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Theorem cnmpt1plusg 18038
Description: Continuity of the group sum; analogue of cnmpt12f 17619 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 2387 . . . . . . . . 9  |-  ( Base `  G )  =  (
Base `  G )
53, 4tmdtopon 18032 . . . . . . . 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 17235 . . . . . . 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 1184 . . . . . 6  |-  ( ph  ->  ( x  e.  X  |->  A ) : X --> ( Base `  G )
)
10 eqid 2387 . . . . . . 7  |-  ( x  e.  X  |->  A )  =  ( x  e.  X  |->  A )
1110fmpt 5829 . . . . . 6  |-  ( A. x  e.  X  A  e.  ( Base `  G
)  <->  ( x  e.  X  |->  A ) : X --> ( Base `  G
) )
129, 11sylibr 204 . . . . 5  |-  ( ph  ->  A. x  e.  X  A  e.  ( Base `  G ) )
1312r19.21bi 2747 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  A  e.  ( Base `  G
) )
14 cnmpt1plusg.b . . . . . . 7  |-  ( ph  ->  ( x  e.  X  |->  B )  e.  ( K  Cn  J ) )
15 cnf2 17235 . . . . . . 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 1184 . . . . . 6  |-  ( ph  ->  ( x  e.  X  |->  B ) : X --> ( Base `  G )
)
17 eqid 2387 . . . . . . 7  |-  ( x  e.  X  |->  B )  =  ( x  e.  X  |->  B )
1817fmpt 5829 . . . . . 6  |-  ( A. x  e.  X  B  e.  ( Base `  G
)  <->  ( x  e.  X  |->  B ) : X --> ( Base `  G
) )
1916, 18sylibr 204 . . . . 5  |-  ( ph  ->  A. x  e.  X  B  e.  ( Base `  G ) )
2019r19.21bi 2747 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  B  e.  ( Base `  G
) )
21 cnmpt1plusg.p . . . . 5  |-  .+  =  ( +g  `  G )
22 eqid 2387 . . . . 5  |-  ( + f `  G )  =  ( + f `  G )
234, 21, 22plusfval 14630 . . . 4  |-  ( ( A  e.  ( Base `  G )  /\  B  e.  ( Base `  G
) )  ->  ( A ( + f `  G ) B )  =  ( A  .+  B ) )
2413, 20, 23syl2anc 643 . . 3  |-  ( (
ph  /\  x  e.  X )  ->  ( A ( + f `  G ) B )  =  ( A  .+  B ) )
2524mpteq2dva 4236 . 2  |-  ( ph  ->  ( x  e.  X  |->  ( A ( + f `  G ) B ) )  =  ( x  e.  X  |->  ( A  .+  B
) ) )
263, 22tmdcn 18034 . . . 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 17619 . 2  |-  ( ph  ->  ( x  e.  X  |->  ( A ( + f `  G ) B ) )  e.  ( K  Cn  J
) )
2925, 28eqeltrrd 2462 1  |-  ( ph  ->  ( x  e.  X  |->  ( A  .+  B
) )  e.  ( K  Cn  J ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   A.wral 2649    e. cmpt 4207   -->wf 5390   ` cfv 5394  (class class class)co 6020   Basecbs 13396   +g cplusg 13456   TopOpenctopn 13576   + fcplusf 14614  TopOnctopon 16882    Cn ccn 17210    tX ctx 17513  TopMndctmd 18021
This theorem is referenced by:  tmdmulg  18043  tmdgsum  18046  tmdlactcn  18053  clsnsg  18060  tgpt0  18069  cnmpt1mulr  18132
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-map 6956  df-topgen 13594  df-plusf 14618  df-top 16886  df-bases 16888  df-topon 16889  df-topsp 16890  df-cn 17213  df-tx 17515  df-tmd 18023
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