Step |
Hyp |
Ref |
Expression |
1 |
|
dsmmelbas.p |
⊢ 𝑃 = ( 𝑆 Xs 𝑅 ) |
2 |
|
dsmmelbas.c |
⊢ 𝐶 = ( 𝑆 ⊕m 𝑅 ) |
3 |
|
dsmmelbas.b |
⊢ 𝐵 = ( Base ‘ 𝑃 ) |
4 |
|
dsmmelbas.h |
⊢ 𝐻 = ( Base ‘ 𝐶 ) |
5 |
|
dsmmelbas.i |
⊢ ( 𝜑 → 𝐼 ∈ 𝑉 ) |
6 |
|
dsmmelbas.r |
⊢ ( 𝜑 → 𝑅 Fn 𝐼 ) |
7 |
2
|
fveq2i |
⊢ ( Base ‘ 𝐶 ) = ( Base ‘ ( 𝑆 ⊕m 𝑅 ) ) |
8 |
4 7
|
eqtri |
⊢ 𝐻 = ( Base ‘ ( 𝑆 ⊕m 𝑅 ) ) |
9 |
|
fnex |
⊢ ( ( 𝑅 Fn 𝐼 ∧ 𝐼 ∈ 𝑉 ) → 𝑅 ∈ V ) |
10 |
6 5 9
|
syl2anc |
⊢ ( 𝜑 → 𝑅 ∈ V ) |
11 |
|
eqid |
⊢ { 𝑏 ∈ ( Base ‘ ( 𝑆 Xs 𝑅 ) ) ∣ { 𝑎 ∈ dom 𝑅 ∣ ( 𝑏 ‘ 𝑎 ) ≠ ( 0g ‘ ( 𝑅 ‘ 𝑎 ) ) } ∈ Fin } = { 𝑏 ∈ ( Base ‘ ( 𝑆 Xs 𝑅 ) ) ∣ { 𝑎 ∈ dom 𝑅 ∣ ( 𝑏 ‘ 𝑎 ) ≠ ( 0g ‘ ( 𝑅 ‘ 𝑎 ) ) } ∈ Fin } |
12 |
11
|
dsmmbase |
⊢ ( 𝑅 ∈ V → { 𝑏 ∈ ( Base ‘ ( 𝑆 Xs 𝑅 ) ) ∣ { 𝑎 ∈ dom 𝑅 ∣ ( 𝑏 ‘ 𝑎 ) ≠ ( 0g ‘ ( 𝑅 ‘ 𝑎 ) ) } ∈ Fin } = ( Base ‘ ( 𝑆 ⊕m 𝑅 ) ) ) |
13 |
10 12
|
syl |
⊢ ( 𝜑 → { 𝑏 ∈ ( Base ‘ ( 𝑆 Xs 𝑅 ) ) ∣ { 𝑎 ∈ dom 𝑅 ∣ ( 𝑏 ‘ 𝑎 ) ≠ ( 0g ‘ ( 𝑅 ‘ 𝑎 ) ) } ∈ Fin } = ( Base ‘ ( 𝑆 ⊕m 𝑅 ) ) ) |
14 |
8 13
|
eqtr4id |
⊢ ( 𝜑 → 𝐻 = { 𝑏 ∈ ( Base ‘ ( 𝑆 Xs 𝑅 ) ) ∣ { 𝑎 ∈ dom 𝑅 ∣ ( 𝑏 ‘ 𝑎 ) ≠ ( 0g ‘ ( 𝑅 ‘ 𝑎 ) ) } ∈ Fin } ) |
15 |
14
|
eleq2d |
⊢ ( 𝜑 → ( 𝑋 ∈ 𝐻 ↔ 𝑋 ∈ { 𝑏 ∈ ( Base ‘ ( 𝑆 Xs 𝑅 ) ) ∣ { 𝑎 ∈ dom 𝑅 ∣ ( 𝑏 ‘ 𝑎 ) ≠ ( 0g ‘ ( 𝑅 ‘ 𝑎 ) ) } ∈ Fin } ) ) |
16 |
|
fveq1 |
⊢ ( 𝑏 = 𝑋 → ( 𝑏 ‘ 𝑎 ) = ( 𝑋 ‘ 𝑎 ) ) |
17 |
16
|
neeq1d |
⊢ ( 𝑏 = 𝑋 → ( ( 𝑏 ‘ 𝑎 ) ≠ ( 0g ‘ ( 𝑅 ‘ 𝑎 ) ) ↔ ( 𝑋 ‘ 𝑎 ) ≠ ( 0g ‘ ( 𝑅 ‘ 𝑎 ) ) ) ) |
18 |
17
|
rabbidv |
⊢ ( 𝑏 = 𝑋 → { 𝑎 ∈ dom 𝑅 ∣ ( 𝑏 ‘ 𝑎 ) ≠ ( 0g ‘ ( 𝑅 ‘ 𝑎 ) ) } = { 𝑎 ∈ dom 𝑅 ∣ ( 𝑋 ‘ 𝑎 ) ≠ ( 0g ‘ ( 𝑅 ‘ 𝑎 ) ) } ) |
19 |
18
|
eleq1d |
⊢ ( 𝑏 = 𝑋 → ( { 𝑎 ∈ dom 𝑅 ∣ ( 𝑏 ‘ 𝑎 ) ≠ ( 0g ‘ ( 𝑅 ‘ 𝑎 ) ) } ∈ Fin ↔ { 𝑎 ∈ dom 𝑅 ∣ ( 𝑋 ‘ 𝑎 ) ≠ ( 0g ‘ ( 𝑅 ‘ 𝑎 ) ) } ∈ Fin ) ) |
20 |
19
|
elrab |
⊢ ( 𝑋 ∈ { 𝑏 ∈ ( Base ‘ ( 𝑆 Xs 𝑅 ) ) ∣ { 𝑎 ∈ dom 𝑅 ∣ ( 𝑏 ‘ 𝑎 ) ≠ ( 0g ‘ ( 𝑅 ‘ 𝑎 ) ) } ∈ Fin } ↔ ( 𝑋 ∈ ( Base ‘ ( 𝑆 Xs 𝑅 ) ) ∧ { 𝑎 ∈ dom 𝑅 ∣ ( 𝑋 ‘ 𝑎 ) ≠ ( 0g ‘ ( 𝑅 ‘ 𝑎 ) ) } ∈ Fin ) ) |
21 |
1
|
fveq2i |
⊢ ( Base ‘ 𝑃 ) = ( Base ‘ ( 𝑆 Xs 𝑅 ) ) |
22 |
3 21
|
eqtr2i |
⊢ ( Base ‘ ( 𝑆 Xs 𝑅 ) ) = 𝐵 |
23 |
22
|
eleq2i |
⊢ ( 𝑋 ∈ ( Base ‘ ( 𝑆 Xs 𝑅 ) ) ↔ 𝑋 ∈ 𝐵 ) |
24 |
23
|
a1i |
⊢ ( 𝜑 → ( 𝑋 ∈ ( Base ‘ ( 𝑆 Xs 𝑅 ) ) ↔ 𝑋 ∈ 𝐵 ) ) |
25 |
|
fndm |
⊢ ( 𝑅 Fn 𝐼 → dom 𝑅 = 𝐼 ) |
26 |
|
rabeq |
⊢ ( dom 𝑅 = 𝐼 → { 𝑎 ∈ dom 𝑅 ∣ ( 𝑋 ‘ 𝑎 ) ≠ ( 0g ‘ ( 𝑅 ‘ 𝑎 ) ) } = { 𝑎 ∈ 𝐼 ∣ ( 𝑋 ‘ 𝑎 ) ≠ ( 0g ‘ ( 𝑅 ‘ 𝑎 ) ) } ) |
27 |
6 25 26
|
3syl |
⊢ ( 𝜑 → { 𝑎 ∈ dom 𝑅 ∣ ( 𝑋 ‘ 𝑎 ) ≠ ( 0g ‘ ( 𝑅 ‘ 𝑎 ) ) } = { 𝑎 ∈ 𝐼 ∣ ( 𝑋 ‘ 𝑎 ) ≠ ( 0g ‘ ( 𝑅 ‘ 𝑎 ) ) } ) |
28 |
27
|
eleq1d |
⊢ ( 𝜑 → ( { 𝑎 ∈ dom 𝑅 ∣ ( 𝑋 ‘ 𝑎 ) ≠ ( 0g ‘ ( 𝑅 ‘ 𝑎 ) ) } ∈ Fin ↔ { 𝑎 ∈ 𝐼 ∣ ( 𝑋 ‘ 𝑎 ) ≠ ( 0g ‘ ( 𝑅 ‘ 𝑎 ) ) } ∈ Fin ) ) |
29 |
24 28
|
anbi12d |
⊢ ( 𝜑 → ( ( 𝑋 ∈ ( Base ‘ ( 𝑆 Xs 𝑅 ) ) ∧ { 𝑎 ∈ dom 𝑅 ∣ ( 𝑋 ‘ 𝑎 ) ≠ ( 0g ‘ ( 𝑅 ‘ 𝑎 ) ) } ∈ Fin ) ↔ ( 𝑋 ∈ 𝐵 ∧ { 𝑎 ∈ 𝐼 ∣ ( 𝑋 ‘ 𝑎 ) ≠ ( 0g ‘ ( 𝑅 ‘ 𝑎 ) ) } ∈ Fin ) ) ) |
30 |
20 29
|
syl5bb |
⊢ ( 𝜑 → ( 𝑋 ∈ { 𝑏 ∈ ( Base ‘ ( 𝑆 Xs 𝑅 ) ) ∣ { 𝑎 ∈ dom 𝑅 ∣ ( 𝑏 ‘ 𝑎 ) ≠ ( 0g ‘ ( 𝑅 ‘ 𝑎 ) ) } ∈ Fin } ↔ ( 𝑋 ∈ 𝐵 ∧ { 𝑎 ∈ 𝐼 ∣ ( 𝑋 ‘ 𝑎 ) ≠ ( 0g ‘ ( 𝑅 ‘ 𝑎 ) ) } ∈ Fin ) ) ) |
31 |
15 30
|
bitrd |
⊢ ( 𝜑 → ( 𝑋 ∈ 𝐻 ↔ ( 𝑋 ∈ 𝐵 ∧ { 𝑎 ∈ 𝐼 ∣ ( 𝑋 ‘ 𝑎 ) ≠ ( 0g ‘ ( 𝑅 ‘ 𝑎 ) ) } ∈ Fin ) ) ) |