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Theorem mgmhmpropd 42110
Description: Magma homomorphism depends only on the operation of structures. (Contributed by AV, 25-Feb-2020.)
Hypotheses
Ref Expression
mgmhmpropd.a (𝜑𝐵 = (Base‘𝐽))
mgmhmpropd.b (𝜑𝐶 = (Base‘𝐾))
mgmhmpropd.c (𝜑𝐵 = (Base‘𝐿))
mgmhmpropd.d (𝜑𝐶 = (Base‘𝑀))
mgmhmpropd.0 (𝜑𝐵 ≠ ∅)
mgmhmpropd.C (𝜑𝐶 ≠ ∅)
mgmhmpropd.e ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐽)𝑦) = (𝑥(+g𝐿)𝑦))
mgmhmpropd.f ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝑀)𝑦))
Assertion
Ref Expression
mgmhmpropd (𝜑 → (𝐽 MgmHom 𝐾) = (𝐿 MgmHom 𝑀))
Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐶,𝑦   𝑥,𝐽,𝑦   𝑥,𝐿,𝑦   𝜑,𝑥,𝑦   𝑥,𝐾,𝑦   𝑥,𝑀,𝑦

Proof of Theorem mgmhmpropd
Dummy variables 𝑤 𝑧 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mgmhmpropd.e . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐽)𝑦) = (𝑥(+g𝐿)𝑦))
21fveq2d 6233 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑓‘(𝑥(+g𝐽)𝑦)) = (𝑓‘(𝑥(+g𝐿)𝑦)))
32adantlr 751 . . . . . . . . . . . 12 (((𝜑𝑓:𝐵𝐶) ∧ (𝑥𝐵𝑦𝐵)) → (𝑓‘(𝑥(+g𝐽)𝑦)) = (𝑓‘(𝑥(+g𝐿)𝑦)))
4 ffvelrn 6397 . . . . . . . . . . . . . . 15 ((𝑓:𝐵𝐶𝑥𝐵) → (𝑓𝑥) ∈ 𝐶)
5 ffvelrn 6397 . . . . . . . . . . . . . . 15 ((𝑓:𝐵𝐶𝑦𝐵) → (𝑓𝑦) ∈ 𝐶)
64, 5anim12dan 900 . . . . . . . . . . . . . 14 ((𝑓:𝐵𝐶 ∧ (𝑥𝐵𝑦𝐵)) → ((𝑓𝑥) ∈ 𝐶 ∧ (𝑓𝑦) ∈ 𝐶))
7 mgmhmpropd.f . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝑀)𝑦))
87ralrimivva 3000 . . . . . . . . . . . . . . 15 (𝜑 → ∀𝑥𝐶𝑦𝐶 (𝑥(+g𝐾)𝑦) = (𝑥(+g𝑀)𝑦))
9 oveq1 6697 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑤 → (𝑥(+g𝐾)𝑦) = (𝑤(+g𝐾)𝑦))
10 oveq1 6697 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑤 → (𝑥(+g𝑀)𝑦) = (𝑤(+g𝑀)𝑦))
119, 10eqeq12d 2666 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑤 → ((𝑥(+g𝐾)𝑦) = (𝑥(+g𝑀)𝑦) ↔ (𝑤(+g𝐾)𝑦) = (𝑤(+g𝑀)𝑦)))
12 oveq2 6698 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑧 → (𝑤(+g𝐾)𝑦) = (𝑤(+g𝐾)𝑧))
13 oveq2 6698 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑧 → (𝑤(+g𝑀)𝑦) = (𝑤(+g𝑀)𝑧))
1412, 13eqeq12d 2666 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑧 → ((𝑤(+g𝐾)𝑦) = (𝑤(+g𝑀)𝑦) ↔ (𝑤(+g𝐾)𝑧) = (𝑤(+g𝑀)𝑧)))
1511, 14cbvral2v 3209 . . . . . . . . . . . . . . 15 (∀𝑥𝐶𝑦𝐶 (𝑥(+g𝐾)𝑦) = (𝑥(+g𝑀)𝑦) ↔ ∀𝑤𝐶𝑧𝐶 (𝑤(+g𝐾)𝑧) = (𝑤(+g𝑀)𝑧))
168, 15sylib 208 . . . . . . . . . . . . . 14 (𝜑 → ∀𝑤𝐶𝑧𝐶 (𝑤(+g𝐾)𝑧) = (𝑤(+g𝑀)𝑧))
17 oveq1 6697 . . . . . . . . . . . . . . . 16 (𝑤 = (𝑓𝑥) → (𝑤(+g𝐾)𝑧) = ((𝑓𝑥)(+g𝐾)𝑧))
18 oveq1 6697 . . . . . . . . . . . . . . . 16 (𝑤 = (𝑓𝑥) → (𝑤(+g𝑀)𝑧) = ((𝑓𝑥)(+g𝑀)𝑧))
1917, 18eqeq12d 2666 . . . . . . . . . . . . . . 15 (𝑤 = (𝑓𝑥) → ((𝑤(+g𝐾)𝑧) = (𝑤(+g𝑀)𝑧) ↔ ((𝑓𝑥)(+g𝐾)𝑧) = ((𝑓𝑥)(+g𝑀)𝑧)))
20 oveq2 6698 . . . . . . . . . . . . . . . 16 (𝑧 = (𝑓𝑦) → ((𝑓𝑥)(+g𝐾)𝑧) = ((𝑓𝑥)(+g𝐾)(𝑓𝑦)))
21 oveq2 6698 . . . . . . . . . . . . . . . 16 (𝑧 = (𝑓𝑦) → ((𝑓𝑥)(+g𝑀)𝑧) = ((𝑓𝑥)(+g𝑀)(𝑓𝑦)))
2220, 21eqeq12d 2666 . . . . . . . . . . . . . . 15 (𝑧 = (𝑓𝑦) → (((𝑓𝑥)(+g𝐾)𝑧) = ((𝑓𝑥)(+g𝑀)𝑧) ↔ ((𝑓𝑥)(+g𝐾)(𝑓𝑦)) = ((𝑓𝑥)(+g𝑀)(𝑓𝑦))))
2319, 22rspc2va 3354 . . . . . . . . . . . . . 14 ((((𝑓𝑥) ∈ 𝐶 ∧ (𝑓𝑦) ∈ 𝐶) ∧ ∀𝑤𝐶𝑧𝐶 (𝑤(+g𝐾)𝑧) = (𝑤(+g𝑀)𝑧)) → ((𝑓𝑥)(+g𝐾)(𝑓𝑦)) = ((𝑓𝑥)(+g𝑀)(𝑓𝑦)))
246, 16, 23syl2anr 494 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑓:𝐵𝐶 ∧ (𝑥𝐵𝑦𝐵))) → ((𝑓𝑥)(+g𝐾)(𝑓𝑦)) = ((𝑓𝑥)(+g𝑀)(𝑓𝑦)))
2524anassrs 681 . . . . . . . . . . . 12 (((𝜑𝑓:𝐵𝐶) ∧ (𝑥𝐵𝑦𝐵)) → ((𝑓𝑥)(+g𝐾)(𝑓𝑦)) = ((𝑓𝑥)(+g𝑀)(𝑓𝑦)))
263, 25eqeq12d 2666 . . . . . . . . . . 11 (((𝜑𝑓:𝐵𝐶) ∧ (𝑥𝐵𝑦𝐵)) → ((𝑓‘(𝑥(+g𝐽)𝑦)) = ((𝑓𝑥)(+g𝐾)(𝑓𝑦)) ↔ (𝑓‘(𝑥(+g𝐿)𝑦)) = ((𝑓𝑥)(+g𝑀)(𝑓𝑦))))
27262ralbidva 3017 . . . . . . . . . 10 ((𝜑𝑓:𝐵𝐶) → (∀𝑥𝐵𝑦𝐵 (𝑓‘(𝑥(+g𝐽)𝑦)) = ((𝑓𝑥)(+g𝐾)(𝑓𝑦)) ↔ ∀𝑥𝐵𝑦𝐵 (𝑓‘(𝑥(+g𝐿)𝑦)) = ((𝑓𝑥)(+g𝑀)(𝑓𝑦))))
2827adantrl 752 . . . . . . . . 9 ((𝜑 ∧ ((𝐽 ∈ Mgm ∧ 𝐾 ∈ Mgm) ∧ 𝑓:𝐵𝐶)) → (∀𝑥𝐵𝑦𝐵 (𝑓‘(𝑥(+g𝐽)𝑦)) = ((𝑓𝑥)(+g𝐾)(𝑓𝑦)) ↔ ∀𝑥𝐵𝑦𝐵 (𝑓‘(𝑥(+g𝐿)𝑦)) = ((𝑓𝑥)(+g𝑀)(𝑓𝑦))))
29 mgmhmpropd.a . . . . . . . . . . 11 (𝜑𝐵 = (Base‘𝐽))
30 raleq 3168 . . . . . . . . . . . 12 (𝐵 = (Base‘𝐽) → (∀𝑦𝐵 (𝑓‘(𝑥(+g𝐽)𝑦)) = ((𝑓𝑥)(+g𝐾)(𝑓𝑦)) ↔ ∀𝑦 ∈ (Base‘𝐽)(𝑓‘(𝑥(+g𝐽)𝑦)) = ((𝑓𝑥)(+g𝐾)(𝑓𝑦))))
3130raleqbi1dv 3176 . . . . . . . . . . 11 (𝐵 = (Base‘𝐽) → (∀𝑥𝐵𝑦𝐵 (𝑓‘(𝑥(+g𝐽)𝑦)) = ((𝑓𝑥)(+g𝐾)(𝑓𝑦)) ↔ ∀𝑥 ∈ (Base‘𝐽)∀𝑦 ∈ (Base‘𝐽)(𝑓‘(𝑥(+g𝐽)𝑦)) = ((𝑓𝑥)(+g𝐾)(𝑓𝑦))))
3229, 31syl 17 . . . . . . . . . 10 (𝜑 → (∀𝑥𝐵𝑦𝐵 (𝑓‘(𝑥(+g𝐽)𝑦)) = ((𝑓𝑥)(+g𝐾)(𝑓𝑦)) ↔ ∀𝑥 ∈ (Base‘𝐽)∀𝑦 ∈ (Base‘𝐽)(𝑓‘(𝑥(+g𝐽)𝑦)) = ((𝑓𝑥)(+g𝐾)(𝑓𝑦))))
3332adantr 480 . . . . . . . . 9 ((𝜑 ∧ ((𝐽 ∈ Mgm ∧ 𝐾 ∈ Mgm) ∧ 𝑓:𝐵𝐶)) → (∀𝑥𝐵𝑦𝐵 (𝑓‘(𝑥(+g𝐽)𝑦)) = ((𝑓𝑥)(+g𝐾)(𝑓𝑦)) ↔ ∀𝑥 ∈ (Base‘𝐽)∀𝑦 ∈ (Base‘𝐽)(𝑓‘(𝑥(+g𝐽)𝑦)) = ((𝑓𝑥)(+g𝐾)(𝑓𝑦))))
34 mgmhmpropd.c . . . . . . . . . . 11 (𝜑𝐵 = (Base‘𝐿))
35 raleq 3168 . . . . . . . . . . . 12 (𝐵 = (Base‘𝐿) → (∀𝑦𝐵 (𝑓‘(𝑥(+g𝐿)𝑦)) = ((𝑓𝑥)(+g𝑀)(𝑓𝑦)) ↔ ∀𝑦 ∈ (Base‘𝐿)(𝑓‘(𝑥(+g𝐿)𝑦)) = ((𝑓𝑥)(+g𝑀)(𝑓𝑦))))
3635raleqbi1dv 3176 . . . . . . . . . . 11 (𝐵 = (Base‘𝐿) → (∀𝑥𝐵𝑦𝐵 (𝑓‘(𝑥(+g𝐿)𝑦)) = ((𝑓𝑥)(+g𝑀)(𝑓𝑦)) ↔ ∀𝑥 ∈ (Base‘𝐿)∀𝑦 ∈ (Base‘𝐿)(𝑓‘(𝑥(+g𝐿)𝑦)) = ((𝑓𝑥)(+g𝑀)(𝑓𝑦))))
3734, 36syl 17 . . . . . . . . . 10 (𝜑 → (∀𝑥𝐵𝑦𝐵 (𝑓‘(𝑥(+g𝐿)𝑦)) = ((𝑓𝑥)(+g𝑀)(𝑓𝑦)) ↔ ∀𝑥 ∈ (Base‘𝐿)∀𝑦 ∈ (Base‘𝐿)(𝑓‘(𝑥(+g𝐿)𝑦)) = ((𝑓𝑥)(+g𝑀)(𝑓𝑦))))
3837adantr 480 . . . . . . . . 9 ((𝜑 ∧ ((𝐽 ∈ Mgm ∧ 𝐾 ∈ Mgm) ∧ 𝑓:𝐵𝐶)) → (∀𝑥𝐵𝑦𝐵 (𝑓‘(𝑥(+g𝐿)𝑦)) = ((𝑓𝑥)(+g𝑀)(𝑓𝑦)) ↔ ∀𝑥 ∈ (Base‘𝐿)∀𝑦 ∈ (Base‘𝐿)(𝑓‘(𝑥(+g𝐿)𝑦)) = ((𝑓𝑥)(+g𝑀)(𝑓𝑦))))
3928, 33, 383bitr3d 298 . . . . . . . 8 ((𝜑 ∧ ((𝐽 ∈ Mgm ∧ 𝐾 ∈ Mgm) ∧ 𝑓:𝐵𝐶)) → (∀𝑥 ∈ (Base‘𝐽)∀𝑦 ∈ (Base‘𝐽)(𝑓‘(𝑥(+g𝐽)𝑦)) = ((𝑓𝑥)(+g𝐾)(𝑓𝑦)) ↔ ∀𝑥 ∈ (Base‘𝐿)∀𝑦 ∈ (Base‘𝐿)(𝑓‘(𝑥(+g𝐿)𝑦)) = ((𝑓𝑥)(+g𝑀)(𝑓𝑦))))
4039anassrs 681 . . . . . . 7 (((𝜑 ∧ (𝐽 ∈ Mgm ∧ 𝐾 ∈ Mgm)) ∧ 𝑓:𝐵𝐶) → (∀𝑥 ∈ (Base‘𝐽)∀𝑦 ∈ (Base‘𝐽)(𝑓‘(𝑥(+g𝐽)𝑦)) = ((𝑓𝑥)(+g𝐾)(𝑓𝑦)) ↔ ∀𝑥 ∈ (Base‘𝐿)∀𝑦 ∈ (Base‘𝐿)(𝑓‘(𝑥(+g𝐿)𝑦)) = ((𝑓𝑥)(+g𝑀)(𝑓𝑦))))
4140pm5.32da 674 . . . . . 6 ((𝜑 ∧ (𝐽 ∈ Mgm ∧ 𝐾 ∈ Mgm)) → ((𝑓:𝐵𝐶 ∧ ∀𝑥 ∈ (Base‘𝐽)∀𝑦 ∈ (Base‘𝐽)(𝑓‘(𝑥(+g𝐽)𝑦)) = ((𝑓𝑥)(+g𝐾)(𝑓𝑦))) ↔ (𝑓:𝐵𝐶 ∧ ∀𝑥 ∈ (Base‘𝐿)∀𝑦 ∈ (Base‘𝐿)(𝑓‘(𝑥(+g𝐿)𝑦)) = ((𝑓𝑥)(+g𝑀)(𝑓𝑦)))))
42 mgmhmpropd.b . . . . . . . . 9 (𝜑𝐶 = (Base‘𝐾))
4329, 42feq23d 6078 . . . . . . . 8 (𝜑 → (𝑓:𝐵𝐶𝑓:(Base‘𝐽)⟶(Base‘𝐾)))
4443adantr 480 . . . . . . 7 ((𝜑 ∧ (𝐽 ∈ Mgm ∧ 𝐾 ∈ Mgm)) → (𝑓:𝐵𝐶𝑓:(Base‘𝐽)⟶(Base‘𝐾)))
4544anbi1d 741 . . . . . 6 ((𝜑 ∧ (𝐽 ∈ Mgm ∧ 𝐾 ∈ Mgm)) → ((𝑓:𝐵𝐶 ∧ ∀𝑥 ∈ (Base‘𝐽)∀𝑦 ∈ (Base‘𝐽)(𝑓‘(𝑥(+g𝐽)𝑦)) = ((𝑓𝑥)(+g𝐾)(𝑓𝑦))) ↔ (𝑓:(Base‘𝐽)⟶(Base‘𝐾) ∧ ∀𝑥 ∈ (Base‘𝐽)∀𝑦 ∈ (Base‘𝐽)(𝑓‘(𝑥(+g𝐽)𝑦)) = ((𝑓𝑥)(+g𝐾)(𝑓𝑦)))))
46 mgmhmpropd.d . . . . . . . . 9 (𝜑𝐶 = (Base‘𝑀))
4734, 46feq23d 6078 . . . . . . . 8 (𝜑 → (𝑓:𝐵𝐶𝑓:(Base‘𝐿)⟶(Base‘𝑀)))
4847adantr 480 . . . . . . 7 ((𝜑 ∧ (𝐽 ∈ Mgm ∧ 𝐾 ∈ Mgm)) → (𝑓:𝐵𝐶𝑓:(Base‘𝐿)⟶(Base‘𝑀)))
4948anbi1d 741 . . . . . 6 ((𝜑 ∧ (𝐽 ∈ Mgm ∧ 𝐾 ∈ Mgm)) → ((𝑓:𝐵𝐶 ∧ ∀𝑥 ∈ (Base‘𝐿)∀𝑦 ∈ (Base‘𝐿)(𝑓‘(𝑥(+g𝐿)𝑦)) = ((𝑓𝑥)(+g𝑀)(𝑓𝑦))) ↔ (𝑓:(Base‘𝐿)⟶(Base‘𝑀) ∧ ∀𝑥 ∈ (Base‘𝐿)∀𝑦 ∈ (Base‘𝐿)(𝑓‘(𝑥(+g𝐿)𝑦)) = ((𝑓𝑥)(+g𝑀)(𝑓𝑦)))))
5041, 45, 493bitr3d 298 . . . . 5 ((𝜑 ∧ (𝐽 ∈ Mgm ∧ 𝐾 ∈ Mgm)) → ((𝑓:(Base‘𝐽)⟶(Base‘𝐾) ∧ ∀𝑥 ∈ (Base‘𝐽)∀𝑦 ∈ (Base‘𝐽)(𝑓‘(𝑥(+g𝐽)𝑦)) = ((𝑓𝑥)(+g𝐾)(𝑓𝑦))) ↔ (𝑓:(Base‘𝐿)⟶(Base‘𝑀) ∧ ∀𝑥 ∈ (Base‘𝐿)∀𝑦 ∈ (Base‘𝐿)(𝑓‘(𝑥(+g𝐿)𝑦)) = ((𝑓𝑥)(+g𝑀)(𝑓𝑦)))))
5150pm5.32da 674 . . . 4 (𝜑 → (((𝐽 ∈ Mgm ∧ 𝐾 ∈ Mgm) ∧ (𝑓:(Base‘𝐽)⟶(Base‘𝐾) ∧ ∀𝑥 ∈ (Base‘𝐽)∀𝑦 ∈ (Base‘𝐽)(𝑓‘(𝑥(+g𝐽)𝑦)) = ((𝑓𝑥)(+g𝐾)(𝑓𝑦)))) ↔ ((𝐽 ∈ Mgm ∧ 𝐾 ∈ Mgm) ∧ (𝑓:(Base‘𝐿)⟶(Base‘𝑀) ∧ ∀𝑥 ∈ (Base‘𝐿)∀𝑦 ∈ (Base‘𝐿)(𝑓‘(𝑥(+g𝐿)𝑦)) = ((𝑓𝑥)(+g𝑀)(𝑓𝑦))))))
52 mgmhmpropd.0 . . . . . . 7 (𝜑𝐵 ≠ ∅)
5329, 34, 52, 1mgmpropd 42100 . . . . . 6 (𝜑 → (𝐽 ∈ Mgm ↔ 𝐿 ∈ Mgm))
54 mgmhmpropd.C . . . . . . 7 (𝜑𝐶 ≠ ∅)
5542, 46, 54, 7mgmpropd 42100 . . . . . 6 (𝜑 → (𝐾 ∈ Mgm ↔ 𝑀 ∈ Mgm))
5653, 55anbi12d 747 . . . . 5 (𝜑 → ((𝐽 ∈ Mgm ∧ 𝐾 ∈ Mgm) ↔ (𝐿 ∈ Mgm ∧ 𝑀 ∈ Mgm)))
5756anbi1d 741 . . . 4 (𝜑 → (((𝐽 ∈ Mgm ∧ 𝐾 ∈ Mgm) ∧ (𝑓:(Base‘𝐿)⟶(Base‘𝑀) ∧ ∀𝑥 ∈ (Base‘𝐿)∀𝑦 ∈ (Base‘𝐿)(𝑓‘(𝑥(+g𝐿)𝑦)) = ((𝑓𝑥)(+g𝑀)(𝑓𝑦)))) ↔ ((𝐿 ∈ Mgm ∧ 𝑀 ∈ Mgm) ∧ (𝑓:(Base‘𝐿)⟶(Base‘𝑀) ∧ ∀𝑥 ∈ (Base‘𝐿)∀𝑦 ∈ (Base‘𝐿)(𝑓‘(𝑥(+g𝐿)𝑦)) = ((𝑓𝑥)(+g𝑀)(𝑓𝑦))))))
5851, 57bitrd 268 . . 3 (𝜑 → (((𝐽 ∈ Mgm ∧ 𝐾 ∈ Mgm) ∧ (𝑓:(Base‘𝐽)⟶(Base‘𝐾) ∧ ∀𝑥 ∈ (Base‘𝐽)∀𝑦 ∈ (Base‘𝐽)(𝑓‘(𝑥(+g𝐽)𝑦)) = ((𝑓𝑥)(+g𝐾)(𝑓𝑦)))) ↔ ((𝐿 ∈ Mgm ∧ 𝑀 ∈ Mgm) ∧ (𝑓:(Base‘𝐿)⟶(Base‘𝑀) ∧ ∀𝑥 ∈ (Base‘𝐿)∀𝑦 ∈ (Base‘𝐿)(𝑓‘(𝑥(+g𝐿)𝑦)) = ((𝑓𝑥)(+g𝑀)(𝑓𝑦))))))
59 eqid 2651 . . . 4 (Base‘𝐽) = (Base‘𝐽)
60 eqid 2651 . . . 4 (Base‘𝐾) = (Base‘𝐾)
61 eqid 2651 . . . 4 (+g𝐽) = (+g𝐽)
62 eqid 2651 . . . 4 (+g𝐾) = (+g𝐾)
6359, 60, 61, 62ismgmhm 42108 . . 3 (𝑓 ∈ (𝐽 MgmHom 𝐾) ↔ ((𝐽 ∈ Mgm ∧ 𝐾 ∈ Mgm) ∧ (𝑓:(Base‘𝐽)⟶(Base‘𝐾) ∧ ∀𝑥 ∈ (Base‘𝐽)∀𝑦 ∈ (Base‘𝐽)(𝑓‘(𝑥(+g𝐽)𝑦)) = ((𝑓𝑥)(+g𝐾)(𝑓𝑦)))))
64 eqid 2651 . . . 4 (Base‘𝐿) = (Base‘𝐿)
65 eqid 2651 . . . 4 (Base‘𝑀) = (Base‘𝑀)
66 eqid 2651 . . . 4 (+g𝐿) = (+g𝐿)
67 eqid 2651 . . . 4 (+g𝑀) = (+g𝑀)
6864, 65, 66, 67ismgmhm 42108 . . 3 (𝑓 ∈ (𝐿 MgmHom 𝑀) ↔ ((𝐿 ∈ Mgm ∧ 𝑀 ∈ Mgm) ∧ (𝑓:(Base‘𝐿)⟶(Base‘𝑀) ∧ ∀𝑥 ∈ (Base‘𝐿)∀𝑦 ∈ (Base‘𝐿)(𝑓‘(𝑥(+g𝐿)𝑦)) = ((𝑓𝑥)(+g𝑀)(𝑓𝑦)))))
6958, 63, 683bitr4g 303 . 2 (𝜑 → (𝑓 ∈ (𝐽 MgmHom 𝐾) ↔ 𝑓 ∈ (𝐿 MgmHom 𝑀)))
7069eqrdv 2649 1 (𝜑 → (𝐽 MgmHom 𝐾) = (𝐿 MgmHom 𝑀))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  wne 2823  wral 2941  c0 3948  wf 5922  cfv 5926  (class class class)co 6690  Basecbs 15904  +gcplusg 15988  Mgmcmgm 17287   MgmHom cmgmhm 42102
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-map 7901  df-mgm 17289  df-mgmhm 42104
This theorem is referenced by: (None)
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