Proof of Theorem ismndo2
Step | Hyp | Ref
| Expression |
1 | | ismndo2.1 |
. . . 4
⊢ 𝑋 = ran 𝐺 |
2 | | mndomgmid 36029 |
. . . . 5
⊢ (𝐺 ∈ MndOp → 𝐺 ∈ (Magma ∩ ExId
)) |
3 | | rngopidOLD 36011 |
. . . . 5
⊢ (𝐺 ∈ (Magma ∩ ExId )
→ ran 𝐺 = dom dom
𝐺) |
4 | 2, 3 | syl 17 |
. . . 4
⊢ (𝐺 ∈ MndOp → ran 𝐺 = dom dom 𝐺) |
5 | 1, 4 | eqtrid 2790 |
. . 3
⊢ (𝐺 ∈ MndOp → 𝑋 = dom dom 𝐺) |
6 | 5 | a1i 11 |
. 2
⊢ (𝐺 ∈ 𝐴 → (𝐺 ∈ MndOp → 𝑋 = dom dom 𝐺)) |
7 | | fdm 6609 |
. . . . . 6
⊢ (𝐺:(𝑋 × 𝑋)⟶𝑋 → dom 𝐺 = (𝑋 × 𝑋)) |
8 | 7 | dmeqd 5814 |
. . . . 5
⊢ (𝐺:(𝑋 × 𝑋)⟶𝑋 → dom dom 𝐺 = dom (𝑋 × 𝑋)) |
9 | | dmxpid 5839 |
. . . . 5
⊢ dom
(𝑋 × 𝑋) = 𝑋 |
10 | 8, 9 | eqtr2di 2795 |
. . . 4
⊢ (𝐺:(𝑋 × 𝑋)⟶𝑋 → 𝑋 = dom dom 𝐺) |
11 | 10 | 3ad2ant1 1132 |
. . 3
⊢ ((𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ∧ ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦)) → 𝑋 = dom dom 𝐺) |
12 | 11 | a1i 11 |
. 2
⊢ (𝐺 ∈ 𝐴 → ((𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ∧ ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦)) → 𝑋 = dom dom 𝐺)) |
13 | | eqid 2738 |
. . . 4
⊢ dom dom
𝐺 = dom dom 𝐺 |
14 | 13 | ismndo1 36031 |
. . 3
⊢ (𝐺 ∈ 𝐴 → (𝐺 ∈ MndOp ↔ (𝐺:(dom dom 𝐺 × dom dom 𝐺)⟶dom dom 𝐺 ∧ ∀𝑥 ∈ dom dom 𝐺∀𝑦 ∈ dom dom 𝐺∀𝑧 ∈ dom dom 𝐺((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ∧ ∃𝑥 ∈ dom dom 𝐺∀𝑦 ∈ dom dom 𝐺((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦)))) |
15 | | xpid11 5841 |
. . . . . . 7
⊢ ((𝑋 × 𝑋) = (dom dom 𝐺 × dom dom 𝐺) ↔ 𝑋 = dom dom 𝐺) |
16 | 15 | biimpri 227 |
. . . . . 6
⊢ (𝑋 = dom dom 𝐺 → (𝑋 × 𝑋) = (dom dom 𝐺 × dom dom 𝐺)) |
17 | | feq23 6584 |
. . . . . 6
⊢ (((𝑋 × 𝑋) = (dom dom 𝐺 × dom dom 𝐺) ∧ 𝑋 = dom dom 𝐺) → (𝐺:(𝑋 × 𝑋)⟶𝑋 ↔ 𝐺:(dom dom 𝐺 × dom dom 𝐺)⟶dom dom 𝐺)) |
18 | 16, 17 | mpancom 685 |
. . . . 5
⊢ (𝑋 = dom dom 𝐺 → (𝐺:(𝑋 × 𝑋)⟶𝑋 ↔ 𝐺:(dom dom 𝐺 × dom dom 𝐺)⟶dom dom 𝐺)) |
19 | | raleq 3342 |
. . . . . . 7
⊢ (𝑋 = dom dom 𝐺 → (∀𝑧 ∈ 𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ↔ ∀𝑧 ∈ dom dom 𝐺((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)))) |
20 | 19 | raleqbi1dv 3340 |
. . . . . 6
⊢ (𝑋 = dom dom 𝐺 → (∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ↔ ∀𝑦 ∈ dom dom 𝐺∀𝑧 ∈ dom dom 𝐺((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)))) |
21 | 20 | raleqbi1dv 3340 |
. . . . 5
⊢ (𝑋 = dom dom 𝐺 → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ↔ ∀𝑥 ∈ dom dom 𝐺∀𝑦 ∈ dom dom 𝐺∀𝑧 ∈ dom dom 𝐺((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)))) |
22 | | raleq 3342 |
. . . . . 6
⊢ (𝑋 = dom dom 𝐺 → (∀𝑦 ∈ 𝑋 ((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦) ↔ ∀𝑦 ∈ dom dom 𝐺((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦))) |
23 | 22 | rexeqbi1dv 3341 |
. . . . 5
⊢ (𝑋 = dom dom 𝐺 → (∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦) ↔ ∃𝑥 ∈ dom dom 𝐺∀𝑦 ∈ dom dom 𝐺((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦))) |
24 | 18, 21, 23 | 3anbi123d 1435 |
. . . 4
⊢ (𝑋 = dom dom 𝐺 → ((𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ∧ ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦)) ↔ (𝐺:(dom dom 𝐺 × dom dom 𝐺)⟶dom dom 𝐺 ∧ ∀𝑥 ∈ dom dom 𝐺∀𝑦 ∈ dom dom 𝐺∀𝑧 ∈ dom dom 𝐺((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ∧ ∃𝑥 ∈ dom dom 𝐺∀𝑦 ∈ dom dom 𝐺((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦)))) |
25 | 24 | bibi2d 343 |
. . 3
⊢ (𝑋 = dom dom 𝐺 → ((𝐺 ∈ MndOp ↔ (𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ∧ ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦))) ↔ (𝐺 ∈ MndOp ↔ (𝐺:(dom dom 𝐺 × dom dom 𝐺)⟶dom dom 𝐺 ∧ ∀𝑥 ∈ dom dom 𝐺∀𝑦 ∈ dom dom 𝐺∀𝑧 ∈ dom dom 𝐺((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ∧ ∃𝑥 ∈ dom dom 𝐺∀𝑦 ∈ dom dom 𝐺((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦))))) |
26 | 14, 25 | syl5ibrcom 246 |
. 2
⊢ (𝐺 ∈ 𝐴 → (𝑋 = dom dom 𝐺 → (𝐺 ∈ MndOp ↔ (𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ∧ ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦))))) |
27 | 6, 12, 26 | pm5.21ndd 381 |
1
⊢ (𝐺 ∈ 𝐴 → (𝐺 ∈ MndOp ↔ (𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ∧ ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦)))) |