MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ghmplusg Structured version   Visualization version   GIF version

Theorem ghmplusg 19879
Description: The pointwise sum of two linear functions is linear. (Contributed by Stefan O'Rear, 5-Sep-2015.)
Hypothesis
Ref Expression
ghmplusg.p + = (+g𝑁)
Assertion
Ref Expression
ghmplusg ((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) → (𝐹f + 𝐺) ∈ (𝑀 GrpHom 𝑁))

Proof of Theorem ghmplusg
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2735 . 2 (Base‘𝑀) = (Base‘𝑀)
2 eqid 2735 . 2 (Base‘𝑁) = (Base‘𝑁)
3 eqid 2735 . 2 (+g𝑀) = (+g𝑀)
4 ghmplusg.p . 2 + = (+g𝑁)
5 ghmgrp1 19249 . . 3 (𝐺 ∈ (𝑀 GrpHom 𝑁) → 𝑀 ∈ Grp)
653ad2ant3 1134 . 2 ((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) → 𝑀 ∈ Grp)
7 ghmgrp2 19250 . . 3 (𝐺 ∈ (𝑀 GrpHom 𝑁) → 𝑁 ∈ Grp)
873ad2ant3 1134 . 2 ((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) → 𝑁 ∈ Grp)
92, 4grpcl 18972 . . . . 5 ((𝑁 ∈ Grp ∧ 𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁)) → (𝑥 + 𝑦) ∈ (Base‘𝑁))
1093expb 1119 . . . 4 ((𝑁 ∈ Grp ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → (𝑥 + 𝑦) ∈ (Base‘𝑁))
118, 10sylan 580 . . 3 (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → (𝑥 + 𝑦) ∈ (Base‘𝑁))
121, 2ghmf 19251 . . . 4 (𝐹 ∈ (𝑀 GrpHom 𝑁) → 𝐹:(Base‘𝑀)⟶(Base‘𝑁))
13123ad2ant2 1133 . . 3 ((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) → 𝐹:(Base‘𝑀)⟶(Base‘𝑁))
141, 2ghmf 19251 . . . 4 (𝐺 ∈ (𝑀 GrpHom 𝑁) → 𝐺:(Base‘𝑀)⟶(Base‘𝑁))
15143ad2ant3 1134 . . 3 ((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) → 𝐺:(Base‘𝑀)⟶(Base‘𝑁))
16 fvexd 6922 . . 3 ((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) → (Base‘𝑀) ∈ V)
17 inidm 4235 . . 3 ((Base‘𝑀) ∩ (Base‘𝑀)) = (Base‘𝑀)
1811, 13, 15, 16, 16, 17off 7715 . 2 ((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) → (𝐹f + 𝐺):(Base‘𝑀)⟶(Base‘𝑁))
191, 3, 4ghmlin 19252 . . . . . . 7 ((𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝐹‘(𝑥(+g𝑀)𝑦)) = ((𝐹𝑥) + (𝐹𝑦)))
20193expb 1119 . . . . . 6 ((𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → (𝐹‘(𝑥(+g𝑀)𝑦)) = ((𝐹𝑥) + (𝐹𝑦)))
21203ad2antl2 1185 . . . . 5 (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → (𝐹‘(𝑥(+g𝑀)𝑦)) = ((𝐹𝑥) + (𝐹𝑦)))
221, 3, 4ghmlin 19252 . . . . . . 7 ((𝐺 ∈ (𝑀 GrpHom 𝑁) ∧ 𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝐺‘(𝑥(+g𝑀)𝑦)) = ((𝐺𝑥) + (𝐺𝑦)))
23223expb 1119 . . . . . 6 ((𝐺 ∈ (𝑀 GrpHom 𝑁) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → (𝐺‘(𝑥(+g𝑀)𝑦)) = ((𝐺𝑥) + (𝐺𝑦)))
24233ad2antl3 1186 . . . . 5 (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → (𝐺‘(𝑥(+g𝑀)𝑦)) = ((𝐺𝑥) + (𝐺𝑦)))
2521, 24oveq12d 7449 . . . 4 (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → ((𝐹‘(𝑥(+g𝑀)𝑦)) + (𝐺‘(𝑥(+g𝑀)𝑦))) = (((𝐹𝑥) + (𝐹𝑦)) + ((𝐺𝑥) + (𝐺𝑦))))
26 simpl1 1190 . . . . . 6 (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → 𝑁 ∈ Abel)
27 ablcmn 19820 . . . . . 6 (𝑁 ∈ Abel → 𝑁 ∈ CMnd)
2826, 27syl 17 . . . . 5 (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → 𝑁 ∈ CMnd)
2913ffvelcdmda 7104 . . . . . 6 (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ 𝑥 ∈ (Base‘𝑀)) → (𝐹𝑥) ∈ (Base‘𝑁))
3029adantrr 717 . . . . 5 (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → (𝐹𝑥) ∈ (Base‘𝑁))
3113ffvelcdmda 7104 . . . . . 6 (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝐹𝑦) ∈ (Base‘𝑁))
3231adantrl 716 . . . . 5 (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → (𝐹𝑦) ∈ (Base‘𝑁))
3315ffvelcdmda 7104 . . . . . 6 (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ 𝑥 ∈ (Base‘𝑀)) → (𝐺𝑥) ∈ (Base‘𝑁))
3433adantrr 717 . . . . 5 (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → (𝐺𝑥) ∈ (Base‘𝑁))
3515ffvelcdmda 7104 . . . . . 6 (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝐺𝑦) ∈ (Base‘𝑁))
3635adantrl 716 . . . . 5 (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → (𝐺𝑦) ∈ (Base‘𝑁))
372, 4cmn4 19834 . . . . 5 ((𝑁 ∈ CMnd ∧ ((𝐹𝑥) ∈ (Base‘𝑁) ∧ (𝐹𝑦) ∈ (Base‘𝑁)) ∧ ((𝐺𝑥) ∈ (Base‘𝑁) ∧ (𝐺𝑦) ∈ (Base‘𝑁))) → (((𝐹𝑥) + (𝐹𝑦)) + ((𝐺𝑥) + (𝐺𝑦))) = (((𝐹𝑥) + (𝐺𝑥)) + ((𝐹𝑦) + (𝐺𝑦))))
3828, 30, 32, 34, 36, 37syl122anc 1378 . . . 4 (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → (((𝐹𝑥) + (𝐹𝑦)) + ((𝐺𝑥) + (𝐺𝑦))) = (((𝐹𝑥) + (𝐺𝑥)) + ((𝐹𝑦) + (𝐺𝑦))))
3925, 38eqtrd 2775 . . 3 (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → ((𝐹‘(𝑥(+g𝑀)𝑦)) + (𝐺‘(𝑥(+g𝑀)𝑦))) = (((𝐹𝑥) + (𝐺𝑥)) + ((𝐹𝑦) + (𝐺𝑦))))
4013ffnd 6738 . . . . 5 ((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) → 𝐹 Fn (Base‘𝑀))
4140adantr 480 . . . 4 (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → 𝐹 Fn (Base‘𝑀))
4215ffnd 6738 . . . . 5 ((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) → 𝐺 Fn (Base‘𝑀))
4342adantr 480 . . . 4 (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → 𝐺 Fn (Base‘𝑀))
44 fvexd 6922 . . . 4 (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → (Base‘𝑀) ∈ V)
451, 3grpcl 18972 . . . . . 6 ((𝑀 ∈ Grp ∧ 𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝑥(+g𝑀)𝑦) ∈ (Base‘𝑀))
46453expb 1119 . . . . 5 ((𝑀 ∈ Grp ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → (𝑥(+g𝑀)𝑦) ∈ (Base‘𝑀))
476, 46sylan 580 . . . 4 (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → (𝑥(+g𝑀)𝑦) ∈ (Base‘𝑀))
48 fnfvof 7714 . . . 4 (((𝐹 Fn (Base‘𝑀) ∧ 𝐺 Fn (Base‘𝑀)) ∧ ((Base‘𝑀) ∈ V ∧ (𝑥(+g𝑀)𝑦) ∈ (Base‘𝑀))) → ((𝐹f + 𝐺)‘(𝑥(+g𝑀)𝑦)) = ((𝐹‘(𝑥(+g𝑀)𝑦)) + (𝐺‘(𝑥(+g𝑀)𝑦))))
4941, 43, 44, 47, 48syl22anc 839 . . 3 (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → ((𝐹f + 𝐺)‘(𝑥(+g𝑀)𝑦)) = ((𝐹‘(𝑥(+g𝑀)𝑦)) + (𝐺‘(𝑥(+g𝑀)𝑦))))
50 simprl 771 . . . . 5 (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → 𝑥 ∈ (Base‘𝑀))
51 fnfvof 7714 . . . . 5 (((𝐹 Fn (Base‘𝑀) ∧ 𝐺 Fn (Base‘𝑀)) ∧ ((Base‘𝑀) ∈ V ∧ 𝑥 ∈ (Base‘𝑀))) → ((𝐹f + 𝐺)‘𝑥) = ((𝐹𝑥) + (𝐺𝑥)))
5241, 43, 44, 50, 51syl22anc 839 . . . 4 (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → ((𝐹f + 𝐺)‘𝑥) = ((𝐹𝑥) + (𝐺𝑥)))
53 simprr 773 . . . . 5 (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → 𝑦 ∈ (Base‘𝑀))
54 fnfvof 7714 . . . . 5 (((𝐹 Fn (Base‘𝑀) ∧ 𝐺 Fn (Base‘𝑀)) ∧ ((Base‘𝑀) ∈ V ∧ 𝑦 ∈ (Base‘𝑀))) → ((𝐹f + 𝐺)‘𝑦) = ((𝐹𝑦) + (𝐺𝑦)))
5541, 43, 44, 53, 54syl22anc 839 . . . 4 (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → ((𝐹f + 𝐺)‘𝑦) = ((𝐹𝑦) + (𝐺𝑦)))
5652, 55oveq12d 7449 . . 3 (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → (((𝐹f + 𝐺)‘𝑥) + ((𝐹f + 𝐺)‘𝑦)) = (((𝐹𝑥) + (𝐺𝑥)) + ((𝐹𝑦) + (𝐺𝑦))))
5739, 49, 563eqtr4d 2785 . 2 (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → ((𝐹f + 𝐺)‘(𝑥(+g𝑀)𝑦)) = (((𝐹f + 𝐺)‘𝑥) + ((𝐹f + 𝐺)‘𝑦)))
581, 2, 3, 4, 6, 8, 18, 57isghmd 19256 1 ((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) → (𝐹f + 𝐺) ∈ (𝑀 GrpHom 𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1537  wcel 2106  Vcvv 3478   Fn wfn 6558  wf 6559  cfv 6563  (class class class)co 7431  f cof 7695  Basecbs 17245  +gcplusg 17298  Grpcgrp 18964   GrpHom cghm 19243  CMndccmn 19813  Abelcabl 19814
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-1st 8013  df-2nd 8014  df-map 8867  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-grp 18967  df-ghm 19244  df-cmn 19815  df-abl 19816
This theorem is referenced by:  lmhmplusg  21061  nmotri  24776  nghmplusg  24777
  Copyright terms: Public domain W3C validator