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Theorem cnmpt1plusg 17770
Description: Continuity of the group sum; analogue of cnmpt12f 17360 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 2283 . . . . . . . . 9  |-  ( Base `  G )  =  (
Base `  G )
53, 4tmdtopon 17764 . . . . . . . 8  |-  ( G  e. TopMnd  ->  J  e.  (TopOn `  ( Base `  G
) ) )
62, 5syl 15 . . . . . . 7  |-  ( ph  ->  J  e.  (TopOn `  ( Base `  G )
) )
7 cnmpt1plusg.a . . . . . . 7  |-  ( ph  ->  ( x  e.  X  |->  A )  e.  ( K  Cn  J ) )
8 cnf2 16979 . . . . . . 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 1182 . . . . . 6  |-  ( ph  ->  ( x  e.  X  |->  A ) : X --> ( Base `  G )
)
10 eqid 2283 . . . . . . 7  |-  ( x  e.  X  |->  A )  =  ( x  e.  X  |->  A )
1110fmpt 5681 . . . . . 6  |-  ( A. x  e.  X  A  e.  ( Base `  G
)  <->  ( x  e.  X  |->  A ) : X --> ( Base `  G
) )
129, 11sylibr 203 . . . . 5  |-  ( ph  ->  A. x  e.  X  A  e.  ( Base `  G ) )
1312r19.21bi 2641 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  A  e.  ( Base `  G
) )
14 cnmpt1plusg.b . . . . . . 7  |-  ( ph  ->  ( x  e.  X  |->  B )  e.  ( K  Cn  J ) )
15 cnf2 16979 . . . . . . 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 1182 . . . . . 6  |-  ( ph  ->  ( x  e.  X  |->  B ) : X --> ( Base `  G )
)
17 eqid 2283 . . . . . . 7  |-  ( x  e.  X  |->  B )  =  ( x  e.  X  |->  B )
1817fmpt 5681 . . . . . 6  |-  ( A. x  e.  X  B  e.  ( Base `  G
)  <->  ( x  e.  X  |->  B ) : X --> ( Base `  G
) )
1916, 18sylibr 203 . . . . 5  |-  ( ph  ->  A. x  e.  X  B  e.  ( Base `  G ) )
2019r19.21bi 2641 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  B  e.  ( Base `  G
) )
21 cnmpt1plusg.p . . . . 5  |-  .+  =  ( +g  `  G )
22 eqid 2283 . . . . 5  |-  ( + f `  G )  =  ( + f `  G )
234, 21, 22plusfval 14380 . . . 4  |-  ( ( A  e.  ( Base `  G )  /\  B  e.  ( Base `  G
) )  ->  ( A ( + f `  G ) B )  =  ( A  .+  B ) )
2413, 20, 23syl2anc 642 . . 3  |-  ( (
ph  /\  x  e.  X )  ->  ( A ( + f `  G ) B )  =  ( A  .+  B ) )
2524mpteq2dva 4106 . 2  |-  ( ph  ->  ( x  e.  X  |->  ( A ( + f `  G ) B ) )  =  ( x  e.  X  |->  ( A  .+  B
) ) )
263, 22tmdcn 17766 . . . 4  |-  ( G  e. TopMnd  ->  ( + f `  G )  e.  ( ( J  tX  J
)  Cn  J ) )
272, 26syl 15 . . 3  |-  ( ph  ->  ( + f `  G )  e.  ( ( J  tX  J
)  Cn  J ) )
281, 7, 14, 27cnmpt12f 17360 . 2  |-  ( ph  ->  ( x  e.  X  |->  ( A ( + f `  G ) B ) )  e.  ( K  Cn  J
) )
2925, 28eqeltrrd 2358 1  |-  ( ph  ->  ( x  e.  X  |->  ( A  .+  B
) )  e.  ( K  Cn  J ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543    e. cmpt 4077   -->wf 5251   ` cfv 5255  (class class class)co 5858   Basecbs 13148   +g cplusg 13208   TopOpenctopn 13326   + fcplusf 14364  TopOnctopon 16632    Cn ccn 16954    tX ctx 17255  TopMndctmd 17753
This theorem is referenced by:  tmdmulg  17775  tmdgsum  17778  tmdlactcn  17785  clsnsg  17792  tgpt0  17801  cnmpt1mulr  17864
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-map 6774  df-topgen 13344  df-plusf 14368  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-cn 16957  df-tx 17257  df-tmd 17755
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