Proof of Theorem gastacos
| Step | Hyp | Ref
| Expression |
| 1 | | gasta.1 |
. . . . . . 7
⊢ 𝑋 = (Base‘𝐺) |
| 2 | | gasta.2 |
. . . . . . 7
⊢ 𝐻 = {𝑢 ∈ 𝑋 ∣ (𝑢 ⊕ 𝐴) = 𝐴} |
| 3 | 1, 2 | gastacl 19327 |
. . . . . 6
⊢ (( ⊕ ∈
(𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) → 𝐻 ∈ (SubGrp‘𝐺)) |
| 4 | 3 | adantr 480 |
. . . . 5
⊢ ((( ⊕ ∈
(𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → 𝐻 ∈ (SubGrp‘𝐺)) |
| 5 | | subgrcl 19149 |
. . . . 5
⊢ (𝐻 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp) |
| 6 | 4, 5 | syl 17 |
. . . 4
⊢ ((( ⊕ ∈
(𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → 𝐺 ∈ Grp) |
| 7 | 1 | subgss 19145 |
. . . . 5
⊢ (𝐻 ∈ (SubGrp‘𝐺) → 𝐻 ⊆ 𝑋) |
| 8 | 4, 7 | syl 17 |
. . . 4
⊢ ((( ⊕ ∈
(𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → 𝐻 ⊆ 𝑋) |
| 9 | | eqid 2737 |
. . . . 5
⊢
(invg‘𝐺) = (invg‘𝐺) |
| 10 | | eqid 2737 |
. . . . 5
⊢
(+g‘𝐺) = (+g‘𝐺) |
| 11 | | orbsta.r |
. . . . 5
⊢ ∼ =
(𝐺 ~QG
𝐻) |
| 12 | 1, 9, 10, 11 | eqgval 19195 |
. . . 4
⊢ ((𝐺 ∈ Grp ∧ 𝐻 ⊆ 𝑋) → (𝐵 ∼ 𝐶 ↔ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ∧ (((invg‘𝐺)‘𝐵)(+g‘𝐺)𝐶) ∈ 𝐻))) |
| 13 | 6, 8, 12 | syl2anc 584 |
. . 3
⊢ ((( ⊕ ∈
(𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐵 ∼ 𝐶 ↔ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ∧ (((invg‘𝐺)‘𝐵)(+g‘𝐺)𝐶) ∈ 𝐻))) |
| 14 | | df-3an 1089 |
. . 3
⊢ ((𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ∧ (((invg‘𝐺)‘𝐵)(+g‘𝐺)𝐶) ∈ 𝐻) ↔ ((𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) ∧ (((invg‘𝐺)‘𝐵)(+g‘𝐺)𝐶) ∈ 𝐻)) |
| 15 | 13, 14 | bitrdi 287 |
. 2
⊢ ((( ⊕ ∈
(𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐵 ∼ 𝐶 ↔ ((𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) ∧ (((invg‘𝐺)‘𝐵)(+g‘𝐺)𝐶) ∈ 𝐻))) |
| 16 | | simpr 484 |
. . 3
⊢ ((( ⊕ ∈
(𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) |
| 17 | 16 | biantrurd 532 |
. 2
⊢ ((( ⊕ ∈
(𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((((invg‘𝐺)‘𝐵)(+g‘𝐺)𝐶) ∈ 𝐻 ↔ ((𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) ∧ (((invg‘𝐺)‘𝐵)(+g‘𝐺)𝐶) ∈ 𝐻))) |
| 18 | | simpll 767 |
. . . . 5
⊢ ((( ⊕ ∈
(𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ⊕ ∈ (𝐺 GrpAct 𝑌)) |
| 19 | | simprl 771 |
. . . . . 6
⊢ ((( ⊕ ∈
(𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → 𝐵 ∈ 𝑋) |
| 20 | 1, 9 | grpinvcl 19005 |
. . . . . 6
⊢ ((𝐺 ∈ Grp ∧ 𝐵 ∈ 𝑋) → ((invg‘𝐺)‘𝐵) ∈ 𝑋) |
| 21 | 6, 19, 20 | syl2anc 584 |
. . . . 5
⊢ ((( ⊕ ∈
(𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((invg‘𝐺)‘𝐵) ∈ 𝑋) |
| 22 | | simprr 773 |
. . . . 5
⊢ ((( ⊕ ∈
(𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → 𝐶 ∈ 𝑋) |
| 23 | | simplr 769 |
. . . . 5
⊢ ((( ⊕ ∈
(𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → 𝐴 ∈ 𝑌) |
| 24 | 1, 10 | gaass 19315 |
. . . . 5
⊢ (( ⊕ ∈
(𝐺 GrpAct 𝑌) ∧ (((invg‘𝐺)‘𝐵) ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌)) → ((((invg‘𝐺)‘𝐵)(+g‘𝐺)𝐶) ⊕ 𝐴) = (((invg‘𝐺)‘𝐵) ⊕ (𝐶 ⊕ 𝐴))) |
| 25 | 18, 21, 22, 23, 24 | syl13anc 1374 |
. . . 4
⊢ ((( ⊕ ∈
(𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((((invg‘𝐺)‘𝐵)(+g‘𝐺)𝐶) ⊕ 𝐴) = (((invg‘𝐺)‘𝐵) ⊕ (𝐶 ⊕ 𝐴))) |
| 26 | 25 | eqeq1d 2739 |
. . 3
⊢ ((( ⊕ ∈
(𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (((((invg‘𝐺)‘𝐵)(+g‘𝐺)𝐶) ⊕ 𝐴) = 𝐴 ↔ (((invg‘𝐺)‘𝐵) ⊕ (𝐶 ⊕ 𝐴)) = 𝐴)) |
| 27 | 1, 10 | grpcl 18959 |
. . . . 5
⊢ ((𝐺 ∈ Grp ∧
((invg‘𝐺)‘𝐵) ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (((invg‘𝐺)‘𝐵)(+g‘𝐺)𝐶) ∈ 𝑋) |
| 28 | 6, 21, 22, 27 | syl3anc 1373 |
. . . 4
⊢ ((( ⊕ ∈
(𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (((invg‘𝐺)‘𝐵)(+g‘𝐺)𝐶) ∈ 𝑋) |
| 29 | | oveq1 7438 |
. . . . . . 7
⊢ (𝑢 =
(((invg‘𝐺)‘𝐵)(+g‘𝐺)𝐶) → (𝑢 ⊕ 𝐴) = ((((invg‘𝐺)‘𝐵)(+g‘𝐺)𝐶) ⊕ 𝐴)) |
| 30 | 29 | eqeq1d 2739 |
. . . . . 6
⊢ (𝑢 =
(((invg‘𝐺)‘𝐵)(+g‘𝐺)𝐶) → ((𝑢 ⊕ 𝐴) = 𝐴 ↔ ((((invg‘𝐺)‘𝐵)(+g‘𝐺)𝐶) ⊕ 𝐴) = 𝐴)) |
| 31 | 30, 2 | elrab2 3695 |
. . . . 5
⊢
((((invg‘𝐺)‘𝐵)(+g‘𝐺)𝐶) ∈ 𝐻 ↔ ((((invg‘𝐺)‘𝐵)(+g‘𝐺)𝐶) ∈ 𝑋 ∧ ((((invg‘𝐺)‘𝐵)(+g‘𝐺)𝐶) ⊕ 𝐴) = 𝐴)) |
| 32 | 31 | baib 535 |
. . . 4
⊢
((((invg‘𝐺)‘𝐵)(+g‘𝐺)𝐶) ∈ 𝑋 → ((((invg‘𝐺)‘𝐵)(+g‘𝐺)𝐶) ∈ 𝐻 ↔ ((((invg‘𝐺)‘𝐵)(+g‘𝐺)𝐶) ⊕ 𝐴) = 𝐴)) |
| 33 | 28, 32 | syl 17 |
. . 3
⊢ ((( ⊕ ∈
(𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((((invg‘𝐺)‘𝐵)(+g‘𝐺)𝐶) ∈ 𝐻 ↔ ((((invg‘𝐺)‘𝐵)(+g‘𝐺)𝐶) ⊕ 𝐴) = 𝐴)) |
| 34 | 1 | gaf 19313 |
. . . . . 6
⊢ ( ⊕ ∈
(𝐺 GrpAct 𝑌) → ⊕ :(𝑋 × 𝑌)⟶𝑌) |
| 35 | 18, 34 | syl 17 |
. . . . 5
⊢ ((( ⊕ ∈
(𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ⊕ :(𝑋 × 𝑌)⟶𝑌) |
| 36 | 35, 22, 23 | fovcdmd 7605 |
. . . 4
⊢ ((( ⊕ ∈
(𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐶 ⊕ 𝐴) ∈ 𝑌) |
| 37 | 1, 9 | gacan 19323 |
. . . 4
⊢ (( ⊕ ∈
(𝐺 GrpAct 𝑌) ∧ (𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌 ∧ (𝐶 ⊕ 𝐴) ∈ 𝑌)) → ((𝐵 ⊕ 𝐴) = (𝐶 ⊕ 𝐴) ↔ (((invg‘𝐺)‘𝐵) ⊕ (𝐶 ⊕ 𝐴)) = 𝐴)) |
| 38 | 18, 19, 23, 36, 37 | syl13anc 1374 |
. . 3
⊢ ((( ⊕ ∈
(𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐵 ⊕ 𝐴) = (𝐶 ⊕ 𝐴) ↔ (((invg‘𝐺)‘𝐵) ⊕ (𝐶 ⊕ 𝐴)) = 𝐴)) |
| 39 | 26, 33, 38 | 3bitr4d 311 |
. 2
⊢ ((( ⊕ ∈
(𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((((invg‘𝐺)‘𝐵)(+g‘𝐺)𝐶) ∈ 𝐻 ↔ (𝐵 ⊕ 𝐴) = (𝐶 ⊕ 𝐴))) |
| 40 | 15, 17, 39 | 3bitr2d 307 |
1
⊢ ((( ⊕ ∈
(𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐵 ∼ 𝐶 ↔ (𝐵 ⊕ 𝐴) = (𝐶 ⊕ 𝐴))) |