Theorem ofaddmndmap 42122

Description: The function operation applied to the addition for functions (with the same domain) into a monoid is a function (with the same domain) into the monoid. (Contributed by AV, 6-Apr-2019.)

Hypotheses

Ref	Expression
ofaddmndmap.r	|- R = ( Base ` M )
ofaddmndmap.p	|- .+ = ( +g ` M )

Assertion

Ref	Expression
ofaddmndmap	|- ( ( M e. Mnd /\ V e. Y /\ ( A e. ( R ^m V ) /\ B e. ( R ^m V ) ) ) -> ( A oF .+ B ) e. ( R ^m V ) )

Proof of Theorem ofaddmndmap

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl1 1064	. . . 4 |- ( ( ( M e. Mnd /\ V e. Y /\ ( A e. ( R ^m V ) /\ B e. ( R ^m V ) ) ) /\ ( x e. R /\ y e. R ) ) -> M e. Mnd )
2		simprl 794	. . . 4 |- ( ( ( M e. Mnd /\ V e. Y /\ ( A e. ( R ^m V ) /\ B e. ( R ^m V ) ) ) /\ ( x e. R /\ y e. R ) ) -> x e. R )
3		simprr 796	. . . 4 |- ( ( ( M e. Mnd /\ V e. Y /\ ( A e. ( R ^m V ) /\ B e. ( R ^m V ) ) ) /\ ( x e. R /\ y e. R ) ) -> y e. R )
4		ofaddmndmap.r	. . . . 5 |- R = ( Base ` M )
5		ofaddmndmap.p	. . . . 5 |- .+ = ( +g ` M )
6	4, 5	mndcl 17301	. . . 4 |- ( ( M e. Mnd /\ x e. R /\ y e. R ) -> ( x .+ y ) e. R )
7	1, 2, 3, 6	syl3anc 1326	. . 3 |- ( ( ( M e. Mnd /\ V e. Y /\ ( A e. ( R ^m V ) /\ B e. ( R ^m V ) ) ) /\ ( x e. R /\ y e. R ) ) -> ( x .+ y ) e. R )
8		elmapi 7879	. . . . 5 |- ( A e. ( R ^m V ) -> A : V --> R )
9	8	adantr 481	. . . 4 |- ( ( A e. ( R ^m V ) /\ B e. ( R ^m V ) ) -> A : V --> R )
10	9	3ad2ant3 1084	. . 3 |- ( ( M e. Mnd /\ V e. Y /\ ( A e. ( R ^m V ) /\ B e. ( R ^m V ) ) ) -> A : V --> R )
11		elmapi 7879	. . . . 5 |- ( B e. ( R ^m V ) -> B : V --> R )
12	11	adantl 482	. . . 4 |- ( ( A e. ( R ^m V ) /\ B e. ( R ^m V ) ) -> B : V --> R )
13	12	3ad2ant3 1084	. . 3 |- ( ( M e. Mnd /\ V e. Y /\ ( A e. ( R ^m V ) /\ B e. ( R ^m V ) ) ) -> B : V --> R )
14		simp2 1062	. . 3 |- ( ( M e. Mnd /\ V e. Y /\ ( A e. ( R ^m V ) /\ B e. ( R ^m V ) ) ) -> V e. Y )
15		inidm 3822	. . 3 |- ( V i^i V ) = V
16	7, 10, 13, 14, 14, 15	off 6912	. 2 |- ( ( M e. Mnd /\ V e. Y /\ ( A e. ( R ^m V ) /\ B e. ( R ^m V ) ) ) -> ( A oF .+ B ) : V --> R )
17		fvex 6201	. . . 4 |- ( Base ` M ) e. _V
18	4, 17	eqeltri 2697	. . 3 |- R e. _V
19		elmapg 7870	. . 3 |- ( ( R e. _V /\ V e. Y ) -> ( ( A oF .+ B ) e. ( R ^m V ) <-> ( A oF .+ B ) : V --> R ) )
20	18, 14, 19	sylancr 695	. 2 |- ( ( M e. Mnd /\ V e. Y /\ ( A e. ( R ^m V ) /\ B e. ( R ^m V ) ) ) -> ( ( A oF .+ B ) e. ( R ^m V ) <-> ( A oF .+ B ) : V --> R ) )
21	16, 20	mpbird 247	1 |- ( ( M e. Mnd /\ V e. Y /\ ( A e. ( R ^m V ) /\ B e. ( R ^m V ) ) ) -> ( A oF .+ B ) e. ( R ^m V ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   _Vcvv 3200   -->wf 5884   ` cfv 5888  (class class class)co 6650   oFcof 6895   ^m cmap 7857   Basecbs 15857   +g cplusg 15941   Mndcmnd 17294

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-rep 4771  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-reu 2919  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-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-1st 7168  df-2nd 7169  df-map 7859  df-mgm 17242  df-sgrp 17284  df-mnd 17295

This theorem is referenced by:  lincsumcl  42220