Step | Hyp | Ref
| Expression |
1 | | ghmgrp.f |
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦))) |
2 | | ghmgrp.x |
. . 3
⊢ 𝑋 = (Base‘𝐺) |
3 | | ghmgrp.y |
. . 3
⊢ 𝑌 = (Base‘𝐻) |
4 | | ghmgrp.p |
. . 3
⊢ + =
(+g‘𝐺) |
5 | | ghmgrp.q |
. . 3
⊢ ⨣ =
(+g‘𝐻) |
6 | | ghmgrp.1 |
. . 3
⊢ (𝜑 → 𝐹:𝑋–onto→𝑌) |
7 | | ghmgrp.3 |
. . . 4
⊢ (𝜑 → 𝐺 ∈ Grp) |
8 | 7 | grpmndd 18589 |
. . 3
⊢ (𝜑 → 𝐺 ∈ Mnd) |
9 | 1, 2, 3, 4, 5, 6, 8 | mhmmnd 18697 |
. 2
⊢ (𝜑 → 𝐻 ∈ Mnd) |
10 | | fof 6688 |
. . . . . . . 8
⊢ (𝐹:𝑋–onto→𝑌 → 𝐹:𝑋⟶𝑌) |
11 | 6, 10 | syl 17 |
. . . . . . 7
⊢ (𝜑 → 𝐹:𝑋⟶𝑌) |
12 | 11 | ad3antrrr 727 |
. . . . . 6
⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → 𝐹:𝑋⟶𝑌) |
13 | 7 | ad3antrrr 727 |
. . . . . . 7
⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → 𝐺 ∈ Grp) |
14 | | simplr 766 |
. . . . . . 7
⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → 𝑖 ∈ 𝑋) |
15 | | eqid 2738 |
. . . . . . . 8
⊢
(invg‘𝐺) = (invg‘𝐺) |
16 | 2, 15 | grpinvcl 18627 |
. . . . . . 7
⊢ ((𝐺 ∈ Grp ∧ 𝑖 ∈ 𝑋) → ((invg‘𝐺)‘𝑖) ∈ 𝑋) |
17 | 13, 14, 16 | syl2anc 584 |
. . . . . 6
⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → ((invg‘𝐺)‘𝑖) ∈ 𝑋) |
18 | 12, 17 | ffvelrnd 6962 |
. . . . 5
⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → (𝐹‘((invg‘𝐺)‘𝑖)) ∈ 𝑌) |
19 | 1 | 3adant1r 1176 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦))) |
20 | 7, 16 | sylan 580 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((invg‘𝐺)‘𝑖) ∈ 𝑋) |
21 | | simpr 485 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → 𝑖 ∈ 𝑋) |
22 | 19, 20, 21 | mhmlem 18695 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐹‘(((invg‘𝐺)‘𝑖) + 𝑖)) = ((𝐹‘((invg‘𝐺)‘𝑖)) ⨣ (𝐹‘𝑖))) |
23 | 22 | ad4ant13 748 |
. . . . . 6
⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → (𝐹‘(((invg‘𝐺)‘𝑖) + 𝑖)) = ((𝐹‘((invg‘𝐺)‘𝑖)) ⨣ (𝐹‘𝑖))) |
24 | | eqid 2738 |
. . . . . . . . . 10
⊢
(0g‘𝐺) = (0g‘𝐺) |
25 | 2, 4, 24, 15 | grplinv 18628 |
. . . . . . . . 9
⊢ ((𝐺 ∈ Grp ∧ 𝑖 ∈ 𝑋) → (((invg‘𝐺)‘𝑖) + 𝑖) = (0g‘𝐺)) |
26 | 25 | fveq2d 6778 |
. . . . . . . 8
⊢ ((𝐺 ∈ Grp ∧ 𝑖 ∈ 𝑋) → (𝐹‘(((invg‘𝐺)‘𝑖) + 𝑖)) = (𝐹‘(0g‘𝐺))) |
27 | 13, 14, 26 | syl2anc 584 |
. . . . . . 7
⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → (𝐹‘(((invg‘𝐺)‘𝑖) + 𝑖)) = (𝐹‘(0g‘𝐺))) |
28 | 1, 2, 3, 4, 5, 6, 8, 24 | mhmid 18696 |
. . . . . . . 8
⊢ (𝜑 → (𝐹‘(0g‘𝐺)) = (0g‘𝐻)) |
29 | 28 | ad3antrrr 727 |
. . . . . . 7
⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → (𝐹‘(0g‘𝐺)) = (0g‘𝐻)) |
30 | 27, 29 | eqtrd 2778 |
. . . . . 6
⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → (𝐹‘(((invg‘𝐺)‘𝑖) + 𝑖)) = (0g‘𝐻)) |
31 | | simpr 485 |
. . . . . . 7
⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → (𝐹‘𝑖) = 𝑎) |
32 | 31 | oveq2d 7291 |
. . . . . 6
⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → ((𝐹‘((invg‘𝐺)‘𝑖)) ⨣ (𝐹‘𝑖)) = ((𝐹‘((invg‘𝐺)‘𝑖)) ⨣ 𝑎)) |
33 | 23, 30, 32 | 3eqtr3rd 2787 |
. . . . 5
⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → ((𝐹‘((invg‘𝐺)‘𝑖)) ⨣ 𝑎) = (0g‘𝐻)) |
34 | | oveq1 7282 |
. . . . . . 7
⊢ (𝑓 = (𝐹‘((invg‘𝐺)‘𝑖)) → (𝑓 ⨣ 𝑎) = ((𝐹‘((invg‘𝐺)‘𝑖)) ⨣ 𝑎)) |
35 | 34 | eqeq1d 2740 |
. . . . . 6
⊢ (𝑓 = (𝐹‘((invg‘𝐺)‘𝑖)) → ((𝑓 ⨣ 𝑎) = (0g‘𝐻) ↔ ((𝐹‘((invg‘𝐺)‘𝑖)) ⨣ 𝑎) = (0g‘𝐻))) |
36 | 35 | rspcev 3561 |
. . . . 5
⊢ (((𝐹‘((invg‘𝐺)‘𝑖)) ∈ 𝑌 ∧ ((𝐹‘((invg‘𝐺)‘𝑖)) ⨣ 𝑎) = (0g‘𝐻)) → ∃𝑓 ∈ 𝑌 (𝑓 ⨣ 𝑎) = (0g‘𝐻)) |
37 | 18, 33, 36 | syl2anc 584 |
. . . 4
⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → ∃𝑓 ∈ 𝑌 (𝑓 ⨣ 𝑎) = (0g‘𝐻)) |
38 | | foelrni 6831 |
. . . . 5
⊢ ((𝐹:𝑋–onto→𝑌 ∧ 𝑎 ∈ 𝑌) → ∃𝑖 ∈ 𝑋 (𝐹‘𝑖) = 𝑎) |
39 | 6, 38 | sylan 580 |
. . . 4
⊢ ((𝜑 ∧ 𝑎 ∈ 𝑌) → ∃𝑖 ∈ 𝑋 (𝐹‘𝑖) = 𝑎) |
40 | 37, 39 | r19.29a 3218 |
. . 3
⊢ ((𝜑 ∧ 𝑎 ∈ 𝑌) → ∃𝑓 ∈ 𝑌 (𝑓 ⨣ 𝑎) = (0g‘𝐻)) |
41 | 40 | ralrimiva 3103 |
. 2
⊢ (𝜑 → ∀𝑎 ∈ 𝑌 ∃𝑓 ∈ 𝑌 (𝑓 ⨣ 𝑎) = (0g‘𝐻)) |
42 | | eqid 2738 |
. . 3
⊢
(0g‘𝐻) = (0g‘𝐻) |
43 | 3, 5, 42 | isgrp 18583 |
. 2
⊢ (𝐻 ∈ Grp ↔ (𝐻 ∈ Mnd ∧ ∀𝑎 ∈ 𝑌 ∃𝑓 ∈ 𝑌 (𝑓 ⨣ 𝑎) = (0g‘𝐻))) |
44 | 9, 41, 43 | sylanbrc 583 |
1
⊢ (𝜑 → 𝐻 ∈ Grp) |