# Metamath Proof Explorer

## Theorem m2detleiblem4

Description: Lemma 4 for m2detleib . (Contributed by AV, 20-Dec-2018) (Proof shortened by AV, 2-Jan-2019)

Ref Expression
Hypotheses m2detleiblem2.n 𝑁 = { 1 , 2 }
m2detleiblem2.p 𝑃 = ( Base ‘ ( SymGrp ‘ 𝑁 ) )
m2detleiblem2.a 𝐴 = ( 𝑁 Mat 𝑅 )
m2detleiblem2.b 𝐵 = ( Base ‘ 𝐴 )
m2detleiblem2.g 𝐺 = ( mulGrp ‘ 𝑅 )
m2detleiblem3.m · = ( +g𝐺 )
Assertion m2detleiblem4 ( ( 𝑅 ∈ Ring ∧ 𝑄 = { ⟨ 1 , 2 ⟩ , ⟨ 2 , 1 ⟩ } ∧ 𝑀𝐵 ) → ( 𝐺 Σg ( 𝑛𝑁 ↦ ( ( 𝑄𝑛 ) 𝑀 𝑛 ) ) ) = ( ( 2 𝑀 1 ) · ( 1 𝑀 2 ) ) )

### Proof

Step Hyp Ref Expression
1 m2detleiblem2.n 𝑁 = { 1 , 2 }
2 m2detleiblem2.p 𝑃 = ( Base ‘ ( SymGrp ‘ 𝑁 ) )
3 m2detleiblem2.a 𝐴 = ( 𝑁 Mat 𝑅 )
4 m2detleiblem2.b 𝐵 = ( Base ‘ 𝐴 )
5 m2detleiblem2.g 𝐺 = ( mulGrp ‘ 𝑅 )
6 m2detleiblem3.m · = ( +g𝐺 )
7 eqid ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 )
8 5 7 mgpbas ( Base ‘ 𝑅 ) = ( Base ‘ 𝐺 )
9 5 fvexi 𝐺 ∈ V
10 9 a1i ( ( 𝑅 ∈ Ring ∧ 𝑄 = { ⟨ 1 , 2 ⟩ , ⟨ 2 , 1 ⟩ } ∧ 𝑀𝐵 ) → 𝐺 ∈ V )
11 1ex 1 ∈ V
12 2nn 2 ∈ ℕ
13 prex { ⟨ 1 , 2 ⟩ , ⟨ 2 , 1 ⟩ } ∈ V
14 13 prid2 { ⟨ 1 , 2 ⟩ , ⟨ 2 , 1 ⟩ } ∈ { { ⟨ 1 , 1 ⟩ , ⟨ 2 , 2 ⟩ } , { ⟨ 1 , 2 ⟩ , ⟨ 2 , 1 ⟩ } }
15 eqid ( SymGrp ‘ 𝑁 ) = ( SymGrp ‘ 𝑁 )
16 15 2 1 symg2bas ( ( 1 ∈ V ∧ 2 ∈ ℕ ) → 𝑃 = { { ⟨ 1 , 1 ⟩ , ⟨ 2 , 2 ⟩ } , { ⟨ 1 , 2 ⟩ , ⟨ 2 , 1 ⟩ } } )
17 14 16 eleqtrrid ( ( 1 ∈ V ∧ 2 ∈ ℕ ) → { ⟨ 1 , 2 ⟩ , ⟨ 2 , 1 ⟩ } ∈ 𝑃 )
18 11 12 17 mp2an { ⟨ 1 , 2 ⟩ , ⟨ 2 , 1 ⟩ } ∈ 𝑃
19 eleq1 ( 𝑄 = { ⟨ 1 , 2 ⟩ , ⟨ 2 , 1 ⟩ } → ( 𝑄𝑃 ↔ { ⟨ 1 , 2 ⟩ , ⟨ 2 , 1 ⟩ } ∈ 𝑃 ) )
20 18 19 mpbiri ( 𝑄 = { ⟨ 1 , 2 ⟩ , ⟨ 2 , 1 ⟩ } → 𝑄𝑃 )
21 1 oveq1i ( 𝑁 Mat 𝑅 ) = ( { 1 , 2 } Mat 𝑅 )
22 3 21 eqtri 𝐴 = ( { 1 , 2 } Mat 𝑅 )
23 1 fveq2i ( SymGrp ‘ 𝑁 ) = ( SymGrp ‘ { 1 , 2 } )
24 23 fveq2i ( Base ‘ ( SymGrp ‘ 𝑁 ) ) = ( Base ‘ ( SymGrp ‘ { 1 , 2 } ) )
25 2 24 eqtri 𝑃 = ( Base ‘ ( SymGrp ‘ { 1 , 2 } ) )
26 22 4 25 matepmcl ( ( 𝑅 ∈ Ring ∧ 𝑄𝑃𝑀𝐵 ) → ∀ 𝑛 ∈ { 1 , 2 } ( ( 𝑄𝑛 ) 𝑀 𝑛 ) ∈ ( Base ‘ 𝑅 ) )
27 20 26 syl3an2 ( ( 𝑅 ∈ Ring ∧ 𝑄 = { ⟨ 1 , 2 ⟩ , ⟨ 2 , 1 ⟩ } ∧ 𝑀𝐵 ) → ∀ 𝑛 ∈ { 1 , 2 } ( ( 𝑄𝑛 ) 𝑀 𝑛 ) ∈ ( Base ‘ 𝑅 ) )
28 1 mpteq1i ( 𝑛𝑁 ↦ ( ( 𝑄𝑛 ) 𝑀 𝑛 ) ) = ( 𝑛 ∈ { 1 , 2 } ↦ ( ( 𝑄𝑛 ) 𝑀 𝑛 ) )
29 28 fmpt ( ∀ 𝑛 ∈ { 1 , 2 } ( ( 𝑄𝑛 ) 𝑀 𝑛 ) ∈ ( Base ‘ 𝑅 ) ↔ ( 𝑛𝑁 ↦ ( ( 𝑄𝑛 ) 𝑀 𝑛 ) ) : { 1 , 2 } ⟶ ( Base ‘ 𝑅 ) )
30 27 29 sylib ( ( 𝑅 ∈ Ring ∧ 𝑄 = { ⟨ 1 , 2 ⟩ , ⟨ 2 , 1 ⟩ } ∧ 𝑀𝐵 ) → ( 𝑛𝑁 ↦ ( ( 𝑄𝑛 ) 𝑀 𝑛 ) ) : { 1 , 2 } ⟶ ( Base ‘ 𝑅 ) )
31 8 6 10 30 gsumpr12val ( ( 𝑅 ∈ Ring ∧ 𝑄 = { ⟨ 1 , 2 ⟩ , ⟨ 2 , 1 ⟩ } ∧ 𝑀𝐵 ) → ( 𝐺 Σg ( 𝑛𝑁 ↦ ( ( 𝑄𝑛 ) 𝑀 𝑛 ) ) ) = ( ( ( 𝑛𝑁 ↦ ( ( 𝑄𝑛 ) 𝑀 𝑛 ) ) ‘ 1 ) · ( ( 𝑛𝑁 ↦ ( ( 𝑄𝑛 ) 𝑀 𝑛 ) ) ‘ 2 ) ) )
32 11 prid1 1 ∈ { 1 , 2 }
33 32 1 eleqtrri 1 ∈ 𝑁
34 3 4 2 matepmcl ( ( 𝑅 ∈ Ring ∧ 𝑄𝑃𝑀𝐵 ) → ∀ 𝑛𝑁 ( ( 𝑄𝑛 ) 𝑀 𝑛 ) ∈ ( Base ‘ 𝑅 ) )
35 20 34 syl3an2 ( ( 𝑅 ∈ Ring ∧ 𝑄 = { ⟨ 1 , 2 ⟩ , ⟨ 2 , 1 ⟩ } ∧ 𝑀𝐵 ) → ∀ 𝑛𝑁 ( ( 𝑄𝑛 ) 𝑀 𝑛 ) ∈ ( Base ‘ 𝑅 ) )
36 fveq2 ( 𝑛 = 1 → ( 𝑄𝑛 ) = ( 𝑄 ‘ 1 ) )
37 id ( 𝑛 = 1 → 𝑛 = 1 )
38 36 37 oveq12d ( 𝑛 = 1 → ( ( 𝑄𝑛 ) 𝑀 𝑛 ) = ( ( 𝑄 ‘ 1 ) 𝑀 1 ) )
39 38 eleq1d ( 𝑛 = 1 → ( ( ( 𝑄𝑛 ) 𝑀 𝑛 ) ∈ ( Base ‘ 𝑅 ) ↔ ( ( 𝑄 ‘ 1 ) 𝑀 1 ) ∈ ( Base ‘ 𝑅 ) ) )
40 39 rspcva ( ( 1 ∈ 𝑁 ∧ ∀ 𝑛𝑁 ( ( 𝑄𝑛 ) 𝑀 𝑛 ) ∈ ( Base ‘ 𝑅 ) ) → ( ( 𝑄 ‘ 1 ) 𝑀 1 ) ∈ ( Base ‘ 𝑅 ) )
41 33 35 40 sylancr ( ( 𝑅 ∈ Ring ∧ 𝑄 = { ⟨ 1 , 2 ⟩ , ⟨ 2 , 1 ⟩ } ∧ 𝑀𝐵 ) → ( ( 𝑄 ‘ 1 ) 𝑀 1 ) ∈ ( Base ‘ 𝑅 ) )
42 eqid ( 𝑛𝑁 ↦ ( ( 𝑄𝑛 ) 𝑀 𝑛 ) ) = ( 𝑛𝑁 ↦ ( ( 𝑄𝑛 ) 𝑀 𝑛 ) )
43 38 42 fvmptg ( ( 1 ∈ 𝑁 ∧ ( ( 𝑄 ‘ 1 ) 𝑀 1 ) ∈ ( Base ‘ 𝑅 ) ) → ( ( 𝑛𝑁 ↦ ( ( 𝑄𝑛 ) 𝑀 𝑛 ) ) ‘ 1 ) = ( ( 𝑄 ‘ 1 ) 𝑀 1 ) )
44 33 41 43 sylancr ( ( 𝑅 ∈ Ring ∧ 𝑄 = { ⟨ 1 , 2 ⟩ , ⟨ 2 , 1 ⟩ } ∧ 𝑀𝐵 ) → ( ( 𝑛𝑁 ↦ ( ( 𝑄𝑛 ) 𝑀 𝑛 ) ) ‘ 1 ) = ( ( 𝑄 ‘ 1 ) 𝑀 1 ) )
45 fveq1 ( 𝑄 = { ⟨ 1 , 2 ⟩ , ⟨ 2 , 1 ⟩ } → ( 𝑄 ‘ 1 ) = ( { ⟨ 1 , 2 ⟩ , ⟨ 2 , 1 ⟩ } ‘ 1 ) )
46 1ne2 1 ≠ 2
47 2ex 2 ∈ V
48 11 47 fvpr1 ( 1 ≠ 2 → ( { ⟨ 1 , 2 ⟩ , ⟨ 2 , 1 ⟩ } ‘ 1 ) = 2 )
49 46 48 ax-mp ( { ⟨ 1 , 2 ⟩ , ⟨ 2 , 1 ⟩ } ‘ 1 ) = 2
50 45 49 eqtrdi ( 𝑄 = { ⟨ 1 , 2 ⟩ , ⟨ 2 , 1 ⟩ } → ( 𝑄 ‘ 1 ) = 2 )
51 50 3ad2ant2 ( ( 𝑅 ∈ Ring ∧ 𝑄 = { ⟨ 1 , 2 ⟩ , ⟨ 2 , 1 ⟩ } ∧ 𝑀𝐵 ) → ( 𝑄 ‘ 1 ) = 2 )
52 51 oveq1d ( ( 𝑅 ∈ Ring ∧ 𝑄 = { ⟨ 1 , 2 ⟩ , ⟨ 2 , 1 ⟩ } ∧ 𝑀𝐵 ) → ( ( 𝑄 ‘ 1 ) 𝑀 1 ) = ( 2 𝑀 1 ) )
53 44 52 eqtrd ( ( 𝑅 ∈ Ring ∧ 𝑄 = { ⟨ 1 , 2 ⟩ , ⟨ 2 , 1 ⟩ } ∧ 𝑀𝐵 ) → ( ( 𝑛𝑁 ↦ ( ( 𝑄𝑛 ) 𝑀 𝑛 ) ) ‘ 1 ) = ( 2 𝑀 1 ) )
54 47 prid2 2 ∈ { 1 , 2 }
55 54 1 eleqtrri 2 ∈ 𝑁
56 fveq2 ( 𝑛 = 2 → ( 𝑄𝑛 ) = ( 𝑄 ‘ 2 ) )
57 id ( 𝑛 = 2 → 𝑛 = 2 )
58 56 57 oveq12d ( 𝑛 = 2 → ( ( 𝑄𝑛 ) 𝑀 𝑛 ) = ( ( 𝑄 ‘ 2 ) 𝑀 2 ) )
59 58 eleq1d ( 𝑛 = 2 → ( ( ( 𝑄𝑛 ) 𝑀 𝑛 ) ∈ ( Base ‘ 𝑅 ) ↔ ( ( 𝑄 ‘ 2 ) 𝑀 2 ) ∈ ( Base ‘ 𝑅 ) ) )
60 59 rspcva ( ( 2 ∈ 𝑁 ∧ ∀ 𝑛𝑁 ( ( 𝑄𝑛 ) 𝑀 𝑛 ) ∈ ( Base ‘ 𝑅 ) ) → ( ( 𝑄 ‘ 2 ) 𝑀 2 ) ∈ ( Base ‘ 𝑅 ) )
61 55 35 60 sylancr ( ( 𝑅 ∈ Ring ∧ 𝑄 = { ⟨ 1 , 2 ⟩ , ⟨ 2 , 1 ⟩ } ∧ 𝑀𝐵 ) → ( ( 𝑄 ‘ 2 ) 𝑀 2 ) ∈ ( Base ‘ 𝑅 ) )
62 58 42 fvmptg ( ( 2 ∈ 𝑁 ∧ ( ( 𝑄 ‘ 2 ) 𝑀 2 ) ∈ ( Base ‘ 𝑅 ) ) → ( ( 𝑛𝑁 ↦ ( ( 𝑄𝑛 ) 𝑀 𝑛 ) ) ‘ 2 ) = ( ( 𝑄 ‘ 2 ) 𝑀 2 ) )
63 55 61 62 sylancr ( ( 𝑅 ∈ Ring ∧ 𝑄 = { ⟨ 1 , 2 ⟩ , ⟨ 2 , 1 ⟩ } ∧ 𝑀𝐵 ) → ( ( 𝑛𝑁 ↦ ( ( 𝑄𝑛 ) 𝑀 𝑛 ) ) ‘ 2 ) = ( ( 𝑄 ‘ 2 ) 𝑀 2 ) )
64 fveq1 ( 𝑄 = { ⟨ 1 , 2 ⟩ , ⟨ 2 , 1 ⟩ } → ( 𝑄 ‘ 2 ) = ( { ⟨ 1 , 2 ⟩ , ⟨ 2 , 1 ⟩ } ‘ 2 ) )
65 47 11 fvpr2 ( 1 ≠ 2 → ( { ⟨ 1 , 2 ⟩ , ⟨ 2 , 1 ⟩ } ‘ 2 ) = 1 )
66 46 65 ax-mp ( { ⟨ 1 , 2 ⟩ , ⟨ 2 , 1 ⟩ } ‘ 2 ) = 1
67 64 66 eqtrdi ( 𝑄 = { ⟨ 1 , 2 ⟩ , ⟨ 2 , 1 ⟩ } → ( 𝑄 ‘ 2 ) = 1 )
68 67 3ad2ant2 ( ( 𝑅 ∈ Ring ∧ 𝑄 = { ⟨ 1 , 2 ⟩ , ⟨ 2 , 1 ⟩ } ∧ 𝑀𝐵 ) → ( 𝑄 ‘ 2 ) = 1 )
69 68 oveq1d ( ( 𝑅 ∈ Ring ∧ 𝑄 = { ⟨ 1 , 2 ⟩ , ⟨ 2 , 1 ⟩ } ∧ 𝑀𝐵 ) → ( ( 𝑄 ‘ 2 ) 𝑀 2 ) = ( 1 𝑀 2 ) )
70 63 69 eqtrd ( ( 𝑅 ∈ Ring ∧ 𝑄 = { ⟨ 1 , 2 ⟩ , ⟨ 2 , 1 ⟩ } ∧ 𝑀𝐵 ) → ( ( 𝑛𝑁 ↦ ( ( 𝑄𝑛 ) 𝑀 𝑛 ) ) ‘ 2 ) = ( 1 𝑀 2 ) )
71 53 70 oveq12d ( ( 𝑅 ∈ Ring ∧ 𝑄 = { ⟨ 1 , 2 ⟩ , ⟨ 2 , 1 ⟩ } ∧ 𝑀𝐵 ) → ( ( ( 𝑛𝑁 ↦ ( ( 𝑄𝑛 ) 𝑀 𝑛 ) ) ‘ 1 ) · ( ( 𝑛𝑁 ↦ ( ( 𝑄𝑛 ) 𝑀 𝑛 ) ) ‘ 2 ) ) = ( ( 2 𝑀 1 ) · ( 1 𝑀 2 ) ) )
72 31 71 eqtrd ( ( 𝑅 ∈ Ring ∧ 𝑄 = { ⟨ 1 , 2 ⟩ , ⟨ 2 , 1 ⟩ } ∧ 𝑀𝐵 ) → ( 𝐺 Σg ( 𝑛𝑁 ↦ ( ( 𝑄𝑛 ) 𝑀 𝑛 ) ) ) = ( ( 2 𝑀 1 ) · ( 1 𝑀 2 ) ) )