| Step | Hyp | Ref
| Expression |
| 1 | | fracerl.3 |
. . . . 5
⊢ ∼ =
(𝑅 ~RL
(RLReg‘𝑅)) |
| 2 | | fracerl.1 |
. . . . . 6
⊢ 𝐵 = (Base‘𝑅) |
| 3 | | eqid 2737 |
. . . . . 6
⊢
(0g‘𝑅) = (0g‘𝑅) |
| 4 | | fracerl.2 |
. . . . . 6
⊢ · =
(.r‘𝑅) |
| 5 | | eqid 2737 |
. . . . . 6
⊢
(-g‘𝑅) = (-g‘𝑅) |
| 6 | | eqid 2737 |
. . . . . 6
⊢ (𝐵 × (RLReg‘𝑅)) = (𝐵 × (RLReg‘𝑅)) |
| 7 | | eqid 2737 |
. . . . . 6
⊢
{〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ (𝐵 × (RLReg‘𝑅)) ∧ 𝑏 ∈ (𝐵 × (RLReg‘𝑅))) ∧ ∃𝑡 ∈ (RLReg‘𝑅)(𝑡 · (((1st
‘𝑎) ·
(2nd ‘𝑏))(-g‘𝑅)((1st ‘𝑏) · (2nd
‘𝑎)))) =
(0g‘𝑅))} =
{〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ (𝐵 × (RLReg‘𝑅)) ∧ 𝑏 ∈ (𝐵 × (RLReg‘𝑅))) ∧ ∃𝑡 ∈ (RLReg‘𝑅)(𝑡 · (((1st
‘𝑎) ·
(2nd ‘𝑏))(-g‘𝑅)((1st ‘𝑏) · (2nd
‘𝑎)))) =
(0g‘𝑅))} |
| 8 | | eqid 2737 |
. . . . . . . 8
⊢
(RLReg‘𝑅) =
(RLReg‘𝑅) |
| 9 | 8, 2 | rrgss 20702 |
. . . . . . 7
⊢
(RLReg‘𝑅)
⊆ 𝐵 |
| 10 | 9 | a1i 11 |
. . . . . 6
⊢ (𝜑 → (RLReg‘𝑅) ⊆ 𝐵) |
| 11 | 2, 3, 4, 5, 6, 7, 10 | erlval 33262 |
. . . . 5
⊢ (𝜑 → (𝑅 ~RL (RLReg‘𝑅)) = {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ (𝐵 × (RLReg‘𝑅)) ∧ 𝑏 ∈ (𝐵 × (RLReg‘𝑅))) ∧ ∃𝑡 ∈ (RLReg‘𝑅)(𝑡 · (((1st
‘𝑎) ·
(2nd ‘𝑏))(-g‘𝑅)((1st ‘𝑏) · (2nd
‘𝑎)))) =
(0g‘𝑅))}) |
| 12 | 1, 11 | eqtrid 2789 |
. . . 4
⊢ (𝜑 → ∼ = {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ (𝐵 × (RLReg‘𝑅)) ∧ 𝑏 ∈ (𝐵 × (RLReg‘𝑅))) ∧ ∃𝑡 ∈ (RLReg‘𝑅)(𝑡 · (((1st
‘𝑎) ·
(2nd ‘𝑏))(-g‘𝑅)((1st ‘𝑏) · (2nd
‘𝑎)))) =
(0g‘𝑅))}) |
| 13 | | simprl 771 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑎 = 〈𝐸, 𝐹〉 ∧ 𝑏 = 〈𝐺, 𝐻〉)) → 𝑎 = 〈𝐸, 𝐹〉) |
| 14 | 13 | fveq2d 6910 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑎 = 〈𝐸, 𝐹〉 ∧ 𝑏 = 〈𝐺, 𝐻〉)) → (1st ‘𝑎) = (1st
‘〈𝐸, 𝐹〉)) |
| 15 | | fracerl.5 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐸 ∈ 𝐵) |
| 16 | | fracerl.7 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐹 ∈ (RLReg‘𝑅)) |
| 17 | 16 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑎 = 〈𝐸, 𝐹〉 ∧ 𝑏 = 〈𝐺, 𝐻〉)) → 𝐹 ∈ (RLReg‘𝑅)) |
| 18 | | op1stg 8026 |
. . . . . . . . . . 11
⊢ ((𝐸 ∈ 𝐵 ∧ 𝐹 ∈ (RLReg‘𝑅)) → (1st ‘〈𝐸, 𝐹〉) = 𝐸) |
| 19 | 15, 17, 18 | syl2an2r 685 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑎 = 〈𝐸, 𝐹〉 ∧ 𝑏 = 〈𝐺, 𝐻〉)) → (1st
‘〈𝐸, 𝐹〉) = 𝐸) |
| 20 | 14, 19 | eqtrd 2777 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑎 = 〈𝐸, 𝐹〉 ∧ 𝑏 = 〈𝐺, 𝐻〉)) → (1st ‘𝑎) = 𝐸) |
| 21 | | simprr 773 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑎 = 〈𝐸, 𝐹〉 ∧ 𝑏 = 〈𝐺, 𝐻〉)) → 𝑏 = 〈𝐺, 𝐻〉) |
| 22 | 21 | fveq2d 6910 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑎 = 〈𝐸, 𝐹〉 ∧ 𝑏 = 〈𝐺, 𝐻〉)) → (2nd ‘𝑏) = (2nd
‘〈𝐺, 𝐻〉)) |
| 23 | | fracerl.6 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐺 ∈ 𝐵) |
| 24 | | fracerl.8 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐻 ∈ (RLReg‘𝑅)) |
| 25 | 24 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑎 = 〈𝐸, 𝐹〉 ∧ 𝑏 = 〈𝐺, 𝐻〉)) → 𝐻 ∈ (RLReg‘𝑅)) |
| 26 | | op2ndg 8027 |
. . . . . . . . . . 11
⊢ ((𝐺 ∈ 𝐵 ∧ 𝐻 ∈ (RLReg‘𝑅)) → (2nd ‘〈𝐺, 𝐻〉) = 𝐻) |
| 27 | 23, 25, 26 | syl2an2r 685 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑎 = 〈𝐸, 𝐹〉 ∧ 𝑏 = 〈𝐺, 𝐻〉)) → (2nd
‘〈𝐺, 𝐻〉) = 𝐻) |
| 28 | 22, 27 | eqtrd 2777 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑎 = 〈𝐸, 𝐹〉 ∧ 𝑏 = 〈𝐺, 𝐻〉)) → (2nd ‘𝑏) = 𝐻) |
| 29 | 20, 28 | oveq12d 7449 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑎 = 〈𝐸, 𝐹〉 ∧ 𝑏 = 〈𝐺, 𝐻〉)) → ((1st
‘𝑎) ·
(2nd ‘𝑏))
= (𝐸 · 𝐻)) |
| 30 | 21 | fveq2d 6910 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑎 = 〈𝐸, 𝐹〉 ∧ 𝑏 = 〈𝐺, 𝐻〉)) → (1st ‘𝑏) = (1st
‘〈𝐺, 𝐻〉)) |
| 31 | | op1stg 8026 |
. . . . . . . . . . 11
⊢ ((𝐺 ∈ 𝐵 ∧ 𝐻 ∈ (RLReg‘𝑅)) → (1st ‘〈𝐺, 𝐻〉) = 𝐺) |
| 32 | 23, 25, 31 | syl2an2r 685 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑎 = 〈𝐸, 𝐹〉 ∧ 𝑏 = 〈𝐺, 𝐻〉)) → (1st
‘〈𝐺, 𝐻〉) = 𝐺) |
| 33 | 30, 32 | eqtrd 2777 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑎 = 〈𝐸, 𝐹〉 ∧ 𝑏 = 〈𝐺, 𝐻〉)) → (1st ‘𝑏) = 𝐺) |
| 34 | 13 | fveq2d 6910 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑎 = 〈𝐸, 𝐹〉 ∧ 𝑏 = 〈𝐺, 𝐻〉)) → (2nd ‘𝑎) = (2nd
‘〈𝐸, 𝐹〉)) |
| 35 | | op2ndg 8027 |
. . . . . . . . . . 11
⊢ ((𝐸 ∈ 𝐵 ∧ 𝐹 ∈ (RLReg‘𝑅)) → (2nd ‘〈𝐸, 𝐹〉) = 𝐹) |
| 36 | 15, 17, 35 | syl2an2r 685 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑎 = 〈𝐸, 𝐹〉 ∧ 𝑏 = 〈𝐺, 𝐻〉)) → (2nd
‘〈𝐸, 𝐹〉) = 𝐹) |
| 37 | 34, 36 | eqtrd 2777 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑎 = 〈𝐸, 𝐹〉 ∧ 𝑏 = 〈𝐺, 𝐻〉)) → (2nd ‘𝑎) = 𝐹) |
| 38 | 33, 37 | oveq12d 7449 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑎 = 〈𝐸, 𝐹〉 ∧ 𝑏 = 〈𝐺, 𝐻〉)) → ((1st
‘𝑏) ·
(2nd ‘𝑎))
= (𝐺 · 𝐹)) |
| 39 | 29, 38 | oveq12d 7449 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑎 = 〈𝐸, 𝐹〉 ∧ 𝑏 = 〈𝐺, 𝐻〉)) → (((1st
‘𝑎) ·
(2nd ‘𝑏))(-g‘𝑅)((1st ‘𝑏) · (2nd
‘𝑎))) = ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹))) |
| 40 | 39 | oveq2d 7447 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑎 = 〈𝐸, 𝐹〉 ∧ 𝑏 = 〈𝐺, 𝐻〉)) → (𝑡 · (((1st
‘𝑎) ·
(2nd ‘𝑏))(-g‘𝑅)((1st ‘𝑏) · (2nd
‘𝑎)))) = (𝑡 · ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹)))) |
| 41 | 40 | eqeq1d 2739 |
. . . . 5
⊢ ((𝜑 ∧ (𝑎 = 〈𝐸, 𝐹〉 ∧ 𝑏 = 〈𝐺, 𝐻〉)) → ((𝑡 · (((1st
‘𝑎) ·
(2nd ‘𝑏))(-g‘𝑅)((1st ‘𝑏) · (2nd
‘𝑎)))) =
(0g‘𝑅)
↔ (𝑡 ·
((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹))) = (0g‘𝑅))) |
| 42 | 41 | rexbidv 3179 |
. . . 4
⊢ ((𝜑 ∧ (𝑎 = 〈𝐸, 𝐹〉 ∧ 𝑏 = 〈𝐺, 𝐻〉)) → (∃𝑡 ∈ (RLReg‘𝑅)(𝑡 · (((1st
‘𝑎) ·
(2nd ‘𝑏))(-g‘𝑅)((1st ‘𝑏) · (2nd
‘𝑎)))) =
(0g‘𝑅)
↔ ∃𝑡 ∈
(RLReg‘𝑅)(𝑡 · ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹))) = (0g‘𝑅))) |
| 43 | 12, 42 | brab2d 32619 |
. . 3
⊢ (𝜑 → (〈𝐸, 𝐹〉 ∼ 〈𝐺, 𝐻〉 ↔ ((〈𝐸, 𝐹〉 ∈ (𝐵 × (RLReg‘𝑅)) ∧ 〈𝐺, 𝐻〉 ∈ (𝐵 × (RLReg‘𝑅))) ∧ ∃𝑡 ∈ (RLReg‘𝑅)(𝑡 · ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹))) = (0g‘𝑅)))) |
| 44 | 15, 16 | opelxpd 5724 |
. . . . 5
⊢ (𝜑 → 〈𝐸, 𝐹〉 ∈ (𝐵 × (RLReg‘𝑅))) |
| 45 | 23, 24 | opelxpd 5724 |
. . . . 5
⊢ (𝜑 → 〈𝐺, 𝐻〉 ∈ (𝐵 × (RLReg‘𝑅))) |
| 46 | 44, 45 | jca 511 |
. . . 4
⊢ (𝜑 → (〈𝐸, 𝐹〉 ∈ (𝐵 × (RLReg‘𝑅)) ∧ 〈𝐺, 𝐻〉 ∈ (𝐵 × (RLReg‘𝑅)))) |
| 47 | 46 | biantrurd 532 |
. . 3
⊢ (𝜑 → (∃𝑡 ∈ (RLReg‘𝑅)(𝑡 · ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹))) = (0g‘𝑅) ↔ ((〈𝐸, 𝐹〉 ∈ (𝐵 × (RLReg‘𝑅)) ∧ 〈𝐺, 𝐻〉 ∈ (𝐵 × (RLReg‘𝑅))) ∧ ∃𝑡 ∈ (RLReg‘𝑅)(𝑡 · ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹))) = (0g‘𝑅)))) |
| 48 | | simplr 769 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑡 ∈ (RLReg‘𝑅)) ∧ (𝑡 · ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹))) = (0g‘𝑅)) → 𝑡 ∈ (RLReg‘𝑅)) |
| 49 | | fracerl.4 |
. . . . . . . . 9
⊢ (𝜑 → 𝑅 ∈ CRing) |
| 50 | 49 | crnggrpd 20244 |
. . . . . . . 8
⊢ (𝜑 → 𝑅 ∈ Grp) |
| 51 | 50 | ad2antrr 726 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑡 ∈ (RLReg‘𝑅)) ∧ (𝑡 · ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹))) = (0g‘𝑅)) → 𝑅 ∈ Grp) |
| 52 | 49 | crngringd 20243 |
. . . . . . . . 9
⊢ (𝜑 → 𝑅 ∈ Ring) |
| 53 | 52 | ad2antrr 726 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑡 ∈ (RLReg‘𝑅)) ∧ (𝑡 · ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹))) = (0g‘𝑅)) → 𝑅 ∈ Ring) |
| 54 | 15 | ad2antrr 726 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑡 ∈ (RLReg‘𝑅)) ∧ (𝑡 · ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹))) = (0g‘𝑅)) → 𝐸 ∈ 𝐵) |
| 55 | 9, 24 | sselid 3981 |
. . . . . . . . 9
⊢ (𝜑 → 𝐻 ∈ 𝐵) |
| 56 | 55 | ad2antrr 726 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑡 ∈ (RLReg‘𝑅)) ∧ (𝑡 · ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹))) = (0g‘𝑅)) → 𝐻 ∈ 𝐵) |
| 57 | 2, 4, 53, 54, 56 | ringcld 20257 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑡 ∈ (RLReg‘𝑅)) ∧ (𝑡 · ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹))) = (0g‘𝑅)) → (𝐸 · 𝐻) ∈ 𝐵) |
| 58 | 23 | ad2antrr 726 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑡 ∈ (RLReg‘𝑅)) ∧ (𝑡 · ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹))) = (0g‘𝑅)) → 𝐺 ∈ 𝐵) |
| 59 | 9, 16 | sselid 3981 |
. . . . . . . . 9
⊢ (𝜑 → 𝐹 ∈ 𝐵) |
| 60 | 59 | ad2antrr 726 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑡 ∈ (RLReg‘𝑅)) ∧ (𝑡 · ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹))) = (0g‘𝑅)) → 𝐹 ∈ 𝐵) |
| 61 | 2, 4, 53, 58, 60 | ringcld 20257 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑡 ∈ (RLReg‘𝑅)) ∧ (𝑡 · ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹))) = (0g‘𝑅)) → (𝐺 · 𝐹) ∈ 𝐵) |
| 62 | 2, 5 | grpsubcl 19038 |
. . . . . . 7
⊢ ((𝑅 ∈ Grp ∧ (𝐸 · 𝐻) ∈ 𝐵 ∧ (𝐺 · 𝐹) ∈ 𝐵) → ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹)) ∈ 𝐵) |
| 63 | 51, 57, 61, 62 | syl3anc 1373 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑡 ∈ (RLReg‘𝑅)) ∧ (𝑡 · ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹))) = (0g‘𝑅)) → ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹)) ∈ 𝐵) |
| 64 | | simpr 484 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑡 ∈ (RLReg‘𝑅)) ∧ (𝑡 · ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹))) = (0g‘𝑅)) → (𝑡 · ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹))) = (0g‘𝑅)) |
| 65 | 8, 2, 4, 3 | rrgeq0i 20699 |
. . . . . . 7
⊢ ((𝑡 ∈ (RLReg‘𝑅) ∧ ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹)) ∈ 𝐵) → ((𝑡 · ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹))) = (0g‘𝑅) → ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹)) = (0g‘𝑅))) |
| 66 | 65 | imp 406 |
. . . . . 6
⊢ (((𝑡 ∈ (RLReg‘𝑅) ∧ ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹)) ∈ 𝐵) ∧ (𝑡 · ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹))) = (0g‘𝑅)) → ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹)) = (0g‘𝑅)) |
| 67 | 48, 63, 64, 66 | syl21anc 838 |
. . . . 5
⊢ (((𝜑 ∧ 𝑡 ∈ (RLReg‘𝑅)) ∧ (𝑡 · ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹))) = (0g‘𝑅)) → ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹)) = (0g‘𝑅)) |
| 68 | 67 | r19.29an 3158 |
. . . 4
⊢ ((𝜑 ∧ ∃𝑡 ∈ (RLReg‘𝑅)(𝑡 · ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹))) = (0g‘𝑅)) → ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹)) = (0g‘𝑅)) |
| 69 | | oveq1 7438 |
. . . . . 6
⊢ (𝑡 = (1r‘𝑅) → (𝑡 · ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹))) = ((1r‘𝑅) · ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹)))) |
| 70 | 69 | eqeq1d 2739 |
. . . . 5
⊢ (𝑡 = (1r‘𝑅) → ((𝑡 · ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹))) = (0g‘𝑅) ↔ ((1r‘𝑅) · ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹))) = (0g‘𝑅))) |
| 71 | | eqid 2737 |
. . . . . . 7
⊢
(1r‘𝑅) = (1r‘𝑅) |
| 72 | 71, 8, 52 | 1rrg 33286 |
. . . . . 6
⊢ (𝜑 → (1r‘𝑅) ∈ (RLReg‘𝑅)) |
| 73 | 72 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹)) = (0g‘𝑅)) → (1r‘𝑅) ∈ (RLReg‘𝑅)) |
| 74 | | simpr 484 |
. . . . . . 7
⊢ ((𝜑 ∧ ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹)) = (0g‘𝑅)) → ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹)) = (0g‘𝑅)) |
| 75 | 74 | oveq2d 7447 |
. . . . . 6
⊢ ((𝜑 ∧ ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹)) = (0g‘𝑅)) → ((1r‘𝑅) · ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹))) = ((1r‘𝑅) ·
(0g‘𝑅))) |
| 76 | 2, 71 | ringidcl 20262 |
. . . . . . . . 9
⊢ (𝑅 ∈ Ring →
(1r‘𝑅)
∈ 𝐵) |
| 77 | 52, 76 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → (1r‘𝑅) ∈ 𝐵) |
| 78 | 2, 4, 3, 52, 77 | ringrzd 20293 |
. . . . . . 7
⊢ (𝜑 →
((1r‘𝑅)
·
(0g‘𝑅)) =
(0g‘𝑅)) |
| 79 | 78 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹)) = (0g‘𝑅)) → ((1r‘𝑅) ·
(0g‘𝑅)) =
(0g‘𝑅)) |
| 80 | 75, 79 | eqtrd 2777 |
. . . . 5
⊢ ((𝜑 ∧ ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹)) = (0g‘𝑅)) → ((1r‘𝑅) · ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹))) = (0g‘𝑅)) |
| 81 | 70, 73, 80 | rspcedvdw 3625 |
. . . 4
⊢ ((𝜑 ∧ ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹)) = (0g‘𝑅)) → ∃𝑡 ∈ (RLReg‘𝑅)(𝑡 · ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹))) = (0g‘𝑅)) |
| 82 | 68, 81 | impbida 801 |
. . 3
⊢ (𝜑 → (∃𝑡 ∈ (RLReg‘𝑅)(𝑡 · ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹))) = (0g‘𝑅) ↔ ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹)) = (0g‘𝑅))) |
| 83 | 43, 47, 82 | 3bitr2d 307 |
. 2
⊢ (𝜑 → (〈𝐸, 𝐹〉 ∼ 〈𝐺, 𝐻〉 ↔ ((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹)) = (0g‘𝑅))) |
| 84 | 2, 4, 52, 15, 55 | ringcld 20257 |
. . 3
⊢ (𝜑 → (𝐸 · 𝐻) ∈ 𝐵) |
| 85 | 2, 4, 52, 23, 59 | ringcld 20257 |
. . 3
⊢ (𝜑 → (𝐺 · 𝐹) ∈ 𝐵) |
| 86 | 2, 3, 5 | grpsubeq0 19044 |
. . 3
⊢ ((𝑅 ∈ Grp ∧ (𝐸 · 𝐻) ∈ 𝐵 ∧ (𝐺 · 𝐹) ∈ 𝐵) → (((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹)) = (0g‘𝑅) ↔ (𝐸 · 𝐻) = (𝐺 · 𝐹))) |
| 87 | 50, 84, 85, 86 | syl3anc 1373 |
. 2
⊢ (𝜑 → (((𝐸 · 𝐻)(-g‘𝑅)(𝐺 · 𝐹)) = (0g‘𝑅) ↔ (𝐸 · 𝐻) = (𝐺 · 𝐹))) |
| 88 | 83, 87 | bitrd 279 |
1
⊢ (𝜑 → (〈𝐸, 𝐹〉 ∼ 〈𝐺, 𝐻〉 ↔ (𝐸 · 𝐻) = (𝐺 · 𝐹))) |