Theorem cnmpt1plusg 21891

Description: Continuity of the group sum; analogue of cnmpt12f 21469 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

Step	Hyp	Ref	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 2622	. . . . . . . . 9 |- ( Base ` G ) = ( Base ` G )
5	3, 4	tmdtopon 21885	. . . . . . . 8 |- ( G e. TopMnd -> J e. (TopOn ` ( Base ` G ) ) )
6	2, 5	syl 17	. . . . . . 7 |- ( ph -> J e. (TopOn ` ( Base ` G ) ) )
7		cnmpt1plusg.a	. . . . . . 7 |- ( ph -> ( x e. X |-> A ) e. ( K Cn J ) )
8		cnf2 21053	. . . . . . 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 ) )
9	1, 6, 7, 8	syl3anc 1326	. . . . . 6 |- ( ph -> ( x e. X |-> A ) : X --> ( Base ` G ) )
10		eqid 2622	. . . . . . 7 |- ( x e. X |-> A ) = ( x e. X |-> A )
11	10	fmpt 6381	. . . . . 6 |- ( A. x e. X A e. ( Base ` G ) <-> ( x e. X |-> A ) : X --> ( Base ` G ) )
12	9, 11	sylibr 224	. . . . 5 |- ( ph -> A. x e. X A e. ( Base ` G ) )
13	12	r19.21bi 2932	. . . 4 |- ( ( ph /\ x e. X ) -> A e. ( Base ` G ) )
14		cnmpt1plusg.b	. . . . . . 7 |- ( ph -> ( x e. X |-> B ) e. ( K Cn J ) )
15		cnf2 21053	. . . . . . 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 ) )
16	1, 6, 14, 15	syl3anc 1326	. . . . . 6 |- ( ph -> ( x e. X |-> B ) : X --> ( Base ` G ) )
17		eqid 2622	. . . . . . 7 |- ( x e. X |-> B ) = ( x e. X |-> B )
18	17	fmpt 6381	. . . . . 6 |- ( A. x e. X B e. ( Base ` G ) <-> ( x e. X |-> B ) : X --> ( Base ` G ) )
19	16, 18	sylibr 224	. . . . 5 |- ( ph -> A. x e. X B e. ( Base ` G ) )
20	19	r19.21bi 2932	. . . 4 |- ( ( ph /\ x e. X ) -> B e. ( Base ` G ) )
21		cnmpt1plusg.p	. . . . 5 |- .+ = ( +g ` G )
22		eqid 2622	. . . . 5 |- ( +f ` G ) = ( +f ` G )
23	4, 21, 22	plusfval 17248	. . . 4 |- ( ( A e. ( Base ` G ) /\ B e. ( Base ` G ) ) -> ( A ( +f ` G ) B ) = ( A .+ B ) )
24	13, 20, 23	syl2anc 693	. . 3 |- ( ( ph /\ x e. X ) -> ( A ( +f ` G ) B ) = ( A .+ B ) )
25	24	mpteq2dva 4744	. 2 |- ( ph -> ( x e. X |-> ( A ( +f ` G ) B ) ) = ( x e. X |-> ( A .+ B ) ) )
26	3, 22	tmdcn 21887	. . . 4 |- ( G e. TopMnd -> ( +f ` G ) e. ( ( J tX J ) Cn J ) )
27	2, 26	syl 17	. . 3 |- ( ph -> ( +f ` G ) e. ( ( J tX J ) Cn J ) )
28	1, 7, 14, 27	cnmpt12f 21469	. 2 |- ( ph -> ( x e. X |-> ( A ( +f ` G ) B ) ) e. ( K Cn J ) )
29	25, 28	eqeltrrd 2702	1 |- ( ph -> ( x e. X |-> ( A .+ B ) ) e. ( K Cn J ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   |-> cmpt 4729   -->wf 5884   ` cfv 5888  (class class class)co 6650   Basecbs 15857   +g cplusg 15941   TopOpenctopn 16082   +fcplusf 17239  TopOnctopon 20715   Cn ccn 21028   tX ctx 21363  TopMndctmd 21874

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-map 7859  df-topgen 16104  df-plusf 17241  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cn 21031  df-tx 21365  df-tmd 21876

This theorem is referenced by:  tmdmulg  21896  tmdgsum  21899  tmdlactcn  21906  clsnsg  21913  tgpt0  21922  cnmpt1mulr  21985