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Theorem cnmpt1plusg 18109
Description: Continuity of the group sum; analogue of cnmpt12f 17690 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 2435 . . . . . . . . 9  |-  ( Base `  G )  =  (
Base `  G )
53, 4tmdtopon 18103 . . . . . . . 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 17305 . . . . . . 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 2435 . . . . . . 7  |-  ( x  e.  X  |->  A )  =  ( x  e.  X  |->  A )
1110fmpt 5882 . . . . . 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 2796 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  A  e.  ( Base `  G
) )
14 cnmpt1plusg.b . . . . . . 7  |-  ( ph  ->  ( x  e.  X  |->  B )  e.  ( K  Cn  J ) )
15 cnf2 17305 . . . . . . 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 2435 . . . . . . 7  |-  ( x  e.  X  |->  B )  =  ( x  e.  X  |->  B )
1817fmpt 5882 . . . . . 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 2796 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  B  e.  ( Base `  G
) )
21 cnmpt1plusg.p . . . . 5  |-  .+  =  ( +g  `  G )
22 eqid 2435 . . . . 5  |-  ( + f `  G )  =  ( + f `  G )
234, 21, 22plusfval 14695 . . . 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 4287 . 2  |-  ( ph  ->  ( x  e.  X  |->  ( A ( + f `  G ) B ) )  =  ( x  e.  X  |->  ( A  .+  B
) ) )
263, 22tmdcn 18105 . . . 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 17690 . 2  |-  ( ph  ->  ( x  e.  X  |->  ( A ( + f `  G ) B ) )  e.  ( K  Cn  J
) )
2925, 28eqeltrrd 2510 1  |-  ( ph  ->  ( x  e.  X  |->  ( A  .+  B
) )  e.  ( K  Cn  J ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2697    e. cmpt 4258   -->wf 5442   ` cfv 5446  (class class class)co 6073   Basecbs 13461   +g cplusg 13521   TopOpenctopn 13641   + fcplusf 14679  TopOnctopon 16951    Cn ccn 17280    tX ctx 17584  TopMndctmd 18092
This theorem is referenced by:  tmdmulg  18114  tmdgsum  18117  tmdlactcn  18124  clsnsg  18131  tgpt0  18140  cnmpt1mulr  18203
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-map 7012  df-topgen 13659  df-plusf 14683  df-top 16955  df-bases 16957  df-topon 16958  df-topsp 16959  df-cn 17283  df-tx 17586  df-tmd 18094
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