Metamath Proof Explorer


Theorem resmgmhm

Description: Restriction of a magma homomorphism to a submagma is a homomorphism. (Contributed by AV, 26-Feb-2020)

Ref Expression
Hypothesis resmgmhm.u U = S 𝑠 X
Assertion resmgmhm F S MgmHom T X SubMgm S F X U MgmHom T

Proof

Step Hyp Ref Expression
1 resmgmhm.u U = S 𝑠 X
2 mgmhmrcl F S MgmHom T S Mgm T Mgm
3 2 simprd F S MgmHom T T Mgm
4 1 submgmmgm X SubMgm S U Mgm
5 3 4 anim12ci F S MgmHom T X SubMgm S U Mgm T Mgm
6 eqid Base S = Base S
7 eqid Base T = Base T
8 6 7 mgmhmf F S MgmHom T F : Base S Base T
9 6 submgmss X SubMgm S X Base S
10 fssres F : Base S Base T X Base S F X : X Base T
11 8 9 10 syl2an F S MgmHom T X SubMgm S F X : X Base T
12 9 adantl F S MgmHom T X SubMgm S X Base S
13 1 6 ressbas2 X Base S X = Base U
14 12 13 syl F S MgmHom T X SubMgm S X = Base U
15 14 feq2d F S MgmHom T X SubMgm S F X : X Base T F X : Base U Base T
16 11 15 mpbid F S MgmHom T X SubMgm S F X : Base U Base T
17 simpll F S MgmHom T X SubMgm S x X y X F S MgmHom T
18 9 ad2antlr F S MgmHom T X SubMgm S x X y X X Base S
19 simprl F S MgmHom T X SubMgm S x X y X x X
20 18 19 sseldd F S MgmHom T X SubMgm S x X y X x Base S
21 simprr F S MgmHom T X SubMgm S x X y X y X
22 18 21 sseldd F S MgmHom T X SubMgm S x X y X y Base S
23 eqid + S = + S
24 eqid + T = + T
25 6 23 24 mgmhmlin F S MgmHom T x Base S y Base S F x + S y = F x + T F y
26 17 20 22 25 syl3anc F S MgmHom T X SubMgm S x X y X F x + S y = F x + T F y
27 23 submgmcl X SubMgm S x X y X x + S y X
28 27 3expb X SubMgm S x X y X x + S y X
29 28 adantll F S MgmHom T X SubMgm S x X y X x + S y X
30 fvres x + S y X F X x + S y = F x + S y
31 29 30 syl F S MgmHom T X SubMgm S x X y X F X x + S y = F x + S y
32 fvres x X F X x = F x
33 fvres y X F X y = F y
34 32 33 oveqan12d x X y X F X x + T F X y = F x + T F y
35 34 adantl F S MgmHom T X SubMgm S x X y X F X x + T F X y = F x + T F y
36 26 31 35 3eqtr4d F S MgmHom T X SubMgm S x X y X F X x + S y = F X x + T F X y
37 36 ralrimivva F S MgmHom T X SubMgm S x X y X F X x + S y = F X x + T F X y
38 1 23 ressplusg X SubMgm S + S = + U
39 38 adantl F S MgmHom T X SubMgm S + S = + U
40 39 oveqd F S MgmHom T X SubMgm S x + S y = x + U y
41 40 fveqeq2d F S MgmHom T X SubMgm S F X x + S y = F X x + T F X y F X x + U y = F X x + T F X y
42 14 41 raleqbidv F S MgmHom T X SubMgm S y X F X x + S y = F X x + T F X y y Base U F X x + U y = F X x + T F X y
43 14 42 raleqbidv F S MgmHom T X SubMgm S x X y X F X x + S y = F X x + T F X y x Base U y Base U F X x + U y = F X x + T F X y
44 37 43 mpbid F S MgmHom T X SubMgm S x Base U y Base U F X x + U y = F X x + T F X y
45 16 44 jca F S MgmHom T X SubMgm S F X : Base U Base T x Base U y Base U F X x + U y = F X x + T F X y
46 eqid Base U = Base U
47 eqid + U = + U
48 46 7 47 24 ismgmhm F X U MgmHom T U Mgm T Mgm F X : Base U Base T x Base U y Base U F X x + U y = F X x + T F X y
49 5 45 48 sylanbrc F S MgmHom T X SubMgm S F X U MgmHom T