| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | oveq1 7438 | . . . . . 6
⊢ (𝑥 = 0 → (𝑥 · (𝐼‘𝑋)) = (0 · (𝐼‘𝑋))) | 
| 2 |  | fvoveq1 7454 | . . . . . 6
⊢ (𝑥 = 0 → (𝐼‘(𝑥 · 𝑋)) = (𝐼‘(0 · 𝑋))) | 
| 3 | 1, 2 | eqeq12d 2753 | . . . . 5
⊢ (𝑥 = 0 → ((𝑥 · (𝐼‘𝑋)) = (𝐼‘(𝑥 · 𝑋)) ↔ (0 · (𝐼‘𝑋)) = (𝐼‘(0 · 𝑋)))) | 
| 4 |  | oveq1 7438 | . . . . . 6
⊢ (𝑥 = 𝑦 → (𝑥 · (𝐼‘𝑋)) = (𝑦 · (𝐼‘𝑋))) | 
| 5 |  | fvoveq1 7454 | . . . . . 6
⊢ (𝑥 = 𝑦 → (𝐼‘(𝑥 · 𝑋)) = (𝐼‘(𝑦 · 𝑋))) | 
| 6 | 4, 5 | eqeq12d 2753 | . . . . 5
⊢ (𝑥 = 𝑦 → ((𝑥 · (𝐼‘𝑋)) = (𝐼‘(𝑥 · 𝑋)) ↔ (𝑦 · (𝐼‘𝑋)) = (𝐼‘(𝑦 · 𝑋)))) | 
| 7 |  | oveq1 7438 | . . . . . 6
⊢ (𝑥 = (𝑦 + 1) → (𝑥 · (𝐼‘𝑋)) = ((𝑦 + 1) · (𝐼‘𝑋))) | 
| 8 |  | fvoveq1 7454 | . . . . . 6
⊢ (𝑥 = (𝑦 + 1) → (𝐼‘(𝑥 · 𝑋)) = (𝐼‘((𝑦 + 1) · 𝑋))) | 
| 9 | 7, 8 | eqeq12d 2753 | . . . . 5
⊢ (𝑥 = (𝑦 + 1) → ((𝑥 · (𝐼‘𝑋)) = (𝐼‘(𝑥 · 𝑋)) ↔ ((𝑦 + 1) · (𝐼‘𝑋)) = (𝐼‘((𝑦 + 1) · 𝑋)))) | 
| 10 |  | oveq1 7438 | . . . . . 6
⊢ (𝑥 = -𝑦 → (𝑥 · (𝐼‘𝑋)) = (-𝑦 · (𝐼‘𝑋))) | 
| 11 |  | fvoveq1 7454 | . . . . . 6
⊢ (𝑥 = -𝑦 → (𝐼‘(𝑥 · 𝑋)) = (𝐼‘(-𝑦 · 𝑋))) | 
| 12 | 10, 11 | eqeq12d 2753 | . . . . 5
⊢ (𝑥 = -𝑦 → ((𝑥 · (𝐼‘𝑋)) = (𝐼‘(𝑥 · 𝑋)) ↔ (-𝑦 · (𝐼‘𝑋)) = (𝐼‘(-𝑦 · 𝑋)))) | 
| 13 |  | oveq1 7438 | . . . . . 6
⊢ (𝑥 = 𝑁 → (𝑥 · (𝐼‘𝑋)) = (𝑁 · (𝐼‘𝑋))) | 
| 14 |  | fvoveq1 7454 | . . . . . 6
⊢ (𝑥 = 𝑁 → (𝐼‘(𝑥 · 𝑋)) = (𝐼‘(𝑁 · 𝑋))) | 
| 15 | 13, 14 | eqeq12d 2753 | . . . . 5
⊢ (𝑥 = 𝑁 → ((𝑥 · (𝐼‘𝑋)) = (𝐼‘(𝑥 · 𝑋)) ↔ (𝑁 · (𝐼‘𝑋)) = (𝐼‘(𝑁 · 𝑋)))) | 
| 16 |  | eqid 2737 | . . . . . . . . 9
⊢
(0g‘𝐺) = (0g‘𝐺) | 
| 17 |  | mulginvcom.i | . . . . . . . . 9
⊢ 𝐼 = (invg‘𝐺) | 
| 18 | 16, 17 | grpinvid 19017 | . . . . . . . 8
⊢ (𝐺 ∈ Grp → (𝐼‘(0g‘𝐺)) = (0g‘𝐺)) | 
| 19 | 18 | eqcomd 2743 | . . . . . . 7
⊢ (𝐺 ∈ Grp →
(0g‘𝐺) =
(𝐼‘(0g‘𝐺))) | 
| 20 | 19 | adantr 480 | . . . . . 6
⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (0g‘𝐺) = (𝐼‘(0g‘𝐺))) | 
| 21 |  | mulginvcom.b | . . . . . . . 8
⊢ 𝐵 = (Base‘𝐺) | 
| 22 | 21, 17 | grpinvcl 19005 | . . . . . . 7
⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝐼‘𝑋) ∈ 𝐵) | 
| 23 |  | mulginvcom.t | . . . . . . . 8
⊢  · =
(.g‘𝐺) | 
| 24 | 21, 16, 23 | mulg0 19092 | . . . . . . 7
⊢ ((𝐼‘𝑋) ∈ 𝐵 → (0 · (𝐼‘𝑋)) = (0g‘𝐺)) | 
| 25 | 22, 24 | syl 17 | . . . . . 6
⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (0 · (𝐼‘𝑋)) = (0g‘𝐺)) | 
| 26 | 21, 16, 23 | mulg0 19092 | . . . . . . . 8
⊢ (𝑋 ∈ 𝐵 → (0 · 𝑋) = (0g‘𝐺)) | 
| 27 | 26 | adantl 481 | . . . . . . 7
⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (0 · 𝑋) = (0g‘𝐺)) | 
| 28 | 27 | fveq2d 6910 | . . . . . 6
⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝐼‘(0 · 𝑋)) = (𝐼‘(0g‘𝐺))) | 
| 29 | 20, 25, 28 | 3eqtr4d 2787 | . . . . 5
⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (0 · (𝐼‘𝑋)) = (𝐼‘(0 · 𝑋))) | 
| 30 |  | oveq2 7439 | . . . . . . . . . 10
⊢ ((𝑦 · (𝐼‘𝑋)) = (𝐼‘(𝑦 · 𝑋)) → ((𝐼‘𝑋)(+g‘𝐺)(𝑦 · (𝐼‘𝑋))) = ((𝐼‘𝑋)(+g‘𝐺)(𝐼‘(𝑦 · 𝑋)))) | 
| 31 | 30 | adantl 481 | . . . . . . . . 9
⊢ (((𝐺 ∈ Grp ∧ 𝑦 ∈ ℕ0
∧ 𝑋 ∈ 𝐵) ∧ (𝑦 · (𝐼‘𝑋)) = (𝐼‘(𝑦 · 𝑋))) → ((𝐼‘𝑋)(+g‘𝐺)(𝑦 · (𝐼‘𝑋))) = ((𝐼‘𝑋)(+g‘𝐺)(𝐼‘(𝑦 · 𝑋)))) | 
| 32 |  | grpmnd 18958 | . . . . . . . . . . . . 13
⊢ (𝐺 ∈ Grp → 𝐺 ∈ Mnd) | 
| 33 | 32 | 3ad2ant1 1134 | . . . . . . . . . . . 12
⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ ℕ0
∧ 𝑋 ∈ 𝐵) → 𝐺 ∈ Mnd) | 
| 34 |  | simp2 1138 | . . . . . . . . . . . 12
⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ ℕ0
∧ 𝑋 ∈ 𝐵) → 𝑦 ∈ ℕ0) | 
| 35 | 22 | 3adant2 1132 | . . . . . . . . . . . 12
⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ ℕ0
∧ 𝑋 ∈ 𝐵) → (𝐼‘𝑋) ∈ 𝐵) | 
| 36 |  | eqid 2737 | . . . . . . . . . . . . 13
⊢
(+g‘𝐺) = (+g‘𝐺) | 
| 37 | 21, 23, 36 | mulgnn0p1 19103 | . . . . . . . . . . . 12
⊢ ((𝐺 ∈ Mnd ∧ 𝑦 ∈ ℕ0
∧ (𝐼‘𝑋) ∈ 𝐵) → ((𝑦 + 1) · (𝐼‘𝑋)) = ((𝑦 · (𝐼‘𝑋))(+g‘𝐺)(𝐼‘𝑋))) | 
| 38 | 33, 34, 35, 37 | syl3anc 1373 | . . . . . . . . . . 11
⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ ℕ0
∧ 𝑋 ∈ 𝐵) → ((𝑦 + 1) · (𝐼‘𝑋)) = ((𝑦 · (𝐼‘𝑋))(+g‘𝐺)(𝐼‘𝑋))) | 
| 39 |  | simp1 1137 | . . . . . . . . . . . 12
⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ ℕ0
∧ 𝑋 ∈ 𝐵) → 𝐺 ∈ Grp) | 
| 40 |  | nn0z 12638 | . . . . . . . . . . . . 13
⊢ (𝑦 ∈ ℕ0
→ 𝑦 ∈
ℤ) | 
| 41 | 40 | 3ad2ant2 1135 | . . . . . . . . . . . 12
⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ ℕ0
∧ 𝑋 ∈ 𝐵) → 𝑦 ∈ ℤ) | 
| 42 | 21, 23, 36 | mulgaddcom 19116 | . . . . . . . . . . . 12
⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ ℤ ∧ (𝐼‘𝑋) ∈ 𝐵) → ((𝑦 · (𝐼‘𝑋))(+g‘𝐺)(𝐼‘𝑋)) = ((𝐼‘𝑋)(+g‘𝐺)(𝑦 · (𝐼‘𝑋)))) | 
| 43 | 39, 41, 35, 42 | syl3anc 1373 | . . . . . . . . . . 11
⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ ℕ0
∧ 𝑋 ∈ 𝐵) → ((𝑦 · (𝐼‘𝑋))(+g‘𝐺)(𝐼‘𝑋)) = ((𝐼‘𝑋)(+g‘𝐺)(𝑦 · (𝐼‘𝑋)))) | 
| 44 | 38, 43 | eqtrd 2777 | . . . . . . . . . 10
⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ ℕ0
∧ 𝑋 ∈ 𝐵) → ((𝑦 + 1) · (𝐼‘𝑋)) = ((𝐼‘𝑋)(+g‘𝐺)(𝑦 · (𝐼‘𝑋)))) | 
| 45 | 44 | adantr 480 | . . . . . . . . 9
⊢ (((𝐺 ∈ Grp ∧ 𝑦 ∈ ℕ0
∧ 𝑋 ∈ 𝐵) ∧ (𝑦 · (𝐼‘𝑋)) = (𝐼‘(𝑦 · 𝑋))) → ((𝑦 + 1) · (𝐼‘𝑋)) = ((𝐼‘𝑋)(+g‘𝐺)(𝑦 · (𝐼‘𝑋)))) | 
| 46 | 21, 23, 36 | mulgnn0p1 19103 | . . . . . . . . . . . . 13
⊢ ((𝐺 ∈ Mnd ∧ 𝑦 ∈ ℕ0
∧ 𝑋 ∈ 𝐵) → ((𝑦 + 1) · 𝑋) = ((𝑦 · 𝑋)(+g‘𝐺)𝑋)) | 
| 47 | 32, 46 | syl3an1 1164 | . . . . . . . . . . . 12
⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ ℕ0
∧ 𝑋 ∈ 𝐵) → ((𝑦 + 1) · 𝑋) = ((𝑦 · 𝑋)(+g‘𝐺)𝑋)) | 
| 48 | 47 | fveq2d 6910 | . . . . . . . . . . 11
⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ ℕ0
∧ 𝑋 ∈ 𝐵) → (𝐼‘((𝑦 + 1) · 𝑋)) = (𝐼‘((𝑦 · 𝑋)(+g‘𝐺)𝑋))) | 
| 49 | 21, 23 | mulgcl 19109 | . . . . . . . . . . . . 13
⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (𝑦 · 𝑋) ∈ 𝐵) | 
| 50 | 40, 49 | syl3an2 1165 | . . . . . . . . . . . 12
⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ ℕ0
∧ 𝑋 ∈ 𝐵) → (𝑦 · 𝑋) ∈ 𝐵) | 
| 51 | 21, 36, 17 | grpinvadd 19036 | . . . . . . . . . . . 12
⊢ ((𝐺 ∈ Grp ∧ (𝑦 · 𝑋) ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → (𝐼‘((𝑦 · 𝑋)(+g‘𝐺)𝑋)) = ((𝐼‘𝑋)(+g‘𝐺)(𝐼‘(𝑦 · 𝑋)))) | 
| 52 | 50, 51 | syld3an2 1413 | . . . . . . . . . . 11
⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ ℕ0
∧ 𝑋 ∈ 𝐵) → (𝐼‘((𝑦 · 𝑋)(+g‘𝐺)𝑋)) = ((𝐼‘𝑋)(+g‘𝐺)(𝐼‘(𝑦 · 𝑋)))) | 
| 53 | 48, 52 | eqtrd 2777 | . . . . . . . . . 10
⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ ℕ0
∧ 𝑋 ∈ 𝐵) → (𝐼‘((𝑦 + 1) · 𝑋)) = ((𝐼‘𝑋)(+g‘𝐺)(𝐼‘(𝑦 · 𝑋)))) | 
| 54 | 53 | adantr 480 | . . . . . . . . 9
⊢ (((𝐺 ∈ Grp ∧ 𝑦 ∈ ℕ0
∧ 𝑋 ∈ 𝐵) ∧ (𝑦 · (𝐼‘𝑋)) = (𝐼‘(𝑦 · 𝑋))) → (𝐼‘((𝑦 + 1) · 𝑋)) = ((𝐼‘𝑋)(+g‘𝐺)(𝐼‘(𝑦 · 𝑋)))) | 
| 55 | 31, 45, 54 | 3eqtr4d 2787 | . . . . . . . 8
⊢ (((𝐺 ∈ Grp ∧ 𝑦 ∈ ℕ0
∧ 𝑋 ∈ 𝐵) ∧ (𝑦 · (𝐼‘𝑋)) = (𝐼‘(𝑦 · 𝑋))) → ((𝑦 + 1) · (𝐼‘𝑋)) = (𝐼‘((𝑦 + 1) · 𝑋))) | 
| 56 | 55 | 3exp1 1353 | . . . . . . 7
⊢ (𝐺 ∈ Grp → (𝑦 ∈ ℕ0
→ (𝑋 ∈ 𝐵 → ((𝑦 · (𝐼‘𝑋)) = (𝐼‘(𝑦 · 𝑋)) → ((𝑦 + 1) · (𝐼‘𝑋)) = (𝐼‘((𝑦 + 1) · 𝑋)))))) | 
| 57 | 56 | com23 86 | . . . . . 6
⊢ (𝐺 ∈ Grp → (𝑋 ∈ 𝐵 → (𝑦 ∈ ℕ0 → ((𝑦 · (𝐼‘𝑋)) = (𝐼‘(𝑦 · 𝑋)) → ((𝑦 + 1) · (𝐼‘𝑋)) = (𝐼‘((𝑦 + 1) · 𝑋)))))) | 
| 58 | 57 | imp 406 | . . . . 5
⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑦 ∈ ℕ0 → ((𝑦 · (𝐼‘𝑋)) = (𝐼‘(𝑦 · 𝑋)) → ((𝑦 + 1) · (𝐼‘𝑋)) = (𝐼‘((𝑦 + 1) · 𝑋))))) | 
| 59 |  | nnz 12634 | . . . . . 6
⊢ (𝑦 ∈ ℕ → 𝑦 ∈
ℤ) | 
| 60 | 22 | 3adant2 1132 | . . . . . . . . . . . 12
⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (𝐼‘𝑋) ∈ 𝐵) | 
| 61 | 21, 23, 17 | mulgneg 19110 | . . . . . . . . . . . 12
⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ ℤ ∧ (𝐼‘𝑋) ∈ 𝐵) → (-𝑦 · (𝐼‘𝑋)) = (𝐼‘(𝑦 · (𝐼‘𝑋)))) | 
| 62 | 60, 61 | syld3an3 1411 | . . . . . . . . . . 11
⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (-𝑦 · (𝐼‘𝑋)) = (𝐼‘(𝑦 · (𝐼‘𝑋)))) | 
| 63 | 62 | adantr 480 | . . . . . . . . . 10
⊢ (((𝐺 ∈ Grp ∧ 𝑦 ∈ ℤ ∧ 𝑋 ∈ 𝐵) ∧ (𝑦 · (𝐼‘𝑋)) = (𝐼‘(𝑦 · 𝑋))) → (-𝑦 · (𝐼‘𝑋)) = (𝐼‘(𝑦 · (𝐼‘𝑋)))) | 
| 64 | 21, 23, 17 | mulgneg 19110 | . . . . . . . . . . . . 13
⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (-𝑦 · 𝑋) = (𝐼‘(𝑦 · 𝑋))) | 
| 65 | 64 | adantr 480 | . . . . . . . . . . . 12
⊢ (((𝐺 ∈ Grp ∧ 𝑦 ∈ ℤ ∧ 𝑋 ∈ 𝐵) ∧ (𝑦 · (𝐼‘𝑋)) = (𝐼‘(𝑦 · 𝑋))) → (-𝑦 · 𝑋) = (𝐼‘(𝑦 · 𝑋))) | 
| 66 |  | simpr 484 | . . . . . . . . . . . 12
⊢ (((𝐺 ∈ Grp ∧ 𝑦 ∈ ℤ ∧ 𝑋 ∈ 𝐵) ∧ (𝑦 · (𝐼‘𝑋)) = (𝐼‘(𝑦 · 𝑋))) → (𝑦 · (𝐼‘𝑋)) = (𝐼‘(𝑦 · 𝑋))) | 
| 67 | 65, 66 | eqtr4d 2780 | . . . . . . . . . . 11
⊢ (((𝐺 ∈ Grp ∧ 𝑦 ∈ ℤ ∧ 𝑋 ∈ 𝐵) ∧ (𝑦 · (𝐼‘𝑋)) = (𝐼‘(𝑦 · 𝑋))) → (-𝑦 · 𝑋) = (𝑦 · (𝐼‘𝑋))) | 
| 68 | 67 | fveq2d 6910 | . . . . . . . . . 10
⊢ (((𝐺 ∈ Grp ∧ 𝑦 ∈ ℤ ∧ 𝑋 ∈ 𝐵) ∧ (𝑦 · (𝐼‘𝑋)) = (𝐼‘(𝑦 · 𝑋))) → (𝐼‘(-𝑦 · 𝑋)) = (𝐼‘(𝑦 · (𝐼‘𝑋)))) | 
| 69 | 63, 68 | eqtr4d 2780 | . . . . . . . . 9
⊢ (((𝐺 ∈ Grp ∧ 𝑦 ∈ ℤ ∧ 𝑋 ∈ 𝐵) ∧ (𝑦 · (𝐼‘𝑋)) = (𝐼‘(𝑦 · 𝑋))) → (-𝑦 · (𝐼‘𝑋)) = (𝐼‘(-𝑦 · 𝑋))) | 
| 70 | 69 | 3exp1 1353 | . . . . . . . 8
⊢ (𝐺 ∈ Grp → (𝑦 ∈ ℤ → (𝑋 ∈ 𝐵 → ((𝑦 · (𝐼‘𝑋)) = (𝐼‘(𝑦 · 𝑋)) → (-𝑦 · (𝐼‘𝑋)) = (𝐼‘(-𝑦 · 𝑋)))))) | 
| 71 | 70 | com23 86 | . . . . . . 7
⊢ (𝐺 ∈ Grp → (𝑋 ∈ 𝐵 → (𝑦 ∈ ℤ → ((𝑦 · (𝐼‘𝑋)) = (𝐼‘(𝑦 · 𝑋)) → (-𝑦 · (𝐼‘𝑋)) = (𝐼‘(-𝑦 · 𝑋)))))) | 
| 72 | 71 | imp 406 | . . . . . 6
⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑦 ∈ ℤ → ((𝑦 · (𝐼‘𝑋)) = (𝐼‘(𝑦 · 𝑋)) → (-𝑦 · (𝐼‘𝑋)) = (𝐼‘(-𝑦 · 𝑋))))) | 
| 73 | 59, 72 | syl5 34 | . . . . 5
⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑦 ∈ ℕ → ((𝑦 · (𝐼‘𝑋)) = (𝐼‘(𝑦 · 𝑋)) → (-𝑦 · (𝐼‘𝑋)) = (𝐼‘(-𝑦 · 𝑋))))) | 
| 74 | 3, 6, 9, 12, 15, 29, 58, 73 | zindd 12719 | . . . 4
⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑁 ∈ ℤ → (𝑁 · (𝐼‘𝑋)) = (𝐼‘(𝑁 · 𝑋)))) | 
| 75 | 74 | ex 412 | . . 3
⊢ (𝐺 ∈ Grp → (𝑋 ∈ 𝐵 → (𝑁 ∈ ℤ → (𝑁 · (𝐼‘𝑋)) = (𝐼‘(𝑁 · 𝑋))))) | 
| 76 | 75 | com23 86 | . 2
⊢ (𝐺 ∈ Grp → (𝑁 ∈ ℤ → (𝑋 ∈ 𝐵 → (𝑁 · (𝐼‘𝑋)) = (𝐼‘(𝑁 · 𝑋))))) | 
| 77 | 76 | 3imp 1111 | 1
⊢ ((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (𝑁 · (𝐼‘𝑋)) = (𝐼‘(𝑁 · 𝑋))) |