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Theorem cnmpt1plusg 19633
Description: Continuity of the group sum; analogue of cnmpt12f 19214 which cannot be used directly because 
+g is not a function. (Contributed by Mario Carneiro, 23-Aug-2015.)
Hypotheses
Ref Expression
tgpcn.j  |-  J  =  ( TopOpen `  G )
cnmpt1plusg.p  |-  .+  =  ( +g  `  G )
cnmpt1plusg.g  |-  ( ph  ->  G  e. TopMnd )
cnmpt1plusg.k  |-  ( ph  ->  K  e.  (TopOn `  X ) )
cnmpt1plusg.a  |-  ( ph  ->  ( x  e.  X  |->  A )  e.  ( K  Cn  J ) )
cnmpt1plusg.b  |-  ( ph  ->  ( x  e.  X  |->  B )  e.  ( K  Cn  J ) )
Assertion
Ref Expression
cnmpt1plusg  |-  ( ph  ->  ( x  e.  X  |->  ( A  .+  B
) )  e.  ( K  Cn  J ) )
Distinct variable groups:    x, G    x, J    x, K    ph, x    x, X
Allowed substitution hints:    A( x)    B( x)    .+ ( x)

Proof of Theorem cnmpt1plusg
StepHypRef Expression
1 cnmpt1plusg.k . . . . . . 7  |-  ( ph  ->  K  e.  (TopOn `  X ) )
2 cnmpt1plusg.g . . . . . . . 8  |-  ( ph  ->  G  e. TopMnd )
3 tgpcn.j . . . . . . . . 9  |-  J  =  ( TopOpen `  G )
4 eqid 2438 . . . . . . . . 9  |-  ( Base `  G )  =  (
Base `  G )
53, 4tmdtopon 19627 . . . . . . . 8  |-  ( G  e. TopMnd  ->  J  e.  (TopOn `  ( Base `  G
) ) )
62, 5syl 16 . . . . . . 7  |-  ( ph  ->  J  e.  (TopOn `  ( Base `  G )
) )
7 cnmpt1plusg.a . . . . . . 7  |-  ( ph  ->  ( x  e.  X  |->  A )  e.  ( K  Cn  J ) )
8 cnf2 18828 . . . . . . 7  |-  ( ( K  e.  (TopOn `  X )  /\  J  e.  (TopOn `  ( Base `  G ) )  /\  ( x  e.  X  |->  A )  e.  ( K  Cn  J ) )  ->  ( x  e.  X  |->  A ) : X --> ( Base `  G ) )
91, 6, 7, 8syl3anc 1218 . . . . . 6  |-  ( ph  ->  ( x  e.  X  |->  A ) : X --> ( Base `  G )
)
10 eqid 2438 . . . . . . 7  |-  ( x  e.  X  |->  A )  =  ( x  e.  X  |->  A )
1110fmpt 5859 . . . . . 6  |-  ( A. x  e.  X  A  e.  ( Base `  G
)  <->  ( x  e.  X  |->  A ) : X --> ( Base `  G
) )
129, 11sylibr 212 . . . . 5  |-  ( ph  ->  A. x  e.  X  A  e.  ( Base `  G ) )
1312r19.21bi 2809 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  A  e.  ( Base `  G
) )
14 cnmpt1plusg.b . . . . . . 7  |-  ( ph  ->  ( x  e.  X  |->  B )  e.  ( K  Cn  J ) )
15 cnf2 18828 . . . . . . 7  |-  ( ( K  e.  (TopOn `  X )  /\  J  e.  (TopOn `  ( Base `  G ) )  /\  ( x  e.  X  |->  B )  e.  ( K  Cn  J ) )  ->  ( x  e.  X  |->  B ) : X --> ( Base `  G ) )
161, 6, 14, 15syl3anc 1218 . . . . . 6  |-  ( ph  ->  ( x  e.  X  |->  B ) : X --> ( Base `  G )
)
17 eqid 2438 . . . . . . 7  |-  ( x  e.  X  |->  B )  =  ( x  e.  X  |->  B )
1817fmpt 5859 . . . . . 6  |-  ( A. x  e.  X  B  e.  ( Base `  G
)  <->  ( x  e.  X  |->  B ) : X --> ( Base `  G
) )
1916, 18sylibr 212 . . . . 5  |-  ( ph  ->  A. x  e.  X  B  e.  ( Base `  G ) )
2019r19.21bi 2809 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  B  e.  ( Base `  G
) )
21 cnmpt1plusg.p . . . . 5  |-  .+  =  ( +g  `  G )
22 eqid 2438 . . . . 5  |-  ( +f `  G )  =  ( +f `  G )
234, 21, 22plusfval 15420 . . . 4  |-  ( ( A  e.  ( Base `  G )  /\  B  e.  ( Base `  G
) )  ->  ( A ( +f `  G ) B )  =  ( A  .+  B ) )
2413, 20, 23syl2anc 661 . . 3  |-  ( (
ph  /\  x  e.  X )  ->  ( A ( +f `  G ) B )  =  ( A  .+  B ) )
2524mpteq2dva 4373 . 2  |-  ( ph  ->  ( x  e.  X  |->  ( A ( +f `  G ) B ) )  =  ( x  e.  X  |->  ( A  .+  B
) ) )
263, 22tmdcn 19629 . . . 4  |-  ( G  e. TopMnd  ->  ( +f `  G )  e.  ( ( J  tX  J
)  Cn  J ) )
272, 26syl 16 . . 3  |-  ( ph  ->  ( +f `  G )  e.  ( ( J  tX  J
)  Cn  J ) )
281, 7, 14, 27cnmpt12f 19214 . 2  |-  ( ph  ->  ( x  e.  X  |->  ( A ( +f `  G ) B ) )  e.  ( K  Cn  J
) )
2925, 28eqeltrrd 2513 1  |-  ( ph  ->  ( x  e.  X  |->  ( A  .+  B
) )  e.  ( K  Cn  J ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2710    e. cmpt 4345   -->wf 5409   ` cfv 5413  (class class class)co 6086   Basecbs 14166   +g cplusg 14230   TopOpenctopn 14352   +fcplusf 15404  TopOnctopon 18474    Cn ccn 18803    tX ctx 19108  TopMndctmd 19616
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-fv 5421  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-1st 6572  df-2nd 6573  df-map 7208  df-topgen 14374  df-plusf 15408  df-top 18478  df-bases 18480  df-topon 18481  df-topsp 18482  df-cn 18806  df-tx 19110  df-tmd 19618
This theorem is referenced by:  tmdmulg  19638  tmdgsum  19641  tmdlactcn  19648  clsnsg  19655  tgpt0  19664  cnmpt1mulr  19731
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