Step |
Hyp |
Ref |
Expression |
1 |
|
rngcrescrhm.u |
⊢ ( 𝜑 → 𝑈 ∈ 𝑉 ) |
2 |
|
rngcrescrhm.c |
⊢ 𝐶 = ( RngCat ‘ 𝑈 ) |
3 |
|
rngcrescrhm.r |
⊢ ( 𝜑 → 𝑅 = ( Ring ∩ 𝑈 ) ) |
4 |
|
rngcrescrhm.h |
⊢ 𝐻 = ( RingHom ↾ ( 𝑅 × 𝑅 ) ) |
5 |
3
|
eleq2d |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝑅 ↔ 𝑥 ∈ ( Ring ∩ 𝑈 ) ) ) |
6 |
|
elinel1 |
⊢ ( 𝑥 ∈ ( Ring ∩ 𝑈 ) → 𝑥 ∈ Ring ) |
7 |
5 6
|
syl6bi |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝑅 → 𝑥 ∈ Ring ) ) |
8 |
7
|
imp |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑅 ) → 𝑥 ∈ Ring ) |
9 |
|
eqid |
⊢ ( Base ‘ 𝑥 ) = ( Base ‘ 𝑥 ) |
10 |
9
|
idrhm |
⊢ ( 𝑥 ∈ Ring → ( I ↾ ( Base ‘ 𝑥 ) ) ∈ ( 𝑥 RingHom 𝑥 ) ) |
11 |
8 10
|
syl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑅 ) → ( I ↾ ( Base ‘ 𝑥 ) ) ∈ ( 𝑥 RingHom 𝑥 ) ) |
12 |
|
eqid |
⊢ ( Base ‘ 𝐶 ) = ( Base ‘ 𝐶 ) |
13 |
2
|
eqcomi |
⊢ ( RngCat ‘ 𝑈 ) = 𝐶 |
14 |
13
|
fveq2i |
⊢ ( Id ‘ ( RngCat ‘ 𝑈 ) ) = ( Id ‘ 𝐶 ) |
15 |
1
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑅 ) → 𝑈 ∈ 𝑉 ) |
16 |
|
incom |
⊢ ( Ring ∩ 𝑈 ) = ( 𝑈 ∩ Ring ) |
17 |
|
ringssrng |
⊢ Ring ⊆ Rng |
18 |
|
sslin |
⊢ ( Ring ⊆ Rng → ( 𝑈 ∩ Ring ) ⊆ ( 𝑈 ∩ Rng ) ) |
19 |
17 18
|
mp1i |
⊢ ( 𝜑 → ( 𝑈 ∩ Ring ) ⊆ ( 𝑈 ∩ Rng ) ) |
20 |
16 19
|
eqsstrid |
⊢ ( 𝜑 → ( Ring ∩ 𝑈 ) ⊆ ( 𝑈 ∩ Rng ) ) |
21 |
2 12 1
|
rngcbas |
⊢ ( 𝜑 → ( Base ‘ 𝐶 ) = ( 𝑈 ∩ Rng ) ) |
22 |
20 3 21
|
3sstr4d |
⊢ ( 𝜑 → 𝑅 ⊆ ( Base ‘ 𝐶 ) ) |
23 |
22
|
sselda |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑅 ) → 𝑥 ∈ ( Base ‘ 𝐶 ) ) |
24 |
2 12 14 15 23 9
|
rngcid |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑅 ) → ( ( Id ‘ ( RngCat ‘ 𝑈 ) ) ‘ 𝑥 ) = ( I ↾ ( Base ‘ 𝑥 ) ) ) |
25 |
1 2 3 4
|
rhmsubclem2 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑅 ∧ 𝑥 ∈ 𝑅 ) → ( 𝑥 𝐻 𝑥 ) = ( 𝑥 RingHom 𝑥 ) ) |
26 |
25
|
3anidm23 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑅 ) → ( 𝑥 𝐻 𝑥 ) = ( 𝑥 RingHom 𝑥 ) ) |
27 |
11 24 26
|
3eltr4d |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑅 ) → ( ( Id ‘ ( RngCat ‘ 𝑈 ) ) ‘ 𝑥 ) ∈ ( 𝑥 𝐻 𝑥 ) ) |