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Theorem mgmhmeql 46183
Description: The equalizer of two magma homomorphisms is a submagma. (Contributed by AV, 27-Feb-2020.)
Assertion
Ref Expression
mgmhmeql ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) → dom (𝐹𝐺) ∈ (SubMgm‘𝑆))

Proof of Theorem mgmhmeql
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2733 . . . . . 6 (Base‘𝑆) = (Base‘𝑆)
2 eqid 2733 . . . . . 6 (Base‘𝑇) = (Base‘𝑇)
31, 2mgmhmf 46164 . . . . 5 (𝐹 ∈ (𝑆 MgmHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
43adantr 482 . . . 4 ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
54ffnd 6670 . . 3 ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) → 𝐹 Fn (Base‘𝑆))
61, 2mgmhmf 46164 . . . . 5 (𝐺 ∈ (𝑆 MgmHom 𝑇) → 𝐺:(Base‘𝑆)⟶(Base‘𝑇))
76adantl 483 . . . 4 ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) → 𝐺:(Base‘𝑆)⟶(Base‘𝑇))
87ffnd 6670 . . 3 ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) → 𝐺 Fn (Base‘𝑆))
9 fndmin 6996 . . 3 ((𝐹 Fn (Base‘𝑆) ∧ 𝐺 Fn (Base‘𝑆)) → dom (𝐹𝐺) = {𝑧 ∈ (Base‘𝑆) ∣ (𝐹𝑧) = (𝐺𝑧)})
105, 8, 9syl2anc 585 . 2 ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) → dom (𝐹𝐺) = {𝑧 ∈ (Base‘𝑆) ∣ (𝐹𝑧) = (𝐺𝑧)})
11 ssrab2 4038 . . . 4 {𝑧 ∈ (Base‘𝑆) ∣ (𝐹𝑧) = (𝐺𝑧)} ⊆ (Base‘𝑆)
1211a1i 11 . . 3 ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) → {𝑧 ∈ (Base‘𝑆) ∣ (𝐹𝑧) = (𝐺𝑧)} ⊆ (Base‘𝑆))
13 mgmhmrcl 46161 . . . . . . . . . . . . . 14 (𝐹 ∈ (𝑆 MgmHom 𝑇) → (𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm))
1413simpld 496 . . . . . . . . . . . . 13 (𝐹 ∈ (𝑆 MgmHom 𝑇) → 𝑆 ∈ Mgm)
1514adantr 482 . . . . . . . . . . . 12 ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) → 𝑆 ∈ Mgm)
1615ad2antrr 725 . . . . . . . . . . 11 ((((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹𝑥) = (𝐺𝑥))) ∧ (𝑦 ∈ (Base‘𝑆) ∧ (𝐹𝑦) = (𝐺𝑦))) → 𝑆 ∈ Mgm)
17 simplrl 776 . . . . . . . . . . 11 ((((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹𝑥) = (𝐺𝑥))) ∧ (𝑦 ∈ (Base‘𝑆) ∧ (𝐹𝑦) = (𝐺𝑦))) → 𝑥 ∈ (Base‘𝑆))
18 simprl 770 . . . . . . . . . . 11 ((((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹𝑥) = (𝐺𝑥))) ∧ (𝑦 ∈ (Base‘𝑆) ∧ (𝐹𝑦) = (𝐺𝑦))) → 𝑦 ∈ (Base‘𝑆))
19 eqid 2733 . . . . . . . . . . . 12 (+g𝑆) = (+g𝑆)
201, 19mgmcl 18505 . . . . . . . . . . 11 ((𝑆 ∈ Mgm ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝑥(+g𝑆)𝑦) ∈ (Base‘𝑆))
2116, 17, 18, 20syl3anc 1372 . . . . . . . . . 10 ((((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹𝑥) = (𝐺𝑥))) ∧ (𝑦 ∈ (Base‘𝑆) ∧ (𝐹𝑦) = (𝐺𝑦))) → (𝑥(+g𝑆)𝑦) ∈ (Base‘𝑆))
22 simplrr 777 . . . . . . . . . . . 12 ((((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹𝑥) = (𝐺𝑥))) ∧ (𝑦 ∈ (Base‘𝑆) ∧ (𝐹𝑦) = (𝐺𝑦))) → (𝐹𝑥) = (𝐺𝑥))
23 simprr 772 . . . . . . . . . . . 12 ((((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹𝑥) = (𝐺𝑥))) ∧ (𝑦 ∈ (Base‘𝑆) ∧ (𝐹𝑦) = (𝐺𝑦))) → (𝐹𝑦) = (𝐺𝑦))
2422, 23oveq12d 7376 . . . . . . . . . . 11 ((((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹𝑥) = (𝐺𝑥))) ∧ (𝑦 ∈ (Base‘𝑆) ∧ (𝐹𝑦) = (𝐺𝑦))) → ((𝐹𝑥)(+g𝑇)(𝐹𝑦)) = ((𝐺𝑥)(+g𝑇)(𝐺𝑦)))
25 simplll 774 . . . . . . . . . . . 12 ((((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹𝑥) = (𝐺𝑥))) ∧ (𝑦 ∈ (Base‘𝑆) ∧ (𝐹𝑦) = (𝐺𝑦))) → 𝐹 ∈ (𝑆 MgmHom 𝑇))
26 eqid 2733 . . . . . . . . . . . . 13 (+g𝑇) = (+g𝑇)
271, 19, 26mgmhmlin 46166 . . . . . . . . . . . 12 ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝐹‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑥)(+g𝑇)(𝐹𝑦)))
2825, 17, 18, 27syl3anc 1372 . . . . . . . . . . 11 ((((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹𝑥) = (𝐺𝑥))) ∧ (𝑦 ∈ (Base‘𝑆) ∧ (𝐹𝑦) = (𝐺𝑦))) → (𝐹‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑥)(+g𝑇)(𝐹𝑦)))
29 simpllr 775 . . . . . . . . . . . 12 ((((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹𝑥) = (𝐺𝑥))) ∧ (𝑦 ∈ (Base‘𝑆) ∧ (𝐹𝑦) = (𝐺𝑦))) → 𝐺 ∈ (𝑆 MgmHom 𝑇))
301, 19, 26mgmhmlin 46166 . . . . . . . . . . . 12 ((𝐺 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝐺‘(𝑥(+g𝑆)𝑦)) = ((𝐺𝑥)(+g𝑇)(𝐺𝑦)))
3129, 17, 18, 30syl3anc 1372 . . . . . . . . . . 11 ((((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹𝑥) = (𝐺𝑥))) ∧ (𝑦 ∈ (Base‘𝑆) ∧ (𝐹𝑦) = (𝐺𝑦))) → (𝐺‘(𝑥(+g𝑆)𝑦)) = ((𝐺𝑥)(+g𝑇)(𝐺𝑦)))
3224, 28, 313eqtr4d 2783 . . . . . . . . . 10 ((((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹𝑥) = (𝐺𝑥))) ∧ (𝑦 ∈ (Base‘𝑆) ∧ (𝐹𝑦) = (𝐺𝑦))) → (𝐹‘(𝑥(+g𝑆)𝑦)) = (𝐺‘(𝑥(+g𝑆)𝑦)))
33 fveq2 6843 . . . . . . . . . . . 12 (𝑧 = (𝑥(+g𝑆)𝑦) → (𝐹𝑧) = (𝐹‘(𝑥(+g𝑆)𝑦)))
34 fveq2 6843 . . . . . . . . . . . 12 (𝑧 = (𝑥(+g𝑆)𝑦) → (𝐺𝑧) = (𝐺‘(𝑥(+g𝑆)𝑦)))
3533, 34eqeq12d 2749 . . . . . . . . . . 11 (𝑧 = (𝑥(+g𝑆)𝑦) → ((𝐹𝑧) = (𝐺𝑧) ↔ (𝐹‘(𝑥(+g𝑆)𝑦)) = (𝐺‘(𝑥(+g𝑆)𝑦))))
3635elrab 3646 . . . . . . . . . 10 ((𝑥(+g𝑆)𝑦) ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹𝑧) = (𝐺𝑧)} ↔ ((𝑥(+g𝑆)𝑦) ∈ (Base‘𝑆) ∧ (𝐹‘(𝑥(+g𝑆)𝑦)) = (𝐺‘(𝑥(+g𝑆)𝑦))))
3721, 32, 36sylanbrc 584 . . . . . . . . 9 ((((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹𝑥) = (𝐺𝑥))) ∧ (𝑦 ∈ (Base‘𝑆) ∧ (𝐹𝑦) = (𝐺𝑦))) → (𝑥(+g𝑆)𝑦) ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹𝑧) = (𝐺𝑧)})
3837expr 458 . . . . . . . 8 ((((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹𝑥) = (𝐺𝑥))) ∧ 𝑦 ∈ (Base‘𝑆)) → ((𝐹𝑦) = (𝐺𝑦) → (𝑥(+g𝑆)𝑦) ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹𝑧) = (𝐺𝑧)}))
3938ralrimiva 3140 . . . . . . 7 (((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹𝑥) = (𝐺𝑥))) → ∀𝑦 ∈ (Base‘𝑆)((𝐹𝑦) = (𝐺𝑦) → (𝑥(+g𝑆)𝑦) ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹𝑧) = (𝐺𝑧)}))
40 fveq2 6843 . . . . . . . . 9 (𝑧 = 𝑦 → (𝐹𝑧) = (𝐹𝑦))
41 fveq2 6843 . . . . . . . . 9 (𝑧 = 𝑦 → (𝐺𝑧) = (𝐺𝑦))
4240, 41eqeq12d 2749 . . . . . . . 8 (𝑧 = 𝑦 → ((𝐹𝑧) = (𝐺𝑧) ↔ (𝐹𝑦) = (𝐺𝑦)))
4342ralrab 3652 . . . . . . 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)}))
4439, 43sylibr 233 . . . . . 6 (((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ (𝐹𝑥) = (𝐺𝑥))) → ∀𝑦 ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹𝑧) = (𝐺𝑧)} (𝑥(+g𝑆)𝑦) ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹𝑧) = (𝐺𝑧)})
4544expr 458 . . . . 5 (((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) ∧ 𝑥 ∈ (Base‘𝑆)) → ((𝐹𝑥) = (𝐺𝑥) → ∀𝑦 ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹𝑧) = (𝐺𝑧)} (𝑥(+g𝑆)𝑦) ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹𝑧) = (𝐺𝑧)}))
4645ralrimiva 3140 . . . 4 ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) → ∀𝑥 ∈ (Base‘𝑆)((𝐹𝑥) = (𝐺𝑥) → ∀𝑦 ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹𝑧) = (𝐺𝑧)} (𝑥(+g𝑆)𝑦) ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹𝑧) = (𝐺𝑧)}))
47 fveq2 6843 . . . . . 6 (𝑧 = 𝑥 → (𝐹𝑧) = (𝐹𝑥))
48 fveq2 6843 . . . . . 6 (𝑧 = 𝑥 → (𝐺𝑧) = (𝐺𝑥))
4947, 48eqeq12d 2749 . . . . 5 (𝑧 = 𝑥 → ((𝐹𝑧) = (𝐺𝑧) ↔ (𝐹𝑥) = (𝐺𝑥)))
5049ralrab 3652 . . . 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)}))
5146, 50sylibr 233 . . 3 ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) → ∀𝑥 ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹𝑧) = (𝐺𝑧)}∀𝑦 ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹𝑧) = (𝐺𝑧)} (𝑥(+g𝑆)𝑦) ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹𝑧) = (𝐺𝑧)})
521, 19issubmgm 46169 . . . 4 (𝑆 ∈ Mgm → ({𝑧 ∈ (Base‘𝑆) ∣ (𝐹𝑧) = (𝐺𝑧)} ∈ (SubMgm‘𝑆) ↔ ({𝑧 ∈ (Base‘𝑆) ∣ (𝐹𝑧) = (𝐺𝑧)} ⊆ (Base‘𝑆) ∧ ∀𝑥 ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹𝑧) = (𝐺𝑧)}∀𝑦 ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹𝑧) = (𝐺𝑧)} (𝑥(+g𝑆)𝑦) ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹𝑧) = (𝐺𝑧)})))
5315, 52syl 17 . . 3 ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) → ({𝑧 ∈ (Base‘𝑆) ∣ (𝐹𝑧) = (𝐺𝑧)} ∈ (SubMgm‘𝑆) ↔ ({𝑧 ∈ (Base‘𝑆) ∣ (𝐹𝑧) = (𝐺𝑧)} ⊆ (Base‘𝑆) ∧ ∀𝑥 ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹𝑧) = (𝐺𝑧)}∀𝑦 ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹𝑧) = (𝐺𝑧)} (𝑥(+g𝑆)𝑦) ∈ {𝑧 ∈ (Base‘𝑆) ∣ (𝐹𝑧) = (𝐺𝑧)})))
5412, 51, 53mpbir2and 712 . 2 ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) → {𝑧 ∈ (Base‘𝑆) ∣ (𝐹𝑧) = (𝐺𝑧)} ∈ (SubMgm‘𝑆))
5510, 54eqeltrd 2834 1 ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) → dom (𝐹𝐺) ∈ (SubMgm‘𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  wral 3061  {crab 3406  cin 3910  wss 3911  dom cdm 5634   Fn wfn 6492  wf 6493  cfv 6497  (class class class)co 7358  Basecbs 17088  +gcplusg 17138  Mgmcmgm 18500   MgmHom cmgmhm 46157  SubMgmcsubmgm 46158
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-sbc 3741  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363  df-map 8770  df-mgm 18502  df-mgmhm 46159  df-submgm 46160
This theorem is referenced by: (None)
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