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Theorem mhmeql 18464
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 2738 . . . . . 6 (Base‘𝑆) = (Base‘𝑆)
2 eqid 2738 . . . . . 6 (Base‘𝑇) = (Base‘𝑇)
31, 2mhmf 18435 . . . . 5 (𝐹 ∈ (𝑆 MndHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
43adantr 481 . . . 4 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
54ffnd 6601 . . 3 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → 𝐹 Fn (Base‘𝑆))
61, 2mhmf 18435 . . . . 5 (𝐺 ∈ (𝑆 MndHom 𝑇) → 𝐺:(Base‘𝑆)⟶(Base‘𝑇))
76adantl 482 . . . 4 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → 𝐺:(Base‘𝑆)⟶(Base‘𝑇))
87ffnd 6601 . . 3 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → 𝐺 Fn (Base‘𝑆))
9 fndmin 6922 . . 3 ((𝐹 Fn (Base‘𝑆) ∧ 𝐺 Fn (Base‘𝑆)) → dom (𝐹𝐺) = {𝑧 ∈ (Base‘𝑆) ∣ (𝐹𝑧) = (𝐺𝑧)})
105, 8, 9syl2anc 584 . 2 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → dom (𝐹𝐺) = {𝑧 ∈ (Base‘𝑆) ∣ (𝐹𝑧) = (𝐺𝑧)})
11 ssrab2 4013 . . . 4 {𝑧 ∈ (Base‘𝑆) ∣ (𝐹𝑧) = (𝐺𝑧)} ⊆ (Base‘𝑆)
1211a1i 11 . . 3 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → {𝑧 ∈ (Base‘𝑆) ∣ (𝐹𝑧) = (𝐺𝑧)} ⊆ (Base‘𝑆))
13 fveq2 6774 . . . . 5 (𝑧 = (0g𝑆) → (𝐹𝑧) = (𝐹‘(0g𝑆)))
14 fveq2 6774 . . . . 5 (𝑧 = (0g𝑆) → (𝐺𝑧) = (𝐺‘(0g𝑆)))
1513, 14eqeq12d 2754 . . . 4 (𝑧 = (0g𝑆) → ((𝐹𝑧) = (𝐺𝑧) ↔ (𝐹‘(0g𝑆)) = (𝐺‘(0g𝑆))))
16 mhmrcl1 18433 . . . . . 6 (𝐹 ∈ (𝑆 MndHom 𝑇) → 𝑆 ∈ Mnd)
1716adantr 481 . . . . 5 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → 𝑆 ∈ Mnd)
18 eqid 2738 . . . . . 6 (0g𝑆) = (0g𝑆)
191, 18mndidcl 18400 . . . . 5 (𝑆 ∈ Mnd → (0g𝑆) ∈ (Base‘𝑆))
2017, 19syl 17 . . . 4 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → (0g𝑆) ∈ (Base‘𝑆))
21 eqid 2738 . . . . . . 7 (0g𝑇) = (0g𝑇)
2218, 21mhm0 18438 . . . . . 6 (𝐹 ∈ (𝑆 MndHom 𝑇) → (𝐹‘(0g𝑆)) = (0g𝑇))
2322adantr 481 . . . . 5 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → (𝐹‘(0g𝑆)) = (0g𝑇))
2418, 21mhm0 18438 . . . . . 6 (𝐺 ∈ (𝑆 MndHom 𝑇) → (𝐺‘(0g𝑆)) = (0g𝑇))
2524adantl 482 . . . . 5 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → (𝐺‘(0g𝑆)) = (0g𝑇))
2623, 25eqtr4d 2781 . . . 4 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → (𝐹‘(0g𝑆)) = (𝐺‘(0g𝑆)))
2715, 20, 26elrabd 3626 . . 3 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → (0g𝑆) ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹𝑧) = (𝐺𝑧)})
28 fveq2 6774 . . . . . . . . . . 11 (𝑧 = (𝑥(+g𝑆)𝑦) → (𝐹𝑧) = (𝐹‘(𝑥(+g𝑆)𝑦)))
29 fveq2 6774 . . . . . . . . . . 11 (𝑧 = (𝑥(+g𝑆)𝑦) → (𝐺𝑧) = (𝐺‘(𝑥(+g𝑆)𝑦)))
3028, 29eqeq12d 2754 . . . . . . . . . 10 (𝑧 = (𝑥(+g𝑆)𝑦) → ((𝐹𝑧) = (𝐺𝑧) ↔ (𝐹‘(𝑥(+g𝑆)𝑦)) = (𝐺‘(𝑥(+g𝑆)𝑦))))
3117ad2antrr 723 . . . . . . . . . . 11 ((((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹𝑥) = (𝐺𝑥))) ∧ (𝑦 ∈ (Base‘𝑆) ∧ (𝐹𝑦) = (𝐺𝑦))) → 𝑆 ∈ Mnd)
32 simplrl 774 . . . . . . . . . . 11 ((((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹𝑥) = (𝐺𝑥))) ∧ (𝑦 ∈ (Base‘𝑆) ∧ (𝐹𝑦) = (𝐺𝑦))) → 𝑥 ∈ (Base‘𝑆))
33 simprl 768 . . . . . . . . . . 11 ((((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹𝑥) = (𝐺𝑥))) ∧ (𝑦 ∈ (Base‘𝑆) ∧ (𝐹𝑦) = (𝐺𝑦))) → 𝑦 ∈ (Base‘𝑆))
34 eqid 2738 . . . . . . . . . . . 12 (+g𝑆) = (+g𝑆)
351, 34mndcl 18393 . . . . . . . . . . 11 ((𝑆 ∈ Mnd ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝑥(+g𝑆)𝑦) ∈ (Base‘𝑆))
3631, 32, 33, 35syl3anc 1370 . . . . . . . . . 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 772 . . . . . . . . . . . 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 2738 . . . . . . . . . . . . 13 (+g𝑇) = (+g𝑇)
391, 34, 38mhmlin 18437 . . . . . . . . . . . 12 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝐹‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑥)(+g𝑇)(𝐹𝑦)))
4037, 32, 33, 39syl3anc 1370 . . . . . . . . . . 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 773 . . . . . . . . . . . . 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 18437 . . . . . . . . . . . . 13 ((𝐺 ∈ (𝑆 MndHom 𝑇) ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝐺‘(𝑥(+g𝑆)𝑦)) = ((𝐺𝑥)(+g𝑇)(𝐺𝑦)))
4341, 32, 33, 42syl3anc 1370 . . . . . . . . . . . 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 775 . . . . . . . . . . . . 13 ((((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹𝑥) = (𝐺𝑥))) ∧ (𝑦 ∈ (Base‘𝑆) ∧ (𝐹𝑦) = (𝐺𝑦))) → (𝐹𝑥) = (𝐺𝑥))
45 simprr 770 . . . . . . . . . . . . 13 ((((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹𝑥) = (𝐺𝑥))) ∧ (𝑦 ∈ (Base‘𝑆) ∧ (𝐹𝑦) = (𝐺𝑦))) → (𝐹𝑦) = (𝐺𝑦))
4644, 45oveq12d 7293 . . . . . . . . . . . 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 2781 . . . . . . . . . . 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 2781 . . . . . . . . . 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)))
4930, 36, 48elrabd 3626 . . . . . . . . 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)})
5049expr 457 . . . . . . . 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)}))
5150ralrimiva 3103 . . . . . . 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)}))
52 fveq2 6774 . . . . . . . . 9 (𝑧 = 𝑦 → (𝐹𝑧) = (𝐹𝑦))
53 fveq2 6774 . . . . . . . . 9 (𝑧 = 𝑦 → (𝐺𝑧) = (𝐺𝑦))
5452, 53eqeq12d 2754 . . . . . . . 8 (𝑧 = 𝑦 → ((𝐹𝑧) = (𝐺𝑧) ↔ (𝐹𝑦) = (𝐺𝑦)))
5554ralrab 3630 . . . . . . 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)}))
5651, 55sylibr 233 . . . . . 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)})
5756expr 457 . . . . 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)}))
5857ralrimiva 3103 . . . 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)}))
59 fveq2 6774 . . . . . 6 (𝑧 = 𝑥 → (𝐹𝑧) = (𝐹𝑥))
60 fveq2 6774 . . . . . 6 (𝑧 = 𝑥 → (𝐺𝑧) = (𝐺𝑥))
6159, 60eqeq12d 2754 . . . . 5 (𝑧 = 𝑥 → ((𝐹𝑧) = (𝐺𝑧) ↔ (𝐹𝑥) = (𝐺𝑥)))
6261ralrab 3630 . . . 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)}))
6358, 62sylibr 233 . . 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)})
641, 18, 34issubm 18442 . . . 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)})))
6517, 64syl 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)})))
6612, 27, 63, 65mpbir3and 1341 . 2 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) → {𝑧 ∈ (Base‘𝑆) ∣ (𝐹𝑧) = (𝐺𝑧)} ∈ (SubMnd‘𝑆))
6710, 66eqeltrd 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 205  wa 396  w3a 1086   = wceq 1539  wcel 2106  wral 3064  {crab 3068  cin 3886  wss 3887  dom cdm 5589   Fn wfn 6428  wf 6429  cfv 6433  (class class class)co 7275  Basecbs 16912  +gcplusg 16962  0gc0g 17150  Mndcmnd 18385   MndHom cmhm 18428  SubMndcsubmnd 18429
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-map 8617  df-0g 17152  df-mgm 18326  df-sgrp 18375  df-mnd 18386  df-mhm 18430  df-submnd 18431
This theorem is referenced by:  ghmeql  18857  rhmeql  20054
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