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Theorem ghmplusg 18462
 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 2769 . 2 (Base‘𝑀) = (Base‘𝑀)
2 eqid 2769 . 2 (Base‘𝑁) = (Base‘𝑁)
3 eqid 2769 . 2 (+g𝑀) = (+g𝑀)
4 ghmplusg.p . 2 + = (+g𝑁)
5 ghmgrp1 17876 . . 3 (𝐺 ∈ (𝑀 GrpHom 𝑁) → 𝑀 ∈ Grp)
653ad2ant3 1127 . 2 ((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) → 𝑀 ∈ Grp)
7 ghmgrp2 17877 . . 3 (𝐺 ∈ (𝑀 GrpHom 𝑁) → 𝑁 ∈ Grp)
873ad2ant3 1127 . 2 ((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) → 𝑁 ∈ Grp)
92, 4grpcl 17644 . . . . 5 ((𝑁 ∈ Grp ∧ 𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁)) → (𝑥 + 𝑦) ∈ (Base‘𝑁))
1093expb 1111 . . . 4 ((𝑁 ∈ Grp ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → (𝑥 + 𝑦) ∈ (Base‘𝑁))
118, 10sylan 490 . . 3 (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → (𝑥 + 𝑦) ∈ (Base‘𝑁))
121, 2ghmf 17878 . . . 4 (𝐹 ∈ (𝑀 GrpHom 𝑁) → 𝐹:(Base‘𝑀)⟶(Base‘𝑁))
13123ad2ant2 1126 . . 3 ((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) → 𝐹:(Base‘𝑀)⟶(Base‘𝑁))
141, 2ghmf 17878 . . . 4 (𝐺 ∈ (𝑀 GrpHom 𝑁) → 𝐺:(Base‘𝑀)⟶(Base‘𝑁))
15143ad2ant3 1127 . . 3 ((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) → 𝐺:(Base‘𝑀)⟶(Base‘𝑁))
16 fvexd 6343 . . 3 ((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) → (Base‘𝑀) ∈ V)
17 inidm 3968 . . 3 ((Base‘𝑀) ∩ (Base‘𝑀)) = (Base‘𝑀)
1811, 13, 15, 16, 16, 17off 7057 . 2 ((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) → (𝐹𝑓 + 𝐺):(Base‘𝑀)⟶(Base‘𝑁))
191, 3, 4ghmlin 17879 . . . . . . 7 ((𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝐹‘(𝑥(+g𝑀)𝑦)) = ((𝐹𝑥) + (𝐹𝑦)))
20193expb 1111 . . . . . 6 ((𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → (𝐹‘(𝑥(+g𝑀)𝑦)) = ((𝐹𝑥) + (𝐹𝑦)))
21203ad2antl2 1199 . . . . 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 17879 . . . . . . 7 ((𝐺 ∈ (𝑀 GrpHom 𝑁) ∧ 𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝐺‘(𝑥(+g𝑀)𝑦)) = ((𝐺𝑥) + (𝐺𝑦)))
23223expb 1111 . . . . . 6 ((𝐺 ∈ (𝑀 GrpHom 𝑁) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → (𝐺‘(𝑥(+g𝑀)𝑦)) = ((𝐺𝑥) + (𝐺𝑦)))
24233ad2antl3 1200 . . . . 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 6809 . . . 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 1225 . . . . . 6 (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → 𝑁 ∈ Abel)
27 ablcmn 18412 . . . . . 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 6501 . . . . . 6 (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ 𝑥 ∈ (Base‘𝑀)) → (𝐹𝑥) ∈ (Base‘𝑁))
3029adantrr 752 . . . . 5 (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → (𝐹𝑥) ∈ (Base‘𝑁))
3113ffvelrnda 6501 . . . . . 6 (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝐹𝑦) ∈ (Base‘𝑁))
3231adantrl 751 . . . . 5 (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → (𝐹𝑦) ∈ (Base‘𝑁))
3315ffvelrnda 6501 . . . . . 6 (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ 𝑥 ∈ (Base‘𝑀)) → (𝐺𝑥) ∈ (Base‘𝑁))
3433adantrr 752 . . . . 5 (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → (𝐺𝑥) ∈ (Base‘𝑁))
3515ffvelrnda 6501 . . . . . 6 (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝐺𝑦) ∈ (Base‘𝑁))
3635adantrl 751 . . . . 5 (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → (𝐺𝑦) ∈ (Base‘𝑁))
372, 4cmn4 18425 . . . . 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 1483 . . . 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 2803 . . 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))))
40 ffn 6184 . . . . . 6 (𝐹:(Base‘𝑀)⟶(Base‘𝑁) → 𝐹 Fn (Base‘𝑀))
4113, 40syl 17 . . . . 5 ((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) → 𝐹 Fn (Base‘𝑀))
4241adantr 473 . . . 4 (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → 𝐹 Fn (Base‘𝑀))
43 ffn 6184 . . . . . 6 (𝐺:(Base‘𝑀)⟶(Base‘𝑁) → 𝐺 Fn (Base‘𝑀))
4415, 43syl 17 . . . . 5 ((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) → 𝐺 Fn (Base‘𝑀))
4544adantr 473 . . . 4 (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → 𝐺 Fn (Base‘𝑀))
46 fvexd 6343 . . . 4 (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → (Base‘𝑀) ∈ V)
471, 3grpcl 17644 . . . . . 6 ((𝑀 ∈ Grp ∧ 𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝑥(+g𝑀)𝑦) ∈ (Base‘𝑀))
48473expb 1111 . . . . 5 ((𝑀 ∈ Grp ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → (𝑥(+g𝑀)𝑦) ∈ (Base‘𝑀))
496, 48sylan 490 . . . 4 (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → (𝑥(+g𝑀)𝑦) ∈ (Base‘𝑀))
50 fnfvof 7056 . . . 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))))
5142, 45, 46, 49, 50syl22anc 1475 . . 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))))
52 simprl 808 . . . . 5 (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → 𝑥 ∈ (Base‘𝑀))
53 fnfvof 7056 . . . . 5 (((𝐹 Fn (Base‘𝑀) ∧ 𝐺 Fn (Base‘𝑀)) ∧ ((Base‘𝑀) ∈ V ∧ 𝑥 ∈ (Base‘𝑀))) → ((𝐹𝑓 + 𝐺)‘𝑥) = ((𝐹𝑥) + (𝐺𝑥)))
5442, 45, 46, 52, 53syl22anc 1475 . . . 4 (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → ((𝐹𝑓 + 𝐺)‘𝑥) = ((𝐹𝑥) + (𝐺𝑥)))
55 simprr 810 . . . . 5 (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → 𝑦 ∈ (Base‘𝑀))
56 fnfvof 7056 . . . . 5 (((𝐹 Fn (Base‘𝑀) ∧ 𝐺 Fn (Base‘𝑀)) ∧ ((Base‘𝑀) ∈ V ∧ 𝑦 ∈ (Base‘𝑀))) → ((𝐹𝑓 + 𝐺)‘𝑦) = ((𝐹𝑦) + (𝐺𝑦)))
5742, 45, 46, 55, 56syl22anc 1475 . . . 4 (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → ((𝐹𝑓 + 𝐺)‘𝑦) = ((𝐹𝑦) + (𝐺𝑦)))
5854, 57oveq12d 6809 . . 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))))
5939, 51, 583eqtr4d 2813 . 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)))
601, 2, 3, 4, 6, 8, 18, 59isghmd 17883 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 383   ∧ w3a 1069   = wceq 1629   ∈ wcel 2143  Vcvv 3348   Fn wfn 6025  ⟶wf 6026  ‘cfv 6030  (class class class)co 6791   ∘𝑓 cof 7040  Basecbs 16070  +gcplusg 16155  Grpcgrp 17636   GrpHom cghm 17871  CMndccmn 18406  Abelcabl 18407 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1868  ax-4 1883  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2145  ax-9 2152  ax-10 2172  ax-11 2188  ax-12 2201  ax-13 2406  ax-ext 2749  ax-rep 4901  ax-sep 4911  ax-nul 4919  ax-pow 4970  ax-pr 5033  ax-un 7094 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1071  df-tru 1632  df-ex 1851  df-nf 1856  df-sb 2048  df-eu 2620  df-mo 2621  df-clab 2756  df-cleq 2762  df-clel 2765  df-nfc 2900  df-ne 2942  df-ral 3064  df-rex 3065  df-reu 3066  df-rab 3068  df-v 3350  df-sbc 3585  df-csb 3680  df-dif 3723  df-un 3725  df-in 3727  df-ss 3734  df-nul 4061  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4572  df-iun 4653  df-br 4784  df-opab 4844  df-mpt 4861  df-id 5156  df-xp 5254  df-rel 5255  df-cnv 5256  df-co 5257  df-dm 5258  df-rn 5259  df-res 5260  df-ima 5261  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-ov 6794  df-oprab 6795  df-mpt2 6796  df-of 7042  df-mgm 17456  df-sgrp 17498  df-mnd 17509  df-grp 17639  df-ghm 17872  df-cmn 18408  df-abl 18409 This theorem is referenced by:  lmhmplusg  19263  nmotri  22769  nghmplusg  22770
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