Proof of Theorem galcan
| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | oveq2 7439 | . . 3
⊢ ((𝐴 ⊕ 𝐵) = (𝐴 ⊕ 𝐶) → (((invg‘𝐺)‘𝐴) ⊕ (𝐴 ⊕ 𝐵)) = (((invg‘𝐺)‘𝐴) ⊕ (𝐴 ⊕ 𝐶))) | 
| 2 |  | simpl 482 | . . . . . . . 8
⊢ (( ⊕ ∈
(𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → ⊕ ∈ (𝐺 GrpAct 𝑌)) | 
| 3 |  | gagrp 19310 | . . . . . . . 8
⊢ ( ⊕ ∈
(𝐺 GrpAct 𝑌) → 𝐺 ∈ Grp) | 
| 4 | 2, 3 | syl 17 | . . . . . . 7
⊢ (( ⊕ ∈
(𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → 𝐺 ∈ Grp) | 
| 5 |  | simpr1 1195 | . . . . . . 7
⊢ (( ⊕ ∈
(𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → 𝐴 ∈ 𝑋) | 
| 6 |  | galcan.1 | . . . . . . . 8
⊢ 𝑋 = (Base‘𝐺) | 
| 7 |  | eqid 2737 | . . . . . . . 8
⊢
(+g‘𝐺) = (+g‘𝐺) | 
| 8 |  | eqid 2737 | . . . . . . . 8
⊢
(0g‘𝐺) = (0g‘𝐺) | 
| 9 |  | eqid 2737 | . . . . . . . 8
⊢
(invg‘𝐺) = (invg‘𝐺) | 
| 10 | 6, 7, 8, 9 | grplinv 19007 | . . . . . . 7
⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (((invg‘𝐺)‘𝐴)(+g‘𝐺)𝐴) = (0g‘𝐺)) | 
| 11 | 4, 5, 10 | syl2anc 584 | . . . . . 6
⊢ (( ⊕ ∈
(𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → (((invg‘𝐺)‘𝐴)(+g‘𝐺)𝐴) = (0g‘𝐺)) | 
| 12 | 11 | oveq1d 7446 | . . . . 5
⊢ (( ⊕ ∈
(𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → ((((invg‘𝐺)‘𝐴)(+g‘𝐺)𝐴) ⊕ 𝐵) = ((0g‘𝐺) ⊕ 𝐵)) | 
| 13 | 6, 9 | grpinvcl 19005 | . . . . . . 7
⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → ((invg‘𝐺)‘𝐴) ∈ 𝑋) | 
| 14 | 4, 5, 13 | syl2anc 584 | . . . . . 6
⊢ (( ⊕ ∈
(𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → ((invg‘𝐺)‘𝐴) ∈ 𝑋) | 
| 15 |  | simpr2 1196 | . . . . . 6
⊢ (( ⊕ ∈
(𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → 𝐵 ∈ 𝑌) | 
| 16 | 6, 7 | gaass 19315 | . . . . . 6
⊢ (( ⊕ ∈
(𝐺 GrpAct 𝑌) ∧ (((invg‘𝐺)‘𝐴) ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → ((((invg‘𝐺)‘𝐴)(+g‘𝐺)𝐴) ⊕ 𝐵) = (((invg‘𝐺)‘𝐴) ⊕ (𝐴 ⊕ 𝐵))) | 
| 17 | 2, 14, 5, 15, 16 | syl13anc 1374 | . . . . 5
⊢ (( ⊕ ∈
(𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → ((((invg‘𝐺)‘𝐴)(+g‘𝐺)𝐴) ⊕ 𝐵) = (((invg‘𝐺)‘𝐴) ⊕ (𝐴 ⊕ 𝐵))) | 
| 18 | 8 | gagrpid 19312 | . . . . . 6
⊢ (( ⊕ ∈
(𝐺 GrpAct 𝑌) ∧ 𝐵 ∈ 𝑌) → ((0g‘𝐺) ⊕ 𝐵) = 𝐵) | 
| 19 | 2, 15, 18 | syl2anc 584 | . . . . 5
⊢ (( ⊕ ∈
(𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → ((0g‘𝐺) ⊕ 𝐵) = 𝐵) | 
| 20 | 12, 17, 19 | 3eqtr3d 2785 | . . . 4
⊢ (( ⊕ ∈
(𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → (((invg‘𝐺)‘𝐴) ⊕ (𝐴 ⊕ 𝐵)) = 𝐵) | 
| 21 | 11 | oveq1d 7446 | . . . . 5
⊢ (( ⊕ ∈
(𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → ((((invg‘𝐺)‘𝐴)(+g‘𝐺)𝐴) ⊕ 𝐶) = ((0g‘𝐺) ⊕ 𝐶)) | 
| 22 |  | simpr3 1197 | . . . . . 6
⊢ (( ⊕ ∈
(𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → 𝐶 ∈ 𝑌) | 
| 23 | 6, 7 | gaass 19315 | . . . . . 6
⊢ (( ⊕ ∈
(𝐺 GrpAct 𝑌) ∧ (((invg‘𝐺)‘𝐴) ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑌)) → ((((invg‘𝐺)‘𝐴)(+g‘𝐺)𝐴) ⊕ 𝐶) = (((invg‘𝐺)‘𝐴) ⊕ (𝐴 ⊕ 𝐶))) | 
| 24 | 2, 14, 5, 22, 23 | syl13anc 1374 | . . . . 5
⊢ (( ⊕ ∈
(𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → ((((invg‘𝐺)‘𝐴)(+g‘𝐺)𝐴) ⊕ 𝐶) = (((invg‘𝐺)‘𝐴) ⊕ (𝐴 ⊕ 𝐶))) | 
| 25 | 8 | gagrpid 19312 | . . . . . 6
⊢ (( ⊕ ∈
(𝐺 GrpAct 𝑌) ∧ 𝐶 ∈ 𝑌) → ((0g‘𝐺) ⊕ 𝐶) = 𝐶) | 
| 26 | 2, 22, 25 | syl2anc 584 | . . . . 5
⊢ (( ⊕ ∈
(𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → ((0g‘𝐺) ⊕ 𝐶) = 𝐶) | 
| 27 | 21, 24, 26 | 3eqtr3d 2785 | . . . 4
⊢ (( ⊕ ∈
(𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → (((invg‘𝐺)‘𝐴) ⊕ (𝐴 ⊕ 𝐶)) = 𝐶) | 
| 28 | 20, 27 | eqeq12d 2753 | . . 3
⊢ (( ⊕ ∈
(𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → ((((invg‘𝐺)‘𝐴) ⊕ (𝐴 ⊕ 𝐵)) = (((invg‘𝐺)‘𝐴) ⊕ (𝐴 ⊕ 𝐶)) ↔ 𝐵 = 𝐶)) | 
| 29 | 1, 28 | imbitrid 244 | . 2
⊢ (( ⊕ ∈
(𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → ((𝐴 ⊕ 𝐵) = (𝐴 ⊕ 𝐶) → 𝐵 = 𝐶)) | 
| 30 |  | oveq2 7439 | . 2
⊢ (𝐵 = 𝐶 → (𝐴 ⊕ 𝐵) = (𝐴 ⊕ 𝐶)) | 
| 31 | 29, 30 | impbid1 225 | 1
⊢ (( ⊕ ∈
(𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → ((𝐴 ⊕ 𝐵) = (𝐴 ⊕ 𝐶) ↔ 𝐵 = 𝐶)) |