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

Theorem ghmplusg 18564
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 2799 . 2 (Base‘𝑀) = (Base‘𝑀)
2 eqid 2799 . 2 (Base‘𝑁) = (Base‘𝑁)
3 eqid 2799 . 2 (+g𝑀) = (+g𝑀)
4 ghmplusg.p . 2 + = (+g𝑁)
5 ghmgrp1 17975 . . 3 (𝐺 ∈ (𝑀 GrpHom 𝑁) → 𝑀 ∈ Grp)
653ad2ant3 1166 . 2 ((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) → 𝑀 ∈ Grp)
7 ghmgrp2 17976 . . 3 (𝐺 ∈ (𝑀 GrpHom 𝑁) → 𝑁 ∈ Grp)
873ad2ant3 1166 . 2 ((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) → 𝑁 ∈ Grp)
92, 4grpcl 17746 . . . . 5 ((𝑁 ∈ Grp ∧ 𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁)) → (𝑥 + 𝑦) ∈ (Base‘𝑁))
1093expb 1150 . . . 4 ((𝑁 ∈ Grp ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → (𝑥 + 𝑦) ∈ (Base‘𝑁))
118, 10sylan 576 . . 3 (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → (𝑥 + 𝑦) ∈ (Base‘𝑁))
121, 2ghmf 17977 . . . 4 (𝐹 ∈ (𝑀 GrpHom 𝑁) → 𝐹:(Base‘𝑀)⟶(Base‘𝑁))
13123ad2ant2 1165 . . 3 ((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) → 𝐹:(Base‘𝑀)⟶(Base‘𝑁))
141, 2ghmf 17977 . . . 4 (𝐺 ∈ (𝑀 GrpHom 𝑁) → 𝐺:(Base‘𝑀)⟶(Base‘𝑁))
15143ad2ant3 1166 . . 3 ((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) → 𝐺:(Base‘𝑀)⟶(Base‘𝑁))
16 fvexd 6426 . . 3 ((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) → (Base‘𝑀) ∈ V)
17 inidm 4018 . . 3 ((Base‘𝑀) ∩ (Base‘𝑀)) = (Base‘𝑀)
1811, 13, 15, 16, 16, 17off 7146 . 2 ((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) → (𝐹𝑓 + 𝐺):(Base‘𝑀)⟶(Base‘𝑁))
191, 3, 4ghmlin 17978 . . . . . . 7 ((𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝐹‘(𝑥(+g𝑀)𝑦)) = ((𝐹𝑥) + (𝐹𝑦)))
20193expb 1150 . . . . . 6 ((𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → (𝐹‘(𝑥(+g𝑀)𝑦)) = ((𝐹𝑥) + (𝐹𝑦)))
21203ad2antl2 1238 . . . . 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 17978 . . . . . . 7 ((𝐺 ∈ (𝑀 GrpHom 𝑁) ∧ 𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝐺‘(𝑥(+g𝑀)𝑦)) = ((𝐺𝑥) + (𝐺𝑦)))
23223expb 1150 . . . . . 6 ((𝐺 ∈ (𝑀 GrpHom 𝑁) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → (𝐺‘(𝑥(+g𝑀)𝑦)) = ((𝐺𝑥) + (𝐺𝑦)))
24233ad2antl3 1239 . . . . 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 6896 . . . 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 1243 . . . . . 6 (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → 𝑁 ∈ Abel)
27 ablcmn 18514 . . . . . 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)
2913ffvelrnda 6585 . . . . . 6 (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ 𝑥 ∈ (Base‘𝑀)) → (𝐹𝑥) ∈ (Base‘𝑁))
3029adantrr 709 . . . . 5 (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → (𝐹𝑥) ∈ (Base‘𝑁))
3113ffvelrnda 6585 . . . . . 6 (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝐹𝑦) ∈ (Base‘𝑁))
3231adantrl 708 . . . . 5 (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → (𝐹𝑦) ∈ (Base‘𝑁))
3315ffvelrnda 6585 . . . . . 6 (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ 𝑥 ∈ (Base‘𝑀)) → (𝐺𝑥) ∈ (Base‘𝑁))
3433adantrr 709 . . . . 5 (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → (𝐺𝑥) ∈ (Base‘𝑁))
3515ffvelrnda 6585 . . . . . 6 (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝐺𝑦) ∈ (Base‘𝑁))
3635adantrl 708 . . . . 5 (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → (𝐺𝑦) ∈ (Base‘𝑁))
372, 4cmn4 18527 . . . . 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 1499 . . . 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 2833 . . 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 6257 . . . . 5 ((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) → 𝐹 Fn (Base‘𝑀))
4140adantr 473 . . . 4 (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → 𝐹 Fn (Base‘𝑀))
4215ffnd 6257 . . . . 5 ((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) → 𝐺 Fn (Base‘𝑀))
4342adantr 473 . . . 4 (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → 𝐺 Fn (Base‘𝑀))
44 fvexd 6426 . . . 4 (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → (Base‘𝑀) ∈ V)
451, 3grpcl 17746 . . . . . 6 ((𝑀 ∈ Grp ∧ 𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝑥(+g𝑀)𝑦) ∈ (Base‘𝑀))
46453expb 1150 . . . . 5 ((𝑀 ∈ Grp ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → (𝑥(+g𝑀)𝑦) ∈ (Base‘𝑀))
476, 46sylan 576 . . . 4 (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → (𝑥(+g𝑀)𝑦) ∈ (Base‘𝑀))
48 fnfvof 7145 . . . 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 868 . . 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 788 . . . . 5 (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → 𝑥 ∈ (Base‘𝑀))
51 fnfvof 7145 . . . . 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 868 . . . 4 (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → ((𝐹𝑓 + 𝐺)‘𝑥) = ((𝐹𝑥) + (𝐺𝑥)))
53 simprr 790 . . . . 5 (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → 𝑦 ∈ (Base‘𝑀))
54 fnfvof 7145 . . . . 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 868 . . . 4 (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → ((𝐹𝑓 + 𝐺)‘𝑦) = ((𝐹𝑦) + (𝐺𝑦)))
5652, 55oveq12d 6896 . . 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 2843 . 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 17982 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 385  w3a 1108   = wceq 1653  wcel 2157  Vcvv 3385   Fn wfn 6096  wf 6097  cfv 6101  (class class class)co 6878  𝑓 cof 7129  Basecbs 16184  +gcplusg 16267  Grpcgrp 17738   GrpHom cghm 17970  CMndccmn 18508  Abelcabl 18509
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-of 7131  df-mgm 17557  df-sgrp 17599  df-mnd 17610  df-grp 17741  df-ghm 17971  df-cmn 18510  df-abl 18511
This theorem is referenced by:  lmhmplusg  19365  nmotri  22871  nghmplusg  22872
  Copyright terms: Public domain W3C validator