Step | Hyp | Ref
| Expression |
1 | | mgm2nsgrp.s |
. . . 4
⊢ 𝑆 = {𝐴, 𝐵} |
2 | 1 | hashprdifel 13844 |
. . 3
⊢
((♯‘𝑆) =
2 → (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵)) |
3 | | 3simpa 1149 |
. . 3
⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆)) |
4 | | mgm2nsgrp.b |
. . . 4
⊢
(Base‘𝑀) =
𝑆 |
5 | | sgrp2nmnd.o |
. . . 4
⊢
(+g‘𝑀) = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ if(𝑥 = 𝐴, 𝐴, 𝐵)) |
6 | 1, 4, 5 | sgrp2nmndlem1 18197 |
. . 3
⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → 𝑀 ∈ Mgm) |
7 | 2, 3, 6 | 3syl 18 |
. 2
⊢
((♯‘𝑆) =
2 → 𝑀 ∈
Mgm) |
8 | | eqid 2738 |
. . . . . . . . . . 11
⊢
(+g‘𝑀) = (+g‘𝑀) |
9 | 1, 4, 5, 8 | sgrp2nmndlem2 18198 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) → (𝐴(+g‘𝑀)𝐴) = 𝐴) |
10 | 9 | oveq1d 7179 |
. . . . . . . . 9
⊢ ((𝐴 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) → ((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝐴) = (𝐴(+g‘𝑀)𝐴)) |
11 | 9 | oveq2d 7180 |
. . . . . . . . 9
⊢ ((𝐴 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) → (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝐴)) = (𝐴(+g‘𝑀)𝐴)) |
12 | 10, 11 | eqtr4d 2776 |
. . . . . . . 8
⊢ ((𝐴 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) → ((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝐴) = (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝐴))) |
13 | 12 | anidms 570 |
. . . . . . 7
⊢ (𝐴 ∈ 𝑆 → ((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝐴) = (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝐴))) |
14 | 13 | 3ad2ant1 1134 |
. . . . . 6
⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → ((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝐴) = (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝐴))) |
15 | 9 | anidms 570 |
. . . . . . . . . 10
⊢ (𝐴 ∈ 𝑆 → (𝐴(+g‘𝑀)𝐴) = 𝐴) |
16 | 15 | adantr 484 |
. . . . . . . . 9
⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴(+g‘𝑀)𝐴) = 𝐴) |
17 | 16 | oveq1d 7179 |
. . . . . . . 8
⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝐵) = (𝐴(+g‘𝑀)𝐵)) |
18 | 1, 4, 5, 8 | sgrp2nmndlem2 18198 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴(+g‘𝑀)𝐵) = 𝐴) |
19 | 18 | oveq2d 7180 |
. . . . . . . . 9
⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝐵)) = (𝐴(+g‘𝑀)𝐴)) |
20 | 16, 19, 18 | 3eqtr4rd 2784 |
. . . . . . . 8
⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴(+g‘𝑀)𝐵) = (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝐵))) |
21 | 17, 20 | eqtrd 2773 |
. . . . . . 7
⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝐵) = (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝐵))) |
22 | 21 | 3adant3 1133 |
. . . . . 6
⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → ((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝐵) = (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝐵))) |
23 | 14, 22 | jca 515 |
. . . . 5
⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → (((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝐴) = (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝐴)) ∧ ((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝐵) = (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝐵)))) |
24 | 18 | 3adant3 1133 |
. . . . . 6
⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → (𝐴(+g‘𝑀)𝐵) = 𝐴) |
25 | 1, 4, 5, 8 | sgrp2nmndlem3 18199 |
. . . . . . 7
⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → (𝐵(+g‘𝑀)𝐴) = 𝐵) |
26 | 25 | oveq2d 7180 |
. . . . . 6
⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → (𝐴(+g‘𝑀)(𝐵(+g‘𝑀)𝐴)) = (𝐴(+g‘𝑀)𝐵)) |
27 | 24 | oveq1d 7179 |
. . . . . . 7
⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → ((𝐴(+g‘𝑀)𝐵)(+g‘𝑀)𝐴) = (𝐴(+g‘𝑀)𝐴)) |
28 | 15 | 3ad2ant1 1134 |
. . . . . . 7
⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → (𝐴(+g‘𝑀)𝐴) = 𝐴) |
29 | 27, 28 | eqtrd 2773 |
. . . . . 6
⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → ((𝐴(+g‘𝑀)𝐵)(+g‘𝑀)𝐴) = 𝐴) |
30 | 24, 26, 29 | 3eqtr4rd 2784 |
. . . . 5
⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → ((𝐴(+g‘𝑀)𝐵)(+g‘𝑀)𝐴) = (𝐴(+g‘𝑀)(𝐵(+g‘𝑀)𝐴))) |
31 | | simp2 1138 |
. . . . . . . 8
⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → 𝐵 ∈ 𝑆) |
32 | 1, 4, 5, 8 | sgrp2nmndlem3 18199 |
. . . . . . . 8
⊢ ((𝐵 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → (𝐵(+g‘𝑀)𝐵) = 𝐵) |
33 | 31, 32 | syld3an1 1411 |
. . . . . . 7
⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → (𝐵(+g‘𝑀)𝐵) = 𝐵) |
34 | 33 | oveq2d 7180 |
. . . . . 6
⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → (𝐴(+g‘𝑀)(𝐵(+g‘𝑀)𝐵)) = (𝐴(+g‘𝑀)𝐵)) |
35 | 18 | oveq1d 7179 |
. . . . . . . 8
⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ((𝐴(+g‘𝑀)𝐵)(+g‘𝑀)𝐵) = (𝐴(+g‘𝑀)𝐵)) |
36 | 35, 18 | eqtrd 2773 |
. . . . . . 7
⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ((𝐴(+g‘𝑀)𝐵)(+g‘𝑀)𝐵) = 𝐴) |
37 | 36 | 3adant3 1133 |
. . . . . 6
⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → ((𝐴(+g‘𝑀)𝐵)(+g‘𝑀)𝐵) = 𝐴) |
38 | 24, 34, 37 | 3eqtr4rd 2784 |
. . . . 5
⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → ((𝐴(+g‘𝑀)𝐵)(+g‘𝑀)𝐵) = (𝐴(+g‘𝑀)(𝐵(+g‘𝑀)𝐵))) |
39 | 23, 30, 38 | jca32 519 |
. . . 4
⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → ((((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝐴) = (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝐴)) ∧ ((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝐵) = (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝐵))) ∧ (((𝐴(+g‘𝑀)𝐵)(+g‘𝑀)𝐴) = (𝐴(+g‘𝑀)(𝐵(+g‘𝑀)𝐴)) ∧ ((𝐴(+g‘𝑀)𝐵)(+g‘𝑀)𝐵) = (𝐴(+g‘𝑀)(𝐵(+g‘𝑀)𝐵))))) |
40 | 25 | oveq1d 7179 |
. . . . . . 7
⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → ((𝐵(+g‘𝑀)𝐴)(+g‘𝑀)𝐴) = (𝐵(+g‘𝑀)𝐴)) |
41 | 28 | oveq2d 7180 |
. . . . . . 7
⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → (𝐵(+g‘𝑀)(𝐴(+g‘𝑀)𝐴)) = (𝐵(+g‘𝑀)𝐴)) |
42 | 40, 41 | eqtr4d 2776 |
. . . . . 6
⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → ((𝐵(+g‘𝑀)𝐴)(+g‘𝑀)𝐴) = (𝐵(+g‘𝑀)(𝐴(+g‘𝑀)𝐴))) |
43 | 24 | oveq2d 7180 |
. . . . . . 7
⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → (𝐵(+g‘𝑀)(𝐴(+g‘𝑀)𝐵)) = (𝐵(+g‘𝑀)𝐴)) |
44 | 25 | oveq1d 7179 |
. . . . . . . 8
⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → ((𝐵(+g‘𝑀)𝐴)(+g‘𝑀)𝐵) = (𝐵(+g‘𝑀)𝐵)) |
45 | 44, 33 | eqtrd 2773 |
. . . . . . 7
⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → ((𝐵(+g‘𝑀)𝐴)(+g‘𝑀)𝐵) = 𝐵) |
46 | 25, 43, 45 | 3eqtr4rd 2784 |
. . . . . 6
⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → ((𝐵(+g‘𝑀)𝐴)(+g‘𝑀)𝐵) = (𝐵(+g‘𝑀)(𝐴(+g‘𝑀)𝐵))) |
47 | 42, 46 | jca 515 |
. . . . 5
⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → (((𝐵(+g‘𝑀)𝐴)(+g‘𝑀)𝐴) = (𝐵(+g‘𝑀)(𝐴(+g‘𝑀)𝐴)) ∧ ((𝐵(+g‘𝑀)𝐴)(+g‘𝑀)𝐵) = (𝐵(+g‘𝑀)(𝐴(+g‘𝑀)𝐵)))) |
48 | 25 | oveq2d 7180 |
. . . . . 6
⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → (𝐵(+g‘𝑀)(𝐵(+g‘𝑀)𝐴)) = (𝐵(+g‘𝑀)𝐵)) |
49 | 33 | oveq1d 7179 |
. . . . . . 7
⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → ((𝐵(+g‘𝑀)𝐵)(+g‘𝑀)𝐴) = (𝐵(+g‘𝑀)𝐴)) |
50 | 49, 25 | eqtrd 2773 |
. . . . . 6
⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → ((𝐵(+g‘𝑀)𝐵)(+g‘𝑀)𝐴) = 𝐵) |
51 | 33, 48, 50 | 3eqtr4rd 2784 |
. . . . 5
⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → ((𝐵(+g‘𝑀)𝐵)(+g‘𝑀)𝐴) = (𝐵(+g‘𝑀)(𝐵(+g‘𝑀)𝐴))) |
52 | 32 | oveq1d 7179 |
. . . . . . 7
⊢ ((𝐵 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → ((𝐵(+g‘𝑀)𝐵)(+g‘𝑀)𝐵) = (𝐵(+g‘𝑀)𝐵)) |
53 | 32 | oveq2d 7180 |
. . . . . . 7
⊢ ((𝐵 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → (𝐵(+g‘𝑀)(𝐵(+g‘𝑀)𝐵)) = (𝐵(+g‘𝑀)𝐵)) |
54 | 52, 53 | eqtr4d 2776 |
. . . . . 6
⊢ ((𝐵 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → ((𝐵(+g‘𝑀)𝐵)(+g‘𝑀)𝐵) = (𝐵(+g‘𝑀)(𝐵(+g‘𝑀)𝐵))) |
55 | 31, 54 | syld3an1 1411 |
. . . . 5
⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → ((𝐵(+g‘𝑀)𝐵)(+g‘𝑀)𝐵) = (𝐵(+g‘𝑀)(𝐵(+g‘𝑀)𝐵))) |
56 | 47, 51, 55 | jca32 519 |
. . . 4
⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → ((((𝐵(+g‘𝑀)𝐴)(+g‘𝑀)𝐴) = (𝐵(+g‘𝑀)(𝐴(+g‘𝑀)𝐴)) ∧ ((𝐵(+g‘𝑀)𝐴)(+g‘𝑀)𝐵) = (𝐵(+g‘𝑀)(𝐴(+g‘𝑀)𝐵))) ∧ (((𝐵(+g‘𝑀)𝐵)(+g‘𝑀)𝐴) = (𝐵(+g‘𝑀)(𝐵(+g‘𝑀)𝐴)) ∧ ((𝐵(+g‘𝑀)𝐵)(+g‘𝑀)𝐵) = (𝐵(+g‘𝑀)(𝐵(+g‘𝑀)𝐵))))) |
57 | | oveq1 7171 |
. . . . . . . . . 10
⊢ (𝑎 = 𝐴 → (𝑎(+g‘𝑀)𝑏) = (𝐴(+g‘𝑀)𝑏)) |
58 | 57 | oveq1d 7179 |
. . . . . . . . 9
⊢ (𝑎 = 𝐴 → ((𝑎(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = ((𝐴(+g‘𝑀)𝑏)(+g‘𝑀)𝑐)) |
59 | | oveq1 7171 |
. . . . . . . . 9
⊢ (𝑎 = 𝐴 → (𝑎(+g‘𝑀)(𝑏(+g‘𝑀)𝑐)) = (𝐴(+g‘𝑀)(𝑏(+g‘𝑀)𝑐))) |
60 | 58, 59 | eqeq12d 2754 |
. . . . . . . 8
⊢ (𝑎 = 𝐴 → (((𝑎(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = (𝑎(+g‘𝑀)(𝑏(+g‘𝑀)𝑐)) ↔ ((𝐴(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = (𝐴(+g‘𝑀)(𝑏(+g‘𝑀)𝑐)))) |
61 | 60 | 2ralbidv 3111 |
. . . . . . 7
⊢ (𝑎 = 𝐴 → (∀𝑏 ∈ {𝐴, 𝐵}∀𝑐 ∈ {𝐴, 𝐵} ((𝑎(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = (𝑎(+g‘𝑀)(𝑏(+g‘𝑀)𝑐)) ↔ ∀𝑏 ∈ {𝐴, 𝐵}∀𝑐 ∈ {𝐴, 𝐵} ((𝐴(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = (𝐴(+g‘𝑀)(𝑏(+g‘𝑀)𝑐)))) |
62 | | oveq1 7171 |
. . . . . . . . . 10
⊢ (𝑎 = 𝐵 → (𝑎(+g‘𝑀)𝑏) = (𝐵(+g‘𝑀)𝑏)) |
63 | 62 | oveq1d 7179 |
. . . . . . . . 9
⊢ (𝑎 = 𝐵 → ((𝑎(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = ((𝐵(+g‘𝑀)𝑏)(+g‘𝑀)𝑐)) |
64 | | oveq1 7171 |
. . . . . . . . 9
⊢ (𝑎 = 𝐵 → (𝑎(+g‘𝑀)(𝑏(+g‘𝑀)𝑐)) = (𝐵(+g‘𝑀)(𝑏(+g‘𝑀)𝑐))) |
65 | 63, 64 | eqeq12d 2754 |
. . . . . . . 8
⊢ (𝑎 = 𝐵 → (((𝑎(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = (𝑎(+g‘𝑀)(𝑏(+g‘𝑀)𝑐)) ↔ ((𝐵(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = (𝐵(+g‘𝑀)(𝑏(+g‘𝑀)𝑐)))) |
66 | 65 | 2ralbidv 3111 |
. . . . . . 7
⊢ (𝑎 = 𝐵 → (∀𝑏 ∈ {𝐴, 𝐵}∀𝑐 ∈ {𝐴, 𝐵} ((𝑎(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = (𝑎(+g‘𝑀)(𝑏(+g‘𝑀)𝑐)) ↔ ∀𝑏 ∈ {𝐴, 𝐵}∀𝑐 ∈ {𝐴, 𝐵} ((𝐵(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = (𝐵(+g‘𝑀)(𝑏(+g‘𝑀)𝑐)))) |
67 | 61, 66 | ralprg 4582 |
. . . . . 6
⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (∀𝑎 ∈ {𝐴, 𝐵}∀𝑏 ∈ {𝐴, 𝐵}∀𝑐 ∈ {𝐴, 𝐵} ((𝑎(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = (𝑎(+g‘𝑀)(𝑏(+g‘𝑀)𝑐)) ↔ (∀𝑏 ∈ {𝐴, 𝐵}∀𝑐 ∈ {𝐴, 𝐵} ((𝐴(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = (𝐴(+g‘𝑀)(𝑏(+g‘𝑀)𝑐)) ∧ ∀𝑏 ∈ {𝐴, 𝐵}∀𝑐 ∈ {𝐴, 𝐵} ((𝐵(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = (𝐵(+g‘𝑀)(𝑏(+g‘𝑀)𝑐))))) |
68 | | oveq2 7172 |
. . . . . . . . . . 11
⊢ (𝑏 = 𝐴 → (𝐴(+g‘𝑀)𝑏) = (𝐴(+g‘𝑀)𝐴)) |
69 | 68 | oveq1d 7179 |
. . . . . . . . . 10
⊢ (𝑏 = 𝐴 → ((𝐴(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = ((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝑐)) |
70 | | oveq1 7171 |
. . . . . . . . . . 11
⊢ (𝑏 = 𝐴 → (𝑏(+g‘𝑀)𝑐) = (𝐴(+g‘𝑀)𝑐)) |
71 | 70 | oveq2d 7180 |
. . . . . . . . . 10
⊢ (𝑏 = 𝐴 → (𝐴(+g‘𝑀)(𝑏(+g‘𝑀)𝑐)) = (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝑐))) |
72 | 69, 71 | eqeq12d 2754 |
. . . . . . . . 9
⊢ (𝑏 = 𝐴 → (((𝐴(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = (𝐴(+g‘𝑀)(𝑏(+g‘𝑀)𝑐)) ↔ ((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝑐) = (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝑐)))) |
73 | 72 | ralbidv 3109 |
. . . . . . . 8
⊢ (𝑏 = 𝐴 → (∀𝑐 ∈ {𝐴, 𝐵} ((𝐴(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = (𝐴(+g‘𝑀)(𝑏(+g‘𝑀)𝑐)) ↔ ∀𝑐 ∈ {𝐴, 𝐵} ((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝑐) = (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝑐)))) |
74 | | oveq2 7172 |
. . . . . . . . . . 11
⊢ (𝑏 = 𝐵 → (𝐴(+g‘𝑀)𝑏) = (𝐴(+g‘𝑀)𝐵)) |
75 | 74 | oveq1d 7179 |
. . . . . . . . . 10
⊢ (𝑏 = 𝐵 → ((𝐴(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = ((𝐴(+g‘𝑀)𝐵)(+g‘𝑀)𝑐)) |
76 | | oveq1 7171 |
. . . . . . . . . . 11
⊢ (𝑏 = 𝐵 → (𝑏(+g‘𝑀)𝑐) = (𝐵(+g‘𝑀)𝑐)) |
77 | 76 | oveq2d 7180 |
. . . . . . . . . 10
⊢ (𝑏 = 𝐵 → (𝐴(+g‘𝑀)(𝑏(+g‘𝑀)𝑐)) = (𝐴(+g‘𝑀)(𝐵(+g‘𝑀)𝑐))) |
78 | 75, 77 | eqeq12d 2754 |
. . . . . . . . 9
⊢ (𝑏 = 𝐵 → (((𝐴(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = (𝐴(+g‘𝑀)(𝑏(+g‘𝑀)𝑐)) ↔ ((𝐴(+g‘𝑀)𝐵)(+g‘𝑀)𝑐) = (𝐴(+g‘𝑀)(𝐵(+g‘𝑀)𝑐)))) |
79 | 78 | ralbidv 3109 |
. . . . . . . 8
⊢ (𝑏 = 𝐵 → (∀𝑐 ∈ {𝐴, 𝐵} ((𝐴(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = (𝐴(+g‘𝑀)(𝑏(+g‘𝑀)𝑐)) ↔ ∀𝑐 ∈ {𝐴, 𝐵} ((𝐴(+g‘𝑀)𝐵)(+g‘𝑀)𝑐) = (𝐴(+g‘𝑀)(𝐵(+g‘𝑀)𝑐)))) |
80 | 73, 79 | ralprg 4582 |
. . . . . . 7
⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (∀𝑏 ∈ {𝐴, 𝐵}∀𝑐 ∈ {𝐴, 𝐵} ((𝐴(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = (𝐴(+g‘𝑀)(𝑏(+g‘𝑀)𝑐)) ↔ (∀𝑐 ∈ {𝐴, 𝐵} ((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝑐) = (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝑐)) ∧ ∀𝑐 ∈ {𝐴, 𝐵} ((𝐴(+g‘𝑀)𝐵)(+g‘𝑀)𝑐) = (𝐴(+g‘𝑀)(𝐵(+g‘𝑀)𝑐))))) |
81 | | oveq2 7172 |
. . . . . . . . . . 11
⊢ (𝑏 = 𝐴 → (𝐵(+g‘𝑀)𝑏) = (𝐵(+g‘𝑀)𝐴)) |
82 | 81 | oveq1d 7179 |
. . . . . . . . . 10
⊢ (𝑏 = 𝐴 → ((𝐵(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = ((𝐵(+g‘𝑀)𝐴)(+g‘𝑀)𝑐)) |
83 | 70 | oveq2d 7180 |
. . . . . . . . . 10
⊢ (𝑏 = 𝐴 → (𝐵(+g‘𝑀)(𝑏(+g‘𝑀)𝑐)) = (𝐵(+g‘𝑀)(𝐴(+g‘𝑀)𝑐))) |
84 | 82, 83 | eqeq12d 2754 |
. . . . . . . . 9
⊢ (𝑏 = 𝐴 → (((𝐵(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = (𝐵(+g‘𝑀)(𝑏(+g‘𝑀)𝑐)) ↔ ((𝐵(+g‘𝑀)𝐴)(+g‘𝑀)𝑐) = (𝐵(+g‘𝑀)(𝐴(+g‘𝑀)𝑐)))) |
85 | 84 | ralbidv 3109 |
. . . . . . . 8
⊢ (𝑏 = 𝐴 → (∀𝑐 ∈ {𝐴, 𝐵} ((𝐵(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = (𝐵(+g‘𝑀)(𝑏(+g‘𝑀)𝑐)) ↔ ∀𝑐 ∈ {𝐴, 𝐵} ((𝐵(+g‘𝑀)𝐴)(+g‘𝑀)𝑐) = (𝐵(+g‘𝑀)(𝐴(+g‘𝑀)𝑐)))) |
86 | | oveq2 7172 |
. . . . . . . . . . 11
⊢ (𝑏 = 𝐵 → (𝐵(+g‘𝑀)𝑏) = (𝐵(+g‘𝑀)𝐵)) |
87 | 86 | oveq1d 7179 |
. . . . . . . . . 10
⊢ (𝑏 = 𝐵 → ((𝐵(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = ((𝐵(+g‘𝑀)𝐵)(+g‘𝑀)𝑐)) |
88 | 76 | oveq2d 7180 |
. . . . . . . . . 10
⊢ (𝑏 = 𝐵 → (𝐵(+g‘𝑀)(𝑏(+g‘𝑀)𝑐)) = (𝐵(+g‘𝑀)(𝐵(+g‘𝑀)𝑐))) |
89 | 87, 88 | eqeq12d 2754 |
. . . . . . . . 9
⊢ (𝑏 = 𝐵 → (((𝐵(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = (𝐵(+g‘𝑀)(𝑏(+g‘𝑀)𝑐)) ↔ ((𝐵(+g‘𝑀)𝐵)(+g‘𝑀)𝑐) = (𝐵(+g‘𝑀)(𝐵(+g‘𝑀)𝑐)))) |
90 | 89 | ralbidv 3109 |
. . . . . . . 8
⊢ (𝑏 = 𝐵 → (∀𝑐 ∈ {𝐴, 𝐵} ((𝐵(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = (𝐵(+g‘𝑀)(𝑏(+g‘𝑀)𝑐)) ↔ ∀𝑐 ∈ {𝐴, 𝐵} ((𝐵(+g‘𝑀)𝐵)(+g‘𝑀)𝑐) = (𝐵(+g‘𝑀)(𝐵(+g‘𝑀)𝑐)))) |
91 | 85, 90 | ralprg 4582 |
. . . . . . 7
⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (∀𝑏 ∈ {𝐴, 𝐵}∀𝑐 ∈ {𝐴, 𝐵} ((𝐵(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = (𝐵(+g‘𝑀)(𝑏(+g‘𝑀)𝑐)) ↔ (∀𝑐 ∈ {𝐴, 𝐵} ((𝐵(+g‘𝑀)𝐴)(+g‘𝑀)𝑐) = (𝐵(+g‘𝑀)(𝐴(+g‘𝑀)𝑐)) ∧ ∀𝑐 ∈ {𝐴, 𝐵} ((𝐵(+g‘𝑀)𝐵)(+g‘𝑀)𝑐) = (𝐵(+g‘𝑀)(𝐵(+g‘𝑀)𝑐))))) |
92 | 80, 91 | anbi12d 634 |
. . . . . 6
⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ((∀𝑏 ∈ {𝐴, 𝐵}∀𝑐 ∈ {𝐴, 𝐵} ((𝐴(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = (𝐴(+g‘𝑀)(𝑏(+g‘𝑀)𝑐)) ∧ ∀𝑏 ∈ {𝐴, 𝐵}∀𝑐 ∈ {𝐴, 𝐵} ((𝐵(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = (𝐵(+g‘𝑀)(𝑏(+g‘𝑀)𝑐))) ↔ ((∀𝑐 ∈ {𝐴, 𝐵} ((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝑐) = (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝑐)) ∧ ∀𝑐 ∈ {𝐴, 𝐵} ((𝐴(+g‘𝑀)𝐵)(+g‘𝑀)𝑐) = (𝐴(+g‘𝑀)(𝐵(+g‘𝑀)𝑐))) ∧ (∀𝑐 ∈ {𝐴, 𝐵} ((𝐵(+g‘𝑀)𝐴)(+g‘𝑀)𝑐) = (𝐵(+g‘𝑀)(𝐴(+g‘𝑀)𝑐)) ∧ ∀𝑐 ∈ {𝐴, 𝐵} ((𝐵(+g‘𝑀)𝐵)(+g‘𝑀)𝑐) = (𝐵(+g‘𝑀)(𝐵(+g‘𝑀)𝑐)))))) |
93 | | oveq2 7172 |
. . . . . . . . . 10
⊢ (𝑐 = 𝐴 → ((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝑐) = ((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝐴)) |
94 | | oveq2 7172 |
. . . . . . . . . . 11
⊢ (𝑐 = 𝐴 → (𝐴(+g‘𝑀)𝑐) = (𝐴(+g‘𝑀)𝐴)) |
95 | 94 | oveq2d 7180 |
. . . . . . . . . 10
⊢ (𝑐 = 𝐴 → (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝑐)) = (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝐴))) |
96 | 93, 95 | eqeq12d 2754 |
. . . . . . . . 9
⊢ (𝑐 = 𝐴 → (((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝑐) = (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝑐)) ↔ ((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝐴) = (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝐴)))) |
97 | | oveq2 7172 |
. . . . . . . . . 10
⊢ (𝑐 = 𝐵 → ((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝑐) = ((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝐵)) |
98 | | oveq2 7172 |
. . . . . . . . . . 11
⊢ (𝑐 = 𝐵 → (𝐴(+g‘𝑀)𝑐) = (𝐴(+g‘𝑀)𝐵)) |
99 | 98 | oveq2d 7180 |
. . . . . . . . . 10
⊢ (𝑐 = 𝐵 → (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝑐)) = (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝐵))) |
100 | 97, 99 | eqeq12d 2754 |
. . . . . . . . 9
⊢ (𝑐 = 𝐵 → (((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝑐) = (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝑐)) ↔ ((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝐵) = (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝐵)))) |
101 | 96, 100 | ralprg 4582 |
. . . . . . . 8
⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (∀𝑐 ∈ {𝐴, 𝐵} ((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝑐) = (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝑐)) ↔ (((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝐴) = (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝐴)) ∧ ((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝐵) = (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝐵))))) |
102 | | oveq2 7172 |
. . . . . . . . . 10
⊢ (𝑐 = 𝐴 → ((𝐴(+g‘𝑀)𝐵)(+g‘𝑀)𝑐) = ((𝐴(+g‘𝑀)𝐵)(+g‘𝑀)𝐴)) |
103 | | oveq2 7172 |
. . . . . . . . . . 11
⊢ (𝑐 = 𝐴 → (𝐵(+g‘𝑀)𝑐) = (𝐵(+g‘𝑀)𝐴)) |
104 | 103 | oveq2d 7180 |
. . . . . . . . . 10
⊢ (𝑐 = 𝐴 → (𝐴(+g‘𝑀)(𝐵(+g‘𝑀)𝑐)) = (𝐴(+g‘𝑀)(𝐵(+g‘𝑀)𝐴))) |
105 | 102, 104 | eqeq12d 2754 |
. . . . . . . . 9
⊢ (𝑐 = 𝐴 → (((𝐴(+g‘𝑀)𝐵)(+g‘𝑀)𝑐) = (𝐴(+g‘𝑀)(𝐵(+g‘𝑀)𝑐)) ↔ ((𝐴(+g‘𝑀)𝐵)(+g‘𝑀)𝐴) = (𝐴(+g‘𝑀)(𝐵(+g‘𝑀)𝐴)))) |
106 | | oveq2 7172 |
. . . . . . . . . 10
⊢ (𝑐 = 𝐵 → ((𝐴(+g‘𝑀)𝐵)(+g‘𝑀)𝑐) = ((𝐴(+g‘𝑀)𝐵)(+g‘𝑀)𝐵)) |
107 | | oveq2 7172 |
. . . . . . . . . . 11
⊢ (𝑐 = 𝐵 → (𝐵(+g‘𝑀)𝑐) = (𝐵(+g‘𝑀)𝐵)) |
108 | 107 | oveq2d 7180 |
. . . . . . . . . 10
⊢ (𝑐 = 𝐵 → (𝐴(+g‘𝑀)(𝐵(+g‘𝑀)𝑐)) = (𝐴(+g‘𝑀)(𝐵(+g‘𝑀)𝐵))) |
109 | 106, 108 | eqeq12d 2754 |
. . . . . . . . 9
⊢ (𝑐 = 𝐵 → (((𝐴(+g‘𝑀)𝐵)(+g‘𝑀)𝑐) = (𝐴(+g‘𝑀)(𝐵(+g‘𝑀)𝑐)) ↔ ((𝐴(+g‘𝑀)𝐵)(+g‘𝑀)𝐵) = (𝐴(+g‘𝑀)(𝐵(+g‘𝑀)𝐵)))) |
110 | 105, 109 | ralprg 4582 |
. . . . . . . 8
⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (∀𝑐 ∈ {𝐴, 𝐵} ((𝐴(+g‘𝑀)𝐵)(+g‘𝑀)𝑐) = (𝐴(+g‘𝑀)(𝐵(+g‘𝑀)𝑐)) ↔ (((𝐴(+g‘𝑀)𝐵)(+g‘𝑀)𝐴) = (𝐴(+g‘𝑀)(𝐵(+g‘𝑀)𝐴)) ∧ ((𝐴(+g‘𝑀)𝐵)(+g‘𝑀)𝐵) = (𝐴(+g‘𝑀)(𝐵(+g‘𝑀)𝐵))))) |
111 | 101, 110 | anbi12d 634 |
. . . . . . 7
⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ((∀𝑐 ∈ {𝐴, 𝐵} ((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝑐) = (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝑐)) ∧ ∀𝑐 ∈ {𝐴, 𝐵} ((𝐴(+g‘𝑀)𝐵)(+g‘𝑀)𝑐) = (𝐴(+g‘𝑀)(𝐵(+g‘𝑀)𝑐))) ↔ ((((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝐴) = (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝐴)) ∧ ((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝐵) = (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝐵))) ∧ (((𝐴(+g‘𝑀)𝐵)(+g‘𝑀)𝐴) = (𝐴(+g‘𝑀)(𝐵(+g‘𝑀)𝐴)) ∧ ((𝐴(+g‘𝑀)𝐵)(+g‘𝑀)𝐵) = (𝐴(+g‘𝑀)(𝐵(+g‘𝑀)𝐵)))))) |
112 | | oveq2 7172 |
. . . . . . . . . 10
⊢ (𝑐 = 𝐴 → ((𝐵(+g‘𝑀)𝐴)(+g‘𝑀)𝑐) = ((𝐵(+g‘𝑀)𝐴)(+g‘𝑀)𝐴)) |
113 | 94 | oveq2d 7180 |
. . . . . . . . . 10
⊢ (𝑐 = 𝐴 → (𝐵(+g‘𝑀)(𝐴(+g‘𝑀)𝑐)) = (𝐵(+g‘𝑀)(𝐴(+g‘𝑀)𝐴))) |
114 | 112, 113 | eqeq12d 2754 |
. . . . . . . . 9
⊢ (𝑐 = 𝐴 → (((𝐵(+g‘𝑀)𝐴)(+g‘𝑀)𝑐) = (𝐵(+g‘𝑀)(𝐴(+g‘𝑀)𝑐)) ↔ ((𝐵(+g‘𝑀)𝐴)(+g‘𝑀)𝐴) = (𝐵(+g‘𝑀)(𝐴(+g‘𝑀)𝐴)))) |
115 | | oveq2 7172 |
. . . . . . . . . 10
⊢ (𝑐 = 𝐵 → ((𝐵(+g‘𝑀)𝐴)(+g‘𝑀)𝑐) = ((𝐵(+g‘𝑀)𝐴)(+g‘𝑀)𝐵)) |
116 | 98 | oveq2d 7180 |
. . . . . . . . . 10
⊢ (𝑐 = 𝐵 → (𝐵(+g‘𝑀)(𝐴(+g‘𝑀)𝑐)) = (𝐵(+g‘𝑀)(𝐴(+g‘𝑀)𝐵))) |
117 | 115, 116 | eqeq12d 2754 |
. . . . . . . . 9
⊢ (𝑐 = 𝐵 → (((𝐵(+g‘𝑀)𝐴)(+g‘𝑀)𝑐) = (𝐵(+g‘𝑀)(𝐴(+g‘𝑀)𝑐)) ↔ ((𝐵(+g‘𝑀)𝐴)(+g‘𝑀)𝐵) = (𝐵(+g‘𝑀)(𝐴(+g‘𝑀)𝐵)))) |
118 | 114, 117 | ralprg 4582 |
. . . . . . . 8
⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (∀𝑐 ∈ {𝐴, 𝐵} ((𝐵(+g‘𝑀)𝐴)(+g‘𝑀)𝑐) = (𝐵(+g‘𝑀)(𝐴(+g‘𝑀)𝑐)) ↔ (((𝐵(+g‘𝑀)𝐴)(+g‘𝑀)𝐴) = (𝐵(+g‘𝑀)(𝐴(+g‘𝑀)𝐴)) ∧ ((𝐵(+g‘𝑀)𝐴)(+g‘𝑀)𝐵) = (𝐵(+g‘𝑀)(𝐴(+g‘𝑀)𝐵))))) |
119 | | oveq2 7172 |
. . . . . . . . . 10
⊢ (𝑐 = 𝐴 → ((𝐵(+g‘𝑀)𝐵)(+g‘𝑀)𝑐) = ((𝐵(+g‘𝑀)𝐵)(+g‘𝑀)𝐴)) |
120 | 103 | oveq2d 7180 |
. . . . . . . . . 10
⊢ (𝑐 = 𝐴 → (𝐵(+g‘𝑀)(𝐵(+g‘𝑀)𝑐)) = (𝐵(+g‘𝑀)(𝐵(+g‘𝑀)𝐴))) |
121 | 119, 120 | eqeq12d 2754 |
. . . . . . . . 9
⊢ (𝑐 = 𝐴 → (((𝐵(+g‘𝑀)𝐵)(+g‘𝑀)𝑐) = (𝐵(+g‘𝑀)(𝐵(+g‘𝑀)𝑐)) ↔ ((𝐵(+g‘𝑀)𝐵)(+g‘𝑀)𝐴) = (𝐵(+g‘𝑀)(𝐵(+g‘𝑀)𝐴)))) |
122 | | oveq2 7172 |
. . . . . . . . . 10
⊢ (𝑐 = 𝐵 → ((𝐵(+g‘𝑀)𝐵)(+g‘𝑀)𝑐) = ((𝐵(+g‘𝑀)𝐵)(+g‘𝑀)𝐵)) |
123 | 107 | oveq2d 7180 |
. . . . . . . . . 10
⊢ (𝑐 = 𝐵 → (𝐵(+g‘𝑀)(𝐵(+g‘𝑀)𝑐)) = (𝐵(+g‘𝑀)(𝐵(+g‘𝑀)𝐵))) |
124 | 122, 123 | eqeq12d 2754 |
. . . . . . . . 9
⊢ (𝑐 = 𝐵 → (((𝐵(+g‘𝑀)𝐵)(+g‘𝑀)𝑐) = (𝐵(+g‘𝑀)(𝐵(+g‘𝑀)𝑐)) ↔ ((𝐵(+g‘𝑀)𝐵)(+g‘𝑀)𝐵) = (𝐵(+g‘𝑀)(𝐵(+g‘𝑀)𝐵)))) |
125 | 121, 124 | ralprg 4582 |
. . . . . . . 8
⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (∀𝑐 ∈ {𝐴, 𝐵} ((𝐵(+g‘𝑀)𝐵)(+g‘𝑀)𝑐) = (𝐵(+g‘𝑀)(𝐵(+g‘𝑀)𝑐)) ↔ (((𝐵(+g‘𝑀)𝐵)(+g‘𝑀)𝐴) = (𝐵(+g‘𝑀)(𝐵(+g‘𝑀)𝐴)) ∧ ((𝐵(+g‘𝑀)𝐵)(+g‘𝑀)𝐵) = (𝐵(+g‘𝑀)(𝐵(+g‘𝑀)𝐵))))) |
126 | 118, 125 | anbi12d 634 |
. . . . . . 7
⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ((∀𝑐 ∈ {𝐴, 𝐵} ((𝐵(+g‘𝑀)𝐴)(+g‘𝑀)𝑐) = (𝐵(+g‘𝑀)(𝐴(+g‘𝑀)𝑐)) ∧ ∀𝑐 ∈ {𝐴, 𝐵} ((𝐵(+g‘𝑀)𝐵)(+g‘𝑀)𝑐) = (𝐵(+g‘𝑀)(𝐵(+g‘𝑀)𝑐))) ↔ ((((𝐵(+g‘𝑀)𝐴)(+g‘𝑀)𝐴) = (𝐵(+g‘𝑀)(𝐴(+g‘𝑀)𝐴)) ∧ ((𝐵(+g‘𝑀)𝐴)(+g‘𝑀)𝐵) = (𝐵(+g‘𝑀)(𝐴(+g‘𝑀)𝐵))) ∧ (((𝐵(+g‘𝑀)𝐵)(+g‘𝑀)𝐴) = (𝐵(+g‘𝑀)(𝐵(+g‘𝑀)𝐴)) ∧ ((𝐵(+g‘𝑀)𝐵)(+g‘𝑀)𝐵) = (𝐵(+g‘𝑀)(𝐵(+g‘𝑀)𝐵)))))) |
127 | 111, 126 | anbi12d 634 |
. . . . . 6
⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (((∀𝑐 ∈ {𝐴, 𝐵} ((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝑐) = (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝑐)) ∧ ∀𝑐 ∈ {𝐴, 𝐵} ((𝐴(+g‘𝑀)𝐵)(+g‘𝑀)𝑐) = (𝐴(+g‘𝑀)(𝐵(+g‘𝑀)𝑐))) ∧ (∀𝑐 ∈ {𝐴, 𝐵} ((𝐵(+g‘𝑀)𝐴)(+g‘𝑀)𝑐) = (𝐵(+g‘𝑀)(𝐴(+g‘𝑀)𝑐)) ∧ ∀𝑐 ∈ {𝐴, 𝐵} ((𝐵(+g‘𝑀)𝐵)(+g‘𝑀)𝑐) = (𝐵(+g‘𝑀)(𝐵(+g‘𝑀)𝑐)))) ↔ (((((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝐴) = (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝐴)) ∧ ((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝐵) = (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝐵))) ∧ (((𝐴(+g‘𝑀)𝐵)(+g‘𝑀)𝐴) = (𝐴(+g‘𝑀)(𝐵(+g‘𝑀)𝐴)) ∧ ((𝐴(+g‘𝑀)𝐵)(+g‘𝑀)𝐵) = (𝐴(+g‘𝑀)(𝐵(+g‘𝑀)𝐵)))) ∧ ((((𝐵(+g‘𝑀)𝐴)(+g‘𝑀)𝐴) = (𝐵(+g‘𝑀)(𝐴(+g‘𝑀)𝐴)) ∧ ((𝐵(+g‘𝑀)𝐴)(+g‘𝑀)𝐵) = (𝐵(+g‘𝑀)(𝐴(+g‘𝑀)𝐵))) ∧ (((𝐵(+g‘𝑀)𝐵)(+g‘𝑀)𝐴) = (𝐵(+g‘𝑀)(𝐵(+g‘𝑀)𝐴)) ∧ ((𝐵(+g‘𝑀)𝐵)(+g‘𝑀)𝐵) = (𝐵(+g‘𝑀)(𝐵(+g‘𝑀)𝐵))))))) |
128 | 67, 92, 127 | 3bitrd 308 |
. . . . 5
⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (∀𝑎 ∈ {𝐴, 𝐵}∀𝑏 ∈ {𝐴, 𝐵}∀𝑐 ∈ {𝐴, 𝐵} ((𝑎(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = (𝑎(+g‘𝑀)(𝑏(+g‘𝑀)𝑐)) ↔ (((((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝐴) = (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝐴)) ∧ ((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝐵) = (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝐵))) ∧ (((𝐴(+g‘𝑀)𝐵)(+g‘𝑀)𝐴) = (𝐴(+g‘𝑀)(𝐵(+g‘𝑀)𝐴)) ∧ ((𝐴(+g‘𝑀)𝐵)(+g‘𝑀)𝐵) = (𝐴(+g‘𝑀)(𝐵(+g‘𝑀)𝐵)))) ∧ ((((𝐵(+g‘𝑀)𝐴)(+g‘𝑀)𝐴) = (𝐵(+g‘𝑀)(𝐴(+g‘𝑀)𝐴)) ∧ ((𝐵(+g‘𝑀)𝐴)(+g‘𝑀)𝐵) = (𝐵(+g‘𝑀)(𝐴(+g‘𝑀)𝐵))) ∧ (((𝐵(+g‘𝑀)𝐵)(+g‘𝑀)𝐴) = (𝐵(+g‘𝑀)(𝐵(+g‘𝑀)𝐴)) ∧ ((𝐵(+g‘𝑀)𝐵)(+g‘𝑀)𝐵) = (𝐵(+g‘𝑀)(𝐵(+g‘𝑀)𝐵))))))) |
129 | 128 | 3adant3 1133 |
. . . 4
⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → (∀𝑎 ∈ {𝐴, 𝐵}∀𝑏 ∈ {𝐴, 𝐵}∀𝑐 ∈ {𝐴, 𝐵} ((𝑎(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = (𝑎(+g‘𝑀)(𝑏(+g‘𝑀)𝑐)) ↔ (((((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝐴) = (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝐴)) ∧ ((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝐵) = (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝐵))) ∧ (((𝐴(+g‘𝑀)𝐵)(+g‘𝑀)𝐴) = (𝐴(+g‘𝑀)(𝐵(+g‘𝑀)𝐴)) ∧ ((𝐴(+g‘𝑀)𝐵)(+g‘𝑀)𝐵) = (𝐴(+g‘𝑀)(𝐵(+g‘𝑀)𝐵)))) ∧ ((((𝐵(+g‘𝑀)𝐴)(+g‘𝑀)𝐴) = (𝐵(+g‘𝑀)(𝐴(+g‘𝑀)𝐴)) ∧ ((𝐵(+g‘𝑀)𝐴)(+g‘𝑀)𝐵) = (𝐵(+g‘𝑀)(𝐴(+g‘𝑀)𝐵))) ∧ (((𝐵(+g‘𝑀)𝐵)(+g‘𝑀)𝐴) = (𝐵(+g‘𝑀)(𝐵(+g‘𝑀)𝐴)) ∧ ((𝐵(+g‘𝑀)𝐵)(+g‘𝑀)𝐵) = (𝐵(+g‘𝑀)(𝐵(+g‘𝑀)𝐵))))))) |
130 | 39, 56, 129 | mpbir2and 713 |
. . 3
⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → ∀𝑎 ∈ {𝐴, 𝐵}∀𝑏 ∈ {𝐴, 𝐵}∀𝑐 ∈ {𝐴, 𝐵} ((𝑎(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = (𝑎(+g‘𝑀)(𝑏(+g‘𝑀)𝑐))) |
131 | 2, 130 | syl 17 |
. 2
⊢
((♯‘𝑆) =
2 → ∀𝑎 ∈
{𝐴, 𝐵}∀𝑏 ∈ {𝐴, 𝐵}∀𝑐 ∈ {𝐴, 𝐵} ((𝑎(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = (𝑎(+g‘𝑀)(𝑏(+g‘𝑀)𝑐))) |
132 | 4, 1 | eqtr2i 2762 |
. . 3
⊢ {𝐴, 𝐵} = (Base‘𝑀) |
133 | 132, 8 | issgrp 18011 |
. 2
⊢ (𝑀 ∈ Smgrp ↔ (𝑀 ∈ Mgm ∧ ∀𝑎 ∈ {𝐴, 𝐵}∀𝑏 ∈ {𝐴, 𝐵}∀𝑐 ∈ {𝐴, 𝐵} ((𝑎(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = (𝑎(+g‘𝑀)(𝑏(+g‘𝑀)𝑐)))) |
134 | 7, 131, 133 | sylanbrc 586 |
1
⊢
((♯‘𝑆) =
2 → 𝑀 ∈
Smgrp) |