Metamath Proof Explorer


Theorem mnd1

Description: The (smallest) structure representing atrivial monoid consists of one element. (Contributed by AV, 28-Apr-2019) (Proof shortened by AV, 11-Feb-2020)

Ref Expression
Hypothesis mnd1.m 𝑀 = { ⟨ ( Base ‘ ndx ) , { 𝐼 } ⟩ , ⟨ ( +g ‘ ndx ) , { ⟨ ⟨ 𝐼 , 𝐼 ⟩ , 𝐼 ⟩ } ⟩ }
Assertion mnd1 ( 𝐼𝑉𝑀 ∈ Mnd )

Proof

Step Hyp Ref Expression
1 mnd1.m 𝑀 = { ⟨ ( Base ‘ ndx ) , { 𝐼 } ⟩ , ⟨ ( +g ‘ ndx ) , { ⟨ ⟨ 𝐼 , 𝐼 ⟩ , 𝐼 ⟩ } ⟩ }
2 1 sgrp1 ( 𝐼𝑉𝑀 ∈ Smgrp )
3 df-ov ( 𝐼 { ⟨ ⟨ 𝐼 , 𝐼 ⟩ , 𝐼 ⟩ } 𝐼 ) = ( { ⟨ ⟨ 𝐼 , 𝐼 ⟩ , 𝐼 ⟩ } ‘ ⟨ 𝐼 , 𝐼 ⟩ )
4 opex 𝐼 , 𝐼 ⟩ ∈ V
5 fvsng ( ( ⟨ 𝐼 , 𝐼 ⟩ ∈ V ∧ 𝐼𝑉 ) → ( { ⟨ ⟨ 𝐼 , 𝐼 ⟩ , 𝐼 ⟩ } ‘ ⟨ 𝐼 , 𝐼 ⟩ ) = 𝐼 )
6 4 5 mpan ( 𝐼𝑉 → ( { ⟨ ⟨ 𝐼 , 𝐼 ⟩ , 𝐼 ⟩ } ‘ ⟨ 𝐼 , 𝐼 ⟩ ) = 𝐼 )
7 3 6 eqtrid ( 𝐼𝑉 → ( 𝐼 { ⟨ ⟨ 𝐼 , 𝐼 ⟩ , 𝐼 ⟩ } 𝐼 ) = 𝐼 )
8 oveq2 ( 𝑦 = 𝐼 → ( 𝐼 { ⟨ ⟨ 𝐼 , 𝐼 ⟩ , 𝐼 ⟩ } 𝑦 ) = ( 𝐼 { ⟨ ⟨ 𝐼 , 𝐼 ⟩ , 𝐼 ⟩ } 𝐼 ) )
9 id ( 𝑦 = 𝐼𝑦 = 𝐼 )
10 8 9 eqeq12d ( 𝑦 = 𝐼 → ( ( 𝐼 { ⟨ ⟨ 𝐼 , 𝐼 ⟩ , 𝐼 ⟩ } 𝑦 ) = 𝑦 ↔ ( 𝐼 { ⟨ ⟨ 𝐼 , 𝐼 ⟩ , 𝐼 ⟩ } 𝐼 ) = 𝐼 ) )
11 oveq1 ( 𝑦 = 𝐼 → ( 𝑦 { ⟨ ⟨ 𝐼 , 𝐼 ⟩ , 𝐼 ⟩ } 𝐼 ) = ( 𝐼 { ⟨ ⟨ 𝐼 , 𝐼 ⟩ , 𝐼 ⟩ } 𝐼 ) )
12 11 9 eqeq12d ( 𝑦 = 𝐼 → ( ( 𝑦 { ⟨ ⟨ 𝐼 , 𝐼 ⟩ , 𝐼 ⟩ } 𝐼 ) = 𝑦 ↔ ( 𝐼 { ⟨ ⟨ 𝐼 , 𝐼 ⟩ , 𝐼 ⟩ } 𝐼 ) = 𝐼 ) )
13 10 12 anbi12d ( 𝑦 = 𝐼 → ( ( ( 𝐼 { ⟨ ⟨ 𝐼 , 𝐼 ⟩ , 𝐼 ⟩ } 𝑦 ) = 𝑦 ∧ ( 𝑦 { ⟨ ⟨ 𝐼 , 𝐼 ⟩ , 𝐼 ⟩ } 𝐼 ) = 𝑦 ) ↔ ( ( 𝐼 { ⟨ ⟨ 𝐼 , 𝐼 ⟩ , 𝐼 ⟩ } 𝐼 ) = 𝐼 ∧ ( 𝐼 { ⟨ ⟨ 𝐼 , 𝐼 ⟩ , 𝐼 ⟩ } 𝐼 ) = 𝐼 ) ) )
14 13 ralsng ( 𝐼𝑉 → ( ∀ 𝑦 ∈ { 𝐼 } ( ( 𝐼 { ⟨ ⟨ 𝐼 , 𝐼 ⟩ , 𝐼 ⟩ } 𝑦 ) = 𝑦 ∧ ( 𝑦 { ⟨ ⟨ 𝐼 , 𝐼 ⟩ , 𝐼 ⟩ } 𝐼 ) = 𝑦 ) ↔ ( ( 𝐼 { ⟨ ⟨ 𝐼 , 𝐼 ⟩ , 𝐼 ⟩ } 𝐼 ) = 𝐼 ∧ ( 𝐼 { ⟨ ⟨ 𝐼 , 𝐼 ⟩ , 𝐼 ⟩ } 𝐼 ) = 𝐼 ) ) )
15 7 7 14 mpbir2and ( 𝐼𝑉 → ∀ 𝑦 ∈ { 𝐼 } ( ( 𝐼 { ⟨ ⟨ 𝐼 , 𝐼 ⟩ , 𝐼 ⟩ } 𝑦 ) = 𝑦 ∧ ( 𝑦 { ⟨ ⟨ 𝐼 , 𝐼 ⟩ , 𝐼 ⟩ } 𝐼 ) = 𝑦 ) )
16 oveq1 ( 𝑥 = 𝐼 → ( 𝑥 { ⟨ ⟨ 𝐼 , 𝐼 ⟩ , 𝐼 ⟩ } 𝑦 ) = ( 𝐼 { ⟨ ⟨ 𝐼 , 𝐼 ⟩ , 𝐼 ⟩ } 𝑦 ) )
17 16 eqeq1d ( 𝑥 = 𝐼 → ( ( 𝑥 { ⟨ ⟨ 𝐼 , 𝐼 ⟩ , 𝐼 ⟩ } 𝑦 ) = 𝑦 ↔ ( 𝐼 { ⟨ ⟨ 𝐼 , 𝐼 ⟩ , 𝐼 ⟩ } 𝑦 ) = 𝑦 ) )
18 17 ovanraleqv ( 𝑥 = 𝐼 → ( ∀ 𝑦 ∈ { 𝐼 } ( ( 𝑥 { ⟨ ⟨ 𝐼 , 𝐼 ⟩ , 𝐼 ⟩ } 𝑦 ) = 𝑦 ∧ ( 𝑦 { ⟨ ⟨ 𝐼 , 𝐼 ⟩ , 𝐼 ⟩ } 𝑥 ) = 𝑦 ) ↔ ∀ 𝑦 ∈ { 𝐼 } ( ( 𝐼 { ⟨ ⟨ 𝐼 , 𝐼 ⟩ , 𝐼 ⟩ } 𝑦 ) = 𝑦 ∧ ( 𝑦 { ⟨ ⟨ 𝐼 , 𝐼 ⟩ , 𝐼 ⟩ } 𝐼 ) = 𝑦 ) ) )
19 18 rexsng ( 𝐼𝑉 → ( ∃ 𝑥 ∈ { 𝐼 } ∀ 𝑦 ∈ { 𝐼 } ( ( 𝑥 { ⟨ ⟨ 𝐼 , 𝐼 ⟩ , 𝐼 ⟩ } 𝑦 ) = 𝑦 ∧ ( 𝑦 { ⟨ ⟨ 𝐼 , 𝐼 ⟩ , 𝐼 ⟩ } 𝑥 ) = 𝑦 ) ↔ ∀ 𝑦 ∈ { 𝐼 } ( ( 𝐼 { ⟨ ⟨ 𝐼 , 𝐼 ⟩ , 𝐼 ⟩ } 𝑦 ) = 𝑦 ∧ ( 𝑦 { ⟨ ⟨ 𝐼 , 𝐼 ⟩ , 𝐼 ⟩ } 𝐼 ) = 𝑦 ) ) )
20 15 19 mpbird ( 𝐼𝑉 → ∃ 𝑥 ∈ { 𝐼 } ∀ 𝑦 ∈ { 𝐼 } ( ( 𝑥 { ⟨ ⟨ 𝐼 , 𝐼 ⟩ , 𝐼 ⟩ } 𝑦 ) = 𝑦 ∧ ( 𝑦 { ⟨ ⟨ 𝐼 , 𝐼 ⟩ , 𝐼 ⟩ } 𝑥 ) = 𝑦 ) )
21 snex { 𝐼 } ∈ V
22 1 grpbase ( { 𝐼 } ∈ V → { 𝐼 } = ( Base ‘ 𝑀 ) )
23 21 22 ax-mp { 𝐼 } = ( Base ‘ 𝑀 )
24 snex { ⟨ ⟨ 𝐼 , 𝐼 ⟩ , 𝐼 ⟩ } ∈ V
25 1 grpplusg ( { ⟨ ⟨ 𝐼 , 𝐼 ⟩ , 𝐼 ⟩ } ∈ V → { ⟨ ⟨ 𝐼 , 𝐼 ⟩ , 𝐼 ⟩ } = ( +g𝑀 ) )
26 24 25 ax-mp { ⟨ ⟨ 𝐼 , 𝐼 ⟩ , 𝐼 ⟩ } = ( +g𝑀 )
27 23 26 ismnddef ( 𝑀 ∈ Mnd ↔ ( 𝑀 ∈ Smgrp ∧ ∃ 𝑥 ∈ { 𝐼 } ∀ 𝑦 ∈ { 𝐼 } ( ( 𝑥 { ⟨ ⟨ 𝐼 , 𝐼 ⟩ , 𝐼 ⟩ } 𝑦 ) = 𝑦 ∧ ( 𝑦 { ⟨ ⟨ 𝐼 , 𝐼 ⟩ , 𝐼 ⟩ } 𝑥 ) = 𝑦 ) ) )
28 2 20 27 sylanbrc ( 𝐼𝑉𝑀 ∈ Mnd )