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Theorem mhmeql 18706
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)) β†’ dom (𝐹 ∩ 𝐺) ∈ (SubMndβ€˜π‘†))
Proof of Theorem mhmeql
Dummy variables π‘₯ 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2732 . . . . . 6 (Baseβ€˜π‘†) = (Baseβ€˜π‘†)
2 eqid 2732 . . . . . 6 (Baseβ€˜π‘‡) = (Baseβ€˜π‘‡)
31, 2mhmf 18676 . . . . 5 (𝐹 ∈ (𝑆 MndHom 𝑇) β†’ 𝐹:(Baseβ€˜π‘†)⟢(Baseβ€˜π‘‡))
43adantr 481 . . . 4 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) β†’ 𝐹:(Baseβ€˜π‘†)⟢(Baseβ€˜π‘‡))
54ffnd 6718 . . 3 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) β†’ 𝐹 Fn (Baseβ€˜π‘†))
61, 2mhmf 18676 . . . . 5 (𝐺 ∈ (𝑆 MndHom 𝑇) β†’ 𝐺:(Baseβ€˜π‘†)⟢(Baseβ€˜π‘‡))
76adantl 482 . . . 4 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) β†’ 𝐺:(Baseβ€˜π‘†)⟢(Baseβ€˜π‘‡))
87ffnd 6718 . . 3 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) β†’ 𝐺 Fn (Baseβ€˜π‘†))
9 fndmin 7046 . . 3 ((𝐹 Fn (Baseβ€˜π‘†) ∧ 𝐺 Fn (Baseβ€˜π‘†)) β†’ dom (𝐹 ∩ 𝐺) = {𝑧 ∈ (Baseβ€˜π‘†) ∣ (πΉβ€˜π‘§) = (πΊβ€˜π‘§)})
105, 8, 9syl2anc 584 . 2 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) β†’ dom (𝐹 ∩ 𝐺) = {𝑧 ∈ (Baseβ€˜π‘†) ∣ (πΉβ€˜π‘§) = (πΊβ€˜π‘§)})
11 ssrab2 4077 . . . 4 {𝑧 ∈ (Baseβ€˜π‘†) ∣ (πΉβ€˜π‘§) = (πΊβ€˜π‘§)} βŠ† (Baseβ€˜π‘†)
1211a1i 11 . . 3 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) β†’ {𝑧 ∈ (Baseβ€˜π‘†) ∣ (πΉβ€˜π‘§) = (πΊβ€˜π‘§)} βŠ† (Baseβ€˜π‘†))
13 fveq2 6891 . . . . 5 (𝑧 = (0gβ€˜π‘†) β†’ (πΉβ€˜π‘§) = (πΉβ€˜(0gβ€˜π‘†)))
14 fveq2 6891 . . . . 5 (𝑧 = (0gβ€˜π‘†) β†’ (πΊβ€˜π‘§) = (πΊβ€˜(0gβ€˜π‘†)))
1513, 14eqeq12d 2748 . . . 4 (𝑧 = (0gβ€˜π‘†) β†’ ((πΉβ€˜π‘§) = (πΊβ€˜π‘§) ↔ (πΉβ€˜(0gβ€˜π‘†)) = (πΊβ€˜(0gβ€˜π‘†))))
16 mhmrcl1 18674 . . . . . 6 (𝐹 ∈ (𝑆 MndHom 𝑇) β†’ 𝑆 ∈ Mnd)
1716adantr 481 . . . . 5 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) β†’ 𝑆 ∈ Mnd)
18 eqid 2732 . . . . . 6 (0gβ€˜π‘†) = (0gβ€˜π‘†)
191, 18mndidcl 18639 . . . . 5 (𝑆 ∈ Mnd β†’ (0gβ€˜π‘†) ∈ (Baseβ€˜π‘†))
2017, 19syl 17 . . . 4 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) β†’ (0gβ€˜π‘†) ∈ (Baseβ€˜π‘†))
21 eqid 2732 . . . . . . 7 (0gβ€˜π‘‡) = (0gβ€˜π‘‡)
2218, 21mhm0 18679 . . . . . 6 (𝐹 ∈ (𝑆 MndHom 𝑇) β†’ (πΉβ€˜(0gβ€˜π‘†)) = (0gβ€˜π‘‡))
2322adantr 481 . . . . 5 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) β†’ (πΉβ€˜(0gβ€˜π‘†)) = (0gβ€˜π‘‡))
2418, 21mhm0 18679 . . . . . 6 (𝐺 ∈ (𝑆 MndHom 𝑇) β†’ (πΊβ€˜(0gβ€˜π‘†)) = (0gβ€˜π‘‡))
2524adantl 482 . . . . 5 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) β†’ (πΊβ€˜(0gβ€˜π‘†)) = (0gβ€˜π‘‡))
2623, 25eqtr4d 2775 . . . 4 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) β†’ (πΉβ€˜(0gβ€˜π‘†)) = (πΊβ€˜(0gβ€˜π‘†)))
2715, 20, 26elrabd 3685 . . 3 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) β†’ (0gβ€˜π‘†) ∈ {𝑧 ∈ (Baseβ€˜π‘†) ∣ (πΉβ€˜π‘§) = (πΊβ€˜π‘§)})
28 fveq2 6891 . . . . . . . . . . 11 (𝑧 = (π‘₯(+gβ€˜π‘†)𝑦) β†’ (πΉβ€˜π‘§) = (πΉβ€˜(π‘₯(+gβ€˜π‘†)𝑦)))
29 fveq2 6891 . . . . . . . . . . 11 (𝑧 = (π‘₯(+gβ€˜π‘†)𝑦) β†’ (πΊβ€˜π‘§) = (πΊβ€˜(π‘₯(+gβ€˜π‘†)𝑦)))
3028, 29eqeq12d 2748 . . . . . . . . . 10 (𝑧 = (π‘₯(+gβ€˜π‘†)𝑦) β†’ ((πΉβ€˜π‘§) = (πΊβ€˜π‘§) ↔ (πΉβ€˜(π‘₯(+gβ€˜π‘†)𝑦)) = (πΊβ€˜(π‘₯(+gβ€˜π‘†)𝑦))))
3117ad2antrr 724 . . . . . . . . . . 11 ((((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ (πΉβ€˜π‘₯) = (πΊβ€˜π‘₯))) ∧ (𝑦 ∈ (Baseβ€˜π‘†) ∧ (πΉβ€˜π‘¦) = (πΊβ€˜π‘¦))) β†’ 𝑆 ∈ Mnd)
32 simplrl 775 . . . . . . . . . . 11 ((((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ (πΉβ€˜π‘₯) = (πΊβ€˜π‘₯))) ∧ (𝑦 ∈ (Baseβ€˜π‘†) ∧ (πΉβ€˜π‘¦) = (πΊβ€˜π‘¦))) β†’ π‘₯ ∈ (Baseβ€˜π‘†))
33 simprl 769 . . . . . . . . . . 11 ((((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ (πΉβ€˜π‘₯) = (πΊβ€˜π‘₯))) ∧ (𝑦 ∈ (Baseβ€˜π‘†) ∧ (πΉβ€˜π‘¦) = (πΊβ€˜π‘¦))) β†’ 𝑦 ∈ (Baseβ€˜π‘†))
34 eqid 2732 . . . . . . . . . . . 12 (+gβ€˜π‘†) = (+gβ€˜π‘†)
351, 34mndcl 18632 . . . . . . . . . . 11 ((𝑆 ∈ Mnd ∧ π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (Baseβ€˜π‘†)) β†’ (π‘₯(+gβ€˜π‘†)𝑦) ∈ (Baseβ€˜π‘†))
3631, 32, 33, 35syl3anc 1371 . . . . . . . . . 10 ((((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ (πΉβ€˜π‘₯) = (πΊβ€˜π‘₯))) ∧ (𝑦 ∈ (Baseβ€˜π‘†) ∧ (πΉβ€˜π‘¦) = (πΊβ€˜π‘¦))) β†’ (π‘₯(+gβ€˜π‘†)𝑦) ∈ (Baseβ€˜π‘†))
37 simplll 773 . . . . . . . . . . . 12 ((((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ (πΉβ€˜π‘₯) = (πΊβ€˜π‘₯))) ∧ (𝑦 ∈ (Baseβ€˜π‘†) ∧ (πΉβ€˜π‘¦) = (πΊβ€˜π‘¦))) β†’ 𝐹 ∈ (𝑆 MndHom 𝑇))
38 eqid 2732 . . . . . . . . . . . . 13 (+gβ€˜π‘‡) = (+gβ€˜π‘‡)
391, 34, 38mhmlin 18678 . . . . . . . . . . . 12 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (Baseβ€˜π‘†)) β†’ (πΉβ€˜(π‘₯(+gβ€˜π‘†)𝑦)) = ((πΉβ€˜π‘₯)(+gβ€˜π‘‡)(πΉβ€˜π‘¦)))
4037, 32, 33, 39syl3anc 1371 . . . . . . . . . . 11 ((((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ (πΉβ€˜π‘₯) = (πΊβ€˜π‘₯))) ∧ (𝑦 ∈ (Baseβ€˜π‘†) ∧ (πΉβ€˜π‘¦) = (πΊβ€˜π‘¦))) β†’ (πΉβ€˜(π‘₯(+gβ€˜π‘†)𝑦)) = ((πΉβ€˜π‘₯)(+gβ€˜π‘‡)(πΉβ€˜π‘¦)))
41 simpllr 774 . . . . . . . . . . . . 13 ((((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ (πΉβ€˜π‘₯) = (πΊβ€˜π‘₯))) ∧ (𝑦 ∈ (Baseβ€˜π‘†) ∧ (πΉβ€˜π‘¦) = (πΊβ€˜π‘¦))) β†’ 𝐺 ∈ (𝑆 MndHom 𝑇))
421, 34, 38mhmlin 18678 . . . . . . . . . . . . 13 ((𝐺 ∈ (𝑆 MndHom 𝑇) ∧ π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (Baseβ€˜π‘†)) β†’ (πΊβ€˜(π‘₯(+gβ€˜π‘†)𝑦)) = ((πΊβ€˜π‘₯)(+gβ€˜π‘‡)(πΊβ€˜π‘¦)))
4341, 32, 33, 42syl3anc 1371 . . . . . . . . . . . 12 ((((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ (πΉβ€˜π‘₯) = (πΊβ€˜π‘₯))) ∧ (𝑦 ∈ (Baseβ€˜π‘†) ∧ (πΉβ€˜π‘¦) = (πΊβ€˜π‘¦))) β†’ (πΊβ€˜(π‘₯(+gβ€˜π‘†)𝑦)) = ((πΊβ€˜π‘₯)(+gβ€˜π‘‡)(πΊβ€˜π‘¦)))
44 simplrr 776 . . . . . . . . . . . . 13 ((((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ (πΉβ€˜π‘₯) = (πΊβ€˜π‘₯))) ∧ (𝑦 ∈ (Baseβ€˜π‘†) ∧ (πΉβ€˜π‘¦) = (πΊβ€˜π‘¦))) β†’ (πΉβ€˜π‘₯) = (πΊβ€˜π‘₯))
45 simprr 771 . . . . . . . . . . . . 13 ((((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ (πΉβ€˜π‘₯) = (πΊβ€˜π‘₯))) ∧ (𝑦 ∈ (Baseβ€˜π‘†) ∧ (πΉβ€˜π‘¦) = (πΊβ€˜π‘¦))) β†’ (πΉβ€˜π‘¦) = (πΊβ€˜π‘¦))
4644, 45oveq12d 7426 . . . . . . . . . . . 12 ((((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ (πΉβ€˜π‘₯) = (πΊβ€˜π‘₯))) ∧ (𝑦 ∈ (Baseβ€˜π‘†) ∧ (πΉβ€˜π‘¦) = (πΊβ€˜π‘¦))) β†’ ((πΉβ€˜π‘₯)(+gβ€˜π‘‡)(πΉβ€˜π‘¦)) = ((πΊβ€˜π‘₯)(+gβ€˜π‘‡)(πΊβ€˜π‘¦)))
4743, 46eqtr4d 2775 . . . . . . . . . . 11 ((((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ (πΉβ€˜π‘₯) = (πΊβ€˜π‘₯))) ∧ (𝑦 ∈ (Baseβ€˜π‘†) ∧ (πΉβ€˜π‘¦) = (πΊβ€˜π‘¦))) β†’ (πΊβ€˜(π‘₯(+gβ€˜π‘†)𝑦)) = ((πΉβ€˜π‘₯)(+gβ€˜π‘‡)(πΉβ€˜π‘¦)))
4840, 47eqtr4d 2775 . . . . . . . . . 10 ((((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ (πΉβ€˜π‘₯) = (πΊβ€˜π‘₯))) ∧ (𝑦 ∈ (Baseβ€˜π‘†) ∧ (πΉβ€˜π‘¦) = (πΊβ€˜π‘¦))) β†’ (πΉβ€˜(π‘₯(+gβ€˜π‘†)𝑦)) = (πΊβ€˜(π‘₯(+gβ€˜π‘†)𝑦)))
4930, 36, 48elrabd 3685 . . . . . . . . 9 ((((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ (πΉβ€˜π‘₯) = (πΊβ€˜π‘₯))) ∧ (𝑦 ∈ (Baseβ€˜π‘†) ∧ (πΉβ€˜π‘¦) = (πΊβ€˜π‘¦))) β†’ (π‘₯(+gβ€˜π‘†)𝑦) ∈ {𝑧 ∈ (Baseβ€˜π‘†) ∣ (πΉβ€˜π‘§) = (πΊβ€˜π‘§)})
5049expr 457 . . . . . . . 8 ((((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ (πΉβ€˜π‘₯) = (πΊβ€˜π‘₯))) ∧ 𝑦 ∈ (Baseβ€˜π‘†)) β†’ ((πΉβ€˜π‘¦) = (πΊβ€˜π‘¦) β†’ (π‘₯(+gβ€˜π‘†)𝑦) ∈ {𝑧 ∈ (Baseβ€˜π‘†) ∣ (πΉβ€˜π‘§) = (πΊβ€˜π‘§)}))
5150ralrimiva 3146 . . . . . . 7 (((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ (πΉβ€˜π‘₯) = (πΊβ€˜π‘₯))) β†’ βˆ€π‘¦ ∈ (Baseβ€˜π‘†)((πΉβ€˜π‘¦) = (πΊβ€˜π‘¦) β†’ (π‘₯(+gβ€˜π‘†)𝑦) ∈ {𝑧 ∈ (Baseβ€˜π‘†) ∣ (πΉβ€˜π‘§) = (πΊβ€˜π‘§)}))
52 fveq2 6891 . . . . . . . . 9 (𝑧 = 𝑦 β†’ (πΉβ€˜π‘§) = (πΉβ€˜π‘¦))
53 fveq2 6891 . . . . . . . . 9 (𝑧 = 𝑦 β†’ (πΊβ€˜π‘§) = (πΊβ€˜π‘¦))
5452, 53eqeq12d 2748 . . . . . . . 8 (𝑧 = 𝑦 β†’ ((πΉβ€˜π‘§) = (πΊβ€˜π‘§) ↔ (πΉβ€˜π‘¦) = (πΊβ€˜π‘¦)))
5554ralrab 3689 . . . . . . 7 (βˆ€π‘¦ ∈ {𝑧 ∈ (Baseβ€˜π‘†) ∣ (πΉβ€˜π‘§) = (πΊβ€˜π‘§)} (π‘₯(+gβ€˜π‘†)𝑦) ∈ {𝑧 ∈ (Baseβ€˜π‘†) ∣ (πΉβ€˜π‘§) = (πΊβ€˜π‘§)} ↔ βˆ€π‘¦ ∈ (Baseβ€˜π‘†)((πΉβ€˜π‘¦) = (πΊβ€˜π‘¦) β†’ (π‘₯(+gβ€˜π‘†)𝑦) ∈ {𝑧 ∈ (Baseβ€˜π‘†) ∣ (πΉβ€˜π‘§) = (πΊβ€˜π‘§)}))
5651, 55sylibr 233 . . . . . 6 (((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ (π‘₯ ∈ (Baseβ€˜π‘†) ∧ (πΉβ€˜π‘₯) = (πΊβ€˜π‘₯))) β†’ βˆ€π‘¦ ∈ {𝑧 ∈ (Baseβ€˜π‘†) ∣ (πΉβ€˜π‘§) = (πΊβ€˜π‘§)} (π‘₯(+gβ€˜π‘†)𝑦) ∈ {𝑧 ∈ (Baseβ€˜π‘†) ∣ (πΉβ€˜π‘§) = (πΊβ€˜π‘§)})
5756expr 457 . . . . 5 (((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) ∧ π‘₯ ∈ (Baseβ€˜π‘†)) β†’ ((πΉβ€˜π‘₯) = (πΊβ€˜π‘₯) β†’ βˆ€π‘¦ ∈ {𝑧 ∈ (Baseβ€˜π‘†) ∣ (πΉβ€˜π‘§) = (πΊβ€˜π‘§)} (π‘₯(+gβ€˜π‘†)𝑦) ∈ {𝑧 ∈ (Baseβ€˜π‘†) ∣ (πΉβ€˜π‘§) = (πΊβ€˜π‘§)}))
5857ralrimiva 3146 . . . 4 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) β†’ βˆ€π‘₯ ∈ (Baseβ€˜π‘†)((πΉβ€˜π‘₯) = (πΊβ€˜π‘₯) β†’ βˆ€π‘¦ ∈ {𝑧 ∈ (Baseβ€˜π‘†) ∣ (πΉβ€˜π‘§) = (πΊβ€˜π‘§)} (π‘₯(+gβ€˜π‘†)𝑦) ∈ {𝑧 ∈ (Baseβ€˜π‘†) ∣ (πΉβ€˜π‘§) = (πΊβ€˜π‘§)}))
59 fveq2 6891 . . . . . 6 (𝑧 = π‘₯ β†’ (πΉβ€˜π‘§) = (πΉβ€˜π‘₯))
60 fveq2 6891 . . . . . 6 (𝑧 = π‘₯ β†’ (πΊβ€˜π‘§) = (πΊβ€˜π‘₯))
6159, 60eqeq12d 2748 . . . . 5 (𝑧 = π‘₯ β†’ ((πΉβ€˜π‘§) = (πΊβ€˜π‘§) ↔ (πΉβ€˜π‘₯) = (πΊβ€˜π‘₯)))
6261ralrab 3689 . . . 4 (βˆ€π‘₯ ∈ {𝑧 ∈ (Baseβ€˜π‘†) ∣ (πΉβ€˜π‘§) = (πΊβ€˜π‘§)}βˆ€π‘¦ ∈ {𝑧 ∈ (Baseβ€˜π‘†) ∣ (πΉβ€˜π‘§) = (πΊβ€˜π‘§)} (π‘₯(+gβ€˜π‘†)𝑦) ∈ {𝑧 ∈ (Baseβ€˜π‘†) ∣ (πΉβ€˜π‘§) = (πΊβ€˜π‘§)} ↔ βˆ€π‘₯ ∈ (Baseβ€˜π‘†)((πΉβ€˜π‘₯) = (πΊβ€˜π‘₯) β†’ βˆ€π‘¦ ∈ {𝑧 ∈ (Baseβ€˜π‘†) ∣ (πΉβ€˜π‘§) = (πΊβ€˜π‘§)} (π‘₯(+gβ€˜π‘†)𝑦) ∈ {𝑧 ∈ (Baseβ€˜π‘†) ∣ (πΉβ€˜π‘§) = (πΊβ€˜π‘§)}))
6358, 62sylibr 233 . . 3 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) β†’ βˆ€π‘₯ ∈ {𝑧 ∈ (Baseβ€˜π‘†) ∣ (πΉβ€˜π‘§) = (πΊβ€˜π‘§)}βˆ€π‘¦ ∈ {𝑧 ∈ (Baseβ€˜π‘†) ∣ (πΉβ€˜π‘§) = (πΊβ€˜π‘§)} (π‘₯(+gβ€˜π‘†)𝑦) ∈ {𝑧 ∈ (Baseβ€˜π‘†) ∣ (πΉβ€˜π‘§) = (πΊβ€˜π‘§)})
641, 18, 34issubm 18683 . . . 4 (𝑆 ∈ Mnd β†’ ({𝑧 ∈ (Baseβ€˜π‘†) ∣ (πΉβ€˜π‘§) = (πΊβ€˜π‘§)} ∈ (SubMndβ€˜π‘†) ↔ ({𝑧 ∈ (Baseβ€˜π‘†) ∣ (πΉβ€˜π‘§) = (πΊβ€˜π‘§)} βŠ† (Baseβ€˜π‘†) ∧ (0gβ€˜π‘†) ∈ {𝑧 ∈ (Baseβ€˜π‘†) ∣ (πΉβ€˜π‘§) = (πΊβ€˜π‘§)} ∧ βˆ€π‘₯ ∈ {𝑧 ∈ (Baseβ€˜π‘†) ∣ (πΉβ€˜π‘§) = (πΊβ€˜π‘§)}βˆ€π‘¦ ∈ {𝑧 ∈ (Baseβ€˜π‘†) ∣ (πΉβ€˜π‘§) = (πΊβ€˜π‘§)} (π‘₯(+gβ€˜π‘†)𝑦) ∈ {𝑧 ∈ (Baseβ€˜π‘†) ∣ (πΉβ€˜π‘§) = (πΊβ€˜π‘§)})))
6517, 64syl 17 . . 3 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) β†’ ({𝑧 ∈ (Baseβ€˜π‘†) ∣ (πΉβ€˜π‘§) = (πΊβ€˜π‘§)} ∈ (SubMndβ€˜π‘†) ↔ ({𝑧 ∈ (Baseβ€˜π‘†) ∣ (πΉβ€˜π‘§) = (πΊβ€˜π‘§)} βŠ† (Baseβ€˜π‘†) ∧ (0gβ€˜π‘†) ∈ {𝑧 ∈ (Baseβ€˜π‘†) ∣ (πΉβ€˜π‘§) = (πΊβ€˜π‘§)} ∧ βˆ€π‘₯ ∈ {𝑧 ∈ (Baseβ€˜π‘†) ∣ (πΉβ€˜π‘§) = (πΊβ€˜π‘§)}βˆ€π‘¦ ∈ {𝑧 ∈ (Baseβ€˜π‘†) ∣ (πΉβ€˜π‘§) = (πΊβ€˜π‘§)} (π‘₯(+gβ€˜π‘†)𝑦) ∈ {𝑧 ∈ (Baseβ€˜π‘†) ∣ (πΉβ€˜π‘§) = (πΊβ€˜π‘§)})))
6612, 27, 63, 65mpbir3and 1342 . 2 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) β†’ {𝑧 ∈ (Baseβ€˜π‘†) ∣ (πΉβ€˜π‘§) = (πΊβ€˜π‘§)} ∈ (SubMndβ€˜π‘†))
6710, 66eqeltrd 2833 1 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) β†’ dom (𝐹 ∩ 𝐺) ∈ (SubMndβ€˜π‘†))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061  {crab 3432   ∩ cin 3947   βŠ† wss 3948  dom cdm 5676   Fn wfn 6538  βŸΆwf 6539  β€˜cfv 6543  (class class class)co 7408  Basecbs 17143  +gcplusg 17196  0gc0g 17384  Mndcmnd 18624   MndHom cmhm 18668  SubMndcsubmnd 18669
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-riota 7364  df-ov 7411  df-oprab 7412  df-mpo 7413  df-map 8821  df-0g 17386  df-mgm 18560  df-sgrp 18609  df-mnd 18625  df-mhm 18670  df-submnd 18671
This theorem is referenced by:  ghmeql  19114  rhmeql  20349
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