Step | Hyp | Ref
| Expression |
1 | | simpllr 772 |
. . . . . . . 8
⊢
((((((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌)) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) ∧ 𝑗 ∈ 𝑋) ∧ (𝐹‘𝑗) = 𝑏) → (𝐹‘𝑖) = 𝑎) |
2 | | simpr 484 |
. . . . . . . 8
⊢
((((((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌)) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) ∧ 𝑗 ∈ 𝑋) ∧ (𝐹‘𝑗) = 𝑏) → (𝐹‘𝑗) = 𝑏) |
3 | 1, 2 | oveq12d 7278 |
. . . . . . 7
⊢
((((((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌)) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) ∧ 𝑗 ∈ 𝑋) ∧ (𝐹‘𝑗) = 𝑏) → ((𝐹‘𝑖) ⨣ (𝐹‘𝑗)) = (𝑎 ⨣ 𝑏)) |
4 | | simp-5l 781 |
. . . . . . . . . 10
⊢
((((((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌)) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) ∧ 𝑗 ∈ 𝑋) ∧ (𝐹‘𝑗) = 𝑏) → 𝜑) |
5 | | ghmgrp.f |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦))) |
6 | 4, 5 | syl3an1 1161 |
. . . . . . . . 9
⊢
(((((((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌)) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) ∧ 𝑗 ∈ 𝑋) ∧ (𝐹‘𝑗) = 𝑏) ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦))) |
7 | | simp-4r 780 |
. . . . . . . . 9
⊢
((((((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌)) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) ∧ 𝑗 ∈ 𝑋) ∧ (𝐹‘𝑗) = 𝑏) → 𝑖 ∈ 𝑋) |
8 | | simplr 765 |
. . . . . . . . 9
⊢
((((((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌)) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) ∧ 𝑗 ∈ 𝑋) ∧ (𝐹‘𝑗) = 𝑏) → 𝑗 ∈ 𝑋) |
9 | 6, 7, 8 | mhmlem 18639 |
. . . . . . . 8
⊢
((((((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌)) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) ∧ 𝑗 ∈ 𝑋) ∧ (𝐹‘𝑗) = 𝑏) → (𝐹‘(𝑖 + 𝑗)) = ((𝐹‘𝑖) ⨣ (𝐹‘𝑗))) |
10 | | ghmgrp.1 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐹:𝑋–onto→𝑌) |
11 | | fof 6677 |
. . . . . . . . . . 11
⊢ (𝐹:𝑋–onto→𝑌 → 𝐹:𝑋⟶𝑌) |
12 | 10, 11 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐹:𝑋⟶𝑌) |
13 | 12 | ad5antr 730 |
. . . . . . . . 9
⊢
((((((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌)) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) ∧ 𝑗 ∈ 𝑋) ∧ (𝐹‘𝑗) = 𝑏) → 𝐹:𝑋⟶𝑌) |
14 | | mhmmnd.3 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐺 ∈ Mnd) |
15 | 14 | ad5antr 730 |
. . . . . . . . . 10
⊢
((((((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌)) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) ∧ 𝑗 ∈ 𝑋) ∧ (𝐹‘𝑗) = 𝑏) → 𝐺 ∈ Mnd) |
16 | | ghmgrp.x |
. . . . . . . . . . 11
⊢ 𝑋 = (Base‘𝐺) |
17 | | ghmgrp.p |
. . . . . . . . . . 11
⊢ + =
(+g‘𝐺) |
18 | 16, 17 | mndcl 18337 |
. . . . . . . . . 10
⊢ ((𝐺 ∈ Mnd ∧ 𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑋) → (𝑖 + 𝑗) ∈ 𝑋) |
19 | 15, 7, 8, 18 | syl3anc 1369 |
. . . . . . . . 9
⊢
((((((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌)) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) ∧ 𝑗 ∈ 𝑋) ∧ (𝐹‘𝑗) = 𝑏) → (𝑖 + 𝑗) ∈ 𝑋) |
20 | 13, 19 | ffvelrnd 6949 |
. . . . . . . 8
⊢
((((((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌)) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) ∧ 𝑗 ∈ 𝑋) ∧ (𝐹‘𝑗) = 𝑏) → (𝐹‘(𝑖 + 𝑗)) ∈ 𝑌) |
21 | 9, 20 | eqeltrrd 2838 |
. . . . . . 7
⊢
((((((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌)) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) ∧ 𝑗 ∈ 𝑋) ∧ (𝐹‘𝑗) = 𝑏) → ((𝐹‘𝑖) ⨣ (𝐹‘𝑗)) ∈ 𝑌) |
22 | 3, 21 | eqeltrrd 2838 |
. . . . . 6
⊢
((((((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌)) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) ∧ 𝑗 ∈ 𝑋) ∧ (𝐹‘𝑗) = 𝑏) → (𝑎 ⨣ 𝑏) ∈ 𝑌) |
23 | | simpr 484 |
. . . . . . . 8
⊢ ((𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌) → 𝑏 ∈ 𝑌) |
24 | | foelrni 6818 |
. . . . . . . 8
⊢ ((𝐹:𝑋–onto→𝑌 ∧ 𝑏 ∈ 𝑌) → ∃𝑗 ∈ 𝑋 (𝐹‘𝑗) = 𝑏) |
25 | 10, 23, 24 | syl2an 595 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌)) → ∃𝑗 ∈ 𝑋 (𝐹‘𝑗) = 𝑏) |
26 | 25 | ad2antrr 722 |
. . . . . 6
⊢ ((((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌)) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → ∃𝑗 ∈ 𝑋 (𝐹‘𝑗) = 𝑏) |
27 | 22, 26 | r19.29a 3216 |
. . . . 5
⊢ ((((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌)) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → (𝑎 ⨣ 𝑏) ∈ 𝑌) |
28 | | simpl 482 |
. . . . . 6
⊢ ((𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌) → 𝑎 ∈ 𝑌) |
29 | | foelrni 6818 |
. . . . . 6
⊢ ((𝐹:𝑋–onto→𝑌 ∧ 𝑎 ∈ 𝑌) → ∃𝑖 ∈ 𝑋 (𝐹‘𝑖) = 𝑎) |
30 | 10, 28, 29 | syl2an 595 |
. . . . 5
⊢ ((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌)) → ∃𝑖 ∈ 𝑋 (𝐹‘𝑖) = 𝑎) |
31 | 27, 30 | r19.29a 3216 |
. . . 4
⊢ ((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌)) → (𝑎 ⨣ 𝑏) ∈ 𝑌) |
32 | | simpll 763 |
. . . . . 6
⊢ (((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌)) ∧ 𝑐 ∈ 𝑌) → 𝜑) |
33 | | simplrl 773 |
. . . . . 6
⊢ (((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌)) ∧ 𝑐 ∈ 𝑌) → 𝑎 ∈ 𝑌) |
34 | | simplrr 774 |
. . . . . 6
⊢ (((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌)) ∧ 𝑐 ∈ 𝑌) → 𝑏 ∈ 𝑌) |
35 | | simpr 484 |
. . . . . 6
⊢ (((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌)) ∧ 𝑐 ∈ 𝑌) → 𝑐 ∈ 𝑌) |
36 | 14 | ad2antrr 722 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌 ∧ 𝑐 ∈ 𝑌)) ∧ 𝑖 ∈ 𝑋) → 𝐺 ∈ Mnd) |
37 | 36 | ad5antr 730 |
. . . . . . . . . . . . . 14
⊢
((((((((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌 ∧ 𝑐 ∈ 𝑌)) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) ∧ 𝑗 ∈ 𝑋) ∧ (𝐹‘𝑗) = 𝑏) ∧ 𝑘 ∈ 𝑋) ∧ (𝐹‘𝑘) = 𝑐) → 𝐺 ∈ Mnd) |
38 | | simp-6r 784 |
. . . . . . . . . . . . . 14
⊢
((((((((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌 ∧ 𝑐 ∈ 𝑌)) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) ∧ 𝑗 ∈ 𝑋) ∧ (𝐹‘𝑗) = 𝑏) ∧ 𝑘 ∈ 𝑋) ∧ (𝐹‘𝑘) = 𝑐) → 𝑖 ∈ 𝑋) |
39 | | simp-4r 780 |
. . . . . . . . . . . . . 14
⊢
((((((((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌 ∧ 𝑐 ∈ 𝑌)) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) ∧ 𝑗 ∈ 𝑋) ∧ (𝐹‘𝑗) = 𝑏) ∧ 𝑘 ∈ 𝑋) ∧ (𝐹‘𝑘) = 𝑐) → 𝑗 ∈ 𝑋) |
40 | | simplr 765 |
. . . . . . . . . . . . . 14
⊢
((((((((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌 ∧ 𝑐 ∈ 𝑌)) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) ∧ 𝑗 ∈ 𝑋) ∧ (𝐹‘𝑗) = 𝑏) ∧ 𝑘 ∈ 𝑋) ∧ (𝐹‘𝑘) = 𝑐) → 𝑘 ∈ 𝑋) |
41 | 16, 17 | mndass 18338 |
. . . . . . . . . . . . . 14
⊢ ((𝐺 ∈ Mnd ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑋 ∧ 𝑘 ∈ 𝑋)) → ((𝑖 + 𝑗) + 𝑘) = (𝑖 + (𝑗 + 𝑘))) |
42 | 37, 38, 39, 40, 41 | syl13anc 1370 |
. . . . . . . . . . . . 13
⊢
((((((((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌 ∧ 𝑐 ∈ 𝑌)) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) ∧ 𝑗 ∈ 𝑋) ∧ (𝐹‘𝑗) = 𝑏) ∧ 𝑘 ∈ 𝑋) ∧ (𝐹‘𝑘) = 𝑐) → ((𝑖 + 𝑗) + 𝑘) = (𝑖 + (𝑗 + 𝑘))) |
43 | 42 | fveq2d 6765 |
. . . . . . . . . . . 12
⊢
((((((((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌 ∧ 𝑐 ∈ 𝑌)) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) ∧ 𝑗 ∈ 𝑋) ∧ (𝐹‘𝑗) = 𝑏) ∧ 𝑘 ∈ 𝑋) ∧ (𝐹‘𝑘) = 𝑐) → (𝐹‘((𝑖 + 𝑗) + 𝑘)) = (𝐹‘(𝑖 + (𝑗 + 𝑘)))) |
44 | | simp-7l 785 |
. . . . . . . . . . . . . 14
⊢
((((((((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌 ∧ 𝑐 ∈ 𝑌)) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) ∧ 𝑗 ∈ 𝑋) ∧ (𝐹‘𝑗) = 𝑏) ∧ 𝑘 ∈ 𝑋) ∧ (𝐹‘𝑘) = 𝑐) → 𝜑) |
45 | 44, 5 | syl3an1 1161 |
. . . . . . . . . . . . 13
⊢
(((((((((𝜑 ∧
(𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌 ∧ 𝑐 ∈ 𝑌)) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) ∧ 𝑗 ∈ 𝑋) ∧ (𝐹‘𝑗) = 𝑏) ∧ 𝑘 ∈ 𝑋) ∧ (𝐹‘𝑘) = 𝑐) ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦))) |
46 | 37, 38, 39, 18 | syl3anc 1369 |
. . . . . . . . . . . . 13
⊢
((((((((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌 ∧ 𝑐 ∈ 𝑌)) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) ∧ 𝑗 ∈ 𝑋) ∧ (𝐹‘𝑗) = 𝑏) ∧ 𝑘 ∈ 𝑋) ∧ (𝐹‘𝑘) = 𝑐) → (𝑖 + 𝑗) ∈ 𝑋) |
47 | 45, 46, 40 | mhmlem 18639 |
. . . . . . . . . . . 12
⊢
((((((((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌 ∧ 𝑐 ∈ 𝑌)) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) ∧ 𝑗 ∈ 𝑋) ∧ (𝐹‘𝑗) = 𝑏) ∧ 𝑘 ∈ 𝑋) ∧ (𝐹‘𝑘) = 𝑐) → (𝐹‘((𝑖 + 𝑗) + 𝑘)) = ((𝐹‘(𝑖 + 𝑗)) ⨣ (𝐹‘𝑘))) |
48 | 16, 17 | mndcl 18337 |
. . . . . . . . . . . . . 14
⊢ ((𝐺 ∈ Mnd ∧ 𝑗 ∈ 𝑋 ∧ 𝑘 ∈ 𝑋) → (𝑗 + 𝑘) ∈ 𝑋) |
49 | 37, 39, 40, 48 | syl3anc 1369 |
. . . . . . . . . . . . 13
⊢
((((((((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌 ∧ 𝑐 ∈ 𝑌)) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) ∧ 𝑗 ∈ 𝑋) ∧ (𝐹‘𝑗) = 𝑏) ∧ 𝑘 ∈ 𝑋) ∧ (𝐹‘𝑘) = 𝑐) → (𝑗 + 𝑘) ∈ 𝑋) |
50 | 45, 38, 49 | mhmlem 18639 |
. . . . . . . . . . . 12
⊢
((((((((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌 ∧ 𝑐 ∈ 𝑌)) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) ∧ 𝑗 ∈ 𝑋) ∧ (𝐹‘𝑗) = 𝑏) ∧ 𝑘 ∈ 𝑋) ∧ (𝐹‘𝑘) = 𝑐) → (𝐹‘(𝑖 + (𝑗 + 𝑘))) = ((𝐹‘𝑖) ⨣ (𝐹‘(𝑗 + 𝑘)))) |
51 | 43, 47, 50 | 3eqtr3d 2785 |
. . . . . . . . . . 11
⊢
((((((((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌 ∧ 𝑐 ∈ 𝑌)) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) ∧ 𝑗 ∈ 𝑋) ∧ (𝐹‘𝑗) = 𝑏) ∧ 𝑘 ∈ 𝑋) ∧ (𝐹‘𝑘) = 𝑐) → ((𝐹‘(𝑖 + 𝑗)) ⨣ (𝐹‘𝑘)) = ((𝐹‘𝑖) ⨣ (𝐹‘(𝑗 + 𝑘)))) |
52 | | simp1 1134 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑋) → 𝜑) |
53 | 52, 5 | syl3an1 1161 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑋) ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦))) |
54 | | simp2 1135 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑋) → 𝑖 ∈ 𝑋) |
55 | | simp3 1136 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑋) → 𝑗 ∈ 𝑋) |
56 | 53, 54, 55 | mhmlem 18639 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑋) → (𝐹‘(𝑖 + 𝑗)) = ((𝐹‘𝑖) ⨣ (𝐹‘𝑗))) |
57 | 44, 38, 39, 56 | syl3anc 1369 |
. . . . . . . . . . . 12
⊢
((((((((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌 ∧ 𝑐 ∈ 𝑌)) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) ∧ 𝑗 ∈ 𝑋) ∧ (𝐹‘𝑗) = 𝑏) ∧ 𝑘 ∈ 𝑋) ∧ (𝐹‘𝑘) = 𝑐) → (𝐹‘(𝑖 + 𝑗)) = ((𝐹‘𝑖) ⨣ (𝐹‘𝑗))) |
58 | 57 | oveq1d 7275 |
. . . . . . . . . . 11
⊢
((((((((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌 ∧ 𝑐 ∈ 𝑌)) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) ∧ 𝑗 ∈ 𝑋) ∧ (𝐹‘𝑗) = 𝑏) ∧ 𝑘 ∈ 𝑋) ∧ (𝐹‘𝑘) = 𝑐) → ((𝐹‘(𝑖 + 𝑗)) ⨣ (𝐹‘𝑘)) = (((𝐹‘𝑖) ⨣ (𝐹‘𝑗)) ⨣ (𝐹‘𝑘))) |
59 | 45, 39, 40 | mhmlem 18639 |
. . . . . . . . . . . 12
⊢
((((((((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌 ∧ 𝑐 ∈ 𝑌)) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) ∧ 𝑗 ∈ 𝑋) ∧ (𝐹‘𝑗) = 𝑏) ∧ 𝑘 ∈ 𝑋) ∧ (𝐹‘𝑘) = 𝑐) → (𝐹‘(𝑗 + 𝑘)) = ((𝐹‘𝑗) ⨣ (𝐹‘𝑘))) |
60 | 59 | oveq2d 7276 |
. . . . . . . . . . 11
⊢
((((((((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌 ∧ 𝑐 ∈ 𝑌)) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) ∧ 𝑗 ∈ 𝑋) ∧ (𝐹‘𝑗) = 𝑏) ∧ 𝑘 ∈ 𝑋) ∧ (𝐹‘𝑘) = 𝑐) → ((𝐹‘𝑖) ⨣ (𝐹‘(𝑗 + 𝑘))) = ((𝐹‘𝑖) ⨣ ((𝐹‘𝑗) ⨣ (𝐹‘𝑘)))) |
61 | 51, 58, 60 | 3eqtr3d 2785 |
. . . . . . . . . 10
⊢
((((((((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌 ∧ 𝑐 ∈ 𝑌)) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) ∧ 𝑗 ∈ 𝑋) ∧ (𝐹‘𝑗) = 𝑏) ∧ 𝑘 ∈ 𝑋) ∧ (𝐹‘𝑘) = 𝑐) → (((𝐹‘𝑖) ⨣ (𝐹‘𝑗)) ⨣ (𝐹‘𝑘)) = ((𝐹‘𝑖) ⨣ ((𝐹‘𝑗) ⨣ (𝐹‘𝑘)))) |
62 | | simp-5r 782 |
. . . . . . . . . . . 12
⊢
((((((((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌 ∧ 𝑐 ∈ 𝑌)) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) ∧ 𝑗 ∈ 𝑋) ∧ (𝐹‘𝑗) = 𝑏) ∧ 𝑘 ∈ 𝑋) ∧ (𝐹‘𝑘) = 𝑐) → (𝐹‘𝑖) = 𝑎) |
63 | | simpllr 772 |
. . . . . . . . . . . 12
⊢
((((((((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌 ∧ 𝑐 ∈ 𝑌)) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) ∧ 𝑗 ∈ 𝑋) ∧ (𝐹‘𝑗) = 𝑏) ∧ 𝑘 ∈ 𝑋) ∧ (𝐹‘𝑘) = 𝑐) → (𝐹‘𝑗) = 𝑏) |
64 | 62, 63 | oveq12d 7278 |
. . . . . . . . . . 11
⊢
((((((((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌 ∧ 𝑐 ∈ 𝑌)) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) ∧ 𝑗 ∈ 𝑋) ∧ (𝐹‘𝑗) = 𝑏) ∧ 𝑘 ∈ 𝑋) ∧ (𝐹‘𝑘) = 𝑐) → ((𝐹‘𝑖) ⨣ (𝐹‘𝑗)) = (𝑎 ⨣ 𝑏)) |
65 | | simpr 484 |
. . . . . . . . . . 11
⊢
((((((((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌 ∧ 𝑐 ∈ 𝑌)) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) ∧ 𝑗 ∈ 𝑋) ∧ (𝐹‘𝑗) = 𝑏) ∧ 𝑘 ∈ 𝑋) ∧ (𝐹‘𝑘) = 𝑐) → (𝐹‘𝑘) = 𝑐) |
66 | 64, 65 | oveq12d 7278 |
. . . . . . . . . 10
⊢
((((((((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌 ∧ 𝑐 ∈ 𝑌)) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) ∧ 𝑗 ∈ 𝑋) ∧ (𝐹‘𝑗) = 𝑏) ∧ 𝑘 ∈ 𝑋) ∧ (𝐹‘𝑘) = 𝑐) → (((𝐹‘𝑖) ⨣ (𝐹‘𝑗)) ⨣ (𝐹‘𝑘)) = ((𝑎 ⨣ 𝑏) ⨣ 𝑐)) |
67 | 63, 65 | oveq12d 7278 |
. . . . . . . . . . 11
⊢
((((((((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌 ∧ 𝑐 ∈ 𝑌)) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) ∧ 𝑗 ∈ 𝑋) ∧ (𝐹‘𝑗) = 𝑏) ∧ 𝑘 ∈ 𝑋) ∧ (𝐹‘𝑘) = 𝑐) → ((𝐹‘𝑗) ⨣ (𝐹‘𝑘)) = (𝑏 ⨣ 𝑐)) |
68 | 62, 67 | oveq12d 7278 |
. . . . . . . . . 10
⊢
((((((((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌 ∧ 𝑐 ∈ 𝑌)) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) ∧ 𝑗 ∈ 𝑋) ∧ (𝐹‘𝑗) = 𝑏) ∧ 𝑘 ∈ 𝑋) ∧ (𝐹‘𝑘) = 𝑐) → ((𝐹‘𝑖) ⨣ ((𝐹‘𝑗) ⨣ (𝐹‘𝑘))) = (𝑎 ⨣ (𝑏 ⨣ 𝑐))) |
69 | 61, 66, 68 | 3eqtr3d 2785 |
. . . . . . . . 9
⊢
((((((((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌 ∧ 𝑐 ∈ 𝑌)) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) ∧ 𝑗 ∈ 𝑋) ∧ (𝐹‘𝑗) = 𝑏) ∧ 𝑘 ∈ 𝑋) ∧ (𝐹‘𝑘) = 𝑐) → ((𝑎 ⨣ 𝑏) ⨣ 𝑐) = (𝑎 ⨣ (𝑏 ⨣ 𝑐))) |
70 | | foelrni 6818 |
. . . . . . . . . . . 12
⊢ ((𝐹:𝑋–onto→𝑌 ∧ 𝑐 ∈ 𝑌) → ∃𝑘 ∈ 𝑋 (𝐹‘𝑘) = 𝑐) |
71 | 10, 70 | sylan 579 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑐 ∈ 𝑌) → ∃𝑘 ∈ 𝑋 (𝐹‘𝑘) = 𝑐) |
72 | 71 | 3ad2antr3 1188 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌 ∧ 𝑐 ∈ 𝑌)) → ∃𝑘 ∈ 𝑋 (𝐹‘𝑘) = 𝑐) |
73 | 72 | ad4antr 728 |
. . . . . . . . 9
⊢
((((((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌 ∧ 𝑐 ∈ 𝑌)) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) ∧ 𝑗 ∈ 𝑋) ∧ (𝐹‘𝑗) = 𝑏) → ∃𝑘 ∈ 𝑋 (𝐹‘𝑘) = 𝑐) |
74 | 69, 73 | r19.29a 3216 |
. . . . . . . 8
⊢
((((((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌 ∧ 𝑐 ∈ 𝑌)) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) ∧ 𝑗 ∈ 𝑋) ∧ (𝐹‘𝑗) = 𝑏) → ((𝑎 ⨣ 𝑏) ⨣ 𝑐) = (𝑎 ⨣ (𝑏 ⨣ 𝑐))) |
75 | 25 | 3adantr3 1169 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌 ∧ 𝑐 ∈ 𝑌)) → ∃𝑗 ∈ 𝑋 (𝐹‘𝑗) = 𝑏) |
76 | 75 | ad2antrr 722 |
. . . . . . . 8
⊢ ((((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌 ∧ 𝑐 ∈ 𝑌)) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → ∃𝑗 ∈ 𝑋 (𝐹‘𝑗) = 𝑏) |
77 | 74, 76 | r19.29a 3216 |
. . . . . . 7
⊢ ((((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌 ∧ 𝑐 ∈ 𝑌)) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → ((𝑎 ⨣ 𝑏) ⨣ 𝑐) = (𝑎 ⨣ (𝑏 ⨣ 𝑐))) |
78 | 30 | 3adantr3 1169 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌 ∧ 𝑐 ∈ 𝑌)) → ∃𝑖 ∈ 𝑋 (𝐹‘𝑖) = 𝑎) |
79 | 77, 78 | r19.29a 3216 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌 ∧ 𝑐 ∈ 𝑌)) → ((𝑎 ⨣ 𝑏) ⨣ 𝑐) = (𝑎 ⨣ (𝑏 ⨣ 𝑐))) |
80 | 32, 33, 34, 35, 79 | syl13anc 1370 |
. . . . 5
⊢ (((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌)) ∧ 𝑐 ∈ 𝑌) → ((𝑎 ⨣ 𝑏) ⨣ 𝑐) = (𝑎 ⨣ (𝑏 ⨣ 𝑐))) |
81 | 80 | ralrimiva 3106 |
. . . 4
⊢ ((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌)) → ∀𝑐 ∈ 𝑌 ((𝑎 ⨣ 𝑏) ⨣ 𝑐) = (𝑎 ⨣ (𝑏 ⨣ 𝑐))) |
82 | 31, 81 | jca 511 |
. . 3
⊢ ((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌)) → ((𝑎 ⨣ 𝑏) ∈ 𝑌 ∧ ∀𝑐 ∈ 𝑌 ((𝑎 ⨣ 𝑏) ⨣ 𝑐) = (𝑎 ⨣ (𝑏 ⨣ 𝑐)))) |
83 | 82 | ralrimivva 3113 |
. 2
⊢ (𝜑 → ∀𝑎 ∈ 𝑌 ∀𝑏 ∈ 𝑌 ((𝑎 ⨣ 𝑏) ∈ 𝑌 ∧ ∀𝑐 ∈ 𝑌 ((𝑎 ⨣ 𝑏) ⨣ 𝑐) = (𝑎 ⨣ (𝑏 ⨣ 𝑐)))) |
84 | | eqid 2737 |
. . . . . 6
⊢
(0g‘𝐺) = (0g‘𝐺) |
85 | 16, 84 | mndidcl 18344 |
. . . . 5
⊢ (𝐺 ∈ Mnd →
(0g‘𝐺)
∈ 𝑋) |
86 | 14, 85 | syl 17 |
. . . 4
⊢ (𝜑 → (0g‘𝐺) ∈ 𝑋) |
87 | 12, 86 | ffvelrnd 6949 |
. . 3
⊢ (𝜑 → (𝐹‘(0g‘𝐺)) ∈ 𝑌) |
88 | | simplll 771 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → 𝜑) |
89 | 88, 5 | syl3an1 1161 |
. . . . . . . . 9
⊢
(((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦))) |
90 | 14 | ad3antrrr 726 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → 𝐺 ∈ Mnd) |
91 | 90, 85 | syl 17 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → (0g‘𝐺) ∈ 𝑋) |
92 | | simplr 765 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → 𝑖 ∈ 𝑋) |
93 | 89, 91, 92 | mhmlem 18639 |
. . . . . . . 8
⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → (𝐹‘((0g‘𝐺) + 𝑖)) = ((𝐹‘(0g‘𝐺)) ⨣ (𝐹‘𝑖))) |
94 | 16, 17, 84 | mndlid 18349 |
. . . . . . . . . 10
⊢ ((𝐺 ∈ Mnd ∧ 𝑖 ∈ 𝑋) → ((0g‘𝐺) + 𝑖) = 𝑖) |
95 | 90, 92, 94 | syl2anc 583 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → ((0g‘𝐺) + 𝑖) = 𝑖) |
96 | 95 | fveq2d 6765 |
. . . . . . . 8
⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → (𝐹‘((0g‘𝐺) + 𝑖)) = (𝐹‘𝑖)) |
97 | 93, 96 | eqtr3d 2779 |
. . . . . . 7
⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → ((𝐹‘(0g‘𝐺)) ⨣ (𝐹‘𝑖)) = (𝐹‘𝑖)) |
98 | | simpr 484 |
. . . . . . . 8
⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → (𝐹‘𝑖) = 𝑎) |
99 | 98 | oveq2d 7276 |
. . . . . . 7
⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → ((𝐹‘(0g‘𝐺)) ⨣ (𝐹‘𝑖)) = ((𝐹‘(0g‘𝐺)) ⨣ 𝑎)) |
100 | 97, 99, 98 | 3eqtr3d 2785 |
. . . . . 6
⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → ((𝐹‘(0g‘𝐺)) ⨣ 𝑎) = 𝑎) |
101 | 89, 92, 91 | mhmlem 18639 |
. . . . . . . 8
⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → (𝐹‘(𝑖 + (0g‘𝐺))) = ((𝐹‘𝑖) ⨣ (𝐹‘(0g‘𝐺)))) |
102 | 16, 17, 84 | mndrid 18350 |
. . . . . . . . . 10
⊢ ((𝐺 ∈ Mnd ∧ 𝑖 ∈ 𝑋) → (𝑖 + (0g‘𝐺)) = 𝑖) |
103 | 90, 92, 102 | syl2anc 583 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → (𝑖 + (0g‘𝐺)) = 𝑖) |
104 | 103 | fveq2d 6765 |
. . . . . . . 8
⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → (𝐹‘(𝑖 + (0g‘𝐺))) = (𝐹‘𝑖)) |
105 | 101, 104 | eqtr3d 2779 |
. . . . . . 7
⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → ((𝐹‘𝑖) ⨣ (𝐹‘(0g‘𝐺))) = (𝐹‘𝑖)) |
106 | 98 | oveq1d 7275 |
. . . . . . 7
⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → ((𝐹‘𝑖) ⨣ (𝐹‘(0g‘𝐺))) = (𝑎 ⨣ (𝐹‘(0g‘𝐺)))) |
107 | 105, 106,
98 | 3eqtr3d 2785 |
. . . . . 6
⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → (𝑎 ⨣ (𝐹‘(0g‘𝐺))) = 𝑎) |
108 | 100, 107 | jca 511 |
. . . . 5
⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → (((𝐹‘(0g‘𝐺)) ⨣ 𝑎) = 𝑎 ∧ (𝑎 ⨣ (𝐹‘(0g‘𝐺))) = 𝑎)) |
109 | 10, 29 | sylan 579 |
. . . . 5
⊢ ((𝜑 ∧ 𝑎 ∈ 𝑌) → ∃𝑖 ∈ 𝑋 (𝐹‘𝑖) = 𝑎) |
110 | 108, 109 | r19.29a 3216 |
. . . 4
⊢ ((𝜑 ∧ 𝑎 ∈ 𝑌) → (((𝐹‘(0g‘𝐺)) ⨣ 𝑎) = 𝑎 ∧ (𝑎 ⨣ (𝐹‘(0g‘𝐺))) = 𝑎)) |
111 | 110 | ralrimiva 3106 |
. . 3
⊢ (𝜑 → ∀𝑎 ∈ 𝑌 (((𝐹‘(0g‘𝐺)) ⨣ 𝑎) = 𝑎 ∧ (𝑎 ⨣ (𝐹‘(0g‘𝐺))) = 𝑎)) |
112 | | oveq1 7267 |
. . . . . 6
⊢ (𝑑 = (𝐹‘(0g‘𝐺)) → (𝑑 ⨣ 𝑎) = ((𝐹‘(0g‘𝐺)) ⨣ 𝑎)) |
113 | 112 | eqeq1d 2739 |
. . . . 5
⊢ (𝑑 = (𝐹‘(0g‘𝐺)) → ((𝑑 ⨣ 𝑎) = 𝑎 ↔ ((𝐹‘(0g‘𝐺)) ⨣ 𝑎) = 𝑎)) |
114 | 113 | ovanraleqv 7284 |
. . . 4
⊢ (𝑑 = (𝐹‘(0g‘𝐺)) → (∀𝑎 ∈ 𝑌 ((𝑑 ⨣ 𝑎) = 𝑎 ∧ (𝑎 ⨣ 𝑑) = 𝑎) ↔ ∀𝑎 ∈ 𝑌 (((𝐹‘(0g‘𝐺)) ⨣ 𝑎) = 𝑎 ∧ (𝑎 ⨣ (𝐹‘(0g‘𝐺))) = 𝑎))) |
115 | 114 | rspcev 3557 |
. . 3
⊢ (((𝐹‘(0g‘𝐺)) ∈ 𝑌 ∧ ∀𝑎 ∈ 𝑌 (((𝐹‘(0g‘𝐺)) ⨣ 𝑎) = 𝑎 ∧ (𝑎 ⨣ (𝐹‘(0g‘𝐺))) = 𝑎)) → ∃𝑑 ∈ 𝑌 ∀𝑎 ∈ 𝑌 ((𝑑 ⨣ 𝑎) = 𝑎 ∧ (𝑎 ⨣ 𝑑) = 𝑎)) |
116 | 87, 111, 115 | syl2anc 583 |
. 2
⊢ (𝜑 → ∃𝑑 ∈ 𝑌 ∀𝑎 ∈ 𝑌 ((𝑑 ⨣ 𝑎) = 𝑎 ∧ (𝑎 ⨣ 𝑑) = 𝑎)) |
117 | | ghmgrp.y |
. . 3
⊢ 𝑌 = (Base‘𝐻) |
118 | | ghmgrp.q |
. . 3
⊢ ⨣ =
(+g‘𝐻) |
119 | 117, 118 | ismnd 18332 |
. 2
⊢ (𝐻 ∈ Mnd ↔
(∀𝑎 ∈ 𝑌 ∀𝑏 ∈ 𝑌 ((𝑎 ⨣ 𝑏) ∈ 𝑌 ∧ ∀𝑐 ∈ 𝑌 ((𝑎 ⨣ 𝑏) ⨣ 𝑐) = (𝑎 ⨣ (𝑏 ⨣ 𝑐))) ∧ ∃𝑑 ∈ 𝑌 ∀𝑎 ∈ 𝑌 ((𝑑 ⨣ 𝑎) = 𝑎 ∧ (𝑎 ⨣ 𝑑) = 𝑎))) |
120 | 83, 116, 119 | sylanbrc 582 |
1
⊢ (𝜑 → 𝐻 ∈ Mnd) |