𝑋) = (𝑋 + (-𝑦 · 𝑋))) | 
| 58 | 57 | 3exp1 1352 | . . . . . . . 8
⊢ (𝐺 ∈ Grp → (𝑦 ∈ ℤ → (𝑋 ∈ 𝐵 → (((𝑦 · 𝑋) + 𝑋) = (𝑋 + (𝑦 · 𝑋)) → ((-𝑦 · 𝑋) + 𝑋) = (𝑋 + (-𝑦 · 𝑋)))))) | 
| 59 | 58 | com23 86 | . . . . . . 7
⊢ (𝐺 ∈ Grp → (𝑋 ∈ 𝐵 → (𝑦 ∈ ℤ → (((𝑦 · 𝑋) + 𝑋) = (𝑋 + (𝑦 · 𝑋)) → ((-𝑦 · 𝑋) + 𝑋) = (𝑋 + (-𝑦 · 𝑋)))))) | 
| 60 | 59 | imp 406 | . . . . . 6
⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑦 ∈ ℤ → (((𝑦 · 𝑋) + 𝑋) = (𝑋 + (𝑦 · 𝑋)) → ((-𝑦 · 𝑋) + 𝑋) = (𝑋 + (-𝑦 · 𝑋))))) | 
| 61 | 56, 60 | syl5 34 | . . . . 5
⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑦 ∈ ℕ → (((𝑦 · 𝑋) + 𝑋) = (𝑋 + (𝑦 · 𝑋)) → ((-𝑦 · 𝑋) + 𝑋) = (𝑋 + (-𝑦 · 𝑋))))) | 
| 62 | 4, 8, 12, 16, 20, 32, 55, 61 | zindd 12721 | . . . 4
⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑁 ∈ ℤ → ((𝑁 · 𝑋) + 𝑋) = (𝑋 + (𝑁 · 𝑋)))) | 
| 63 | 62 | ex 412 | . . 3
⊢ (𝐺 ∈ Grp → (𝑋 ∈ 𝐵 → (𝑁 ∈ ℤ → ((𝑁 · 𝑋) + 𝑋) = (𝑋 + (𝑁 · 𝑋))))) | 
| 64 | 63 | com23 86 | . 2
⊢ (𝐺 ∈ Grp → (𝑁 ∈ ℤ → (𝑋 ∈ 𝐵 → ((𝑁 · 𝑋) + 𝑋) = (𝑋 + (𝑁 · 𝑋))))) | 
| 65 | 64 | 3imp 1110 | 1
⊢ ((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → ((𝑁 · 𝑋) + 𝑋) = (𝑋 + (𝑁 · 𝑋))) |