Step |
Hyp |
Ref |
Expression |
1 |
|
dsmmbas2.p |
⊢ 𝑃 = ( 𝑆 Xs 𝑅 ) |
2 |
|
dsmmbas2.b |
⊢ 𝐵 = { 𝑓 ∈ ( Base ‘ 𝑃 ) ∣ dom ( 𝑓 ∖ ( 0g ∘ 𝑅 ) ) ∈ Fin } |
3 |
1
|
fveq2i |
⊢ ( Base ‘ 𝑃 ) = ( Base ‘ ( 𝑆 Xs 𝑅 ) ) |
4 |
3
|
rabeqi |
⊢ { 𝑓 ∈ ( Base ‘ 𝑃 ) ∣ dom ( 𝑓 ∖ ( 0g ∘ 𝑅 ) ) ∈ Fin } = { 𝑓 ∈ ( Base ‘ ( 𝑆 Xs 𝑅 ) ) ∣ dom ( 𝑓 ∖ ( 0g ∘ 𝑅 ) ) ∈ Fin } |
5 |
|
simpll |
⊢ ( ( ( 𝑅 Fn 𝐼 ∧ 𝐼 ∈ 𝑉 ) ∧ 𝑓 ∈ ( Base ‘ ( 𝑆 Xs 𝑅 ) ) ) → 𝑅 Fn 𝐼 ) |
6 |
|
fvco2 |
⊢ ( ( 𝑅 Fn 𝐼 ∧ 𝑥 ∈ 𝐼 ) → ( ( 0g ∘ 𝑅 ) ‘ 𝑥 ) = ( 0g ‘ ( 𝑅 ‘ 𝑥 ) ) ) |
7 |
5 6
|
sylan |
⊢ ( ( ( ( 𝑅 Fn 𝐼 ∧ 𝐼 ∈ 𝑉 ) ∧ 𝑓 ∈ ( Base ‘ ( 𝑆 Xs 𝑅 ) ) ) ∧ 𝑥 ∈ 𝐼 ) → ( ( 0g ∘ 𝑅 ) ‘ 𝑥 ) = ( 0g ‘ ( 𝑅 ‘ 𝑥 ) ) ) |
8 |
7
|
neeq2d |
⊢ ( ( ( ( 𝑅 Fn 𝐼 ∧ 𝐼 ∈ 𝑉 ) ∧ 𝑓 ∈ ( Base ‘ ( 𝑆 Xs 𝑅 ) ) ) ∧ 𝑥 ∈ 𝐼 ) → ( ( 𝑓 ‘ 𝑥 ) ≠ ( ( 0g ∘ 𝑅 ) ‘ 𝑥 ) ↔ ( 𝑓 ‘ 𝑥 ) ≠ ( 0g ‘ ( 𝑅 ‘ 𝑥 ) ) ) ) |
9 |
8
|
rabbidva |
⊢ ( ( ( 𝑅 Fn 𝐼 ∧ 𝐼 ∈ 𝑉 ) ∧ 𝑓 ∈ ( Base ‘ ( 𝑆 Xs 𝑅 ) ) ) → { 𝑥 ∈ 𝐼 ∣ ( 𝑓 ‘ 𝑥 ) ≠ ( ( 0g ∘ 𝑅 ) ‘ 𝑥 ) } = { 𝑥 ∈ 𝐼 ∣ ( 𝑓 ‘ 𝑥 ) ≠ ( 0g ‘ ( 𝑅 ‘ 𝑥 ) ) } ) |
10 |
|
eqid |
⊢ ( 𝑆 Xs 𝑅 ) = ( 𝑆 Xs 𝑅 ) |
11 |
|
eqid |
⊢ ( Base ‘ ( 𝑆 Xs 𝑅 ) ) = ( Base ‘ ( 𝑆 Xs 𝑅 ) ) |
12 |
|
reldmprds |
⊢ Rel dom Xs |
13 |
10 11 12
|
strov2rcl |
⊢ ( 𝑓 ∈ ( Base ‘ ( 𝑆 Xs 𝑅 ) ) → 𝑆 ∈ V ) |
14 |
13
|
adantl |
⊢ ( ( ( 𝑅 Fn 𝐼 ∧ 𝐼 ∈ 𝑉 ) ∧ 𝑓 ∈ ( Base ‘ ( 𝑆 Xs 𝑅 ) ) ) → 𝑆 ∈ V ) |
15 |
|
simplr |
⊢ ( ( ( 𝑅 Fn 𝐼 ∧ 𝐼 ∈ 𝑉 ) ∧ 𝑓 ∈ ( Base ‘ ( 𝑆 Xs 𝑅 ) ) ) → 𝐼 ∈ 𝑉 ) |
16 |
|
simpr |
⊢ ( ( ( 𝑅 Fn 𝐼 ∧ 𝐼 ∈ 𝑉 ) ∧ 𝑓 ∈ ( Base ‘ ( 𝑆 Xs 𝑅 ) ) ) → 𝑓 ∈ ( Base ‘ ( 𝑆 Xs 𝑅 ) ) ) |
17 |
10 11 14 15 5 16
|
prdsbasfn |
⊢ ( ( ( 𝑅 Fn 𝐼 ∧ 𝐼 ∈ 𝑉 ) ∧ 𝑓 ∈ ( Base ‘ ( 𝑆 Xs 𝑅 ) ) ) → 𝑓 Fn 𝐼 ) |
18 |
|
fn0g |
⊢ 0g Fn V |
19 |
|
dffn2 |
⊢ ( 0g Fn V ↔ 0g : V ⟶ V ) |
20 |
18 19
|
mpbi |
⊢ 0g : V ⟶ V |
21 |
|
dffn2 |
⊢ ( 𝑅 Fn 𝐼 ↔ 𝑅 : 𝐼 ⟶ V ) |
22 |
21
|
biimpi |
⊢ ( 𝑅 Fn 𝐼 → 𝑅 : 𝐼 ⟶ V ) |
23 |
|
fco |
⊢ ( ( 0g : V ⟶ V ∧ 𝑅 : 𝐼 ⟶ V ) → ( 0g ∘ 𝑅 ) : 𝐼 ⟶ V ) |
24 |
20 22 23
|
sylancr |
⊢ ( 𝑅 Fn 𝐼 → ( 0g ∘ 𝑅 ) : 𝐼 ⟶ V ) |
25 |
24
|
ffnd |
⊢ ( 𝑅 Fn 𝐼 → ( 0g ∘ 𝑅 ) Fn 𝐼 ) |
26 |
5 25
|
syl |
⊢ ( ( ( 𝑅 Fn 𝐼 ∧ 𝐼 ∈ 𝑉 ) ∧ 𝑓 ∈ ( Base ‘ ( 𝑆 Xs 𝑅 ) ) ) → ( 0g ∘ 𝑅 ) Fn 𝐼 ) |
27 |
|
fndmdif |
⊢ ( ( 𝑓 Fn 𝐼 ∧ ( 0g ∘ 𝑅 ) Fn 𝐼 ) → dom ( 𝑓 ∖ ( 0g ∘ 𝑅 ) ) = { 𝑥 ∈ 𝐼 ∣ ( 𝑓 ‘ 𝑥 ) ≠ ( ( 0g ∘ 𝑅 ) ‘ 𝑥 ) } ) |
28 |
17 26 27
|
syl2anc |
⊢ ( ( ( 𝑅 Fn 𝐼 ∧ 𝐼 ∈ 𝑉 ) ∧ 𝑓 ∈ ( Base ‘ ( 𝑆 Xs 𝑅 ) ) ) → dom ( 𝑓 ∖ ( 0g ∘ 𝑅 ) ) = { 𝑥 ∈ 𝐼 ∣ ( 𝑓 ‘ 𝑥 ) ≠ ( ( 0g ∘ 𝑅 ) ‘ 𝑥 ) } ) |
29 |
|
fndm |
⊢ ( 𝑅 Fn 𝐼 → dom 𝑅 = 𝐼 ) |
30 |
29
|
rabeqdv |
⊢ ( 𝑅 Fn 𝐼 → { 𝑥 ∈ dom 𝑅 ∣ ( 𝑓 ‘ 𝑥 ) ≠ ( 0g ‘ ( 𝑅 ‘ 𝑥 ) ) } = { 𝑥 ∈ 𝐼 ∣ ( 𝑓 ‘ 𝑥 ) ≠ ( 0g ‘ ( 𝑅 ‘ 𝑥 ) ) } ) |
31 |
5 30
|
syl |
⊢ ( ( ( 𝑅 Fn 𝐼 ∧ 𝐼 ∈ 𝑉 ) ∧ 𝑓 ∈ ( Base ‘ ( 𝑆 Xs 𝑅 ) ) ) → { 𝑥 ∈ dom 𝑅 ∣ ( 𝑓 ‘ 𝑥 ) ≠ ( 0g ‘ ( 𝑅 ‘ 𝑥 ) ) } = { 𝑥 ∈ 𝐼 ∣ ( 𝑓 ‘ 𝑥 ) ≠ ( 0g ‘ ( 𝑅 ‘ 𝑥 ) ) } ) |
32 |
9 28 31
|
3eqtr4d |
⊢ ( ( ( 𝑅 Fn 𝐼 ∧ 𝐼 ∈ 𝑉 ) ∧ 𝑓 ∈ ( Base ‘ ( 𝑆 Xs 𝑅 ) ) ) → dom ( 𝑓 ∖ ( 0g ∘ 𝑅 ) ) = { 𝑥 ∈ dom 𝑅 ∣ ( 𝑓 ‘ 𝑥 ) ≠ ( 0g ‘ ( 𝑅 ‘ 𝑥 ) ) } ) |
33 |
32
|
eleq1d |
⊢ ( ( ( 𝑅 Fn 𝐼 ∧ 𝐼 ∈ 𝑉 ) ∧ 𝑓 ∈ ( Base ‘ ( 𝑆 Xs 𝑅 ) ) ) → ( dom ( 𝑓 ∖ ( 0g ∘ 𝑅 ) ) ∈ Fin ↔ { 𝑥 ∈ dom 𝑅 ∣ ( 𝑓 ‘ 𝑥 ) ≠ ( 0g ‘ ( 𝑅 ‘ 𝑥 ) ) } ∈ Fin ) ) |
34 |
33
|
rabbidva |
⊢ ( ( 𝑅 Fn 𝐼 ∧ 𝐼 ∈ 𝑉 ) → { 𝑓 ∈ ( Base ‘ ( 𝑆 Xs 𝑅 ) ) ∣ dom ( 𝑓 ∖ ( 0g ∘ 𝑅 ) ) ∈ Fin } = { 𝑓 ∈ ( Base ‘ ( 𝑆 Xs 𝑅 ) ) ∣ { 𝑥 ∈ dom 𝑅 ∣ ( 𝑓 ‘ 𝑥 ) ≠ ( 0g ‘ ( 𝑅 ‘ 𝑥 ) ) } ∈ Fin } ) |
35 |
4 34
|
syl5eq |
⊢ ( ( 𝑅 Fn 𝐼 ∧ 𝐼 ∈ 𝑉 ) → { 𝑓 ∈ ( Base ‘ 𝑃 ) ∣ dom ( 𝑓 ∖ ( 0g ∘ 𝑅 ) ) ∈ Fin } = { 𝑓 ∈ ( Base ‘ ( 𝑆 Xs 𝑅 ) ) ∣ { 𝑥 ∈ dom 𝑅 ∣ ( 𝑓 ‘ 𝑥 ) ≠ ( 0g ‘ ( 𝑅 ‘ 𝑥 ) ) } ∈ Fin } ) |
36 |
|
fnex |
⊢ ( ( 𝑅 Fn 𝐼 ∧ 𝐼 ∈ 𝑉 ) → 𝑅 ∈ V ) |
37 |
|
eqid |
⊢ { 𝑓 ∈ ( Base ‘ ( 𝑆 Xs 𝑅 ) ) ∣ { 𝑥 ∈ dom 𝑅 ∣ ( 𝑓 ‘ 𝑥 ) ≠ ( 0g ‘ ( 𝑅 ‘ 𝑥 ) ) } ∈ Fin } = { 𝑓 ∈ ( Base ‘ ( 𝑆 Xs 𝑅 ) ) ∣ { 𝑥 ∈ dom 𝑅 ∣ ( 𝑓 ‘ 𝑥 ) ≠ ( 0g ‘ ( 𝑅 ‘ 𝑥 ) ) } ∈ Fin } |
38 |
37
|
dsmmbase |
⊢ ( 𝑅 ∈ V → { 𝑓 ∈ ( Base ‘ ( 𝑆 Xs 𝑅 ) ) ∣ { 𝑥 ∈ dom 𝑅 ∣ ( 𝑓 ‘ 𝑥 ) ≠ ( 0g ‘ ( 𝑅 ‘ 𝑥 ) ) } ∈ Fin } = ( Base ‘ ( 𝑆 ⊕m 𝑅 ) ) ) |
39 |
36 38
|
syl |
⊢ ( ( 𝑅 Fn 𝐼 ∧ 𝐼 ∈ 𝑉 ) → { 𝑓 ∈ ( Base ‘ ( 𝑆 Xs 𝑅 ) ) ∣ { 𝑥 ∈ dom 𝑅 ∣ ( 𝑓 ‘ 𝑥 ) ≠ ( 0g ‘ ( 𝑅 ‘ 𝑥 ) ) } ∈ Fin } = ( Base ‘ ( 𝑆 ⊕m 𝑅 ) ) ) |
40 |
35 39
|
eqtrd |
⊢ ( ( 𝑅 Fn 𝐼 ∧ 𝐼 ∈ 𝑉 ) → { 𝑓 ∈ ( Base ‘ 𝑃 ) ∣ dom ( 𝑓 ∖ ( 0g ∘ 𝑅 ) ) ∈ Fin } = ( Base ‘ ( 𝑆 ⊕m 𝑅 ) ) ) |
41 |
2 40
|
syl5eq |
⊢ ( ( 𝑅 Fn 𝐼 ∧ 𝐼 ∈ 𝑉 ) → 𝐵 = ( Base ‘ ( 𝑆 ⊕m 𝑅 ) ) ) |