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Theorem mhmpropd 17954
Description: Monoid homomorphism depends only on the monoidal attributes of structures. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 7-Nov-2015.)
Hypotheses
Ref Expression
mhmpropd.a (𝜑𝐵 = (Base‘𝐽))
mhmpropd.b (𝜑𝐶 = (Base‘𝐾))
mhmpropd.c (𝜑𝐵 = (Base‘𝐿))
mhmpropd.d (𝜑𝐶 = (Base‘𝑀))
mhmpropd.e ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐽)𝑦) = (𝑥(+g𝐿)𝑦))
mhmpropd.f ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝑀)𝑦))
Assertion
Ref Expression
mhmpropd (𝜑 → (𝐽 MndHom 𝐾) = (𝐿 MndHom 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 mhmpropd
Dummy variables 𝑤 𝑧 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mhmpropd.e . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐽)𝑦) = (𝑥(+g𝐿)𝑦))
21fveq2d 6667 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑓‘(𝑥(+g𝐽)𝑦)) = (𝑓‘(𝑥(+g𝐿)𝑦)))
32adantlr 713 . . . . . . . . . . . . . 14 (((𝜑𝑓:𝐵𝐶) ∧ (𝑥𝐵𝑦𝐵)) → (𝑓‘(𝑥(+g𝐽)𝑦)) = (𝑓‘(𝑥(+g𝐿)𝑦)))
4 ffvelrn 6842 . . . . . . . . . . . . . . . . 17 ((𝑓:𝐵𝐶𝑥𝐵) → (𝑓𝑥) ∈ 𝐶)
5 ffvelrn 6842 . . . . . . . . . . . . . . . . 17 ((𝑓:𝐵𝐶𝑦𝐵) → (𝑓𝑦) ∈ 𝐶)
64, 5anim12dan 620 . . . . . . . . . . . . . . . 16 ((𝑓:𝐵𝐶 ∧ (𝑥𝐵𝑦𝐵)) → ((𝑓𝑥) ∈ 𝐶 ∧ (𝑓𝑦) ∈ 𝐶))
7 mhmpropd.f . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝑀)𝑦))
87ralrimivva 3189 . . . . . . . . . . . . . . . . 17 (𝜑 → ∀𝑥𝐶𝑦𝐶 (𝑥(+g𝐾)𝑦) = (𝑥(+g𝑀)𝑦))
9 oveq1 7155 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑤 → (𝑥(+g𝐾)𝑦) = (𝑤(+g𝐾)𝑦))
10 oveq1 7155 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑤 → (𝑥(+g𝑀)𝑦) = (𝑤(+g𝑀)𝑦))
119, 10eqeq12d 2835 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑤 → ((𝑥(+g𝐾)𝑦) = (𝑥(+g𝑀)𝑦) ↔ (𝑤(+g𝐾)𝑦) = (𝑤(+g𝑀)𝑦)))
12 oveq2 7156 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑧 → (𝑤(+g𝐾)𝑦) = (𝑤(+g𝐾)𝑧))
13 oveq2 7156 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑧 → (𝑤(+g𝑀)𝑦) = (𝑤(+g𝑀)𝑧))
1412, 13eqeq12d 2835 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑧 → ((𝑤(+g𝐾)𝑦) = (𝑤(+g𝑀)𝑦) ↔ (𝑤(+g𝐾)𝑧) = (𝑤(+g𝑀)𝑧)))
1511, 14cbvral2vw 3460 . . . . . . . . . . . . . . . . 17 (∀𝑥𝐶𝑦𝐶 (𝑥(+g𝐾)𝑦) = (𝑥(+g𝑀)𝑦) ↔ ∀𝑤𝐶𝑧𝐶 (𝑤(+g𝐾)𝑧) = (𝑤(+g𝑀)𝑧))
168, 15sylib 220 . . . . . . . . . . . . . . . 16 (𝜑 → ∀𝑤𝐶𝑧𝐶 (𝑤(+g𝐾)𝑧) = (𝑤(+g𝑀)𝑧))
17 oveq1 7155 . . . . . . . . . . . . . . . . . 18 (𝑤 = (𝑓𝑥) → (𝑤(+g𝐾)𝑧) = ((𝑓𝑥)(+g𝐾)𝑧))
18 oveq1 7155 . . . . . . . . . . . . . . . . . 18 (𝑤 = (𝑓𝑥) → (𝑤(+g𝑀)𝑧) = ((𝑓𝑥)(+g𝑀)𝑧))
1917, 18eqeq12d 2835 . . . . . . . . . . . . . . . . 17 (𝑤 = (𝑓𝑥) → ((𝑤(+g𝐾)𝑧) = (𝑤(+g𝑀)𝑧) ↔ ((𝑓𝑥)(+g𝐾)𝑧) = ((𝑓𝑥)(+g𝑀)𝑧)))
20 oveq2 7156 . . . . . . . . . . . . . . . . . 18 (𝑧 = (𝑓𝑦) → ((𝑓𝑥)(+g𝐾)𝑧) = ((𝑓𝑥)(+g𝐾)(𝑓𝑦)))
21 oveq2 7156 . . . . . . . . . . . . . . . . . 18 (𝑧 = (𝑓𝑦) → ((𝑓𝑥)(+g𝑀)𝑧) = ((𝑓𝑥)(+g𝑀)(𝑓𝑦)))
2220, 21eqeq12d 2835 . . . . . . . . . . . . . . . . 17 (𝑧 = (𝑓𝑦) → (((𝑓𝑥)(+g𝐾)𝑧) = ((𝑓𝑥)(+g𝑀)𝑧) ↔ ((𝑓𝑥)(+g𝐾)(𝑓𝑦)) = ((𝑓𝑥)(+g𝑀)(𝑓𝑦))))
2319, 22rspc2va 3632 . . . . . . . . . . . . . . . 16 ((((𝑓𝑥) ∈ 𝐶 ∧ (𝑓𝑦) ∈ 𝐶) ∧ ∀𝑤𝐶𝑧𝐶 (𝑤(+g𝐾)𝑧) = (𝑤(+g𝑀)𝑧)) → ((𝑓𝑥)(+g𝐾)(𝑓𝑦)) = ((𝑓𝑥)(+g𝑀)(𝑓𝑦)))
246, 16, 23syl2anr 598 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑓:𝐵𝐶 ∧ (𝑥𝐵𝑦𝐵))) → ((𝑓𝑥)(+g𝐾)(𝑓𝑦)) = ((𝑓𝑥)(+g𝑀)(𝑓𝑦)))
2524anassrs 470 . . . . . . . . . . . . . 14 (((𝜑𝑓:𝐵𝐶) ∧ (𝑥𝐵𝑦𝐵)) → ((𝑓𝑥)(+g𝐾)(𝑓𝑦)) = ((𝑓𝑥)(+g𝑀)(𝑓𝑦)))
263, 25eqeq12d 2835 . . . . . . . . . . . . 13 (((𝜑𝑓:𝐵𝐶) ∧ (𝑥𝐵𝑦𝐵)) → ((𝑓‘(𝑥(+g𝐽)𝑦)) = ((𝑓𝑥)(+g𝐾)(𝑓𝑦)) ↔ (𝑓‘(𝑥(+g𝐿)𝑦)) = ((𝑓𝑥)(+g𝑀)(𝑓𝑦))))
27262ralbidva 3196 . . . . . . . . . . . 12 ((𝜑𝑓:𝐵𝐶) → (∀𝑥𝐵𝑦𝐵 (𝑓‘(𝑥(+g𝐽)𝑦)) = ((𝑓𝑥)(+g𝐾)(𝑓𝑦)) ↔ ∀𝑥𝐵𝑦𝐵 (𝑓‘(𝑥(+g𝐿)𝑦)) = ((𝑓𝑥)(+g𝑀)(𝑓𝑦))))
2827adantrl 714 . . . . . . . . . . 11 ((𝜑 ∧ ((𝐽 ∈ Mnd ∧ 𝐾 ∈ Mnd) ∧ 𝑓:𝐵𝐶)) → (∀𝑥𝐵𝑦𝐵 (𝑓‘(𝑥(+g𝐽)𝑦)) = ((𝑓𝑥)(+g𝐾)(𝑓𝑦)) ↔ ∀𝑥𝐵𝑦𝐵 (𝑓‘(𝑥(+g𝐿)𝑦)) = ((𝑓𝑥)(+g𝑀)(𝑓𝑦))))
29 mhmpropd.a . . . . . . . . . . . . 13 (𝜑𝐵 = (Base‘𝐽))
30 raleq 3404 . . . . . . . . . . . . . 14 (𝐵 = (Base‘𝐽) → (∀𝑦𝐵 (𝑓‘(𝑥(+g𝐽)𝑦)) = ((𝑓𝑥)(+g𝐾)(𝑓𝑦)) ↔ ∀𝑦 ∈ (Base‘𝐽)(𝑓‘(𝑥(+g𝐽)𝑦)) = ((𝑓𝑥)(+g𝐾)(𝑓𝑦))))
3130raleqbi1dv 3402 . . . . . . . . . . . . 13 (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 . . . . . . . . . . . 12 (𝜑 → (∀𝑥𝐵𝑦𝐵 (𝑓‘(𝑥(+g𝐽)𝑦)) = ((𝑓𝑥)(+g𝐾)(𝑓𝑦)) ↔ ∀𝑥 ∈ (Base‘𝐽)∀𝑦 ∈ (Base‘𝐽)(𝑓‘(𝑥(+g𝐽)𝑦)) = ((𝑓𝑥)(+g𝐾)(𝑓𝑦))))
3332adantr 483 . . . . . . . . . . 11 ((𝜑 ∧ ((𝐽 ∈ Mnd ∧ 𝐾 ∈ Mnd) ∧ 𝑓:𝐵𝐶)) → (∀𝑥𝐵𝑦𝐵 (𝑓‘(𝑥(+g𝐽)𝑦)) = ((𝑓𝑥)(+g𝐾)(𝑓𝑦)) ↔ ∀𝑥 ∈ (Base‘𝐽)∀𝑦 ∈ (Base‘𝐽)(𝑓‘(𝑥(+g𝐽)𝑦)) = ((𝑓𝑥)(+g𝐾)(𝑓𝑦))))
34 mhmpropd.c . . . . . . . . . . . . 13 (𝜑𝐵 = (Base‘𝐿))
35 raleq 3404 . . . . . . . . . . . . . 14 (𝐵 = (Base‘𝐿) → (∀𝑦𝐵 (𝑓‘(𝑥(+g𝐿)𝑦)) = ((𝑓𝑥)(+g𝑀)(𝑓𝑦)) ↔ ∀𝑦 ∈ (Base‘𝐿)(𝑓‘(𝑥(+g𝐿)𝑦)) = ((𝑓𝑥)(+g𝑀)(𝑓𝑦))))
3635raleqbi1dv 3402 . . . . . . . . . . . . 13 (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 . . . . . . . . . . . 12 (𝜑 → (∀𝑥𝐵𝑦𝐵 (𝑓‘(𝑥(+g𝐿)𝑦)) = ((𝑓𝑥)(+g𝑀)(𝑓𝑦)) ↔ ∀𝑥 ∈ (Base‘𝐿)∀𝑦 ∈ (Base‘𝐿)(𝑓‘(𝑥(+g𝐿)𝑦)) = ((𝑓𝑥)(+g𝑀)(𝑓𝑦))))
3837adantr 483 . . . . . . . . . . 11 ((𝜑 ∧ ((𝐽 ∈ Mnd ∧ 𝐾 ∈ Mnd) ∧ 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 311 . . . . . . . . . 10 ((𝜑 ∧ ((𝐽 ∈ Mnd ∧ 𝐾 ∈ Mnd) ∧ 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))))
4029adantr 483 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝐽 ∈ Mnd ∧ 𝐾 ∈ Mnd) ∧ 𝑓:𝐵𝐶)) → 𝐵 = (Base‘𝐽))
4134adantr 483 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝐽 ∈ Mnd ∧ 𝐾 ∈ Mnd) ∧ 𝑓:𝐵𝐶)) → 𝐵 = (Base‘𝐿))
421adantlr 713 . . . . . . . . . . . . 13 (((𝜑 ∧ ((𝐽 ∈ Mnd ∧ 𝐾 ∈ Mnd) ∧ 𝑓:𝐵𝐶)) ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐽)𝑦) = (𝑥(+g𝐿)𝑦))
4340, 41, 42grpidpropd 17864 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝐽 ∈ Mnd ∧ 𝐾 ∈ Mnd) ∧ 𝑓:𝐵𝐶)) → (0g𝐽) = (0g𝐿))
4443fveq2d 6667 . . . . . . . . . . 11 ((𝜑 ∧ ((𝐽 ∈ Mnd ∧ 𝐾 ∈ Mnd) ∧ 𝑓:𝐵𝐶)) → (𝑓‘(0g𝐽)) = (𝑓‘(0g𝐿)))
45 mhmpropd.b . . . . . . . . . . . . 13 (𝜑𝐶 = (Base‘𝐾))
4645adantr 483 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝐽 ∈ Mnd ∧ 𝐾 ∈ Mnd) ∧ 𝑓:𝐵𝐶)) → 𝐶 = (Base‘𝐾))
47 mhmpropd.d . . . . . . . . . . . . 13 (𝜑𝐶 = (Base‘𝑀))
4847adantr 483 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝐽 ∈ Mnd ∧ 𝐾 ∈ Mnd) ∧ 𝑓:𝐵𝐶)) → 𝐶 = (Base‘𝑀))
497adantlr 713 . . . . . . . . . . . 12 (((𝜑 ∧ ((𝐽 ∈ Mnd ∧ 𝐾 ∈ Mnd) ∧ 𝑓:𝐵𝐶)) ∧ (𝑥𝐶𝑦𝐶)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝑀)𝑦))
5046, 48, 49grpidpropd 17864 . . . . . . . . . . 11 ((𝜑 ∧ ((𝐽 ∈ Mnd ∧ 𝐾 ∈ Mnd) ∧ 𝑓:𝐵𝐶)) → (0g𝐾) = (0g𝑀))
5144, 50eqeq12d 2835 . . . . . . . . . 10 ((𝜑 ∧ ((𝐽 ∈ Mnd ∧ 𝐾 ∈ Mnd) ∧ 𝑓:𝐵𝐶)) → ((𝑓‘(0g𝐽)) = (0g𝐾) ↔ (𝑓‘(0g𝐿)) = (0g𝑀)))
5239, 51anbi12d 632 . . . . . . . . 9 ((𝜑 ∧ ((𝐽 ∈ Mnd ∧ 𝐾 ∈ Mnd) ∧ 𝑓:𝐵𝐶)) → ((∀𝑥 ∈ (Base‘𝐽)∀𝑦 ∈ (Base‘𝐽)(𝑓‘(𝑥(+g𝐽)𝑦)) = ((𝑓𝑥)(+g𝐾)(𝑓𝑦)) ∧ (𝑓‘(0g𝐽)) = (0g𝐾)) ↔ (∀𝑥 ∈ (Base‘𝐿)∀𝑦 ∈ (Base‘𝐿)(𝑓‘(𝑥(+g𝐿)𝑦)) = ((𝑓𝑥)(+g𝑀)(𝑓𝑦)) ∧ (𝑓‘(0g𝐿)) = (0g𝑀))))
5352anassrs 470 . . . . . . . 8 (((𝜑 ∧ (𝐽 ∈ Mnd ∧ 𝐾 ∈ Mnd)) ∧ 𝑓:𝐵𝐶) → ((∀𝑥 ∈ (Base‘𝐽)∀𝑦 ∈ (Base‘𝐽)(𝑓‘(𝑥(+g𝐽)𝑦)) = ((𝑓𝑥)(+g𝐾)(𝑓𝑦)) ∧ (𝑓‘(0g𝐽)) = (0g𝐾)) ↔ (∀𝑥 ∈ (Base‘𝐿)∀𝑦 ∈ (Base‘𝐿)(𝑓‘(𝑥(+g𝐿)𝑦)) = ((𝑓𝑥)(+g𝑀)(𝑓𝑦)) ∧ (𝑓‘(0g𝐿)) = (0g𝑀))))
5453pm5.32da 581 . . . . . . 7 ((𝜑 ∧ (𝐽 ∈ Mnd ∧ 𝐾 ∈ Mnd)) → ((𝑓:𝐵𝐶 ∧ (∀𝑥 ∈ (Base‘𝐽)∀𝑦 ∈ (Base‘𝐽)(𝑓‘(𝑥(+g𝐽)𝑦)) = ((𝑓𝑥)(+g𝐾)(𝑓𝑦)) ∧ (𝑓‘(0g𝐽)) = (0g𝐾))) ↔ (𝑓:𝐵𝐶 ∧ (∀𝑥 ∈ (Base‘𝐿)∀𝑦 ∈ (Base‘𝐿)(𝑓‘(𝑥(+g𝐿)𝑦)) = ((𝑓𝑥)(+g𝑀)(𝑓𝑦)) ∧ (𝑓‘(0g𝐿)) = (0g𝑀)))))
5529, 45feq23d 6502 . . . . . . . . 9 (𝜑 → (𝑓:𝐵𝐶𝑓:(Base‘𝐽)⟶(Base‘𝐾)))
5655adantr 483 . . . . . . . 8 ((𝜑 ∧ (𝐽 ∈ Mnd ∧ 𝐾 ∈ Mnd)) → (𝑓:𝐵𝐶𝑓:(Base‘𝐽)⟶(Base‘𝐾)))
5756anbi1d 631 . . . . . . 7 ((𝜑 ∧ (𝐽 ∈ Mnd ∧ 𝐾 ∈ Mnd)) → ((𝑓:𝐵𝐶 ∧ (∀𝑥 ∈ (Base‘𝐽)∀𝑦 ∈ (Base‘𝐽)(𝑓‘(𝑥(+g𝐽)𝑦)) = ((𝑓𝑥)(+g𝐾)(𝑓𝑦)) ∧ (𝑓‘(0g𝐽)) = (0g𝐾))) ↔ (𝑓:(Base‘𝐽)⟶(Base‘𝐾) ∧ (∀𝑥 ∈ (Base‘𝐽)∀𝑦 ∈ (Base‘𝐽)(𝑓‘(𝑥(+g𝐽)𝑦)) = ((𝑓𝑥)(+g𝐾)(𝑓𝑦)) ∧ (𝑓‘(0g𝐽)) = (0g𝐾)))))
5834, 47feq23d 6502 . . . . . . . . 9 (𝜑 → (𝑓:𝐵𝐶𝑓:(Base‘𝐿)⟶(Base‘𝑀)))
5958adantr 483 . . . . . . . 8 ((𝜑 ∧ (𝐽 ∈ Mnd ∧ 𝐾 ∈ Mnd)) → (𝑓:𝐵𝐶𝑓:(Base‘𝐿)⟶(Base‘𝑀)))
6059anbi1d 631 . . . . . . 7 ((𝜑 ∧ (𝐽 ∈ Mnd ∧ 𝐾 ∈ Mnd)) → ((𝑓:𝐵𝐶 ∧ (∀𝑥 ∈ (Base‘𝐿)∀𝑦 ∈ (Base‘𝐿)(𝑓‘(𝑥(+g𝐿)𝑦)) = ((𝑓𝑥)(+g𝑀)(𝑓𝑦)) ∧ (𝑓‘(0g𝐿)) = (0g𝑀))) ↔ (𝑓:(Base‘𝐿)⟶(Base‘𝑀) ∧ (∀𝑥 ∈ (Base‘𝐿)∀𝑦 ∈ (Base‘𝐿)(𝑓‘(𝑥(+g𝐿)𝑦)) = ((𝑓𝑥)(+g𝑀)(𝑓𝑦)) ∧ (𝑓‘(0g𝐿)) = (0g𝑀)))))
6154, 57, 603bitr3d 311 . . . . . 6 ((𝜑 ∧ (𝐽 ∈ Mnd ∧ 𝐾 ∈ Mnd)) → ((𝑓:(Base‘𝐽)⟶(Base‘𝐾) ∧ (∀𝑥 ∈ (Base‘𝐽)∀𝑦 ∈ (Base‘𝐽)(𝑓‘(𝑥(+g𝐽)𝑦)) = ((𝑓𝑥)(+g𝐾)(𝑓𝑦)) ∧ (𝑓‘(0g𝐽)) = (0g𝐾))) ↔ (𝑓:(Base‘𝐿)⟶(Base‘𝑀) ∧ (∀𝑥 ∈ (Base‘𝐿)∀𝑦 ∈ (Base‘𝐿)(𝑓‘(𝑥(+g𝐿)𝑦)) = ((𝑓𝑥)(+g𝑀)(𝑓𝑦)) ∧ (𝑓‘(0g𝐿)) = (0g𝑀)))))
62 3anass 1090 . . . . . 6 ((𝑓:(Base‘𝐽)⟶(Base‘𝐾) ∧ ∀𝑥 ∈ (Base‘𝐽)∀𝑦 ∈ (Base‘𝐽)(𝑓‘(𝑥(+g𝐽)𝑦)) = ((𝑓𝑥)(+g𝐾)(𝑓𝑦)) ∧ (𝑓‘(0g𝐽)) = (0g𝐾)) ↔ (𝑓:(Base‘𝐽)⟶(Base‘𝐾) ∧ (∀𝑥 ∈ (Base‘𝐽)∀𝑦 ∈ (Base‘𝐽)(𝑓‘(𝑥(+g𝐽)𝑦)) = ((𝑓𝑥)(+g𝐾)(𝑓𝑦)) ∧ (𝑓‘(0g𝐽)) = (0g𝐾))))
63 3anass 1090 . . . . . 6 ((𝑓:(Base‘𝐿)⟶(Base‘𝑀) ∧ ∀𝑥 ∈ (Base‘𝐿)∀𝑦 ∈ (Base‘𝐿)(𝑓‘(𝑥(+g𝐿)𝑦)) = ((𝑓𝑥)(+g𝑀)(𝑓𝑦)) ∧ (𝑓‘(0g𝐿)) = (0g𝑀)) ↔ (𝑓:(Base‘𝐿)⟶(Base‘𝑀) ∧ (∀𝑥 ∈ (Base‘𝐿)∀𝑦 ∈ (Base‘𝐿)(𝑓‘(𝑥(+g𝐿)𝑦)) = ((𝑓𝑥)(+g𝑀)(𝑓𝑦)) ∧ (𝑓‘(0g𝐿)) = (0g𝑀))))
6461, 62, 633bitr4g 316 . . . . 5 ((𝜑 ∧ (𝐽 ∈ Mnd ∧ 𝐾 ∈ Mnd)) → ((𝑓:(Base‘𝐽)⟶(Base‘𝐾) ∧ ∀𝑥 ∈ (Base‘𝐽)∀𝑦 ∈ (Base‘𝐽)(𝑓‘(𝑥(+g𝐽)𝑦)) = ((𝑓𝑥)(+g𝐾)(𝑓𝑦)) ∧ (𝑓‘(0g𝐽)) = (0g𝐾)) ↔ (𝑓:(Base‘𝐿)⟶(Base‘𝑀) ∧ ∀𝑥 ∈ (Base‘𝐿)∀𝑦 ∈ (Base‘𝐿)(𝑓‘(𝑥(+g𝐿)𝑦)) = ((𝑓𝑥)(+g𝑀)(𝑓𝑦)) ∧ (𝑓‘(0g𝐿)) = (0g𝑀))))
6564pm5.32da 581 . . . 4 (𝜑 → (((𝐽 ∈ Mnd ∧ 𝐾 ∈ Mnd) ∧ (𝑓:(Base‘𝐽)⟶(Base‘𝐾) ∧ ∀𝑥 ∈ (Base‘𝐽)∀𝑦 ∈ (Base‘𝐽)(𝑓‘(𝑥(+g𝐽)𝑦)) = ((𝑓𝑥)(+g𝐾)(𝑓𝑦)) ∧ (𝑓‘(0g𝐽)) = (0g𝐾))) ↔ ((𝐽 ∈ Mnd ∧ 𝐾 ∈ Mnd) ∧ (𝑓:(Base‘𝐿)⟶(Base‘𝑀) ∧ ∀𝑥 ∈ (Base‘𝐿)∀𝑦 ∈ (Base‘𝐿)(𝑓‘(𝑥(+g𝐿)𝑦)) = ((𝑓𝑥)(+g𝑀)(𝑓𝑦)) ∧ (𝑓‘(0g𝐿)) = (0g𝑀)))))
6629, 34, 1mndpropd 17928 . . . . . 6 (𝜑 → (𝐽 ∈ Mnd ↔ 𝐿 ∈ Mnd))
6745, 47, 7mndpropd 17928 . . . . . 6 (𝜑 → (𝐾 ∈ Mnd ↔ 𝑀 ∈ Mnd))
6866, 67anbi12d 632 . . . . 5 (𝜑 → ((𝐽 ∈ Mnd ∧ 𝐾 ∈ Mnd) ↔ (𝐿 ∈ Mnd ∧ 𝑀 ∈ Mnd)))
6968anbi1d 631 . . . 4 (𝜑 → (((𝐽 ∈ Mnd ∧ 𝐾 ∈ Mnd) ∧ (𝑓:(Base‘𝐿)⟶(Base‘𝑀) ∧ ∀𝑥 ∈ (Base‘𝐿)∀𝑦 ∈ (Base‘𝐿)(𝑓‘(𝑥(+g𝐿)𝑦)) = ((𝑓𝑥)(+g𝑀)(𝑓𝑦)) ∧ (𝑓‘(0g𝐿)) = (0g𝑀))) ↔ ((𝐿 ∈ Mnd ∧ 𝑀 ∈ Mnd) ∧ (𝑓:(Base‘𝐿)⟶(Base‘𝑀) ∧ ∀𝑥 ∈ (Base‘𝐿)∀𝑦 ∈ (Base‘𝐿)(𝑓‘(𝑥(+g𝐿)𝑦)) = ((𝑓𝑥)(+g𝑀)(𝑓𝑦)) ∧ (𝑓‘(0g𝐿)) = (0g𝑀)))))
7065, 69bitrd 281 . . 3 (𝜑 → (((𝐽 ∈ Mnd ∧ 𝐾 ∈ Mnd) ∧ (𝑓:(Base‘𝐽)⟶(Base‘𝐾) ∧ ∀𝑥 ∈ (Base‘𝐽)∀𝑦 ∈ (Base‘𝐽)(𝑓‘(𝑥(+g𝐽)𝑦)) = ((𝑓𝑥)(+g𝐾)(𝑓𝑦)) ∧ (𝑓‘(0g𝐽)) = (0g𝐾))) ↔ ((𝐿 ∈ Mnd ∧ 𝑀 ∈ Mnd) ∧ (𝑓:(Base‘𝐿)⟶(Base‘𝑀) ∧ ∀𝑥 ∈ (Base‘𝐿)∀𝑦 ∈ (Base‘𝐿)(𝑓‘(𝑥(+g𝐿)𝑦)) = ((𝑓𝑥)(+g𝑀)(𝑓𝑦)) ∧ (𝑓‘(0g𝐿)) = (0g𝑀)))))
71 eqid 2819 . . . 4 (Base‘𝐽) = (Base‘𝐽)
72 eqid 2819 . . . 4 (Base‘𝐾) = (Base‘𝐾)
73 eqid 2819 . . . 4 (+g𝐽) = (+g𝐽)
74 eqid 2819 . . . 4 (+g𝐾) = (+g𝐾)
75 eqid 2819 . . . 4 (0g𝐽) = (0g𝐽)
76 eqid 2819 . . . 4 (0g𝐾) = (0g𝐾)
7771, 72, 73, 74, 75, 76ismhm 17950 . . 3 (𝑓 ∈ (𝐽 MndHom 𝐾) ↔ ((𝐽 ∈ Mnd ∧ 𝐾 ∈ Mnd) ∧ (𝑓:(Base‘𝐽)⟶(Base‘𝐾) ∧ ∀𝑥 ∈ (Base‘𝐽)∀𝑦 ∈ (Base‘𝐽)(𝑓‘(𝑥(+g𝐽)𝑦)) = ((𝑓𝑥)(+g𝐾)(𝑓𝑦)) ∧ (𝑓‘(0g𝐽)) = (0g𝐾))))
78 eqid 2819 . . . 4 (Base‘𝐿) = (Base‘𝐿)
79 eqid 2819 . . . 4 (Base‘𝑀) = (Base‘𝑀)
80 eqid 2819 . . . 4 (+g𝐿) = (+g𝐿)
81 eqid 2819 . . . 4 (+g𝑀) = (+g𝑀)
82 eqid 2819 . . . 4 (0g𝐿) = (0g𝐿)
83 eqid 2819 . . . 4 (0g𝑀) = (0g𝑀)
8478, 79, 80, 81, 82, 83ismhm 17950 . . 3 (𝑓 ∈ (𝐿 MndHom 𝑀) ↔ ((𝐿 ∈ Mnd ∧ 𝑀 ∈ Mnd) ∧ (𝑓:(Base‘𝐿)⟶(Base‘𝑀) ∧ ∀𝑥 ∈ (Base‘𝐿)∀𝑦 ∈ (Base‘𝐿)(𝑓‘(𝑥(+g𝐿)𝑦)) = ((𝑓𝑥)(+g𝑀)(𝑓𝑦)) ∧ (𝑓‘(0g𝐿)) = (0g𝑀))))
8570, 77, 843bitr4g 316 . 2 (𝜑 → (𝑓 ∈ (𝐽 MndHom 𝐾) ↔ 𝑓 ∈ (𝐿 MndHom 𝑀)))
8685eqrdv 2817 1 (𝜑 → (𝐽 MndHom 𝐾) = (𝐿 MndHom 𝑀))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  w3a 1082   = wceq 1531  wcel 2108  wral 3136  wf 6344  cfv 6348  (class class class)co 7148  Basecbs 16475  +gcplusg 16557  0gc0g 16705  Mndcmnd 17903   MndHom cmhm 17946
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356  df-ov 7151  df-oprab 7152  df-mpo 7153  df-map 8400  df-0g 16707  df-mgm 17844  df-sgrp 17893  df-mnd 17904  df-mhm 17948
This theorem is referenced by:  ghmpropd  18388  pwsco1rhm  19482  pwsco2rhm  19483  pwsdiagrhm  19561  rhmpropd  19563
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