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Theorem ghmplusg 18445
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 2756 . 2 (Base‘𝑀) = (Base‘𝑀)
2 eqid 2756 . 2 (Base‘𝑁) = (Base‘𝑁)
3 eqid 2756 . 2 (+g𝑀) = (+g𝑀)
4 ghmplusg.p . 2 + = (+g𝑁)
5 ghmgrp1 17859 . . 3 (𝐺 ∈ (𝑀 GrpHom 𝑁) → 𝑀 ∈ Grp)
653ad2ant3 1130 . 2 ((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) → 𝑀 ∈ Grp)
7 ghmgrp2 17860 . . 3 (𝐺 ∈ (𝑀 GrpHom 𝑁) → 𝑁 ∈ Grp)
873ad2ant3 1130 . 2 ((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) → 𝑁 ∈ Grp)
92, 4grpcl 17627 . . . . 5 ((𝑁 ∈ Grp ∧ 𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁)) → (𝑥 + 𝑦) ∈ (Base‘𝑁))
1093expb 1114 . . . 4 ((𝑁 ∈ Grp ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → (𝑥 + 𝑦) ∈ (Base‘𝑁))
118, 10sylan 489 . . 3 (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ (𝑥 ∈ (Base‘𝑁) ∧ 𝑦 ∈ (Base‘𝑁))) → (𝑥 + 𝑦) ∈ (Base‘𝑁))
121, 2ghmf 17861 . . . 4 (𝐹 ∈ (𝑀 GrpHom 𝑁) → 𝐹:(Base‘𝑀)⟶(Base‘𝑁))
13123ad2ant2 1129 . . 3 ((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) → 𝐹:(Base‘𝑀)⟶(Base‘𝑁))
141, 2ghmf 17861 . . . 4 (𝐺 ∈ (𝑀 GrpHom 𝑁) → 𝐺:(Base‘𝑀)⟶(Base‘𝑁))
15143ad2ant3 1130 . . 3 ((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) → 𝐺:(Base‘𝑀)⟶(Base‘𝑁))
16 fvexd 6360 . . 3 ((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) → (Base‘𝑀) ∈ V)
17 inidm 3961 . . 3 ((Base‘𝑀) ∩ (Base‘𝑀)) = (Base‘𝑀)
1811, 13, 15, 16, 16, 17off 7073 . 2 ((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) → (𝐹𝑓 + 𝐺):(Base‘𝑀)⟶(Base‘𝑁))
191, 3, 4ghmlin 17862 . . . . . . 7 ((𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝐹‘(𝑥(+g𝑀)𝑦)) = ((𝐹𝑥) + (𝐹𝑦)))
20193expb 1114 . . . . . 6 ((𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → (𝐹‘(𝑥(+g𝑀)𝑦)) = ((𝐹𝑥) + (𝐹𝑦)))
21203ad2antl2 1202 . . . . 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 17862 . . . . . . 7 ((𝐺 ∈ (𝑀 GrpHom 𝑁) ∧ 𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝐺‘(𝑥(+g𝑀)𝑦)) = ((𝐺𝑥) + (𝐺𝑦)))
23223expb 1114 . . . . . 6 ((𝐺 ∈ (𝑀 GrpHom 𝑁) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → (𝐺‘(𝑥(+g𝑀)𝑦)) = ((𝐺𝑥) + (𝐺𝑦)))
24233ad2antl3 1203 . . . . 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 6827 . . . 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 1228 . . . . . 6 (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → 𝑁 ∈ Abel)
27 ablcmn 18395 . . . . . 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 6518 . . . . . 6 (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ 𝑥 ∈ (Base‘𝑀)) → (𝐹𝑥) ∈ (Base‘𝑁))
3029adantrr 755 . . . . 5 (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → (𝐹𝑥) ∈ (Base‘𝑁))
3113ffvelrnda 6518 . . . . . 6 (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝐹𝑦) ∈ (Base‘𝑁))
3231adantrl 754 . . . . 5 (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → (𝐹𝑦) ∈ (Base‘𝑁))
3315ffvelrnda 6518 . . . . . 6 (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ 𝑥 ∈ (Base‘𝑀)) → (𝐺𝑥) ∈ (Base‘𝑁))
3433adantrr 755 . . . . 5 (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → (𝐺𝑥) ∈ (Base‘𝑁))
3515ffvelrnda 6518 . . . . . 6 (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝐺𝑦) ∈ (Base‘𝑁))
3635adantrl 754 . . . . 5 (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → (𝐺𝑦) ∈ (Base‘𝑁))
372, 4cmn4 18408 . . . . 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 1486 . . . 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 2790 . . 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 6202 . . . . . 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 472 . . . 4 (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → 𝐹 Fn (Base‘𝑀))
43 ffn 6202 . . . . . 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 472 . . . 4 (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → 𝐺 Fn (Base‘𝑀))
46 fvexd 6360 . . . 4 (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → (Base‘𝑀) ∈ V)
471, 3grpcl 17627 . . . . . 6 ((𝑀 ∈ Grp ∧ 𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝑥(+g𝑀)𝑦) ∈ (Base‘𝑀))
48473expb 1114 . . . . 5 ((𝑀 ∈ Grp ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → (𝑥(+g𝑀)𝑦) ∈ (Base‘𝑀))
496, 48sylan 489 . . . 4 (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → (𝑥(+g𝑀)𝑦) ∈ (Base‘𝑀))
50 fnfvof 7072 . . . 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 1478 . . 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 811 . . . . 5 (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → 𝑥 ∈ (Base‘𝑀))
53 fnfvof 7072 . . . . 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 1478 . . . 4 (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → ((𝐹𝑓 + 𝐺)‘𝑥) = ((𝐹𝑥) + (𝐺𝑥)))
55 simprr 813 . . . . 5 (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → 𝑦 ∈ (Base‘𝑀))
56 fnfvof 7072 . . . . 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 1478 . . . 4 (((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) ∧ (𝑥 ∈ (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀))) → ((𝐹𝑓 + 𝐺)‘𝑦) = ((𝐹𝑦) + (𝐺𝑦)))
5854, 57oveq12d 6827 . . 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 2800 . 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 17866 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 1072   = wceq 1628  wcel 2135  Vcvv 3336   Fn wfn 6040  wf 6041  cfv 6045  (class class class)co 6809  𝑓 cof 7056  Basecbs 16055  +gcplusg 16139  Grpcgrp 17619   GrpHom cghm 17854  CMndccmn 18389  Abelcabl 18390
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1867  ax-4 1882  ax-5 1984  ax-6 2050  ax-7 2086  ax-8 2137  ax-9 2144  ax-10 2164  ax-11 2179  ax-12 2192  ax-13 2387  ax-ext 2736  ax-rep 4919  ax-sep 4929  ax-nul 4937  ax-pow 4988  ax-pr 5051  ax-un 7110
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1631  df-ex 1850  df-nf 1855  df-sb 2043  df-eu 2607  df-mo 2608  df-clab 2743  df-cleq 2749  df-clel 2752  df-nfc 2887  df-ne 2929  df-ral 3051  df-rex 3052  df-reu 3053  df-rab 3055  df-v 3338  df-sbc 3573  df-csb 3671  df-dif 3714  df-un 3716  df-in 3718  df-ss 3725  df-nul 4055  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4585  df-iun 4670  df-br 4801  df-opab 4861  df-mpt 4878  df-id 5170  df-xp 5268  df-rel 5269  df-cnv 5270  df-co 5271  df-dm 5272  df-rn 5273  df-res 5274  df-ima 5275  df-iota 6008  df-fun 6047  df-fn 6048  df-f 6049  df-f1 6050  df-fo 6051  df-f1o 6052  df-fv 6053  df-ov 6812  df-oprab 6813  df-mpt2 6814  df-of 7058  df-mgm 17439  df-sgrp 17481  df-mnd 17492  df-grp 17622  df-ghm 17855  df-cmn 18391  df-abl 18392
This theorem is referenced by:  lmhmplusg  19242  nmotri  22740  nghmplusg  22741
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