Step |
Hyp |
Ref |
Expression |
1 |
|
psrvsca.s |
⊢ 𝑆 = ( 𝐼 mPwSer 𝑅 ) |
2 |
|
psrvsca.n |
⊢ ∙ = ( ·𝑠 ‘ 𝑆 ) |
3 |
|
psrvsca.k |
⊢ 𝐾 = ( Base ‘ 𝑅 ) |
4 |
|
psrvsca.b |
⊢ 𝐵 = ( Base ‘ 𝑆 ) |
5 |
|
psrvsca.m |
⊢ · = ( .r ‘ 𝑅 ) |
6 |
|
psrvsca.d |
⊢ 𝐷 = { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } |
7 |
|
eqid |
⊢ ( +g ‘ 𝑅 ) = ( +g ‘ 𝑅 ) |
8 |
|
eqid |
⊢ ( TopOpen ‘ 𝑅 ) = ( TopOpen ‘ 𝑅 ) |
9 |
|
simpl |
⊢ ( ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) → 𝐼 ∈ V ) |
10 |
1 3 6 4 9
|
psrbas |
⊢ ( ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) → 𝐵 = ( 𝐾 ↑m 𝐷 ) ) |
11 |
|
eqid |
⊢ ( +g ‘ 𝑆 ) = ( +g ‘ 𝑆 ) |
12 |
1 4 7 11
|
psrplusg |
⊢ ( +g ‘ 𝑆 ) = ( ∘f ( +g ‘ 𝑅 ) ↾ ( 𝐵 × 𝐵 ) ) |
13 |
|
eqid |
⊢ ( .r ‘ 𝑆 ) = ( .r ‘ 𝑆 ) |
14 |
1 4 5 13 6
|
psrmulr |
⊢ ( .r ‘ 𝑆 ) = ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ ( 𝑘 ∈ 𝐷 ↦ ( 𝑅 Σg ( 𝑥 ∈ { 𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑘 } ↦ ( ( 𝑓 ‘ 𝑥 ) · ( 𝑔 ‘ ( 𝑘 ∘f − 𝑥 ) ) ) ) ) ) ) |
15 |
|
eqid |
⊢ ( 𝑥 ∈ 𝐾 , 𝑓 ∈ 𝐵 ↦ ( ( 𝐷 × { 𝑥 } ) ∘f · 𝑓 ) ) = ( 𝑥 ∈ 𝐾 , 𝑓 ∈ 𝐵 ↦ ( ( 𝐷 × { 𝑥 } ) ∘f · 𝑓 ) ) |
16 |
|
eqidd |
⊢ ( ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) → ( ∏t ‘ ( 𝐷 × { ( TopOpen ‘ 𝑅 ) } ) ) = ( ∏t ‘ ( 𝐷 × { ( TopOpen ‘ 𝑅 ) } ) ) ) |
17 |
|
simpr |
⊢ ( ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) → 𝑅 ∈ V ) |
18 |
1 3 7 5 8 6 10 12 14 15 16 9 17
|
psrval |
⊢ ( ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) → 𝑆 = ( { 〈 ( Base ‘ ndx ) , 𝐵 〉 , 〈 ( +g ‘ ndx ) , ( +g ‘ 𝑆 ) 〉 , 〈 ( .r ‘ ndx ) , ( .r ‘ 𝑆 ) 〉 } ∪ { 〈 ( Scalar ‘ ndx ) , 𝑅 〉 , 〈 ( ·𝑠 ‘ ndx ) , ( 𝑥 ∈ 𝐾 , 𝑓 ∈ 𝐵 ↦ ( ( 𝐷 × { 𝑥 } ) ∘f · 𝑓 ) ) 〉 , 〈 ( TopSet ‘ ndx ) , ( ∏t ‘ ( 𝐷 × { ( TopOpen ‘ 𝑅 ) } ) ) 〉 } ) ) |
19 |
18
|
fveq2d |
⊢ ( ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) → ( ·𝑠 ‘ 𝑆 ) = ( ·𝑠 ‘ ( { 〈 ( Base ‘ ndx ) , 𝐵 〉 , 〈 ( +g ‘ ndx ) , ( +g ‘ 𝑆 ) 〉 , 〈 ( .r ‘ ndx ) , ( .r ‘ 𝑆 ) 〉 } ∪ { 〈 ( Scalar ‘ ndx ) , 𝑅 〉 , 〈 ( ·𝑠 ‘ ndx ) , ( 𝑥 ∈ 𝐾 , 𝑓 ∈ 𝐵 ↦ ( ( 𝐷 × { 𝑥 } ) ∘f · 𝑓 ) ) 〉 , 〈 ( TopSet ‘ ndx ) , ( ∏t ‘ ( 𝐷 × { ( TopOpen ‘ 𝑅 ) } ) ) 〉 } ) ) ) |
20 |
3
|
fvexi |
⊢ 𝐾 ∈ V |
21 |
4
|
fvexi |
⊢ 𝐵 ∈ V |
22 |
20 21
|
mpoex |
⊢ ( 𝑥 ∈ 𝐾 , 𝑓 ∈ 𝐵 ↦ ( ( 𝐷 × { 𝑥 } ) ∘f · 𝑓 ) ) ∈ V |
23 |
|
psrvalstr |
⊢ ( { 〈 ( Base ‘ ndx ) , 𝐵 〉 , 〈 ( +g ‘ ndx ) , ( +g ‘ 𝑆 ) 〉 , 〈 ( .r ‘ ndx ) , ( .r ‘ 𝑆 ) 〉 } ∪ { 〈 ( Scalar ‘ ndx ) , 𝑅 〉 , 〈 ( ·𝑠 ‘ ndx ) , ( 𝑥 ∈ 𝐾 , 𝑓 ∈ 𝐵 ↦ ( ( 𝐷 × { 𝑥 } ) ∘f · 𝑓 ) ) 〉 , 〈 ( TopSet ‘ ndx ) , ( ∏t ‘ ( 𝐷 × { ( TopOpen ‘ 𝑅 ) } ) ) 〉 } ) Struct 〈 1 , 9 〉 |
24 |
|
vscaid |
⊢ ·𝑠 = Slot ( ·𝑠 ‘ ndx ) |
25 |
|
snsstp2 |
⊢ { 〈 ( ·𝑠 ‘ ndx ) , ( 𝑥 ∈ 𝐾 , 𝑓 ∈ 𝐵 ↦ ( ( 𝐷 × { 𝑥 } ) ∘f · 𝑓 ) ) 〉 } ⊆ { 〈 ( Scalar ‘ ndx ) , 𝑅 〉 , 〈 ( ·𝑠 ‘ ndx ) , ( 𝑥 ∈ 𝐾 , 𝑓 ∈ 𝐵 ↦ ( ( 𝐷 × { 𝑥 } ) ∘f · 𝑓 ) ) 〉 , 〈 ( TopSet ‘ ndx ) , ( ∏t ‘ ( 𝐷 × { ( TopOpen ‘ 𝑅 ) } ) ) 〉 } |
26 |
|
ssun2 |
⊢ { 〈 ( Scalar ‘ ndx ) , 𝑅 〉 , 〈 ( ·𝑠 ‘ ndx ) , ( 𝑥 ∈ 𝐾 , 𝑓 ∈ 𝐵 ↦ ( ( 𝐷 × { 𝑥 } ) ∘f · 𝑓 ) ) 〉 , 〈 ( TopSet ‘ ndx ) , ( ∏t ‘ ( 𝐷 × { ( TopOpen ‘ 𝑅 ) } ) ) 〉 } ⊆ ( { 〈 ( Base ‘ ndx ) , 𝐵 〉 , 〈 ( +g ‘ ndx ) , ( +g ‘ 𝑆 ) 〉 , 〈 ( .r ‘ ndx ) , ( .r ‘ 𝑆 ) 〉 } ∪ { 〈 ( Scalar ‘ ndx ) , 𝑅 〉 , 〈 ( ·𝑠 ‘ ndx ) , ( 𝑥 ∈ 𝐾 , 𝑓 ∈ 𝐵 ↦ ( ( 𝐷 × { 𝑥 } ) ∘f · 𝑓 ) ) 〉 , 〈 ( TopSet ‘ ndx ) , ( ∏t ‘ ( 𝐷 × { ( TopOpen ‘ 𝑅 ) } ) ) 〉 } ) |
27 |
25 26
|
sstri |
⊢ { 〈 ( ·𝑠 ‘ ndx ) , ( 𝑥 ∈ 𝐾 , 𝑓 ∈ 𝐵 ↦ ( ( 𝐷 × { 𝑥 } ) ∘f · 𝑓 ) ) 〉 } ⊆ ( { 〈 ( Base ‘ ndx ) , 𝐵 〉 , 〈 ( +g ‘ ndx ) , ( +g ‘ 𝑆 ) 〉 , 〈 ( .r ‘ ndx ) , ( .r ‘ 𝑆 ) 〉 } ∪ { 〈 ( Scalar ‘ ndx ) , 𝑅 〉 , 〈 ( ·𝑠 ‘ ndx ) , ( 𝑥 ∈ 𝐾 , 𝑓 ∈ 𝐵 ↦ ( ( 𝐷 × { 𝑥 } ) ∘f · 𝑓 ) ) 〉 , 〈 ( TopSet ‘ ndx ) , ( ∏t ‘ ( 𝐷 × { ( TopOpen ‘ 𝑅 ) } ) ) 〉 } ) |
28 |
23 24 27
|
strfv |
⊢ ( ( 𝑥 ∈ 𝐾 , 𝑓 ∈ 𝐵 ↦ ( ( 𝐷 × { 𝑥 } ) ∘f · 𝑓 ) ) ∈ V → ( 𝑥 ∈ 𝐾 , 𝑓 ∈ 𝐵 ↦ ( ( 𝐷 × { 𝑥 } ) ∘f · 𝑓 ) ) = ( ·𝑠 ‘ ( { 〈 ( Base ‘ ndx ) , 𝐵 〉 , 〈 ( +g ‘ ndx ) , ( +g ‘ 𝑆 ) 〉 , 〈 ( .r ‘ ndx ) , ( .r ‘ 𝑆 ) 〉 } ∪ { 〈 ( Scalar ‘ ndx ) , 𝑅 〉 , 〈 ( ·𝑠 ‘ ndx ) , ( 𝑥 ∈ 𝐾 , 𝑓 ∈ 𝐵 ↦ ( ( 𝐷 × { 𝑥 } ) ∘f · 𝑓 ) ) 〉 , 〈 ( TopSet ‘ ndx ) , ( ∏t ‘ ( 𝐷 × { ( TopOpen ‘ 𝑅 ) } ) ) 〉 } ) ) ) |
29 |
22 28
|
ax-mp |
⊢ ( 𝑥 ∈ 𝐾 , 𝑓 ∈ 𝐵 ↦ ( ( 𝐷 × { 𝑥 } ) ∘f · 𝑓 ) ) = ( ·𝑠 ‘ ( { 〈 ( Base ‘ ndx ) , 𝐵 〉 , 〈 ( +g ‘ ndx ) , ( +g ‘ 𝑆 ) 〉 , 〈 ( .r ‘ ndx ) , ( .r ‘ 𝑆 ) 〉 } ∪ { 〈 ( Scalar ‘ ndx ) , 𝑅 〉 , 〈 ( ·𝑠 ‘ ndx ) , ( 𝑥 ∈ 𝐾 , 𝑓 ∈ 𝐵 ↦ ( ( 𝐷 × { 𝑥 } ) ∘f · 𝑓 ) ) 〉 , 〈 ( TopSet ‘ ndx ) , ( ∏t ‘ ( 𝐷 × { ( TopOpen ‘ 𝑅 ) } ) ) 〉 } ) ) |
30 |
19 2 29
|
3eqtr4g |
⊢ ( ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) → ∙ = ( 𝑥 ∈ 𝐾 , 𝑓 ∈ 𝐵 ↦ ( ( 𝐷 × { 𝑥 } ) ∘f · 𝑓 ) ) ) |
31 |
|
eqid |
⊢ ∅ = ∅ |
32 |
|
fn0 |
⊢ ( ∅ Fn ∅ ↔ ∅ = ∅ ) |
33 |
31 32
|
mpbir |
⊢ ∅ Fn ∅ |
34 |
|
reldmpsr |
⊢ Rel dom mPwSer |
35 |
34
|
ovprc |
⊢ ( ¬ ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) → ( 𝐼 mPwSer 𝑅 ) = ∅ ) |
36 |
1 35
|
syl5eq |
⊢ ( ¬ ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) → 𝑆 = ∅ ) |
37 |
36
|
fveq2d |
⊢ ( ¬ ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) → ( ·𝑠 ‘ 𝑆 ) = ( ·𝑠 ‘ ∅ ) ) |
38 |
|
df-vsca |
⊢ ·𝑠 = Slot 6 |
39 |
38
|
str0 |
⊢ ∅ = ( ·𝑠 ‘ ∅ ) |
40 |
37 2 39
|
3eqtr4g |
⊢ ( ¬ ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) → ∙ = ∅ ) |
41 |
34 1 4
|
elbasov |
⊢ ( 𝑓 ∈ 𝐵 → ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) ) |
42 |
41
|
con3i |
⊢ ( ¬ ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) → ¬ 𝑓 ∈ 𝐵 ) |
43 |
42
|
eq0rdv |
⊢ ( ¬ ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) → 𝐵 = ∅ ) |
44 |
43
|
xpeq2d |
⊢ ( ¬ ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) → ( 𝐾 × 𝐵 ) = ( 𝐾 × ∅ ) ) |
45 |
|
xp0 |
⊢ ( 𝐾 × ∅ ) = ∅ |
46 |
44 45
|
eqtrdi |
⊢ ( ¬ ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) → ( 𝐾 × 𝐵 ) = ∅ ) |
47 |
40 46
|
fneq12d |
⊢ ( ¬ ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) → ( ∙ Fn ( 𝐾 × 𝐵 ) ↔ ∅ Fn ∅ ) ) |
48 |
33 47
|
mpbiri |
⊢ ( ¬ ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) → ∙ Fn ( 𝐾 × 𝐵 ) ) |
49 |
|
fnov |
⊢ ( ∙ Fn ( 𝐾 × 𝐵 ) ↔ ∙ = ( 𝑥 ∈ 𝐾 , 𝑓 ∈ 𝐵 ↦ ( 𝑥 ∙ 𝑓 ) ) ) |
50 |
48 49
|
sylib |
⊢ ( ¬ ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) → ∙ = ( 𝑥 ∈ 𝐾 , 𝑓 ∈ 𝐵 ↦ ( 𝑥 ∙ 𝑓 ) ) ) |
51 |
42
|
pm2.21d |
⊢ ( ¬ ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) → ( 𝑓 ∈ 𝐵 → ( ( 𝐷 × { 𝑥 } ) ∘f · 𝑓 ) = ( 𝑥 ∙ 𝑓 ) ) ) |
52 |
51
|
a1d |
⊢ ( ¬ ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) → ( 𝑥 ∈ 𝐾 → ( 𝑓 ∈ 𝐵 → ( ( 𝐷 × { 𝑥 } ) ∘f · 𝑓 ) = ( 𝑥 ∙ 𝑓 ) ) ) ) |
53 |
52
|
3imp |
⊢ ( ( ¬ ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) ∧ 𝑥 ∈ 𝐾 ∧ 𝑓 ∈ 𝐵 ) → ( ( 𝐷 × { 𝑥 } ) ∘f · 𝑓 ) = ( 𝑥 ∙ 𝑓 ) ) |
54 |
53
|
mpoeq3dva |
⊢ ( ¬ ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) → ( 𝑥 ∈ 𝐾 , 𝑓 ∈ 𝐵 ↦ ( ( 𝐷 × { 𝑥 } ) ∘f · 𝑓 ) ) = ( 𝑥 ∈ 𝐾 , 𝑓 ∈ 𝐵 ↦ ( 𝑥 ∙ 𝑓 ) ) ) |
55 |
50 54
|
eqtr4d |
⊢ ( ¬ ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) → ∙ = ( 𝑥 ∈ 𝐾 , 𝑓 ∈ 𝐵 ↦ ( ( 𝐷 × { 𝑥 } ) ∘f · 𝑓 ) ) ) |
56 |
30 55
|
pm2.61i |
⊢ ∙ = ( 𝑥 ∈ 𝐾 , 𝑓 ∈ 𝐵 ↦ ( ( 𝐷 × { 𝑥 } ) ∘f · 𝑓 ) ) |