Step |
Hyp |
Ref |
Expression |
1 |
|
mgcoval.1 |
⊢ 𝐴 = ( Base ‘ 𝑉 ) |
2 |
|
mgcoval.2 |
⊢ 𝐵 = ( Base ‘ 𝑊 ) |
3 |
|
mgcoval.3 |
⊢ ≤ = ( le ‘ 𝑉 ) |
4 |
|
mgcoval.4 |
⊢ ≲ = ( le ‘ 𝑊 ) |
5 |
|
df-mgc |
⊢ MGalConn = ( 𝑣 ∈ V , 𝑤 ∈ V ↦ ⦋ ( Base ‘ 𝑣 ) / 𝑎 ⦌ ⦋ ( Base ‘ 𝑤 ) / 𝑏 ⦌ { 〈 𝑓 , 𝑔 〉 ∣ ( ( 𝑓 ∈ ( 𝑏 ↑m 𝑎 ) ∧ 𝑔 ∈ ( 𝑎 ↑m 𝑏 ) ) ∧ ∀ 𝑥 ∈ 𝑎 ∀ 𝑦 ∈ 𝑏 ( ( 𝑓 ‘ 𝑥 ) ( le ‘ 𝑤 ) 𝑦 ↔ 𝑥 ( le ‘ 𝑣 ) ( 𝑔 ‘ 𝑦 ) ) ) } ) |
6 |
5
|
a1i |
⊢ ( ( 𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌 ) → MGalConn = ( 𝑣 ∈ V , 𝑤 ∈ V ↦ ⦋ ( Base ‘ 𝑣 ) / 𝑎 ⦌ ⦋ ( Base ‘ 𝑤 ) / 𝑏 ⦌ { 〈 𝑓 , 𝑔 〉 ∣ ( ( 𝑓 ∈ ( 𝑏 ↑m 𝑎 ) ∧ 𝑔 ∈ ( 𝑎 ↑m 𝑏 ) ) ∧ ∀ 𝑥 ∈ 𝑎 ∀ 𝑦 ∈ 𝑏 ( ( 𝑓 ‘ 𝑥 ) ( le ‘ 𝑤 ) 𝑦 ↔ 𝑥 ( le ‘ 𝑣 ) ( 𝑔 ‘ 𝑦 ) ) ) } ) ) |
7 |
|
fvexd |
⊢ ( ( ( 𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌 ) ∧ ( 𝑣 = 𝑉 ∧ 𝑤 = 𝑊 ) ) → ( Base ‘ 𝑣 ) ∈ V ) |
8 |
|
simprl |
⊢ ( ( ( 𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌 ) ∧ ( 𝑣 = 𝑉 ∧ 𝑤 = 𝑊 ) ) → 𝑣 = 𝑉 ) |
9 |
8
|
fveq2d |
⊢ ( ( ( 𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌 ) ∧ ( 𝑣 = 𝑉 ∧ 𝑤 = 𝑊 ) ) → ( Base ‘ 𝑣 ) = ( Base ‘ 𝑉 ) ) |
10 |
9 1
|
eqtr4di |
⊢ ( ( ( 𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌 ) ∧ ( 𝑣 = 𝑉 ∧ 𝑤 = 𝑊 ) ) → ( Base ‘ 𝑣 ) = 𝐴 ) |
11 |
|
fvexd |
⊢ ( ( ( ( 𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌 ) ∧ ( 𝑣 = 𝑉 ∧ 𝑤 = 𝑊 ) ) ∧ 𝑎 = 𝐴 ) → ( Base ‘ 𝑤 ) ∈ V ) |
12 |
|
simplrr |
⊢ ( ( ( ( 𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌 ) ∧ ( 𝑣 = 𝑉 ∧ 𝑤 = 𝑊 ) ) ∧ 𝑎 = 𝐴 ) → 𝑤 = 𝑊 ) |
13 |
12
|
fveq2d |
⊢ ( ( ( ( 𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌 ) ∧ ( 𝑣 = 𝑉 ∧ 𝑤 = 𝑊 ) ) ∧ 𝑎 = 𝐴 ) → ( Base ‘ 𝑤 ) = ( Base ‘ 𝑊 ) ) |
14 |
13 2
|
eqtr4di |
⊢ ( ( ( ( 𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌 ) ∧ ( 𝑣 = 𝑉 ∧ 𝑤 = 𝑊 ) ) ∧ 𝑎 = 𝐴 ) → ( Base ‘ 𝑤 ) = 𝐵 ) |
15 |
|
simpr |
⊢ ( ( ( ( ( 𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌 ) ∧ ( 𝑣 = 𝑉 ∧ 𝑤 = 𝑊 ) ) ∧ 𝑎 = 𝐴 ) ∧ 𝑏 = 𝐵 ) → 𝑏 = 𝐵 ) |
16 |
|
simplr |
⊢ ( ( ( ( ( 𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌 ) ∧ ( 𝑣 = 𝑉 ∧ 𝑤 = 𝑊 ) ) ∧ 𝑎 = 𝐴 ) ∧ 𝑏 = 𝐵 ) → 𝑎 = 𝐴 ) |
17 |
15 16
|
oveq12d |
⊢ ( ( ( ( ( 𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌 ) ∧ ( 𝑣 = 𝑉 ∧ 𝑤 = 𝑊 ) ) ∧ 𝑎 = 𝐴 ) ∧ 𝑏 = 𝐵 ) → ( 𝑏 ↑m 𝑎 ) = ( 𝐵 ↑m 𝐴 ) ) |
18 |
17
|
eleq2d |
⊢ ( ( ( ( ( 𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌 ) ∧ ( 𝑣 = 𝑉 ∧ 𝑤 = 𝑊 ) ) ∧ 𝑎 = 𝐴 ) ∧ 𝑏 = 𝐵 ) → ( 𝑓 ∈ ( 𝑏 ↑m 𝑎 ) ↔ 𝑓 ∈ ( 𝐵 ↑m 𝐴 ) ) ) |
19 |
16 15
|
oveq12d |
⊢ ( ( ( ( ( 𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌 ) ∧ ( 𝑣 = 𝑉 ∧ 𝑤 = 𝑊 ) ) ∧ 𝑎 = 𝐴 ) ∧ 𝑏 = 𝐵 ) → ( 𝑎 ↑m 𝑏 ) = ( 𝐴 ↑m 𝐵 ) ) |
20 |
19
|
eleq2d |
⊢ ( ( ( ( ( 𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌 ) ∧ ( 𝑣 = 𝑉 ∧ 𝑤 = 𝑊 ) ) ∧ 𝑎 = 𝐴 ) ∧ 𝑏 = 𝐵 ) → ( 𝑔 ∈ ( 𝑎 ↑m 𝑏 ) ↔ 𝑔 ∈ ( 𝐴 ↑m 𝐵 ) ) ) |
21 |
18 20
|
anbi12d |
⊢ ( ( ( ( ( 𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌 ) ∧ ( 𝑣 = 𝑉 ∧ 𝑤 = 𝑊 ) ) ∧ 𝑎 = 𝐴 ) ∧ 𝑏 = 𝐵 ) → ( ( 𝑓 ∈ ( 𝑏 ↑m 𝑎 ) ∧ 𝑔 ∈ ( 𝑎 ↑m 𝑏 ) ) ↔ ( 𝑓 ∈ ( 𝐵 ↑m 𝐴 ) ∧ 𝑔 ∈ ( 𝐴 ↑m 𝐵 ) ) ) ) |
22 |
12
|
adantr |
⊢ ( ( ( ( ( 𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌 ) ∧ ( 𝑣 = 𝑉 ∧ 𝑤 = 𝑊 ) ) ∧ 𝑎 = 𝐴 ) ∧ 𝑏 = 𝐵 ) → 𝑤 = 𝑊 ) |
23 |
22
|
fveq2d |
⊢ ( ( ( ( ( 𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌 ) ∧ ( 𝑣 = 𝑉 ∧ 𝑤 = 𝑊 ) ) ∧ 𝑎 = 𝐴 ) ∧ 𝑏 = 𝐵 ) → ( le ‘ 𝑤 ) = ( le ‘ 𝑊 ) ) |
24 |
23 4
|
eqtr4di |
⊢ ( ( ( ( ( 𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌 ) ∧ ( 𝑣 = 𝑉 ∧ 𝑤 = 𝑊 ) ) ∧ 𝑎 = 𝐴 ) ∧ 𝑏 = 𝐵 ) → ( le ‘ 𝑤 ) = ≲ ) |
25 |
24
|
breqd |
⊢ ( ( ( ( ( 𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌 ) ∧ ( 𝑣 = 𝑉 ∧ 𝑤 = 𝑊 ) ) ∧ 𝑎 = 𝐴 ) ∧ 𝑏 = 𝐵 ) → ( ( 𝑓 ‘ 𝑥 ) ( le ‘ 𝑤 ) 𝑦 ↔ ( 𝑓 ‘ 𝑥 ) ≲ 𝑦 ) ) |
26 |
8
|
ad2antrr |
⊢ ( ( ( ( ( 𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌 ) ∧ ( 𝑣 = 𝑉 ∧ 𝑤 = 𝑊 ) ) ∧ 𝑎 = 𝐴 ) ∧ 𝑏 = 𝐵 ) → 𝑣 = 𝑉 ) |
27 |
26
|
fveq2d |
⊢ ( ( ( ( ( 𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌 ) ∧ ( 𝑣 = 𝑉 ∧ 𝑤 = 𝑊 ) ) ∧ 𝑎 = 𝐴 ) ∧ 𝑏 = 𝐵 ) → ( le ‘ 𝑣 ) = ( le ‘ 𝑉 ) ) |
28 |
27 3
|
eqtr4di |
⊢ ( ( ( ( ( 𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌 ) ∧ ( 𝑣 = 𝑉 ∧ 𝑤 = 𝑊 ) ) ∧ 𝑎 = 𝐴 ) ∧ 𝑏 = 𝐵 ) → ( le ‘ 𝑣 ) = ≤ ) |
29 |
28
|
breqd |
⊢ ( ( ( ( ( 𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌 ) ∧ ( 𝑣 = 𝑉 ∧ 𝑤 = 𝑊 ) ) ∧ 𝑎 = 𝐴 ) ∧ 𝑏 = 𝐵 ) → ( 𝑥 ( le ‘ 𝑣 ) ( 𝑔 ‘ 𝑦 ) ↔ 𝑥 ≤ ( 𝑔 ‘ 𝑦 ) ) ) |
30 |
25 29
|
bibi12d |
⊢ ( ( ( ( ( 𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌 ) ∧ ( 𝑣 = 𝑉 ∧ 𝑤 = 𝑊 ) ) ∧ 𝑎 = 𝐴 ) ∧ 𝑏 = 𝐵 ) → ( ( ( 𝑓 ‘ 𝑥 ) ( le ‘ 𝑤 ) 𝑦 ↔ 𝑥 ( le ‘ 𝑣 ) ( 𝑔 ‘ 𝑦 ) ) ↔ ( ( 𝑓 ‘ 𝑥 ) ≲ 𝑦 ↔ 𝑥 ≤ ( 𝑔 ‘ 𝑦 ) ) ) ) |
31 |
15 30
|
raleqbidv |
⊢ ( ( ( ( ( 𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌 ) ∧ ( 𝑣 = 𝑉 ∧ 𝑤 = 𝑊 ) ) ∧ 𝑎 = 𝐴 ) ∧ 𝑏 = 𝐵 ) → ( ∀ 𝑦 ∈ 𝑏 ( ( 𝑓 ‘ 𝑥 ) ( le ‘ 𝑤 ) 𝑦 ↔ 𝑥 ( le ‘ 𝑣 ) ( 𝑔 ‘ 𝑦 ) ) ↔ ∀ 𝑦 ∈ 𝐵 ( ( 𝑓 ‘ 𝑥 ) ≲ 𝑦 ↔ 𝑥 ≤ ( 𝑔 ‘ 𝑦 ) ) ) ) |
32 |
16 31
|
raleqbidv |
⊢ ( ( ( ( ( 𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌 ) ∧ ( 𝑣 = 𝑉 ∧ 𝑤 = 𝑊 ) ) ∧ 𝑎 = 𝐴 ) ∧ 𝑏 = 𝐵 ) → ( ∀ 𝑥 ∈ 𝑎 ∀ 𝑦 ∈ 𝑏 ( ( 𝑓 ‘ 𝑥 ) ( le ‘ 𝑤 ) 𝑦 ↔ 𝑥 ( le ‘ 𝑣 ) ( 𝑔 ‘ 𝑦 ) ) ↔ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ( ( 𝑓 ‘ 𝑥 ) ≲ 𝑦 ↔ 𝑥 ≤ ( 𝑔 ‘ 𝑦 ) ) ) ) |
33 |
21 32
|
anbi12d |
⊢ ( ( ( ( ( 𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌 ) ∧ ( 𝑣 = 𝑉 ∧ 𝑤 = 𝑊 ) ) ∧ 𝑎 = 𝐴 ) ∧ 𝑏 = 𝐵 ) → ( ( ( 𝑓 ∈ ( 𝑏 ↑m 𝑎 ) ∧ 𝑔 ∈ ( 𝑎 ↑m 𝑏 ) ) ∧ ∀ 𝑥 ∈ 𝑎 ∀ 𝑦 ∈ 𝑏 ( ( 𝑓 ‘ 𝑥 ) ( le ‘ 𝑤 ) 𝑦 ↔ 𝑥 ( le ‘ 𝑣 ) ( 𝑔 ‘ 𝑦 ) ) ) ↔ ( ( 𝑓 ∈ ( 𝐵 ↑m 𝐴 ) ∧ 𝑔 ∈ ( 𝐴 ↑m 𝐵 ) ) ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ( ( 𝑓 ‘ 𝑥 ) ≲ 𝑦 ↔ 𝑥 ≤ ( 𝑔 ‘ 𝑦 ) ) ) ) ) |
34 |
33
|
opabbidv |
⊢ ( ( ( ( ( 𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌 ) ∧ ( 𝑣 = 𝑉 ∧ 𝑤 = 𝑊 ) ) ∧ 𝑎 = 𝐴 ) ∧ 𝑏 = 𝐵 ) → { 〈 𝑓 , 𝑔 〉 ∣ ( ( 𝑓 ∈ ( 𝑏 ↑m 𝑎 ) ∧ 𝑔 ∈ ( 𝑎 ↑m 𝑏 ) ) ∧ ∀ 𝑥 ∈ 𝑎 ∀ 𝑦 ∈ 𝑏 ( ( 𝑓 ‘ 𝑥 ) ( le ‘ 𝑤 ) 𝑦 ↔ 𝑥 ( le ‘ 𝑣 ) ( 𝑔 ‘ 𝑦 ) ) ) } = { 〈 𝑓 , 𝑔 〉 ∣ ( ( 𝑓 ∈ ( 𝐵 ↑m 𝐴 ) ∧ 𝑔 ∈ ( 𝐴 ↑m 𝐵 ) ) ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ( ( 𝑓 ‘ 𝑥 ) ≲ 𝑦 ↔ 𝑥 ≤ ( 𝑔 ‘ 𝑦 ) ) ) } ) |
35 |
11 14 34
|
csbied2 |
⊢ ( ( ( ( 𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌 ) ∧ ( 𝑣 = 𝑉 ∧ 𝑤 = 𝑊 ) ) ∧ 𝑎 = 𝐴 ) → ⦋ ( Base ‘ 𝑤 ) / 𝑏 ⦌ { 〈 𝑓 , 𝑔 〉 ∣ ( ( 𝑓 ∈ ( 𝑏 ↑m 𝑎 ) ∧ 𝑔 ∈ ( 𝑎 ↑m 𝑏 ) ) ∧ ∀ 𝑥 ∈ 𝑎 ∀ 𝑦 ∈ 𝑏 ( ( 𝑓 ‘ 𝑥 ) ( le ‘ 𝑤 ) 𝑦 ↔ 𝑥 ( le ‘ 𝑣 ) ( 𝑔 ‘ 𝑦 ) ) ) } = { 〈 𝑓 , 𝑔 〉 ∣ ( ( 𝑓 ∈ ( 𝐵 ↑m 𝐴 ) ∧ 𝑔 ∈ ( 𝐴 ↑m 𝐵 ) ) ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ( ( 𝑓 ‘ 𝑥 ) ≲ 𝑦 ↔ 𝑥 ≤ ( 𝑔 ‘ 𝑦 ) ) ) } ) |
36 |
7 10 35
|
csbied2 |
⊢ ( ( ( 𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌 ) ∧ ( 𝑣 = 𝑉 ∧ 𝑤 = 𝑊 ) ) → ⦋ ( Base ‘ 𝑣 ) / 𝑎 ⦌ ⦋ ( Base ‘ 𝑤 ) / 𝑏 ⦌ { 〈 𝑓 , 𝑔 〉 ∣ ( ( 𝑓 ∈ ( 𝑏 ↑m 𝑎 ) ∧ 𝑔 ∈ ( 𝑎 ↑m 𝑏 ) ) ∧ ∀ 𝑥 ∈ 𝑎 ∀ 𝑦 ∈ 𝑏 ( ( 𝑓 ‘ 𝑥 ) ( le ‘ 𝑤 ) 𝑦 ↔ 𝑥 ( le ‘ 𝑣 ) ( 𝑔 ‘ 𝑦 ) ) ) } = { 〈 𝑓 , 𝑔 〉 ∣ ( ( 𝑓 ∈ ( 𝐵 ↑m 𝐴 ) ∧ 𝑔 ∈ ( 𝐴 ↑m 𝐵 ) ) ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ( ( 𝑓 ‘ 𝑥 ) ≲ 𝑦 ↔ 𝑥 ≤ ( 𝑔 ‘ 𝑦 ) ) ) } ) |
37 |
|
simpl |
⊢ ( ( 𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌 ) → 𝑉 ∈ 𝑋 ) |
38 |
37
|
elexd |
⊢ ( ( 𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌 ) → 𝑉 ∈ V ) |
39 |
|
simpr |
⊢ ( ( 𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌 ) → 𝑊 ∈ 𝑌 ) |
40 |
39
|
elexd |
⊢ ( ( 𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌 ) → 𝑊 ∈ V ) |
41 |
|
ovexd |
⊢ ( ( 𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌 ) → ( 𝐵 ↑m 𝐴 ) ∈ V ) |
42 |
|
ovexd |
⊢ ( ( 𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌 ) → ( 𝐴 ↑m 𝐵 ) ∈ V ) |
43 |
|
simprll |
⊢ ( ( ( 𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌 ) ∧ ( ( 𝑓 ∈ ( 𝐵 ↑m 𝐴 ) ∧ 𝑔 ∈ ( 𝐴 ↑m 𝐵 ) ) ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ( ( 𝑓 ‘ 𝑥 ) ≲ 𝑦 ↔ 𝑥 ≤ ( 𝑔 ‘ 𝑦 ) ) ) ) → 𝑓 ∈ ( 𝐵 ↑m 𝐴 ) ) |
44 |
|
simprlr |
⊢ ( ( ( 𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌 ) ∧ ( ( 𝑓 ∈ ( 𝐵 ↑m 𝐴 ) ∧ 𝑔 ∈ ( 𝐴 ↑m 𝐵 ) ) ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ( ( 𝑓 ‘ 𝑥 ) ≲ 𝑦 ↔ 𝑥 ≤ ( 𝑔 ‘ 𝑦 ) ) ) ) → 𝑔 ∈ ( 𝐴 ↑m 𝐵 ) ) |
45 |
41 42 43 44
|
opabex2 |
⊢ ( ( 𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌 ) → { 〈 𝑓 , 𝑔 〉 ∣ ( ( 𝑓 ∈ ( 𝐵 ↑m 𝐴 ) ∧ 𝑔 ∈ ( 𝐴 ↑m 𝐵 ) ) ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ( ( 𝑓 ‘ 𝑥 ) ≲ 𝑦 ↔ 𝑥 ≤ ( 𝑔 ‘ 𝑦 ) ) ) } ∈ V ) |
46 |
6 36 38 40 45
|
ovmpod |
⊢ ( ( 𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌 ) → ( 𝑉 MGalConn 𝑊 ) = { 〈 𝑓 , 𝑔 〉 ∣ ( ( 𝑓 ∈ ( 𝐵 ↑m 𝐴 ) ∧ 𝑔 ∈ ( 𝐴 ↑m 𝐵 ) ) ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ( ( 𝑓 ‘ 𝑥 ) ≲ 𝑦 ↔ 𝑥 ≤ ( 𝑔 ‘ 𝑦 ) ) ) } ) |