Step |
Hyp |
Ref |
Expression |
1 |
|
psrgrp.s |
⊢ 𝑆 = ( 𝐼 mPwSer 𝑅 ) |
2 |
|
psrgrp.i |
⊢ ( 𝜑 → 𝐼 ∈ 𝑉 ) |
3 |
|
psrgrp.r |
⊢ ( 𝜑 → 𝑅 ∈ Grp ) |
4 |
|
psrnegcl.d |
⊢ 𝐷 = { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } |
5 |
|
psrnegcl.i |
⊢ 𝑁 = ( invg ‘ 𝑅 ) |
6 |
|
psrnegcl.b |
⊢ 𝐵 = ( Base ‘ 𝑆 ) |
7 |
|
psrnegcl.z |
⊢ ( 𝜑 → 𝑋 ∈ 𝐵 ) |
8 |
|
psrlinv.o |
⊢ 0 = ( 0g ‘ 𝑅 ) |
9 |
|
psrlinv.p |
⊢ + = ( +g ‘ 𝑆 ) |
10 |
|
ovex |
⊢ ( ℕ0 ↑m 𝐼 ) ∈ V |
11 |
4 10
|
rabex2 |
⊢ 𝐷 ∈ V |
12 |
11
|
a1i |
⊢ ( 𝜑 → 𝐷 ∈ V ) |
13 |
|
fvexd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐷 ) → ( 𝑁 ‘ ( 𝑋 ‘ 𝑥 ) ) ∈ V ) |
14 |
|
eqid |
⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 ) |
15 |
1 14 4 6 7
|
psrelbas |
⊢ ( 𝜑 → 𝑋 : 𝐷 ⟶ ( Base ‘ 𝑅 ) ) |
16 |
15
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐷 ) → ( 𝑋 ‘ 𝑥 ) ∈ ( Base ‘ 𝑅 ) ) |
17 |
15
|
feqmptd |
⊢ ( 𝜑 → 𝑋 = ( 𝑥 ∈ 𝐷 ↦ ( 𝑋 ‘ 𝑥 ) ) ) |
18 |
14 5 3
|
grpinvf1o |
⊢ ( 𝜑 → 𝑁 : ( Base ‘ 𝑅 ) –1-1-onto→ ( Base ‘ 𝑅 ) ) |
19 |
|
f1of |
⊢ ( 𝑁 : ( Base ‘ 𝑅 ) –1-1-onto→ ( Base ‘ 𝑅 ) → 𝑁 : ( Base ‘ 𝑅 ) ⟶ ( Base ‘ 𝑅 ) ) |
20 |
18 19
|
syl |
⊢ ( 𝜑 → 𝑁 : ( Base ‘ 𝑅 ) ⟶ ( Base ‘ 𝑅 ) ) |
21 |
20
|
feqmptd |
⊢ ( 𝜑 → 𝑁 = ( 𝑦 ∈ ( Base ‘ 𝑅 ) ↦ ( 𝑁 ‘ 𝑦 ) ) ) |
22 |
|
fveq2 |
⊢ ( 𝑦 = ( 𝑋 ‘ 𝑥 ) → ( 𝑁 ‘ 𝑦 ) = ( 𝑁 ‘ ( 𝑋 ‘ 𝑥 ) ) ) |
23 |
16 17 21 22
|
fmptco |
⊢ ( 𝜑 → ( 𝑁 ∘ 𝑋 ) = ( 𝑥 ∈ 𝐷 ↦ ( 𝑁 ‘ ( 𝑋 ‘ 𝑥 ) ) ) ) |
24 |
12 13 16 23 17
|
offval2 |
⊢ ( 𝜑 → ( ( 𝑁 ∘ 𝑋 ) ∘f ( +g ‘ 𝑅 ) 𝑋 ) = ( 𝑥 ∈ 𝐷 ↦ ( ( 𝑁 ‘ ( 𝑋 ‘ 𝑥 ) ) ( +g ‘ 𝑅 ) ( 𝑋 ‘ 𝑥 ) ) ) ) |
25 |
|
eqid |
⊢ ( +g ‘ 𝑅 ) = ( +g ‘ 𝑅 ) |
26 |
1 2 3 4 5 6 7
|
psrnegcl |
⊢ ( 𝜑 → ( 𝑁 ∘ 𝑋 ) ∈ 𝐵 ) |
27 |
1 6 25 9 26 7
|
psradd |
⊢ ( 𝜑 → ( ( 𝑁 ∘ 𝑋 ) + 𝑋 ) = ( ( 𝑁 ∘ 𝑋 ) ∘f ( +g ‘ 𝑅 ) 𝑋 ) ) |
28 |
|
fconstmpt |
⊢ ( 𝐷 × { 0 } ) = ( 𝑥 ∈ 𝐷 ↦ 0 ) |
29 |
14 25 8 5
|
grplinv |
⊢ ( ( 𝑅 ∈ Grp ∧ ( 𝑋 ‘ 𝑥 ) ∈ ( Base ‘ 𝑅 ) ) → ( ( 𝑁 ‘ ( 𝑋 ‘ 𝑥 ) ) ( +g ‘ 𝑅 ) ( 𝑋 ‘ 𝑥 ) ) = 0 ) |
30 |
3 16 29
|
syl2an2r |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐷 ) → ( ( 𝑁 ‘ ( 𝑋 ‘ 𝑥 ) ) ( +g ‘ 𝑅 ) ( 𝑋 ‘ 𝑥 ) ) = 0 ) |
31 |
30
|
mpteq2dva |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐷 ↦ ( ( 𝑁 ‘ ( 𝑋 ‘ 𝑥 ) ) ( +g ‘ 𝑅 ) ( 𝑋 ‘ 𝑥 ) ) ) = ( 𝑥 ∈ 𝐷 ↦ 0 ) ) |
32 |
28 31
|
eqtr4id |
⊢ ( 𝜑 → ( 𝐷 × { 0 } ) = ( 𝑥 ∈ 𝐷 ↦ ( ( 𝑁 ‘ ( 𝑋 ‘ 𝑥 ) ) ( +g ‘ 𝑅 ) ( 𝑋 ‘ 𝑥 ) ) ) ) |
33 |
24 27 32
|
3eqtr4d |
⊢ ( 𝜑 → ( ( 𝑁 ∘ 𝑋 ) + 𝑋 ) = ( 𝐷 × { 0 } ) ) |