Step |
Hyp |
Ref |
Expression |
1 |
|
prdsbasmpt.y |
⊢ 𝑌 = ( 𝑆 Xs 𝑅 ) |
2 |
|
prdsbasmpt.b |
⊢ 𝐵 = ( Base ‘ 𝑌 ) |
3 |
|
prdsbasmpt.s |
⊢ ( 𝜑 → 𝑆 ∈ 𝑉 ) |
4 |
|
prdsbasmpt.i |
⊢ ( 𝜑 → 𝐼 ∈ 𝑊 ) |
5 |
|
prdsbasmpt.r |
⊢ ( 𝜑 → 𝑅 Fn 𝐼 ) |
6 |
|
prdsplusgval.f |
⊢ ( 𝜑 → 𝐹 ∈ 𝐵 ) |
7 |
|
prdsplusgval.g |
⊢ ( 𝜑 → 𝐺 ∈ 𝐵 ) |
8 |
|
prdsleval.l |
⊢ ≤ = ( le ‘ 𝑌 ) |
9 |
|
df-br |
⊢ ( 𝐹 ≤ 𝐺 ↔ 〈 𝐹 , 𝐺 〉 ∈ ≤ ) |
10 |
|
fnex |
⊢ ( ( 𝑅 Fn 𝐼 ∧ 𝐼 ∈ 𝑊 ) → 𝑅 ∈ V ) |
11 |
5 4 10
|
syl2anc |
⊢ ( 𝜑 → 𝑅 ∈ V ) |
12 |
5
|
fndmd |
⊢ ( 𝜑 → dom 𝑅 = 𝐼 ) |
13 |
1 3 11 2 12 8
|
prdsle |
⊢ ( 𝜑 → ≤ = { 〈 𝑓 , 𝑔 〉 ∣ ( { 𝑓 , 𝑔 } ⊆ 𝐵 ∧ ∀ 𝑥 ∈ 𝐼 ( 𝑓 ‘ 𝑥 ) ( le ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) } ) |
14 |
|
vex |
⊢ 𝑓 ∈ V |
15 |
|
vex |
⊢ 𝑔 ∈ V |
16 |
14 15
|
prss |
⊢ ( ( 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵 ) ↔ { 𝑓 , 𝑔 } ⊆ 𝐵 ) |
17 |
16
|
anbi1i |
⊢ ( ( ( 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵 ) ∧ ∀ 𝑥 ∈ 𝐼 ( 𝑓 ‘ 𝑥 ) ( le ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ↔ ( { 𝑓 , 𝑔 } ⊆ 𝐵 ∧ ∀ 𝑥 ∈ 𝐼 ( 𝑓 ‘ 𝑥 ) ( le ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) |
18 |
17
|
opabbii |
⊢ { 〈 𝑓 , 𝑔 〉 ∣ ( ( 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵 ) ∧ ∀ 𝑥 ∈ 𝐼 ( 𝑓 ‘ 𝑥 ) ( le ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) } = { 〈 𝑓 , 𝑔 〉 ∣ ( { 𝑓 , 𝑔 } ⊆ 𝐵 ∧ ∀ 𝑥 ∈ 𝐼 ( 𝑓 ‘ 𝑥 ) ( le ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) } |
19 |
13 18
|
eqtr4di |
⊢ ( 𝜑 → ≤ = { 〈 𝑓 , 𝑔 〉 ∣ ( ( 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵 ) ∧ ∀ 𝑥 ∈ 𝐼 ( 𝑓 ‘ 𝑥 ) ( le ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) } ) |
20 |
19
|
eleq2d |
⊢ ( 𝜑 → ( 〈 𝐹 , 𝐺 〉 ∈ ≤ ↔ 〈 𝐹 , 𝐺 〉 ∈ { 〈 𝑓 , 𝑔 〉 ∣ ( ( 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵 ) ∧ ∀ 𝑥 ∈ 𝐼 ( 𝑓 ‘ 𝑥 ) ( le ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) } ) ) |
21 |
9 20
|
syl5bb |
⊢ ( 𝜑 → ( 𝐹 ≤ 𝐺 ↔ 〈 𝐹 , 𝐺 〉 ∈ { 〈 𝑓 , 𝑔 〉 ∣ ( ( 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵 ) ∧ ∀ 𝑥 ∈ 𝐼 ( 𝑓 ‘ 𝑥 ) ( le ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) } ) ) |
22 |
|
fveq1 |
⊢ ( 𝑓 = 𝐹 → ( 𝑓 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑥 ) ) |
23 |
|
fveq1 |
⊢ ( 𝑔 = 𝐺 → ( 𝑔 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑥 ) ) |
24 |
22 23
|
breqan12d |
⊢ ( ( 𝑓 = 𝐹 ∧ 𝑔 = 𝐺 ) → ( ( 𝑓 ‘ 𝑥 ) ( le ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ↔ ( 𝐹 ‘ 𝑥 ) ( le ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝐺 ‘ 𝑥 ) ) ) |
25 |
24
|
ralbidv |
⊢ ( ( 𝑓 = 𝐹 ∧ 𝑔 = 𝐺 ) → ( ∀ 𝑥 ∈ 𝐼 ( 𝑓 ‘ 𝑥 ) ( le ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ↔ ∀ 𝑥 ∈ 𝐼 ( 𝐹 ‘ 𝑥 ) ( le ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝐺 ‘ 𝑥 ) ) ) |
26 |
25
|
opelopab2a |
⊢ ( ( 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵 ) → ( 〈 𝐹 , 𝐺 〉 ∈ { 〈 𝑓 , 𝑔 〉 ∣ ( ( 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵 ) ∧ ∀ 𝑥 ∈ 𝐼 ( 𝑓 ‘ 𝑥 ) ( le ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) } ↔ ∀ 𝑥 ∈ 𝐼 ( 𝐹 ‘ 𝑥 ) ( le ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝐺 ‘ 𝑥 ) ) ) |
27 |
6 7 26
|
syl2anc |
⊢ ( 𝜑 → ( 〈 𝐹 , 𝐺 〉 ∈ { 〈 𝑓 , 𝑔 〉 ∣ ( ( 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵 ) ∧ ∀ 𝑥 ∈ 𝐼 ( 𝑓 ‘ 𝑥 ) ( le ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) } ↔ ∀ 𝑥 ∈ 𝐼 ( 𝐹 ‘ 𝑥 ) ( le ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝐺 ‘ 𝑥 ) ) ) |
28 |
21 27
|
bitrd |
⊢ ( 𝜑 → ( 𝐹 ≤ 𝐺 ↔ ∀ 𝑥 ∈ 𝐼 ( 𝐹 ‘ 𝑥 ) ( le ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝐺 ‘ 𝑥 ) ) ) |