Step |
Hyp |
Ref |
Expression |
1 |
|
df-rnghomo |
⊢ RngHomo = ( 𝑟 ∈ Rng , 𝑠 ∈ Rng ↦ ⦋ ( Base ‘ 𝑟 ) / 𝑣 ⦌ ⦋ ( Base ‘ 𝑠 ) / 𝑤 ⦌ { 𝑓 ∈ ( 𝑤 ↑m 𝑣 ) ∣ ∀ 𝑥 ∈ 𝑣 ∀ 𝑦 ∈ 𝑣 ( ( 𝑓 ‘ ( 𝑥 ( +g ‘ 𝑟 ) 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) ( +g ‘ 𝑠 ) ( 𝑓 ‘ 𝑦 ) ) ∧ ( 𝑓 ‘ ( 𝑥 ( .r ‘ 𝑟 ) 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) ( .r ‘ 𝑠 ) ( 𝑓 ‘ 𝑦 ) ) ) } ) |
2 |
|
ovex |
⊢ ( 𝑤 ↑m 𝑣 ) ∈ V |
3 |
2
|
rabex |
⊢ { 𝑓 ∈ ( 𝑤 ↑m 𝑣 ) ∣ ∀ 𝑥 ∈ 𝑣 ∀ 𝑦 ∈ 𝑣 ( ( 𝑓 ‘ ( 𝑥 ( +g ‘ 𝑟 ) 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) ( +g ‘ 𝑠 ) ( 𝑓 ‘ 𝑦 ) ) ∧ ( 𝑓 ‘ ( 𝑥 ( .r ‘ 𝑟 ) 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) ( .r ‘ 𝑠 ) ( 𝑓 ‘ 𝑦 ) ) ) } ∈ V |
4 |
3
|
csbex |
⊢ ⦋ ( Base ‘ 𝑠 ) / 𝑤 ⦌ { 𝑓 ∈ ( 𝑤 ↑m 𝑣 ) ∣ ∀ 𝑥 ∈ 𝑣 ∀ 𝑦 ∈ 𝑣 ( ( 𝑓 ‘ ( 𝑥 ( +g ‘ 𝑟 ) 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) ( +g ‘ 𝑠 ) ( 𝑓 ‘ 𝑦 ) ) ∧ ( 𝑓 ‘ ( 𝑥 ( .r ‘ 𝑟 ) 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) ( .r ‘ 𝑠 ) ( 𝑓 ‘ 𝑦 ) ) ) } ∈ V |
5 |
4
|
csbex |
⊢ ⦋ ( Base ‘ 𝑟 ) / 𝑣 ⦌ ⦋ ( Base ‘ 𝑠 ) / 𝑤 ⦌ { 𝑓 ∈ ( 𝑤 ↑m 𝑣 ) ∣ ∀ 𝑥 ∈ 𝑣 ∀ 𝑦 ∈ 𝑣 ( ( 𝑓 ‘ ( 𝑥 ( +g ‘ 𝑟 ) 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) ( +g ‘ 𝑠 ) ( 𝑓 ‘ 𝑦 ) ) ∧ ( 𝑓 ‘ ( 𝑥 ( .r ‘ 𝑟 ) 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) ( .r ‘ 𝑠 ) ( 𝑓 ‘ 𝑦 ) ) ) } ∈ V |
6 |
1 5
|
fnmpoi |
⊢ RngHomo Fn ( Rng × Rng ) |