Step | Hyp | Ref
| Expression |
1 | | simp-4r 542 |
. . . . . . . . . 10
⊢
(((((𝐺 ∈ Smgrp
∧ 𝑒 ∈ 𝐵) ∧ 𝑤 ∈ 𝑒) ∧ 𝑒 = 0 ) ∧ 𝑥 ∈ 𝐵) → 𝑒 ∈ 𝐵) |
2 | | simpllr 534 |
. . . . . . . . . . 11
⊢
(((((𝐺 ∈ Smgrp
∧ 𝑒 ∈ 𝐵) ∧ 𝑤 ∈ 𝑒) ∧ 𝑒 = 0 ) ∧ 𝑥 ∈ 𝐵) → 𝑤 ∈ 𝑒) |
3 | 2 | 19.8ad 1591 |
. . . . . . . . . 10
⊢
(((((𝐺 ∈ Smgrp
∧ 𝑒 ∈ 𝐵) ∧ 𝑤 ∈ 𝑒) ∧ 𝑒 = 0 ) ∧ 𝑥 ∈ 𝐵) → ∃𝑤 𝑤 ∈ 𝑒) |
4 | | simplr 528 |
. . . . . . . . . . 11
⊢
(((((𝐺 ∈ Smgrp
∧ 𝑒 ∈ 𝐵) ∧ 𝑤 ∈ 𝑒) ∧ 𝑒 = 0 ) ∧ 𝑥 ∈ 𝐵) → 𝑒 = 0 ) |
5 | | sgrpidmnd.b |
. . . . . . . . . . . . . 14
⊢ 𝐵 = (Base‘𝐺) |
6 | | eqid 2177 |
. . . . . . . . . . . . . 14
⊢
(+g‘𝐺) = (+g‘𝐺) |
7 | | sgrpidmnd.0 |
. . . . . . . . . . . . . 14
⊢ 0 =
(0g‘𝐺) |
8 | 5, 6, 7 | grpidvalg 12797 |
. . . . . . . . . . . . 13
⊢ (𝐺 ∈ Smgrp → 0 =
(℩𝑦(𝑦 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑦(+g‘𝐺)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐺)𝑦) = 𝑥)))) |
9 | 8 | eqeq2d 2189 |
. . . . . . . . . . . 12
⊢ (𝐺 ∈ Smgrp → (𝑒 = 0 ↔ 𝑒 = (℩𝑦(𝑦 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑦(+g‘𝐺)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐺)𝑦) = 𝑥))))) |
10 | 9 | ad4antr 494 |
. . . . . . . . . . 11
⊢
(((((𝐺 ∈ Smgrp
∧ 𝑒 ∈ 𝐵) ∧ 𝑤 ∈ 𝑒) ∧ 𝑒 = 0 ) ∧ 𝑥 ∈ 𝐵) → (𝑒 = 0 ↔ 𝑒 = (℩𝑦(𝑦 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑦(+g‘𝐺)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐺)𝑦) = 𝑥))))) |
11 | 4, 10 | mpbid 147 |
. . . . . . . . . 10
⊢
(((((𝐺 ∈ Smgrp
∧ 𝑒 ∈ 𝐵) ∧ 𝑤 ∈ 𝑒) ∧ 𝑒 = 0 ) ∧ 𝑥 ∈ 𝐵) → 𝑒 = (℩𝑦(𝑦 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑦(+g‘𝐺)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐺)𝑦) = 𝑥)))) |
12 | 1, 3, 11 | 3jca 1177 |
. . . . . . . . 9
⊢
(((((𝐺 ∈ Smgrp
∧ 𝑒 ∈ 𝐵) ∧ 𝑤 ∈ 𝑒) ∧ 𝑒 = 0 ) ∧ 𝑥 ∈ 𝐵) → (𝑒 ∈ 𝐵 ∧ ∃𝑤 𝑤 ∈ 𝑒 ∧ 𝑒 = (℩𝑦(𝑦 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑦(+g‘𝐺)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐺)𝑦) = 𝑥))))) |
13 | | simpr 110 |
. . . . . . . . 9
⊢
(((((𝐺 ∈ Smgrp
∧ 𝑒 ∈ 𝐵) ∧ 𝑤 ∈ 𝑒) ∧ 𝑒 = 0 ) ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵) |
14 | | eleq1w 2238 |
. . . . . . . . . . . 12
⊢ (𝑦 = 𝑒 → (𝑦 ∈ 𝐵 ↔ 𝑒 ∈ 𝐵)) |
15 | | oveq1 5884 |
. . . . . . . . . . . . . 14
⊢ (𝑦 = 𝑒 → (𝑦(+g‘𝐺)𝑥) = (𝑒(+g‘𝐺)𝑥)) |
16 | 15 | eqeq1d 2186 |
. . . . . . . . . . . . 13
⊢ (𝑦 = 𝑒 → ((𝑦(+g‘𝐺)𝑥) = 𝑥 ↔ (𝑒(+g‘𝐺)𝑥) = 𝑥)) |
17 | 16 | ovanraleqv 5901 |
. . . . . . . . . . . 12
⊢ (𝑦 = 𝑒 → (∀𝑥 ∈ 𝐵 ((𝑦(+g‘𝐺)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐺)𝑦) = 𝑥) ↔ ∀𝑥 ∈ 𝐵 ((𝑒(+g‘𝐺)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐺)𝑒) = 𝑥))) |
18 | 14, 17 | anbi12d 473 |
. . . . . . . . . . 11
⊢ (𝑦 = 𝑒 → ((𝑦 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑦(+g‘𝐺)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐺)𝑦) = 𝑥)) ↔ (𝑒 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑒(+g‘𝐺)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐺)𝑒) = 𝑥)))) |
19 | 18 | iotam 5210 |
. . . . . . . . . 10
⊢ ((𝑒 ∈ 𝐵 ∧ ∃𝑤 𝑤 ∈ 𝑒 ∧ 𝑒 = (℩𝑦(𝑦 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑦(+g‘𝐺)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐺)𝑦) = 𝑥)))) → (𝑒 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑒(+g‘𝐺)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐺)𝑒) = 𝑥))) |
20 | | rsp 2524 |
. . . . . . . . . 10
⊢
(∀𝑥 ∈
𝐵 ((𝑒(+g‘𝐺)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐺)𝑒) = 𝑥) → (𝑥 ∈ 𝐵 → ((𝑒(+g‘𝐺)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐺)𝑒) = 𝑥))) |
21 | 19, 20 | simpl2im 386 |
. . . . . . . . 9
⊢ ((𝑒 ∈ 𝐵 ∧ ∃𝑤 𝑤 ∈ 𝑒 ∧ 𝑒 = (℩𝑦(𝑦 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑦(+g‘𝐺)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐺)𝑦) = 𝑥)))) → (𝑥 ∈ 𝐵 → ((𝑒(+g‘𝐺)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐺)𝑒) = 𝑥))) |
22 | 12, 13, 21 | sylc 62 |
. . . . . . . 8
⊢
(((((𝐺 ∈ Smgrp
∧ 𝑒 ∈ 𝐵) ∧ 𝑤 ∈ 𝑒) ∧ 𝑒 = 0 ) ∧ 𝑥 ∈ 𝐵) → ((𝑒(+g‘𝐺)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐺)𝑒) = 𝑥)) |
23 | 22 | ralrimiva 2550 |
. . . . . . 7
⊢ ((((𝐺 ∈ Smgrp ∧ 𝑒 ∈ 𝐵) ∧ 𝑤 ∈ 𝑒) ∧ 𝑒 = 0 ) → ∀𝑥 ∈ 𝐵 ((𝑒(+g‘𝐺)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐺)𝑒) = 𝑥)) |
24 | 23 | exp31 364 |
. . . . . 6
⊢ ((𝐺 ∈ Smgrp ∧ 𝑒 ∈ 𝐵) → (𝑤 ∈ 𝑒 → (𝑒 = 0 → ∀𝑥 ∈ 𝐵 ((𝑒(+g‘𝐺)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐺)𝑒) = 𝑥)))) |
25 | 24 | exlimdv 1819 |
. . . . 5
⊢ ((𝐺 ∈ Smgrp ∧ 𝑒 ∈ 𝐵) → (∃𝑤 𝑤 ∈ 𝑒 → (𝑒 = 0 → ∀𝑥 ∈ 𝐵 ((𝑒(+g‘𝐺)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐺)𝑒) = 𝑥)))) |
26 | 25 | impd 254 |
. . . 4
⊢ ((𝐺 ∈ Smgrp ∧ 𝑒 ∈ 𝐵) → ((∃𝑤 𝑤 ∈ 𝑒 ∧ 𝑒 = 0 ) → ∀𝑥 ∈ 𝐵 ((𝑒(+g‘𝐺)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐺)𝑒) = 𝑥))) |
27 | 26 | reximdva 2579 |
. . 3
⊢ (𝐺 ∈ Smgrp →
(∃𝑒 ∈ 𝐵 (∃𝑤 𝑤 ∈ 𝑒 ∧ 𝑒 = 0 ) → ∃𝑒 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑒(+g‘𝐺)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐺)𝑒) = 𝑥))) |
28 | 27 | imdistani 445 |
. 2
⊢ ((𝐺 ∈ Smgrp ∧ ∃𝑒 ∈ 𝐵 (∃𝑤 𝑤 ∈ 𝑒 ∧ 𝑒 = 0 )) → (𝐺 ∈ Smgrp ∧ ∃𝑒 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑒(+g‘𝐺)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐺)𝑒) = 𝑥))) |
29 | 5, 6 | ismnddef 12824 |
. 2
⊢ (𝐺 ∈ Mnd ↔ (𝐺 ∈ Smgrp ∧ ∃𝑒 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑒(+g‘𝐺)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐺)𝑒) = 𝑥))) |
30 | 28, 29 | sylibr 134 |
1
⊢ ((𝐺 ∈ Smgrp ∧ ∃𝑒 ∈ 𝐵 (∃𝑤 𝑤 ∈ 𝑒 ∧ 𝑒 = 0 )) → 𝐺 ∈ Mnd) |