MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cnmpt1plusg Structured version   Unicode version

Theorem cnmpt1plusg 20712
Description: Continuity of the group sum; analogue of cnmpt12f 20293 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 2457 . . . . . . . . 9  |-  ( Base `  G )  =  (
Base `  G )
53, 4tmdtopon 20706 . . . . . . . 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 19877 . . . . . . 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 1228 . . . . . 6  |-  ( ph  ->  ( x  e.  X  |->  A ) : X --> ( Base `  G )
)
10 eqid 2457 . . . . . . 7  |-  ( x  e.  X  |->  A )  =  ( x  e.  X  |->  A )
1110fmpt 6053 . . . . . 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 2826 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  A  e.  ( Base `  G
) )
14 cnmpt1plusg.b . . . . . . 7  |-  ( ph  ->  ( x  e.  X  |->  B )  e.  ( K  Cn  J ) )
15 cnf2 19877 . . . . . . 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 1228 . . . . . 6  |-  ( ph  ->  ( x  e.  X  |->  B ) : X --> ( Base `  G )
)
17 eqid 2457 . . . . . . 7  |-  ( x  e.  X  |->  B )  =  ( x  e.  X  |->  B )
1817fmpt 6053 . . . . . 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 2826 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  B  e.  ( Base `  G
) )
21 cnmpt1plusg.p . . . . 5  |-  .+  =  ( +g  `  G )
22 eqid 2457 . . . . 5  |-  ( +f `  G )  =  ( +f `  G )
234, 21, 22plusfval 16005 . . . 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 4543 . 2  |-  ( ph  ->  ( x  e.  X  |->  ( A ( +f `  G ) B ) )  =  ( x  e.  X  |->  ( A  .+  B
) ) )
263, 22tmdcn 20708 . . . 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 20293 . 2  |-  ( ph  ->  ( x  e.  X  |->  ( A ( +f `  G ) B ) )  e.  ( K  Cn  J
) )
2925, 28eqeltrrd 2546 1  |-  ( ph  ->  ( x  e.  X  |->  ( A  .+  B
) )  e.  ( K  Cn  J ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1395    e. wcel 1819   A.wral 2807    |-> cmpt 4515   -->wf 5590   ` cfv 5594  (class class class)co 6296   Basecbs 14644   +g cplusg 14712   TopOpenctopn 14839   +fcplusf 15996  TopOnctopon 19522    Cn ccn 19852    tX ctx 20187  TopMndctmd 20695
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800  df-map 7440  df-topgen 14861  df-plusf 15998  df-top 19526  df-bases 19528  df-topon 19529  df-topsp 19530  df-cn 19855  df-tx 20189  df-tmd 20697
This theorem is referenced by:  tmdmulg  20717  tmdgsum  20720  tmdlactcn  20727  clsnsg  20734  tgpt0  20743  cnmpt1mulr  20810
  Copyright terms: Public domain W3C validator