Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  pwsco2mhm Structured version   Visualization version   GIF version

Theorem pwsco2mhm 17989
 Description: Left composition with a monoid homomorphism yields a monoid homomorphism of structure powers. (Contributed by Mario Carneiro, 12-Jun-2015.)
Hypotheses
Ref Expression
pwsco2mhm.y 𝑌 = (𝑅s 𝐴)
pwsco2mhm.z 𝑍 = (𝑆s 𝐴)
pwsco2mhm.b 𝐵 = (Base‘𝑌)
pwsco2mhm.a (𝜑𝐴𝑉)
pwsco2mhm.f (𝜑𝐹 ∈ (𝑅 MndHom 𝑆))
Assertion
Ref Expression
pwsco2mhm (𝜑 → (𝑔𝐵 ↦ (𝐹𝑔)) ∈ (𝑌 MndHom 𝑍))
Distinct variable groups:   𝐵,𝑔   𝑔,𝐹   𝑔,𝑌   𝑔,𝑍   𝜑,𝑔
Allowed substitution hints:   𝐴(𝑔)   𝑅(𝑔)   𝑆(𝑔)   𝑉(𝑔)

Proof of Theorem pwsco2mhm
Dummy variables 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pwsco2mhm.f . . . 4 (𝜑𝐹 ∈ (𝑅 MndHom 𝑆))
2 mhmrcl1 17951 . . . 4 (𝐹 ∈ (𝑅 MndHom 𝑆) → 𝑅 ∈ Mnd)
31, 2syl 17 . . 3 (𝜑𝑅 ∈ Mnd)
4 pwsco2mhm.a . . 3 (𝜑𝐴𝑉)
5 pwsco2mhm.y . . . 4 𝑌 = (𝑅s 𝐴)
65pwsmnd 17938 . . 3 ((𝑅 ∈ Mnd ∧ 𝐴𝑉) → 𝑌 ∈ Mnd)
73, 4, 6syl2anc 586 . 2 (𝜑𝑌 ∈ Mnd)
8 mhmrcl2 17952 . . . 4 (𝐹 ∈ (𝑅 MndHom 𝑆) → 𝑆 ∈ Mnd)
91, 8syl 17 . . 3 (𝜑𝑆 ∈ Mnd)
10 pwsco2mhm.z . . . 4 𝑍 = (𝑆s 𝐴)
1110pwsmnd 17938 . . 3 ((𝑆 ∈ Mnd ∧ 𝐴𝑉) → 𝑍 ∈ Mnd)
129, 4, 11syl2anc 586 . 2 (𝜑𝑍 ∈ Mnd)
13 eqid 2819 . . . . . . . 8 (Base‘𝑅) = (Base‘𝑅)
14 eqid 2819 . . . . . . . 8 (Base‘𝑆) = (Base‘𝑆)
1513, 14mhmf 17953 . . . . . . 7 (𝐹 ∈ (𝑅 MndHom 𝑆) → 𝐹:(Base‘𝑅)⟶(Base‘𝑆))
161, 15syl 17 . . . . . 6 (𝜑𝐹:(Base‘𝑅)⟶(Base‘𝑆))
17 pwsco2mhm.b . . . . . . 7 𝐵 = (Base‘𝑌)
183adantr 483 . . . . . . 7 ((𝜑𝑔𝐵) → 𝑅 ∈ Mnd)
194adantr 483 . . . . . . 7 ((𝜑𝑔𝐵) → 𝐴𝑉)
20 simpr 487 . . . . . . 7 ((𝜑𝑔𝐵) → 𝑔𝐵)
215, 13, 17, 18, 19, 20pwselbas 16754 . . . . . 6 ((𝜑𝑔𝐵) → 𝑔:𝐴⟶(Base‘𝑅))
22 fco 6524 . . . . . 6 ((𝐹:(Base‘𝑅)⟶(Base‘𝑆) ∧ 𝑔:𝐴⟶(Base‘𝑅)) → (𝐹𝑔):𝐴⟶(Base‘𝑆))
2316, 21, 22syl2an2r 683 . . . . 5 ((𝜑𝑔𝐵) → (𝐹𝑔):𝐴⟶(Base‘𝑆))
24 eqid 2819 . . . . . . 7 (Base‘𝑍) = (Base‘𝑍)
2510, 14, 24pwselbasb 16753 . . . . . 6 ((𝑆 ∈ Mnd ∧ 𝐴𝑉) → ((𝐹𝑔) ∈ (Base‘𝑍) ↔ (𝐹𝑔):𝐴⟶(Base‘𝑆)))
269, 19, 25syl2an2r 683 . . . . 5 ((𝜑𝑔𝐵) → ((𝐹𝑔) ∈ (Base‘𝑍) ↔ (𝐹𝑔):𝐴⟶(Base‘𝑆)))
2723, 26mpbird 259 . . . 4 ((𝜑𝑔𝐵) → (𝐹𝑔) ∈ (Base‘𝑍))
2827fmpttd 6872 . . 3 (𝜑 → (𝑔𝐵 ↦ (𝐹𝑔)):𝐵⟶(Base‘𝑍))
291adantr 483 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝐹 ∈ (𝑅 MndHom 𝑆))
3029adantr 483 . . . . . . . . 9 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑤𝐴) → 𝐹 ∈ (𝑅 MndHom 𝑆))
3129, 2syl 17 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝑅 ∈ Mnd)
324adantr 483 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝐴𝑉)
33 simprl 769 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝑥𝐵)
345, 13, 17, 31, 32, 33pwselbas 16754 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝑥:𝐴⟶(Base‘𝑅))
3534ffvelrnda 6844 . . . . . . . . 9 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑤𝐴) → (𝑥𝑤) ∈ (Base‘𝑅))
36 simprr 771 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝑦𝐵)
375, 13, 17, 31, 32, 36pwselbas 16754 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝑦:𝐴⟶(Base‘𝑅))
3837ffvelrnda 6844 . . . . . . . . 9 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑤𝐴) → (𝑦𝑤) ∈ (Base‘𝑅))
39 eqid 2819 . . . . . . . . . 10 (+g𝑅) = (+g𝑅)
40 eqid 2819 . . . . . . . . . 10 (+g𝑆) = (+g𝑆)
4113, 39, 40mhmlin 17955 . . . . . . . . 9 ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ (𝑥𝑤) ∈ (Base‘𝑅) ∧ (𝑦𝑤) ∈ (Base‘𝑅)) → (𝐹‘((𝑥𝑤)(+g𝑅)(𝑦𝑤))) = ((𝐹‘(𝑥𝑤))(+g𝑆)(𝐹‘(𝑦𝑤))))
4230, 35, 38, 41syl3anc 1366 . . . . . . . 8 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑤𝐴) → (𝐹‘((𝑥𝑤)(+g𝑅)(𝑦𝑤))) = ((𝐹‘(𝑥𝑤))(+g𝑆)(𝐹‘(𝑦𝑤))))
4342mpteq2dva 5152 . . . . . . 7 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑤𝐴 ↦ (𝐹‘((𝑥𝑤)(+g𝑅)(𝑦𝑤)))) = (𝑤𝐴 ↦ ((𝐹‘(𝑥𝑤))(+g𝑆)(𝐹‘(𝑦𝑤)))))
44 fvexd 6678 . . . . . . . 8 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑤𝐴) → (𝐹‘(𝑥𝑤)) ∈ V)
45 fvexd 6678 . . . . . . . 8 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑤𝐴) → (𝐹‘(𝑦𝑤)) ∈ V)
4634feqmptd 6726 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝑥 = (𝑤𝐴 ↦ (𝑥𝑤)))
4729, 15syl 17 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝐹:(Base‘𝑅)⟶(Base‘𝑆))
4847feqmptd 6726 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝐹 = (𝑧 ∈ (Base‘𝑅) ↦ (𝐹𝑧)))
49 fveq2 6663 . . . . . . . . 9 (𝑧 = (𝑥𝑤) → (𝐹𝑧) = (𝐹‘(𝑥𝑤)))
5035, 46, 48, 49fmptco 6884 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝐹𝑥) = (𝑤𝐴 ↦ (𝐹‘(𝑥𝑤))))
5137feqmptd 6726 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝑦 = (𝑤𝐴 ↦ (𝑦𝑤)))
52 fveq2 6663 . . . . . . . . 9 (𝑧 = (𝑦𝑤) → (𝐹𝑧) = (𝐹‘(𝑦𝑤)))
5338, 51, 48, 52fmptco 6884 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝐹𝑦) = (𝑤𝐴 ↦ (𝐹‘(𝑦𝑤))))
5432, 44, 45, 50, 53offval2 7418 . . . . . . 7 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ((𝐹𝑥) ∘f (+g𝑆)(𝐹𝑦)) = (𝑤𝐴 ↦ ((𝐹‘(𝑥𝑤))(+g𝑆)(𝐹‘(𝑦𝑤)))))
5543, 54eqtr4d 2857 . . . . . 6 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑤𝐴 ↦ (𝐹‘((𝑥𝑤)(+g𝑅)(𝑦𝑤)))) = ((𝐹𝑥) ∘f (+g𝑆)(𝐹𝑦)))
5631adantr 483 . . . . . . . 8 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑤𝐴) → 𝑅 ∈ Mnd)
5713, 39mndcl 17911 . . . . . . . 8 ((𝑅 ∈ Mnd ∧ (𝑥𝑤) ∈ (Base‘𝑅) ∧ (𝑦𝑤) ∈ (Base‘𝑅)) → ((𝑥𝑤)(+g𝑅)(𝑦𝑤)) ∈ (Base‘𝑅))
5856, 35, 38, 57syl3anc 1366 . . . . . . 7 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑤𝐴) → ((𝑥𝑤)(+g𝑅)(𝑦𝑤)) ∈ (Base‘𝑅))
59 eqid 2819 . . . . . . . . 9 (+g𝑌) = (+g𝑌)
605, 17, 31, 32, 33, 36, 39, 59pwsplusgval 16755 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝑌)𝑦) = (𝑥f (+g𝑅)𝑦))
61 fvexd 6678 . . . . . . . . 9 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑤𝐴) → (𝑥𝑤) ∈ V)
62 fvexd 6678 . . . . . . . . 9 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑤𝐴) → (𝑦𝑤) ∈ V)
6332, 61, 62, 46, 51offval2 7418 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥f (+g𝑅)𝑦) = (𝑤𝐴 ↦ ((𝑥𝑤)(+g𝑅)(𝑦𝑤))))
6460, 63eqtrd 2854 . . . . . . 7 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝑌)𝑦) = (𝑤𝐴 ↦ ((𝑥𝑤)(+g𝑅)(𝑦𝑤))))
65 fveq2 6663 . . . . . . 7 (𝑧 = ((𝑥𝑤)(+g𝑅)(𝑦𝑤)) → (𝐹𝑧) = (𝐹‘((𝑥𝑤)(+g𝑅)(𝑦𝑤))))
6658, 64, 48, 65fmptco 6884 . . . . . 6 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝐹 ∘ (𝑥(+g𝑌)𝑦)) = (𝑤𝐴 ↦ (𝐹‘((𝑥𝑤)(+g𝑅)(𝑦𝑤)))))
6729, 8syl 17 . . . . . . 7 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝑆 ∈ Mnd)
68 fco 6524 . . . . . . . . 9 ((𝐹:(Base‘𝑅)⟶(Base‘𝑆) ∧ 𝑥:𝐴⟶(Base‘𝑅)) → (𝐹𝑥):𝐴⟶(Base‘𝑆))
6947, 34, 68syl2anc 586 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝐹𝑥):𝐴⟶(Base‘𝑆))
7010, 14, 24pwselbasb 16753 . . . . . . . . 9 ((𝑆 ∈ Mnd ∧ 𝐴𝑉) → ((𝐹𝑥) ∈ (Base‘𝑍) ↔ (𝐹𝑥):𝐴⟶(Base‘𝑆)))
7167, 32, 70syl2anc 586 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ((𝐹𝑥) ∈ (Base‘𝑍) ↔ (𝐹𝑥):𝐴⟶(Base‘𝑆)))
7269, 71mpbird 259 . . . . . . 7 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝐹𝑥) ∈ (Base‘𝑍))
73 fco 6524 . . . . . . . . 9 ((𝐹:(Base‘𝑅)⟶(Base‘𝑆) ∧ 𝑦:𝐴⟶(Base‘𝑅)) → (𝐹𝑦):𝐴⟶(Base‘𝑆))
7447, 37, 73syl2anc 586 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝐹𝑦):𝐴⟶(Base‘𝑆))
7510, 14, 24pwselbasb 16753 . . . . . . . . 9 ((𝑆 ∈ Mnd ∧ 𝐴𝑉) → ((𝐹𝑦) ∈ (Base‘𝑍) ↔ (𝐹𝑦):𝐴⟶(Base‘𝑆)))
7667, 32, 75syl2anc 586 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ((𝐹𝑦) ∈ (Base‘𝑍) ↔ (𝐹𝑦):𝐴⟶(Base‘𝑆)))
7774, 76mpbird 259 . . . . . . 7 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝐹𝑦) ∈ (Base‘𝑍))
78 eqid 2819 . . . . . . 7 (+g𝑍) = (+g𝑍)
7910, 24, 67, 32, 72, 77, 40, 78pwsplusgval 16755 . . . . . 6 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ((𝐹𝑥)(+g𝑍)(𝐹𝑦)) = ((𝐹𝑥) ∘f (+g𝑆)(𝐹𝑦)))
8055, 66, 793eqtr4d 2864 . . . . 5 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝐹 ∘ (𝑥(+g𝑌)𝑦)) = ((𝐹𝑥)(+g𝑍)(𝐹𝑦)))
81 eqid 2819 . . . . . 6 (𝑔𝐵 ↦ (𝐹𝑔)) = (𝑔𝐵 ↦ (𝐹𝑔))
82 coeq2 5722 . . . . . 6 (𝑔 = (𝑥(+g𝑌)𝑦) → (𝐹𝑔) = (𝐹 ∘ (𝑥(+g𝑌)𝑦)))
8317, 59mndcl 17911 . . . . . . . 8 ((𝑌 ∈ Mnd ∧ 𝑥𝐵𝑦𝐵) → (𝑥(+g𝑌)𝑦) ∈ 𝐵)
84833expb 1115 . . . . . . 7 ((𝑌 ∈ Mnd ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝑌)𝑦) ∈ 𝐵)
857, 84sylan 582 . . . . . 6 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝑌)𝑦) ∈ 𝐵)
86 coexg 7626 . . . . . . 7 ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ (𝑥(+g𝑌)𝑦) ∈ 𝐵) → (𝐹 ∘ (𝑥(+g𝑌)𝑦)) ∈ V)
871, 85, 86syl2an2r 683 . . . . . 6 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝐹 ∘ (𝑥(+g𝑌)𝑦)) ∈ V)
8881, 82, 85, 87fvmptd3 6784 . . . . 5 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ((𝑔𝐵 ↦ (𝐹𝑔))‘(𝑥(+g𝑌)𝑦)) = (𝐹 ∘ (𝑥(+g𝑌)𝑦)))
89 coeq2 5722 . . . . . . 7 (𝑔 = 𝑥 → (𝐹𝑔) = (𝐹𝑥))
9081, 89, 33, 72fvmptd3 6784 . . . . . 6 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ((𝑔𝐵 ↦ (𝐹𝑔))‘𝑥) = (𝐹𝑥))
91 coeq2 5722 . . . . . . 7 (𝑔 = 𝑦 → (𝐹𝑔) = (𝐹𝑦))
9281, 91, 36, 77fvmptd3 6784 . . . . . 6 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ((𝑔𝐵 ↦ (𝐹𝑔))‘𝑦) = (𝐹𝑦))
9390, 92oveq12d 7166 . . . . 5 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (((𝑔𝐵 ↦ (𝐹𝑔))‘𝑥)(+g𝑍)((𝑔𝐵 ↦ (𝐹𝑔))‘𝑦)) = ((𝐹𝑥)(+g𝑍)(𝐹𝑦)))
9480, 88, 933eqtr4d 2864 . . . 4 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ((𝑔𝐵 ↦ (𝐹𝑔))‘(𝑥(+g𝑌)𝑦)) = (((𝑔𝐵 ↦ (𝐹𝑔))‘𝑥)(+g𝑍)((𝑔𝐵 ↦ (𝐹𝑔))‘𝑦)))
9594ralrimivva 3189 . . 3 (𝜑 → ∀𝑥𝐵𝑦𝐵 ((𝑔𝐵 ↦ (𝐹𝑔))‘(𝑥(+g𝑌)𝑦)) = (((𝑔𝐵 ↦ (𝐹𝑔))‘𝑥)(+g𝑍)((𝑔𝐵 ↦ (𝐹𝑔))‘𝑦)))
96 coeq2 5722 . . . . 5 (𝑔 = (0g𝑌) → (𝐹𝑔) = (𝐹 ∘ (0g𝑌)))
97 eqid 2819 . . . . . . 7 (0g𝑌) = (0g𝑌)
9817, 97mndidcl 17918 . . . . . 6 (𝑌 ∈ Mnd → (0g𝑌) ∈ 𝐵)
997, 98syl 17 . . . . 5 (𝜑 → (0g𝑌) ∈ 𝐵)
100 coexg 7626 . . . . . 6 ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ (0g𝑌) ∈ 𝐵) → (𝐹 ∘ (0g𝑌)) ∈ V)
1011, 99, 100syl2anc 586 . . . . 5 (𝜑 → (𝐹 ∘ (0g𝑌)) ∈ V)
10281, 96, 99, 101fvmptd3 6784 . . . 4 (𝜑 → ((𝑔𝐵 ↦ (𝐹𝑔))‘(0g𝑌)) = (𝐹 ∘ (0g𝑌)))
10316ffnd 6508 . . . . . 6 (𝜑𝐹 Fn (Base‘𝑅))
104 eqid 2819 . . . . . . . 8 (0g𝑅) = (0g𝑅)
10513, 104mndidcl 17918 . . . . . . 7 (𝑅 ∈ Mnd → (0g𝑅) ∈ (Base‘𝑅))
1063, 105syl 17 . . . . . 6 (𝜑 → (0g𝑅) ∈ (Base‘𝑅))
107 fcoconst 6889 . . . . . 6 ((𝐹 Fn (Base‘𝑅) ∧ (0g𝑅) ∈ (Base‘𝑅)) → (𝐹 ∘ (𝐴 × {(0g𝑅)})) = (𝐴 × {(𝐹‘(0g𝑅))}))
108103, 106, 107syl2anc 586 . . . . 5 (𝜑 → (𝐹 ∘ (𝐴 × {(0g𝑅)})) = (𝐴 × {(𝐹‘(0g𝑅))}))
1095, 104pws0g 17939 . . . . . . 7 ((𝑅 ∈ Mnd ∧ 𝐴𝑉) → (𝐴 × {(0g𝑅)}) = (0g𝑌))
1103, 4, 109syl2anc 586 . . . . . 6 (𝜑 → (𝐴 × {(0g𝑅)}) = (0g𝑌))
111110coeq2d 5726 . . . . 5 (𝜑 → (𝐹 ∘ (𝐴 × {(0g𝑅)})) = (𝐹 ∘ (0g𝑌)))
112 eqid 2819 . . . . . . . . 9 (0g𝑆) = (0g𝑆)
113104, 112mhm0 17956 . . . . . . . 8 (𝐹 ∈ (𝑅 MndHom 𝑆) → (𝐹‘(0g𝑅)) = (0g𝑆))
1141, 113syl 17 . . . . . . 7 (𝜑 → (𝐹‘(0g𝑅)) = (0g𝑆))
115114sneqd 4571 . . . . . 6 (𝜑 → {(𝐹‘(0g𝑅))} = {(0g𝑆)})
116115xpeq2d 5578 . . . . 5 (𝜑 → (𝐴 × {(𝐹‘(0g𝑅))}) = (𝐴 × {(0g𝑆)}))
117108, 111, 1163eqtr3d 2862 . . . 4 (𝜑 → (𝐹 ∘ (0g𝑌)) = (𝐴 × {(0g𝑆)}))
11810, 112pws0g 17939 . . . . 5 ((𝑆 ∈ Mnd ∧ 𝐴𝑉) → (𝐴 × {(0g𝑆)}) = (0g𝑍))
1199, 4, 118syl2anc 586 . . . 4 (𝜑 → (𝐴 × {(0g𝑆)}) = (0g𝑍))
120102, 117, 1193eqtrd 2858 . . 3 (𝜑 → ((𝑔𝐵 ↦ (𝐹𝑔))‘(0g𝑌)) = (0g𝑍))
12128, 95, 1203jca 1123 . 2 (𝜑 → ((𝑔𝐵 ↦ (𝐹𝑔)):𝐵⟶(Base‘𝑍) ∧ ∀𝑥𝐵𝑦𝐵 ((𝑔𝐵 ↦ (𝐹𝑔))‘(𝑥(+g𝑌)𝑦)) = (((𝑔𝐵 ↦ (𝐹𝑔))‘𝑥)(+g𝑍)((𝑔𝐵 ↦ (𝐹𝑔))‘𝑦)) ∧ ((𝑔𝐵 ↦ (𝐹𝑔))‘(0g𝑌)) = (0g𝑍)))
122 eqid 2819 . . 3 (0g𝑍) = (0g𝑍)
12317, 24, 59, 78, 97, 122ismhm 17950 . 2 ((𝑔𝐵 ↦ (𝐹𝑔)) ∈ (𝑌 MndHom 𝑍) ↔ ((𝑌 ∈ Mnd ∧ 𝑍 ∈ Mnd) ∧ ((𝑔𝐵 ↦ (𝐹𝑔)):𝐵⟶(Base‘𝑍) ∧ ∀𝑥𝐵𝑦𝐵 ((𝑔𝐵 ↦ (𝐹𝑔))‘(𝑥(+g𝑌)𝑦)) = (((𝑔𝐵 ↦ (𝐹𝑔))‘𝑥)(+g𝑍)((𝑔𝐵 ↦ (𝐹𝑔))‘𝑦)) ∧ ((𝑔𝐵 ↦ (𝐹𝑔))‘(0g𝑌)) = (0g𝑍))))
1247, 12, 121, 123syl21anbrc 1339 1 (𝜑 → (𝑔𝐵 ↦ (𝐹𝑔)) ∈ (𝑌 MndHom 𝑍))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1082   = wceq 1531   ∈ wcel 2108  ∀wral 3136  Vcvv 3493  {csn 4559   ↦ cmpt 5137   × cxp 5546   ∘ ccom 5552   Fn wfn 6343  ⟶wf 6344  ‘cfv 6348  (class class class)co 7148   ∘f cof 7399  Basecbs 16475  +gcplusg 16557  0gc0g 16705   ↑s cpws 16712  Mndcmnd 17903   MndHom cmhm 17946 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1083  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-nel 3122  df-ral 3141  df-rex 3142  df-reu 3143  df-rmo 3144  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-of 7401  df-om 7573  df-1st 7681  df-2nd 7682  df-wrecs 7939  df-recs 8000  df-rdg 8038  df-1o 8094  df-oadd 8098  df-er 8281  df-map 8400  df-ixp 8454  df-en 8502  df-dom 8503  df-sdom 8504  df-fin 8505  df-sup 8898  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-nn 11631  df-2 11692  df-3 11693  df-4 11694  df-5 11695  df-6 11696  df-7 11697  df-8 11698  df-9 11699  df-n0 11890  df-z 11974  df-dec 12091  df-uz 12236  df-fz 12885  df-struct 16477  df-ndx 16478  df-slot 16479  df-base 16481  df-plusg 16570  df-mulr 16571  df-sca 16573  df-vsca 16574  df-ip 16575  df-tset 16576  df-ple 16577  df-ds 16579  df-hom 16581  df-cco 16582  df-0g 16707  df-prds 16713  df-pws 16715  df-mgm 17844  df-sgrp 17893  df-mnd 17904  df-mhm 17948 This theorem is referenced by:  pwsco2rhm  19483
 Copyright terms: Public domain W3C validator