Step |
Hyp |
Ref |
Expression |
1 |
|
tsmsadd.b |
|- B = ( Base ` G ) |
2 |
|
tsmsadd.p |
|- .+ = ( +g ` G ) |
3 |
|
tsmsadd.1 |
|- ( ph -> G e. CMnd ) |
4 |
|
tsmsadd.2 |
|- ( ph -> G e. TopMnd ) |
5 |
|
tsmsadd.a |
|- ( ph -> A e. V ) |
6 |
|
tsmsadd.f |
|- ( ph -> F : A --> B ) |
7 |
|
tsmsadd.h |
|- ( ph -> H : A --> B ) |
8 |
|
tsmsadd.x |
|- ( ph -> X e. ( G tsums F ) ) |
9 |
|
tsmsadd.y |
|- ( ph -> Y e. ( G tsums H ) ) |
10 |
|
tmdtps |
|- ( G e. TopMnd -> G e. TopSp ) |
11 |
4 10
|
syl |
|- ( ph -> G e. TopSp ) |
12 |
1 3 11 5 6
|
tsmscl |
|- ( ph -> ( G tsums F ) C_ B ) |
13 |
12 8
|
sseldd |
|- ( ph -> X e. B ) |
14 |
1 3 11 5 7
|
tsmscl |
|- ( ph -> ( G tsums H ) C_ B ) |
15 |
14 9
|
sseldd |
|- ( ph -> Y e. B ) |
16 |
|
eqid |
|- ( +f ` G ) = ( +f ` G ) |
17 |
1 2 16
|
plusfval |
|- ( ( X e. B /\ Y e. B ) -> ( X ( +f ` G ) Y ) = ( X .+ Y ) ) |
18 |
13 15 17
|
syl2anc |
|- ( ph -> ( X ( +f ` G ) Y ) = ( X .+ Y ) ) |
19 |
|
eqid |
|- ( TopOpen ` G ) = ( TopOpen ` G ) |
20 |
1 19
|
istps |
|- ( G e. TopSp <-> ( TopOpen ` G ) e. ( TopOn ` B ) ) |
21 |
11 20
|
sylib |
|- ( ph -> ( TopOpen ` G ) e. ( TopOn ` B ) ) |
22 |
|
eqid |
|- ( ~P A i^i Fin ) = ( ~P A i^i Fin ) |
23 |
|
eqid |
|- ( y e. ( ~P A i^i Fin ) |-> { z e. ( ~P A i^i Fin ) | y C_ z } ) = ( y e. ( ~P A i^i Fin ) |-> { z e. ( ~P A i^i Fin ) | y C_ z } ) |
24 |
|
eqid |
|- ran ( y e. ( ~P A i^i Fin ) |-> { z e. ( ~P A i^i Fin ) | y C_ z } ) = ran ( y e. ( ~P A i^i Fin ) |-> { z e. ( ~P A i^i Fin ) | y C_ z } ) |
25 |
22 23 24 5
|
tsmsfbas |
|- ( ph -> ran ( y e. ( ~P A i^i Fin ) |-> { z e. ( ~P A i^i Fin ) | y C_ z } ) e. ( fBas ` ( ~P A i^i Fin ) ) ) |
26 |
|
fgcl |
|- ( ran ( y e. ( ~P A i^i Fin ) |-> { z e. ( ~P A i^i Fin ) | y C_ z } ) e. ( fBas ` ( ~P A i^i Fin ) ) -> ( ( ~P A i^i Fin ) filGen ran ( y e. ( ~P A i^i Fin ) |-> { z e. ( ~P A i^i Fin ) | y C_ z } ) ) e. ( Fil ` ( ~P A i^i Fin ) ) ) |
27 |
25 26
|
syl |
|- ( ph -> ( ( ~P A i^i Fin ) filGen ran ( y e. ( ~P A i^i Fin ) |-> { z e. ( ~P A i^i Fin ) | y C_ z } ) ) e. ( Fil ` ( ~P A i^i Fin ) ) ) |
28 |
1 22 3 5 6
|
tsmslem1 |
|- ( ( ph /\ z e. ( ~P A i^i Fin ) ) -> ( G gsum ( F |` z ) ) e. B ) |
29 |
1 22 3 5 7
|
tsmslem1 |
|- ( ( ph /\ z e. ( ~P A i^i Fin ) ) -> ( G gsum ( H |` z ) ) e. B ) |
30 |
1 19 22 24 3 5 6
|
tsmsval |
|- ( ph -> ( G tsums F ) = ( ( ( TopOpen ` G ) fLimf ( ( ~P A i^i Fin ) filGen ran ( y e. ( ~P A i^i Fin ) |-> { z e. ( ~P A i^i Fin ) | y C_ z } ) ) ) ` ( z e. ( ~P A i^i Fin ) |-> ( G gsum ( F |` z ) ) ) ) ) |
31 |
8 30
|
eleqtrd |
|- ( ph -> X e. ( ( ( TopOpen ` G ) fLimf ( ( ~P A i^i Fin ) filGen ran ( y e. ( ~P A i^i Fin ) |-> { z e. ( ~P A i^i Fin ) | y C_ z } ) ) ) ` ( z e. ( ~P A i^i Fin ) |-> ( G gsum ( F |` z ) ) ) ) ) |
32 |
1 19 22 24 3 5 7
|
tsmsval |
|- ( ph -> ( G tsums H ) = ( ( ( TopOpen ` G ) fLimf ( ( ~P A i^i Fin ) filGen ran ( y e. ( ~P A i^i Fin ) |-> { z e. ( ~P A i^i Fin ) | y C_ z } ) ) ) ` ( z e. ( ~P A i^i Fin ) |-> ( G gsum ( H |` z ) ) ) ) ) |
33 |
9 32
|
eleqtrd |
|- ( ph -> Y e. ( ( ( TopOpen ` G ) fLimf ( ( ~P A i^i Fin ) filGen ran ( y e. ( ~P A i^i Fin ) |-> { z e. ( ~P A i^i Fin ) | y C_ z } ) ) ) ` ( z e. ( ~P A i^i Fin ) |-> ( G gsum ( H |` z ) ) ) ) ) |
34 |
19 16
|
tmdcn |
|- ( G e. TopMnd -> ( +f ` G ) e. ( ( ( TopOpen ` G ) tX ( TopOpen ` G ) ) Cn ( TopOpen ` G ) ) ) |
35 |
4 34
|
syl |
|- ( ph -> ( +f ` G ) e. ( ( ( TopOpen ` G ) tX ( TopOpen ` G ) ) Cn ( TopOpen ` G ) ) ) |
36 |
13 15
|
opelxpd |
|- ( ph -> <. X , Y >. e. ( B X. B ) ) |
37 |
|
txtopon |
|- ( ( ( TopOpen ` G ) e. ( TopOn ` B ) /\ ( TopOpen ` G ) e. ( TopOn ` B ) ) -> ( ( TopOpen ` G ) tX ( TopOpen ` G ) ) e. ( TopOn ` ( B X. B ) ) ) |
38 |
21 21 37
|
syl2anc |
|- ( ph -> ( ( TopOpen ` G ) tX ( TopOpen ` G ) ) e. ( TopOn ` ( B X. B ) ) ) |
39 |
|
toponuni |
|- ( ( ( TopOpen ` G ) tX ( TopOpen ` G ) ) e. ( TopOn ` ( B X. B ) ) -> ( B X. B ) = U. ( ( TopOpen ` G ) tX ( TopOpen ` G ) ) ) |
40 |
38 39
|
syl |
|- ( ph -> ( B X. B ) = U. ( ( TopOpen ` G ) tX ( TopOpen ` G ) ) ) |
41 |
36 40
|
eleqtrd |
|- ( ph -> <. X , Y >. e. U. ( ( TopOpen ` G ) tX ( TopOpen ` G ) ) ) |
42 |
|
eqid |
|- U. ( ( TopOpen ` G ) tX ( TopOpen ` G ) ) = U. ( ( TopOpen ` G ) tX ( TopOpen ` G ) ) |
43 |
42
|
cncnpi |
|- ( ( ( +f ` G ) e. ( ( ( TopOpen ` G ) tX ( TopOpen ` G ) ) Cn ( TopOpen ` G ) ) /\ <. X , Y >. e. U. ( ( TopOpen ` G ) tX ( TopOpen ` G ) ) ) -> ( +f ` G ) e. ( ( ( ( TopOpen ` G ) tX ( TopOpen ` G ) ) CnP ( TopOpen ` G ) ) ` <. X , Y >. ) ) |
44 |
35 41 43
|
syl2anc |
|- ( ph -> ( +f ` G ) e. ( ( ( ( TopOpen ` G ) tX ( TopOpen ` G ) ) CnP ( TopOpen ` G ) ) ` <. X , Y >. ) ) |
45 |
21 21 27 28 29 31 33 44
|
flfcnp2 |
|- ( ph -> ( X ( +f ` G ) Y ) e. ( ( ( TopOpen ` G ) fLimf ( ( ~P A i^i Fin ) filGen ran ( y e. ( ~P A i^i Fin ) |-> { z e. ( ~P A i^i Fin ) | y C_ z } ) ) ) ` ( z e. ( ~P A i^i Fin ) |-> ( ( G gsum ( F |` z ) ) ( +f ` G ) ( G gsum ( H |` z ) ) ) ) ) ) |
46 |
18 45
|
eqeltrrd |
|- ( ph -> ( X .+ Y ) e. ( ( ( TopOpen ` G ) fLimf ( ( ~P A i^i Fin ) filGen ran ( y e. ( ~P A i^i Fin ) |-> { z e. ( ~P A i^i Fin ) | y C_ z } ) ) ) ` ( z e. ( ~P A i^i Fin ) |-> ( ( G gsum ( F |` z ) ) ( +f ` G ) ( G gsum ( H |` z ) ) ) ) ) ) |
47 |
|
cmnmnd |
|- ( G e. CMnd -> G e. Mnd ) |
48 |
3 47
|
syl |
|- ( ph -> G e. Mnd ) |
49 |
1 2
|
mndcl |
|- ( ( G e. Mnd /\ x e. B /\ y e. B ) -> ( x .+ y ) e. B ) |
50 |
49
|
3expb |
|- ( ( G e. Mnd /\ ( x e. B /\ y e. B ) ) -> ( x .+ y ) e. B ) |
51 |
48 50
|
sylan |
|- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x .+ y ) e. B ) |
52 |
|
inidm |
|- ( A i^i A ) = A |
53 |
51 6 7 5 5 52
|
off |
|- ( ph -> ( F oF .+ H ) : A --> B ) |
54 |
1 19 22 24 3 5 53
|
tsmsval |
|- ( ph -> ( G tsums ( F oF .+ H ) ) = ( ( ( TopOpen ` G ) fLimf ( ( ~P A i^i Fin ) filGen ran ( y e. ( ~P A i^i Fin ) |-> { z e. ( ~P A i^i Fin ) | y C_ z } ) ) ) ` ( z e. ( ~P A i^i Fin ) |-> ( G gsum ( ( F oF .+ H ) |` z ) ) ) ) ) |
55 |
|
eqid |
|- ( 0g ` G ) = ( 0g ` G ) |
56 |
3
|
adantr |
|- ( ( ph /\ z e. ( ~P A i^i Fin ) ) -> G e. CMnd ) |
57 |
|
elinel2 |
|- ( z e. ( ~P A i^i Fin ) -> z e. Fin ) |
58 |
57
|
adantl |
|- ( ( ph /\ z e. ( ~P A i^i Fin ) ) -> z e. Fin ) |
59 |
|
elfpw |
|- ( z e. ( ~P A i^i Fin ) <-> ( z C_ A /\ z e. Fin ) ) |
60 |
59
|
simplbi |
|- ( z e. ( ~P A i^i Fin ) -> z C_ A ) |
61 |
|
fssres |
|- ( ( F : A --> B /\ z C_ A ) -> ( F |` z ) : z --> B ) |
62 |
6 60 61
|
syl2an |
|- ( ( ph /\ z e. ( ~P A i^i Fin ) ) -> ( F |` z ) : z --> B ) |
63 |
|
fssres |
|- ( ( H : A --> B /\ z C_ A ) -> ( H |` z ) : z --> B ) |
64 |
7 60 63
|
syl2an |
|- ( ( ph /\ z e. ( ~P A i^i Fin ) ) -> ( H |` z ) : z --> B ) |
65 |
|
fvexd |
|- ( ( ph /\ z e. ( ~P A i^i Fin ) ) -> ( 0g ` G ) e. _V ) |
66 |
62 58 65
|
fdmfifsupp |
|- ( ( ph /\ z e. ( ~P A i^i Fin ) ) -> ( F |` z ) finSupp ( 0g ` G ) ) |
67 |
64 58 65
|
fdmfifsupp |
|- ( ( ph /\ z e. ( ~P A i^i Fin ) ) -> ( H |` z ) finSupp ( 0g ` G ) ) |
68 |
1 55 2 56 58 62 64 66 67
|
gsumadd |
|- ( ( ph /\ z e. ( ~P A i^i Fin ) ) -> ( G gsum ( ( F |` z ) oF .+ ( H |` z ) ) ) = ( ( G gsum ( F |` z ) ) .+ ( G gsum ( H |` z ) ) ) ) |
69 |
6 5
|
fexd |
|- ( ph -> F e. _V ) |
70 |
7 5
|
fexd |
|- ( ph -> H e. _V ) |
71 |
|
offres |
|- ( ( F e. _V /\ H e. _V ) -> ( ( F oF .+ H ) |` z ) = ( ( F |` z ) oF .+ ( H |` z ) ) ) |
72 |
69 70 71
|
syl2anc |
|- ( ph -> ( ( F oF .+ H ) |` z ) = ( ( F |` z ) oF .+ ( H |` z ) ) ) |
73 |
72
|
adantr |
|- ( ( ph /\ z e. ( ~P A i^i Fin ) ) -> ( ( F oF .+ H ) |` z ) = ( ( F |` z ) oF .+ ( H |` z ) ) ) |
74 |
73
|
oveq2d |
|- ( ( ph /\ z e. ( ~P A i^i Fin ) ) -> ( G gsum ( ( F oF .+ H ) |` z ) ) = ( G gsum ( ( F |` z ) oF .+ ( H |` z ) ) ) ) |
75 |
1 2 16
|
plusfval |
|- ( ( ( G gsum ( F |` z ) ) e. B /\ ( G gsum ( H |` z ) ) e. B ) -> ( ( G gsum ( F |` z ) ) ( +f ` G ) ( G gsum ( H |` z ) ) ) = ( ( G gsum ( F |` z ) ) .+ ( G gsum ( H |` z ) ) ) ) |
76 |
28 29 75
|
syl2anc |
|- ( ( ph /\ z e. ( ~P A i^i Fin ) ) -> ( ( G gsum ( F |` z ) ) ( +f ` G ) ( G gsum ( H |` z ) ) ) = ( ( G gsum ( F |` z ) ) .+ ( G gsum ( H |` z ) ) ) ) |
77 |
68 74 76
|
3eqtr4d |
|- ( ( ph /\ z e. ( ~P A i^i Fin ) ) -> ( G gsum ( ( F oF .+ H ) |` z ) ) = ( ( G gsum ( F |` z ) ) ( +f ` G ) ( G gsum ( H |` z ) ) ) ) |
78 |
77
|
mpteq2dva |
|- ( ph -> ( z e. ( ~P A i^i Fin ) |-> ( G gsum ( ( F oF .+ H ) |` z ) ) ) = ( z e. ( ~P A i^i Fin ) |-> ( ( G gsum ( F |` z ) ) ( +f ` G ) ( G gsum ( H |` z ) ) ) ) ) |
79 |
78
|
fveq2d |
|- ( ph -> ( ( ( TopOpen ` G ) fLimf ( ( ~P A i^i Fin ) filGen ran ( y e. ( ~P A i^i Fin ) |-> { z e. ( ~P A i^i Fin ) | y C_ z } ) ) ) ` ( z e. ( ~P A i^i Fin ) |-> ( G gsum ( ( F oF .+ H ) |` z ) ) ) ) = ( ( ( TopOpen ` G ) fLimf ( ( ~P A i^i Fin ) filGen ran ( y e. ( ~P A i^i Fin ) |-> { z e. ( ~P A i^i Fin ) | y C_ z } ) ) ) ` ( z e. ( ~P A i^i Fin ) |-> ( ( G gsum ( F |` z ) ) ( +f ` G ) ( G gsum ( H |` z ) ) ) ) ) ) |
80 |
54 79
|
eqtrd |
|- ( ph -> ( G tsums ( F oF .+ H ) ) = ( ( ( TopOpen ` G ) fLimf ( ( ~P A i^i Fin ) filGen ran ( y e. ( ~P A i^i Fin ) |-> { z e. ( ~P A i^i Fin ) | y C_ z } ) ) ) ` ( z e. ( ~P A i^i Fin ) |-> ( ( G gsum ( F |` z ) ) ( +f ` G ) ( G gsum ( H |` z ) ) ) ) ) ) |
81 |
46 80
|
eleqtrrd |
|- ( ph -> ( X .+ Y ) e. ( G tsums ( F oF .+ H ) ) ) |