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Theorem mhmeql 17585
Description: The equalizer of two monoid homomorphisms is a submonoid. (Contributed by Stefan O'Rear, 7-Mar-2015.) (Revised by Mario Carneiro, 6-May-2015.)
Assertion
Ref Expression
mhmeql ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → dom (𝐹𝐺) ∈ (SubMnd‘𝑆))

Proof of Theorem mhmeql
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2760 . . . . . 6 (Base‘𝑆) = (Base‘𝑆)
2 eqid 2760 . . . . . 6 (Base‘𝑇) = (Base‘𝑇)
31, 2mhmf 17561 . . . . 5 (𝐹 ∈ (𝑆 MndHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
43adantr 472 . . . 4 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
5 ffn 6206 . . . 4 (𝐹:(Base‘𝑆)⟶(Base‘𝑇) → 𝐹 Fn (Base‘𝑆))
64, 5syl 17 . . 3 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → 𝐹 Fn (Base‘𝑆))
71, 2mhmf 17561 . . . . 5 (𝐺 ∈ (𝑆 MndHom 𝑇) → 𝐺:(Base‘𝑆)⟶(Base‘𝑇))
87adantl 473 . . . 4 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → 𝐺:(Base‘𝑆)⟶(Base‘𝑇))
9 ffn 6206 . . . 4 (𝐺:(Base‘𝑆)⟶(Base‘𝑇) → 𝐺 Fn (Base‘𝑆))
108, 9syl 17 . . 3 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → 𝐺 Fn (Base‘𝑆))
11 fndmin 6488 . . 3 ((𝐹 Fn (Base‘𝑆) ∧ 𝐺 Fn (Base‘𝑆)) → dom (𝐹𝐺) = {𝑧 ∈ (Base‘𝑆) ∣ (𝐹𝑧) = (𝐺𝑧)})
126, 10, 11syl2anc 696 . 2 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → dom (𝐹𝐺) = {𝑧 ∈ (Base‘𝑆) ∣ (𝐹𝑧) = (𝐺𝑧)})
13 ssrab2 3828 . . . 4 {𝑧 ∈ (Base‘𝑆) ∣ (𝐹𝑧) = (𝐺𝑧)} ⊆ (Base‘𝑆)
1413a1i 11 . . 3 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → {𝑧 ∈ (Base‘𝑆) ∣ (𝐹𝑧) = (𝐺𝑧)} ⊆ (Base‘𝑆))
15 mhmrcl1 17559 . . . . . 6 (𝐹 ∈ (𝑆 MndHom 𝑇) → 𝑆 ∈ Mnd)
1615adantr 472 . . . . 5 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → 𝑆 ∈ Mnd)
17 eqid 2760 . . . . . 6 (0g𝑆) = (0g𝑆)
181, 17mndidcl 17529 . . . . 5 (𝑆 ∈ Mnd → (0g𝑆) ∈ (Base‘𝑆))
1916, 18syl 17 . . . 4 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → (0g𝑆) ∈ (Base‘𝑆))
20 eqid 2760 . . . . . . 7 (0g𝑇) = (0g𝑇)
2117, 20mhm0 17564 . . . . . 6 (𝐹 ∈ (𝑆 MndHom 𝑇) → (𝐹‘(0g𝑆)) = (0g𝑇))
2221adantr 472 . . . . 5 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → (𝐹‘(0g𝑆)) = (0g𝑇))
2317, 20mhm0 17564 . . . . . 6 (𝐺 ∈ (𝑆 MndHom 𝑇) → (𝐺‘(0g𝑆)) = (0g𝑇))
2423adantl 473 . . . . 5 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → (𝐺‘(0g𝑆)) = (0g𝑇))
2522, 24eqtr4d 2797 . . . 4 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → (𝐹‘(0g𝑆)) = (𝐺‘(0g𝑆)))
26 fveq2 6353 . . . . . 6 (𝑧 = (0g𝑆) → (𝐹𝑧) = (𝐹‘(0g𝑆)))
27 fveq2 6353 . . . . . 6 (𝑧 = (0g𝑆) → (𝐺𝑧) = (𝐺‘(0g𝑆)))
2826, 27eqeq12d 2775 . . . . 5 (𝑧 = (0g𝑆) → ((𝐹𝑧) = (𝐺𝑧) ↔ (𝐹‘(0g𝑆)) = (𝐺‘(0g𝑆))))
2928elrab 3504 . . . 4 ((0g𝑆) ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹𝑧) = (𝐺𝑧)} ↔ ((0g𝑆) ∈ (Base‘𝑆) ∧ (𝐹‘(0g𝑆)) = (𝐺‘(0g𝑆))))
3019, 25, 29sylanbrc 701 . . 3 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → (0g𝑆) ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹𝑧) = (𝐺𝑧)})
3116ad2antrr 764 . . . . . . . . . . 11 ((((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹𝑥) = (𝐺𝑥))) ∧ (𝑦 ∈ (Base‘𝑆) ∧ (𝐹𝑦) = (𝐺𝑦))) → 𝑆 ∈ Mnd)
32 simplrl 819 . . . . . . . . . . 11 ((((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹𝑥) = (𝐺𝑥))) ∧ (𝑦 ∈ (Base‘𝑆) ∧ (𝐹𝑦) = (𝐺𝑦))) → 𝑥 ∈ (Base‘𝑆))
33 simprl 811 . . . . . . . . . . 11 ((((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹𝑥) = (𝐺𝑥))) ∧ (𝑦 ∈ (Base‘𝑆) ∧ (𝐹𝑦) = (𝐺𝑦))) → 𝑦 ∈ (Base‘𝑆))
34 eqid 2760 . . . . . . . . . . . 12 (+g𝑆) = (+g𝑆)
351, 34mndcl 17522 . . . . . . . . . . 11 ((𝑆 ∈ Mnd ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝑥(+g𝑆)𝑦) ∈ (Base‘𝑆))
3631, 32, 33, 35syl3anc 1477 . . . . . . . . . 10 ((((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹𝑥) = (𝐺𝑥))) ∧ (𝑦 ∈ (Base‘𝑆) ∧ (𝐹𝑦) = (𝐺𝑦))) → (𝑥(+g𝑆)𝑦) ∈ (Base‘𝑆))
37 simplll 815 . . . . . . . . . . . 12 ((((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹𝑥) = (𝐺𝑥))) ∧ (𝑦 ∈ (Base‘𝑆) ∧ (𝐹𝑦) = (𝐺𝑦))) → 𝐹 ∈ (𝑆 MndHom 𝑇))
38 eqid 2760 . . . . . . . . . . . . 13 (+g𝑇) = (+g𝑇)
391, 34, 38mhmlin 17563 . . . . . . . . . . . 12 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝐹‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑥)(+g𝑇)(𝐹𝑦)))
4037, 32, 33, 39syl3anc 1477 . . . . . . . . . . 11 ((((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹𝑥) = (𝐺𝑥))) ∧ (𝑦 ∈ (Base‘𝑆) ∧ (𝐹𝑦) = (𝐺𝑦))) → (𝐹‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑥)(+g𝑇)(𝐹𝑦)))
41 simpllr 817 . . . . . . . . . . . . 13 ((((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹𝑥) = (𝐺𝑥))) ∧ (𝑦 ∈ (Base‘𝑆) ∧ (𝐹𝑦) = (𝐺𝑦))) → 𝐺 ∈ (𝑆 MndHom 𝑇))
421, 34, 38mhmlin 17563 . . . . . . . . . . . . 13 ((𝐺 ∈ (𝑆 MndHom 𝑇) ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝐺‘(𝑥(+g𝑆)𝑦)) = ((𝐺𝑥)(+g𝑇)(𝐺𝑦)))
4341, 32, 33, 42syl3anc 1477 . . . . . . . . . . . 12 ((((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹𝑥) = (𝐺𝑥))) ∧ (𝑦 ∈ (Base‘𝑆) ∧ (𝐹𝑦) = (𝐺𝑦))) → (𝐺‘(𝑥(+g𝑆)𝑦)) = ((𝐺𝑥)(+g𝑇)(𝐺𝑦)))
44 simplrr 820 . . . . . . . . . . . . 13 ((((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹𝑥) = (𝐺𝑥))) ∧ (𝑦 ∈ (Base‘𝑆) ∧ (𝐹𝑦) = (𝐺𝑦))) → (𝐹𝑥) = (𝐺𝑥))
45 simprr 813 . . . . . . . . . . . . 13 ((((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹𝑥) = (𝐺𝑥))) ∧ (𝑦 ∈ (Base‘𝑆) ∧ (𝐹𝑦) = (𝐺𝑦))) → (𝐹𝑦) = (𝐺𝑦))
4644, 45oveq12d 6832 . . . . . . . . . . . 12 ((((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹𝑥) = (𝐺𝑥))) ∧ (𝑦 ∈ (Base‘𝑆) ∧ (𝐹𝑦) = (𝐺𝑦))) → ((𝐹𝑥)(+g𝑇)(𝐹𝑦)) = ((𝐺𝑥)(+g𝑇)(𝐺𝑦)))
4743, 46eqtr4d 2797 . . . . . . . . . . 11 ((((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹𝑥) = (𝐺𝑥))) ∧ (𝑦 ∈ (Base‘𝑆) ∧ (𝐹𝑦) = (𝐺𝑦))) → (𝐺‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑥)(+g𝑇)(𝐹𝑦)))
4840, 47eqtr4d 2797 . . . . . . . . . 10 ((((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹𝑥) = (𝐺𝑥))) ∧ (𝑦 ∈ (Base‘𝑆) ∧ (𝐹𝑦) = (𝐺𝑦))) → (𝐹‘(𝑥(+g𝑆)𝑦)) = (𝐺‘(𝑥(+g𝑆)𝑦)))
49 fveq2 6353 . . . . . . . . . . . 12 (𝑧 = (𝑥(+g𝑆)𝑦) → (𝐹𝑧) = (𝐹‘(𝑥(+g𝑆)𝑦)))
50 fveq2 6353 . . . . . . . . . . . 12 (𝑧 = (𝑥(+g𝑆)𝑦) → (𝐺𝑧) = (𝐺‘(𝑥(+g𝑆)𝑦)))
5149, 50eqeq12d 2775 . . . . . . . . . . 11 (𝑧 = (𝑥(+g𝑆)𝑦) → ((𝐹𝑧) = (𝐺𝑧) ↔ (𝐹‘(𝑥(+g𝑆)𝑦)) = (𝐺‘(𝑥(+g𝑆)𝑦))))
5251elrab 3504 . . . . . . . . . 10 ((𝑥(+g𝑆)𝑦) ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹𝑧) = (𝐺𝑧)} ↔ ((𝑥(+g𝑆)𝑦) ∈ (Base‘𝑆) ∧ (𝐹‘(𝑥(+g𝑆)𝑦)) = (𝐺‘(𝑥(+g𝑆)𝑦))))
5336, 48, 52sylanbrc 701 . . . . . . . . 9 ((((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹𝑥) = (𝐺𝑥))) ∧ (𝑦 ∈ (Base‘𝑆) ∧ (𝐹𝑦) = (𝐺𝑦))) → (𝑥(+g𝑆)𝑦) ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹𝑧) = (𝐺𝑧)})
5453expr 644 . . . . . . . 8 ((((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹𝑥) = (𝐺𝑥))) ∧ 𝑦 ∈ (Base‘𝑆)) → ((𝐹𝑦) = (𝐺𝑦) → (𝑥(+g𝑆)𝑦) ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹𝑧) = (𝐺𝑧)}))
5554ralrimiva 3104 . . . . . . 7 (((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹𝑥) = (𝐺𝑥))) → ∀𝑦 ∈ (Base‘𝑆)((𝐹𝑦) = (𝐺𝑦) → (𝑥(+g𝑆)𝑦) ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹𝑧) = (𝐺𝑧)}))
56 fveq2 6353 . . . . . . . . 9 (𝑧 = 𝑦 → (𝐹𝑧) = (𝐹𝑦))
57 fveq2 6353 . . . . . . . . 9 (𝑧 = 𝑦 → (𝐺𝑧) = (𝐺𝑦))
5856, 57eqeq12d 2775 . . . . . . . 8 (𝑧 = 𝑦 → ((𝐹𝑧) = (𝐺𝑧) ↔ (𝐹𝑦) = (𝐺𝑦)))
5958ralrab 3509 . . . . . . 7 (∀𝑦 ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹𝑧) = (𝐺𝑧)} (𝑥(+g𝑆)𝑦) ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹𝑧) = (𝐺𝑧)} ↔ ∀𝑦 ∈ (Base‘𝑆)((𝐹𝑦) = (𝐺𝑦) → (𝑥(+g𝑆)𝑦) ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹𝑧) = (𝐺𝑧)}))
6055, 59sylibr 224 . . . . . 6 (((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹𝑥) = (𝐺𝑥))) → ∀𝑦 ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹𝑧) = (𝐺𝑧)} (𝑥(+g𝑆)𝑦) ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹𝑧) = (𝐺𝑧)})
6160expr 644 . . . . 5 (((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ 𝑥 ∈ (Base‘𝑆)) → ((𝐹𝑥) = (𝐺𝑥) → ∀𝑦 ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹𝑧) = (𝐺𝑧)} (𝑥(+g𝑆)𝑦) ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹𝑧) = (𝐺𝑧)}))
6261ralrimiva 3104 . . . 4 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → ∀𝑥 ∈ (Base‘𝑆)((𝐹𝑥) = (𝐺𝑥) → ∀𝑦 ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹𝑧) = (𝐺𝑧)} (𝑥(+g𝑆)𝑦) ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹𝑧) = (𝐺𝑧)}))
63 fveq2 6353 . . . . . 6 (𝑧 = 𝑥 → (𝐹𝑧) = (𝐹𝑥))
64 fveq2 6353 . . . . . 6 (𝑧 = 𝑥 → (𝐺𝑧) = (𝐺𝑥))
6563, 64eqeq12d 2775 . . . . 5 (𝑧 = 𝑥 → ((𝐹𝑧) = (𝐺𝑧) ↔ (𝐹𝑥) = (𝐺𝑥)))
6665ralrab 3509 . . . 4 (∀𝑥 ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹𝑧) = (𝐺𝑧)}∀𝑦 ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹𝑧) = (𝐺𝑧)} (𝑥(+g𝑆)𝑦) ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹𝑧) = (𝐺𝑧)} ↔ ∀𝑥 ∈ (Base‘𝑆)((𝐹𝑥) = (𝐺𝑥) → ∀𝑦 ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹𝑧) = (𝐺𝑧)} (𝑥(+g𝑆)𝑦) ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹𝑧) = (𝐺𝑧)}))
6762, 66sylibr 224 . . 3 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → ∀𝑥 ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹𝑧) = (𝐺𝑧)}∀𝑦 ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹𝑧) = (𝐺𝑧)} (𝑥(+g𝑆)𝑦) ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹𝑧) = (𝐺𝑧)})
681, 17, 34issubm 17568 . . . 4 (𝑆 ∈ Mnd → ({𝑧 ∈ (Base‘𝑆) ∣ (𝐹𝑧) = (𝐺𝑧)} ∈ (SubMnd‘𝑆) ↔ ({𝑧 ∈ (Base‘𝑆) ∣ (𝐹𝑧) = (𝐺𝑧)} ⊆ (Base‘𝑆) ∧ (0g𝑆) ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹𝑧) = (𝐺𝑧)} ∧ ∀𝑥 ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹𝑧) = (𝐺𝑧)}∀𝑦 ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹𝑧) = (𝐺𝑧)} (𝑥(+g𝑆)𝑦) ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹𝑧) = (𝐺𝑧)})))
6916, 68syl 17 . . 3 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → ({𝑧 ∈ (Base‘𝑆) ∣ (𝐹𝑧) = (𝐺𝑧)} ∈ (SubMnd‘𝑆) ↔ ({𝑧 ∈ (Base‘𝑆) ∣ (𝐹𝑧) = (𝐺𝑧)} ⊆ (Base‘𝑆) ∧ (0g𝑆) ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹𝑧) = (𝐺𝑧)} ∧ ∀𝑥 ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹𝑧) = (𝐺𝑧)}∀𝑦 ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹𝑧) = (𝐺𝑧)} (𝑥(+g𝑆)𝑦) ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹𝑧) = (𝐺𝑧)})))
7014, 30, 67, 69mpbir3and 1428 . 2 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → {𝑧 ∈ (Base‘𝑆) ∣ (𝐹𝑧) = (𝐺𝑧)} ∈ (SubMnd‘𝑆))
7112, 70eqeltrd 2839 1 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → dom (𝐹𝐺) ∈ (SubMnd‘𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072   = wceq 1632  wcel 2139  wral 3050  {crab 3054  cin 3714  wss 3715  dom cdm 5266   Fn wfn 6044  wf 6045  cfv 6049  (class class class)co 6814  Basecbs 16079  +gcplusg 16163  0gc0g 16322  Mndcmnd 17515   MndHom cmhm 17554  SubMndcsubmnd 17555
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-fv 6057  df-riota 6775  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-map 8027  df-0g 16324  df-mgm 17463  df-sgrp 17505  df-mnd 17516  df-mhm 17556  df-submnd 17557
This theorem is referenced by:  ghmeql  17904  rhmeql  19032
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