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Theorem mhmeql 12881
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 2177 . . . . . 6 (Baseβ€˜π‘†) = (Baseβ€˜π‘†)
2 eqid 2177 . . . . . 6 (Baseβ€˜π‘‡) = (Baseβ€˜π‘‡)
31, 2mhmf 12861 . . . . 5 (𝐹 ∈ (𝑆 MndHom 𝑇) β†’ 𝐹:(Baseβ€˜π‘†)⟢(Baseβ€˜π‘‡))
43adantr 276 . . . 4 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) β†’ 𝐹:(Baseβ€˜π‘†)⟢(Baseβ€˜π‘‡))
54ffnd 5368 . . 3 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) β†’ 𝐹 Fn (Baseβ€˜π‘†))
61, 2mhmf 12861 . . . . 5 (𝐺 ∈ (𝑆 MndHom 𝑇) β†’ 𝐺:(Baseβ€˜π‘†)⟢(Baseβ€˜π‘‡))
76adantl 277 . . . 4 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) β†’ 𝐺:(Baseβ€˜π‘†)⟢(Baseβ€˜π‘‡))
87ffnd 5368 . . 3 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) β†’ 𝐺 Fn (Baseβ€˜π‘†))
9 fndmin 5625 . . 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 411 . 2 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) β†’ dom (𝐹 ∩ 𝐺) = {𝑧 ∈ (Baseβ€˜π‘†) ∣ (πΉβ€˜π‘§) = (πΊβ€˜π‘§)})
11 ssrab2 3242 . . . 4 {𝑧 ∈ (Baseβ€˜π‘†) ∣ (πΉβ€˜π‘§) = (πΊβ€˜π‘§)} βŠ† (Baseβ€˜π‘†)
1211a1i 9 . . 3 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) β†’ {𝑧 ∈ (Baseβ€˜π‘†) ∣ (πΉβ€˜π‘§) = (πΊβ€˜π‘§)} βŠ† (Baseβ€˜π‘†))
13 fveq2 5517 . . . . 5 (𝑧 = (0gβ€˜π‘†) β†’ (πΉβ€˜π‘§) = (πΉβ€˜(0gβ€˜π‘†)))
14 fveq2 5517 . . . . 5 (𝑧 = (0gβ€˜π‘†) β†’ (πΊβ€˜π‘§) = (πΊβ€˜(0gβ€˜π‘†)))
1513, 14eqeq12d 2192 . . . 4 (𝑧 = (0gβ€˜π‘†) β†’ ((πΉβ€˜π‘§) = (πΊβ€˜π‘§) ↔ (πΉβ€˜(0gβ€˜π‘†)) = (πΊβ€˜(0gβ€˜π‘†))))
16 mhmrcl1 12859 . . . . . 6 (𝐹 ∈ (𝑆 MndHom 𝑇) β†’ 𝑆 ∈ Mnd)
1716adantr 276 . . . . 5 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) β†’ 𝑆 ∈ Mnd)
18 eqid 2177 . . . . . 6 (0gβ€˜π‘†) = (0gβ€˜π‘†)
191, 18mndidcl 12836 . . . . 5 (𝑆 ∈ Mnd β†’ (0gβ€˜π‘†) ∈ (Baseβ€˜π‘†))
2017, 19syl 14 . . . 4 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) β†’ (0gβ€˜π‘†) ∈ (Baseβ€˜π‘†))
21 eqid 2177 . . . . . . 7 (0gβ€˜π‘‡) = (0gβ€˜π‘‡)
2218, 21mhm0 12864 . . . . . 6 (𝐹 ∈ (𝑆 MndHom 𝑇) β†’ (πΉβ€˜(0gβ€˜π‘†)) = (0gβ€˜π‘‡))
2322adantr 276 . . . . 5 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) β†’ (πΉβ€˜(0gβ€˜π‘†)) = (0gβ€˜π‘‡))
2418, 21mhm0 12864 . . . . . 6 (𝐺 ∈ (𝑆 MndHom 𝑇) β†’ (πΊβ€˜(0gβ€˜π‘†)) = (0gβ€˜π‘‡))
2524adantl 277 . . . . 5 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) β†’ (πΊβ€˜(0gβ€˜π‘†)) = (0gβ€˜π‘‡))
2623, 25eqtr4d 2213 . . . 4 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) β†’ (πΉβ€˜(0gβ€˜π‘†)) = (πΊβ€˜(0gβ€˜π‘†)))
2715, 20, 26elrabd 2897 . . 3 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) β†’ (0gβ€˜π‘†) ∈ {𝑧 ∈ (Baseβ€˜π‘†) ∣ (πΉβ€˜π‘§) = (πΊβ€˜π‘§)})
28 fveq2 5517 . . . . . . . . . . 11 (𝑧 = (π‘₯(+gβ€˜π‘†)𝑦) β†’ (πΉβ€˜π‘§) = (πΉβ€˜(π‘₯(+gβ€˜π‘†)𝑦)))
29 fveq2 5517 . . . . . . . . . . 11 (𝑧 = (π‘₯(+gβ€˜π‘†)𝑦) β†’ (πΊβ€˜π‘§) = (πΊβ€˜(π‘₯(+gβ€˜π‘†)𝑦)))
3028, 29eqeq12d 2192 . . . . . . . . . 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 488 . . . . . . . . . . 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 535 . . . . . . . . . . 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 529 . . . . . . . . . . 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 2177 . . . . . . . . . . . 12 (+gβ€˜π‘†) = (+gβ€˜π‘†)
351, 34mndcl 12829 . . . . . . . . . . 11 ((𝑆 ∈ Mnd ∧ π‘₯ ∈ (Baseβ€˜π‘†) ∧ 𝑦 ∈ (Baseβ€˜π‘†)) β†’ (π‘₯(+gβ€˜π‘†)𝑦) ∈ (Baseβ€˜π‘†))
3631, 32, 33, 35syl3anc 1238 . . . . . . . . . 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 533 . . . . . . . . . . . 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 2177 . . . . . . . . . . . . 13 (+gβ€˜π‘‡) = (+gβ€˜π‘‡)
391, 34, 38mhmlin 12863 . . . . . . . . . . . 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 1238 . . . . . . . . . . 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 534 . . . . . . . . . . . . 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 12863 . . . . . . . . . . . . 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 1238 . . . . . . . . . . . 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 536 . . . . . . . . . . . . 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 531 . . . . . . . . . . . . 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 5895 . . . . . . . . . . . 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 2213 . . . . . . . . . . 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 2213 . . . . . . . . . 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 2897 . . . . . . . . 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 375 . . . . . . . 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 2550 . . . . . . 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 5517 . . . . . . . . 9 (𝑧 = 𝑦 β†’ (πΉβ€˜π‘§) = (πΉβ€˜π‘¦))
53 fveq2 5517 . . . . . . . . 9 (𝑧 = 𝑦 β†’ (πΊβ€˜π‘§) = (πΊβ€˜π‘¦))
5452, 53eqeq12d 2192 . . . . . . . 8 (𝑧 = 𝑦 β†’ ((πΉβ€˜π‘§) = (πΊβ€˜π‘§) ↔ (πΉβ€˜π‘¦) = (πΊβ€˜π‘¦)))
5554ralrab 2900 . . . . . . 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 134 . . . . . 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 375 . . . . 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 2550 . . . 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 5517 . . . . . 6 (𝑧 = π‘₯ β†’ (πΉβ€˜π‘§) = (πΉβ€˜π‘₯))
60 fveq2 5517 . . . . . 6 (𝑧 = π‘₯ β†’ (πΊβ€˜π‘§) = (πΊβ€˜π‘₯))
6159, 60eqeq12d 2192 . . . . 5 (𝑧 = π‘₯ β†’ ((πΉβ€˜π‘§) = (πΊβ€˜π‘§) ↔ (πΉβ€˜π‘₯) = (πΊβ€˜π‘₯)))
6261ralrab 2900 . . . 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 134 . . 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 12868 . . . 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 14 . . 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 1180 . 2 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) β†’ {𝑧 ∈ (Baseβ€˜π‘†) ∣ (πΉβ€˜π‘§) = (πΊβ€˜π‘§)} ∈ (SubMndβ€˜π‘†))
6710, 66eqeltrd 2254 1 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝐺 ∈ (𝑆 MndHom 𝑇)) β†’ dom (𝐹 ∩ 𝐺) ∈ (SubMndβ€˜π‘†))
Colors of variables: wff set class
Syntax hints:   β†’ wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 978   = wceq 1353   ∈ wcel 2148  βˆ€wral 2455  {crab 2459   ∩ cin 3130   βŠ† wss 3131  dom cdm 4628   Fn wfn 5213  βŸΆwf 5214  β€˜cfv 5218  (class class class)co 5877  Basecbs 12464  +gcplusg 12538  0gc0g 12710  Mndcmnd 12822   MndHom cmhm 12854  SubMndcsubmnd 12855
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-cnex 7904  ax-resscn 7905  ax-1re 7907  ax-addrcl 7910
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-fv 5226  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-map 6652  df-inn 8922  df-2 8980  df-ndx 12467  df-slot 12468  df-base 12470  df-plusg 12551  df-0g 12712  df-mgm 12780  df-sgrp 12813  df-mnd 12823  df-mhm 12856  df-submnd 12857
This theorem is referenced by: (None)
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