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Metamath Proof Explorer

Theorem symginv

Description: The group inverse in the symmetric group corresponds to the functional inverse. (Contributed by Stefan O'Rear, 24-Aug-2015) (Revised by Mario Carneiro, 2-Sep-2015)

Ref Expression
Hypotheses symggrp.1 ⊢ 𝐺 = ( SymGrp ‘ 𝐴 )
symginv.2 ⊢ 𝐵 = ( Base ‘ 𝐺 )
symginv.3 ⊢ 𝑁 = ( invg ‘ 𝐺 )
Assertion symginv ( 𝐹 ∈ 𝐵 → ( 𝑁 ‘ 𝐹 ) = ◡ 𝐹 )

Proof

Step Hyp Ref Expression
1 symggrp.1 ⊢ 𝐺 = ( SymGrp ‘ 𝐴 )
2 symginv.2 ⊢ 𝐵 = ( Base ‘ 𝐺 )
3 symginv.3 ⊢ 𝑁 = ( invg ‘ 𝐺 )
4 1 2 elsymgbas2 ⊢ ( 𝐹 ∈ 𝐵 → ( 𝐹 ∈ 𝐵 ↔ 𝐹 : 𝐴 –1-1-onto→ 𝐴 ) )
5 4 ibi ⊢ ( 𝐹 ∈ 𝐵 → 𝐹 : 𝐴 –1-1-onto→ 𝐴 )
6 f1ocnv ⊢ ( 𝐹 : 𝐴 –1-1-onto→ 𝐴 → ◡ 𝐹 : 𝐴 –1-1-onto→ 𝐴 )
7 5 6 syl ⊢ ( 𝐹 ∈ 𝐵 → ◡ 𝐹 : 𝐴 –1-1-onto→ 𝐴 )
8 cnvexg ⊢ ( 𝐹 ∈ 𝐵 → ◡ 𝐹 ∈ V )
9 1 2 elsymgbas2 ⊢ ( ◡ 𝐹 ∈ V → ( ◡ 𝐹 ∈ 𝐵 ↔ ◡ 𝐹 : 𝐴 –1-1-onto→ 𝐴 ) )
10 8 9 syl ⊢ ( 𝐹 ∈ 𝐵 → ( ◡ 𝐹 ∈ 𝐵 ↔ ◡ 𝐹 : 𝐴 –1-1-onto→ 𝐴 ) )
11 7 10 mpbird ⊢ ( 𝐹 ∈ 𝐵 → ◡ 𝐹 ∈ 𝐵 )
12 eqid ⊢ ( +g ‘ 𝐺 ) = ( +g ‘ 𝐺 )
13 1 2 12 symgov ⊢ ( ( 𝐹 ∈ 𝐵 ∧ ◡ 𝐹 ∈ 𝐵 ) → ( 𝐹 ( +g ‘ 𝐺 ) ◡ 𝐹 ) = ( 𝐹 ∘ ◡ 𝐹 ) )
14 11 13 mpdan ⊢ ( 𝐹 ∈ 𝐵 → ( 𝐹 ( +g ‘ 𝐺 ) ◡ 𝐹 ) = ( 𝐹 ∘ ◡ 𝐹 ) )
15 f1ococnv2 ⊢ ( 𝐹 : 𝐴 –1-1-onto→ 𝐴 → ( 𝐹 ∘ ◡ 𝐹 ) = ( I ↾ 𝐴 ) )
16 5 15 syl ⊢ ( 𝐹 ∈ 𝐵 → ( 𝐹 ∘ ◡ 𝐹 ) = ( I ↾ 𝐴 ) )
17 1 2 elbasfv ⊢ ( 𝐹 ∈ 𝐵 → 𝐴 ∈ V )
18 1 symgid ⊢ ( 𝐴 ∈ V → ( I ↾ 𝐴 ) = ( 0g ‘ 𝐺 ) )
19 17 18 syl ⊢ ( 𝐹 ∈ 𝐵 → ( I ↾ 𝐴 ) = ( 0g ‘ 𝐺 ) )
20 14 16 19 3eqtrd ⊢ ( 𝐹 ∈ 𝐵 → ( 𝐹 ( +g ‘ 𝐺 ) ◡ 𝐹 ) = ( 0g ‘ 𝐺 ) )
21 1 symggrp ⊢ ( 𝐴 ∈ V → 𝐺 ∈ Grp )
22 17 21 syl ⊢ ( 𝐹 ∈ 𝐵 → 𝐺 ∈ Grp )
23 id ⊢ ( 𝐹 ∈ 𝐵 → 𝐹 ∈ 𝐵 )
24 eqid ⊢ ( 0g ‘ 𝐺 ) = ( 0g ‘ 𝐺 )
25 2 12 24 3 grpinvid1 ⊢ ( ( 𝐺 ∈ Grp ∧ 𝐹 ∈ 𝐵 ∧ ◡ 𝐹 ∈ 𝐵 ) → ( ( 𝑁 ‘ 𝐹 ) = ◡ 𝐹 ↔ ( 𝐹 ( +g ‘ 𝐺 ) ◡ 𝐹 ) = ( 0g ‘ 𝐺 ) ) )
26 22 23 11 25 syl3anc ⊢ ( 𝐹 ∈ 𝐵 → ( ( 𝑁 ‘ 𝐹 ) = ◡ 𝐹 ↔ ( 𝐹 ( +g ‘ 𝐺 ) ◡ 𝐹 ) = ( 0g ‘ 𝐺 ) ) )
27 20 26 mpbird ⊢ ( 𝐹 ∈ 𝐵 → ( 𝑁 ‘ 𝐹 ) = ◡ 𝐹 )