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 |
|
eqid |
⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 ) |
9 |
8 5 3
|
grpinvf1o |
⊢ ( 𝜑 → 𝑁 : ( Base ‘ 𝑅 ) –1-1-onto→ ( Base ‘ 𝑅 ) ) |
10 |
|
f1of |
⊢ ( 𝑁 : ( Base ‘ 𝑅 ) –1-1-onto→ ( Base ‘ 𝑅 ) → 𝑁 : ( Base ‘ 𝑅 ) ⟶ ( Base ‘ 𝑅 ) ) |
11 |
9 10
|
syl |
⊢ ( 𝜑 → 𝑁 : ( Base ‘ 𝑅 ) ⟶ ( Base ‘ 𝑅 ) ) |
12 |
1 8 4 6 7
|
psrelbas |
⊢ ( 𝜑 → 𝑋 : 𝐷 ⟶ ( Base ‘ 𝑅 ) ) |
13 |
|
fco |
⊢ ( ( 𝑁 : ( Base ‘ 𝑅 ) ⟶ ( Base ‘ 𝑅 ) ∧ 𝑋 : 𝐷 ⟶ ( Base ‘ 𝑅 ) ) → ( 𝑁 ∘ 𝑋 ) : 𝐷 ⟶ ( Base ‘ 𝑅 ) ) |
14 |
11 12 13
|
syl2anc |
⊢ ( 𝜑 → ( 𝑁 ∘ 𝑋 ) : 𝐷 ⟶ ( Base ‘ 𝑅 ) ) |
15 |
|
fvex |
⊢ ( Base ‘ 𝑅 ) ∈ V |
16 |
|
ovex |
⊢ ( ℕ0 ↑m 𝐼 ) ∈ V |
17 |
4 16
|
rabex2 |
⊢ 𝐷 ∈ V |
18 |
15 17
|
elmap |
⊢ ( ( 𝑁 ∘ 𝑋 ) ∈ ( ( Base ‘ 𝑅 ) ↑m 𝐷 ) ↔ ( 𝑁 ∘ 𝑋 ) : 𝐷 ⟶ ( Base ‘ 𝑅 ) ) |
19 |
14 18
|
sylibr |
⊢ ( 𝜑 → ( 𝑁 ∘ 𝑋 ) ∈ ( ( Base ‘ 𝑅 ) ↑m 𝐷 ) ) |
20 |
1 8 4 6 2
|
psrbas |
⊢ ( 𝜑 → 𝐵 = ( ( Base ‘ 𝑅 ) ↑m 𝐷 ) ) |
21 |
19 20
|
eleqtrrd |
⊢ ( 𝜑 → ( 𝑁 ∘ 𝑋 ) ∈ 𝐵 ) |