Metamath Proof Explorer


Theorem frgpup2

Description: The evaluation map has the intended behavior on the generators. (Contributed by Mario Carneiro, 2-Oct-2015) (Revised by Mario Carneiro, 28-Feb-2016)

Ref Expression
Hypotheses frgpup.b 𝐵 = ( Base ‘ 𝐻 )
frgpup.n 𝑁 = ( invg𝐻 )
frgpup.t 𝑇 = ( 𝑦𝐼 , 𝑧 ∈ 2o ↦ if ( 𝑧 = ∅ , ( 𝐹𝑦 ) , ( 𝑁 ‘ ( 𝐹𝑦 ) ) ) )
frgpup.h ( 𝜑𝐻 ∈ Grp )
frgpup.i ( 𝜑𝐼𝑉 )
frgpup.a ( 𝜑𝐹 : 𝐼𝐵 )
frgpup.w 𝑊 = ( I ‘ Word ( 𝐼 × 2o ) )
frgpup.r = ( ~FG𝐼 )
frgpup.g 𝐺 = ( freeGrp ‘ 𝐼 )
frgpup.x 𝑋 = ( Base ‘ 𝐺 )
frgpup.e 𝐸 = ran ( 𝑔𝑊 ↦ ⟨ [ 𝑔 ] , ( 𝐻 Σg ( 𝑇𝑔 ) ) ⟩ )
frgpup.u 𝑈 = ( varFGrp𝐼 )
frgpup.y ( 𝜑𝐴𝐼 )
Assertion frgpup2 ( 𝜑 → ( 𝐸 ‘ ( 𝑈𝐴 ) ) = ( 𝐹𝐴 ) )

Proof

Step Hyp Ref Expression
1 frgpup.b 𝐵 = ( Base ‘ 𝐻 )
2 frgpup.n 𝑁 = ( invg𝐻 )
3 frgpup.t 𝑇 = ( 𝑦𝐼 , 𝑧 ∈ 2o ↦ if ( 𝑧 = ∅ , ( 𝐹𝑦 ) , ( 𝑁 ‘ ( 𝐹𝑦 ) ) ) )
4 frgpup.h ( 𝜑𝐻 ∈ Grp )
5 frgpup.i ( 𝜑𝐼𝑉 )
6 frgpup.a ( 𝜑𝐹 : 𝐼𝐵 )
7 frgpup.w 𝑊 = ( I ‘ Word ( 𝐼 × 2o ) )
8 frgpup.r = ( ~FG𝐼 )
9 frgpup.g 𝐺 = ( freeGrp ‘ 𝐼 )
10 frgpup.x 𝑋 = ( Base ‘ 𝐺 )
11 frgpup.e 𝐸 = ran ( 𝑔𝑊 ↦ ⟨ [ 𝑔 ] , ( 𝐻 Σg ( 𝑇𝑔 ) ) ⟩ )
12 frgpup.u 𝑈 = ( varFGrp𝐼 )
13 frgpup.y ( 𝜑𝐴𝐼 )
14 8 12 vrgpval ( ( 𝐼𝑉𝐴𝐼 ) → ( 𝑈𝐴 ) = [ ⟨“ ⟨ 𝐴 , ∅ ⟩ ”⟩ ] )
15 5 13 14 syl2anc ( 𝜑 → ( 𝑈𝐴 ) = [ ⟨“ ⟨ 𝐴 , ∅ ⟩ ”⟩ ] )
16 15 fveq2d ( 𝜑 → ( 𝐸 ‘ ( 𝑈𝐴 ) ) = ( 𝐸 ‘ [ ⟨“ ⟨ 𝐴 , ∅ ⟩ ”⟩ ] ) )
17 0ex ∅ ∈ V
18 17 prid1 ∅ ∈ { ∅ , 1o }
19 df2o3 2o = { ∅ , 1o }
20 18 19 eleqtrri ∅ ∈ 2o
21 opelxpi ( ( 𝐴𝐼 ∧ ∅ ∈ 2o ) → ⟨ 𝐴 , ∅ ⟩ ∈ ( 𝐼 × 2o ) )
22 13 20 21 sylancl ( 𝜑 → ⟨ 𝐴 , ∅ ⟩ ∈ ( 𝐼 × 2o ) )
23 22 s1cld ( 𝜑 → ⟨“ ⟨ 𝐴 , ∅ ⟩ ”⟩ ∈ Word ( 𝐼 × 2o ) )
24 2on 2o ∈ On
25 xpexg ( ( 𝐼𝑉 ∧ 2o ∈ On ) → ( 𝐼 × 2o ) ∈ V )
26 5 24 25 sylancl ( 𝜑 → ( 𝐼 × 2o ) ∈ V )
27 wrdexg ( ( 𝐼 × 2o ) ∈ V → Word ( 𝐼 × 2o ) ∈ V )
28 fvi ( Word ( 𝐼 × 2o ) ∈ V → ( I ‘ Word ( 𝐼 × 2o ) ) = Word ( 𝐼 × 2o ) )
29 26 27 28 3syl ( 𝜑 → ( I ‘ Word ( 𝐼 × 2o ) ) = Word ( 𝐼 × 2o ) )
30 7 29 syl5eq ( 𝜑𝑊 = Word ( 𝐼 × 2o ) )
31 23 30 eleqtrrd ( 𝜑 → ⟨“ ⟨ 𝐴 , ∅ ⟩ ”⟩ ∈ 𝑊 )
32 1 2 3 4 5 6 7 8 9 10 11 frgpupval ( ( 𝜑 ∧ ⟨“ ⟨ 𝐴 , ∅ ⟩ ”⟩ ∈ 𝑊 ) → ( 𝐸 ‘ [ ⟨“ ⟨ 𝐴 , ∅ ⟩ ”⟩ ] ) = ( 𝐻 Σg ( 𝑇 ∘ ⟨“ ⟨ 𝐴 , ∅ ⟩ ”⟩ ) ) )
33 31 32 mpdan ( 𝜑 → ( 𝐸 ‘ [ ⟨“ ⟨ 𝐴 , ∅ ⟩ ”⟩ ] ) = ( 𝐻 Σg ( 𝑇 ∘ ⟨“ ⟨ 𝐴 , ∅ ⟩ ”⟩ ) ) )
34 1 2 3 4 5 6 frgpuptf ( 𝜑𝑇 : ( 𝐼 × 2o ) ⟶ 𝐵 )
35 s1co ( ( ⟨ 𝐴 , ∅ ⟩ ∈ ( 𝐼 × 2o ) ∧ 𝑇 : ( 𝐼 × 2o ) ⟶ 𝐵 ) → ( 𝑇 ∘ ⟨“ ⟨ 𝐴 , ∅ ⟩ ”⟩ ) = ⟨“ ( 𝑇 ‘ ⟨ 𝐴 , ∅ ⟩ ) ”⟩ )
36 22 34 35 syl2anc ( 𝜑 → ( 𝑇 ∘ ⟨“ ⟨ 𝐴 , ∅ ⟩ ”⟩ ) = ⟨“ ( 𝑇 ‘ ⟨ 𝐴 , ∅ ⟩ ) ”⟩ )
37 df-ov ( 𝐴 𝑇 ∅ ) = ( 𝑇 ‘ ⟨ 𝐴 , ∅ ⟩ )
38 iftrue ( 𝑧 = ∅ → if ( 𝑧 = ∅ , ( 𝐹𝑦 ) , ( 𝑁 ‘ ( 𝐹𝑦 ) ) ) = ( 𝐹𝑦 ) )
39 fveq2 ( 𝑦 = 𝐴 → ( 𝐹𝑦 ) = ( 𝐹𝐴 ) )
40 38 39 sylan9eqr ( ( 𝑦 = 𝐴𝑧 = ∅ ) → if ( 𝑧 = ∅ , ( 𝐹𝑦 ) , ( 𝑁 ‘ ( 𝐹𝑦 ) ) ) = ( 𝐹𝐴 ) )
41 fvex ( 𝐹𝐴 ) ∈ V
42 40 3 41 ovmpoa ( ( 𝐴𝐼 ∧ ∅ ∈ 2o ) → ( 𝐴 𝑇 ∅ ) = ( 𝐹𝐴 ) )
43 13 20 42 sylancl ( 𝜑 → ( 𝐴 𝑇 ∅ ) = ( 𝐹𝐴 ) )
44 37 43 eqtr3id ( 𝜑 → ( 𝑇 ‘ ⟨ 𝐴 , ∅ ⟩ ) = ( 𝐹𝐴 ) )
45 44 s1eqd ( 𝜑 → ⟨“ ( 𝑇 ‘ ⟨ 𝐴 , ∅ ⟩ ) ”⟩ = ⟨“ ( 𝐹𝐴 ) ”⟩ )
46 36 45 eqtrd ( 𝜑 → ( 𝑇 ∘ ⟨“ ⟨ 𝐴 , ∅ ⟩ ”⟩ ) = ⟨“ ( 𝐹𝐴 ) ”⟩ )
47 46 oveq2d ( 𝜑 → ( 𝐻 Σg ( 𝑇 ∘ ⟨“ ⟨ 𝐴 , ∅ ⟩ ”⟩ ) ) = ( 𝐻 Σg ⟨“ ( 𝐹𝐴 ) ”⟩ ) )
48 6 13 ffvelrnd ( 𝜑 → ( 𝐹𝐴 ) ∈ 𝐵 )
49 1 gsumws1 ( ( 𝐹𝐴 ) ∈ 𝐵 → ( 𝐻 Σg ⟨“ ( 𝐹𝐴 ) ”⟩ ) = ( 𝐹𝐴 ) )
50 48 49 syl ( 𝜑 → ( 𝐻 Σg ⟨“ ( 𝐹𝐴 ) ”⟩ ) = ( 𝐹𝐴 ) )
51 47 50 eqtrd ( 𝜑 → ( 𝐻 Σg ( 𝑇 ∘ ⟨“ ⟨ 𝐴 , ∅ ⟩ ”⟩ ) ) = ( 𝐹𝐴 ) )
52 16 33 51 3eqtrd ( 𝜑 → ( 𝐸 ‘ ( 𝑈𝐴 ) ) = ( 𝐹𝐴 ) )