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Theorem pwsco2mhm 18789
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 18743 . . . 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 18728 . . 3 ((𝑅 ∈ Mnd ∧ 𝐴𝑉) → 𝑌 ∈ Mnd)
73, 4, 6syl2anc 582 . 2 (𝜑𝑌 ∈ Mnd)
8 mhmrcl2 18744 . . . 4 (𝐹 ∈ (𝑅 MndHom 𝑆) → 𝑆 ∈ Mnd)
91, 8syl 17 . . 3 (𝜑𝑆 ∈ Mnd)
10 pwsco2mhm.z . . . 4 𝑍 = (𝑆s 𝐴)
1110pwsmnd 18728 . . 3 ((𝑆 ∈ Mnd ∧ 𝐴𝑉) → 𝑍 ∈ Mnd)
129, 4, 11syl2anc 582 . 2 (𝜑𝑍 ∈ Mnd)
13 eqid 2725 . . . . . . . 8 (Base‘𝑅) = (Base‘𝑅)
14 eqid 2725 . . . . . . . 8 (Base‘𝑆) = (Base‘𝑆)
1513, 14mhmf 18745 . . . . . . 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 479 . . . . . . 7 ((𝜑𝑔𝐵) → 𝑅 ∈ Mnd)
194adantr 479 . . . . . . 7 ((𝜑𝑔𝐵) → 𝐴𝑉)
20 simpr 483 . . . . . . 7 ((𝜑𝑔𝐵) → 𝑔𝐵)
215, 13, 17, 18, 19, 20pwselbas 17470 . . . . . 6 ((𝜑𝑔𝐵) → 𝑔:𝐴⟶(Base‘𝑅))
22 fco 6742 . . . . . 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 2725 . . . . . . 7 (Base‘𝑍) = (Base‘𝑍)
2510, 14, 24pwselbasb 17469 . . . . . 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 256 . . . 4 ((𝜑𝑔𝐵) → (𝐹𝑔) ∈ (Base‘𝑍))
2827fmpttd 7120 . . 3 (𝜑 → (𝑔𝐵 ↦ (𝐹𝑔)):𝐵⟶(Base‘𝑍))
291adantr 479 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝐹 ∈ (𝑅 MndHom 𝑆))
3029adantr 479 . . . . . . . . 9 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑤𝐴) → 𝐹 ∈ (𝑅 MndHom 𝑆))
3129, 2syl 17 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝑅 ∈ Mnd)
324adantr 479 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝐴𝑉)
33 simprl 769 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝑥𝐵)
345, 13, 17, 31, 32, 33pwselbas 17470 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝑥:𝐴⟶(Base‘𝑅))
3534ffvelcdmda 7089 . . . . . . . . 9 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑤𝐴) → (𝑥𝑤) ∈ (Base‘𝑅))
36 simprr 771 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝑦𝐵)
375, 13, 17, 31, 32, 36pwselbas 17470 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝑦:𝐴⟶(Base‘𝑅))
3837ffvelcdmda 7089 . . . . . . . . 9 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑤𝐴) → (𝑦𝑤) ∈ (Base‘𝑅))
39 eqid 2725 . . . . . . . . . 10 (+g𝑅) = (+g𝑅)
40 eqid 2725 . . . . . . . . . 10 (+g𝑆) = (+g𝑆)
4113, 39, 40mhmlin 18749 . . . . . . . . 9 ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ (𝑥𝑤) ∈ (Base‘𝑅) ∧ (𝑦𝑤) ∈ (Base‘𝑅)) → (𝐹‘((𝑥𝑤)(+g𝑅)(𝑦𝑤))) = ((𝐹‘(𝑥𝑤))(+g𝑆)(𝐹‘(𝑦𝑤))))
4230, 35, 38, 41syl3anc 1368 . . . . . . . 8 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑤𝐴) → (𝐹‘((𝑥𝑤)(+g𝑅)(𝑦𝑤))) = ((𝐹‘(𝑥𝑤))(+g𝑆)(𝐹‘(𝑦𝑤))))
4342mpteq2dva 5243 . . . . . . 7 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑤𝐴 ↦ (𝐹‘((𝑥𝑤)(+g𝑅)(𝑦𝑤)))) = (𝑤𝐴 ↦ ((𝐹‘(𝑥𝑤))(+g𝑆)(𝐹‘(𝑦𝑤)))))
44 fvexd 6907 . . . . . . . 8 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑤𝐴) → (𝐹‘(𝑥𝑤)) ∈ V)
45 fvexd 6907 . . . . . . . 8 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑤𝐴) → (𝐹‘(𝑦𝑤)) ∈ V)
4634feqmptd 6962 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝑥 = (𝑤𝐴 ↦ (𝑥𝑤)))
4729, 15syl 17 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝐹:(Base‘𝑅)⟶(Base‘𝑆))
4847feqmptd 6962 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝐹 = (𝑧 ∈ (Base‘𝑅) ↦ (𝐹𝑧)))
49 fveq2 6892 . . . . . . . . 9 (𝑧 = (𝑥𝑤) → (𝐹𝑧) = (𝐹‘(𝑥𝑤)))
5035, 46, 48, 49fmptco 7134 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝐹𝑥) = (𝑤𝐴 ↦ (𝐹‘(𝑥𝑤))))
5137feqmptd 6962 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝑦 = (𝑤𝐴 ↦ (𝑦𝑤)))
52 fveq2 6892 . . . . . . . . 9 (𝑧 = (𝑦𝑤) → (𝐹𝑧) = (𝐹‘(𝑦𝑤)))
5338, 51, 48, 52fmptco 7134 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝐹𝑦) = (𝑤𝐴 ↦ (𝐹‘(𝑦𝑤))))
5432, 44, 45, 50, 53offval2 7702 . . . . . . 7 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ((𝐹𝑥) ∘f (+g𝑆)(𝐹𝑦)) = (𝑤𝐴 ↦ ((𝐹‘(𝑥𝑤))(+g𝑆)(𝐹‘(𝑦𝑤)))))
5543, 54eqtr4d 2768 . . . . . 6 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑤𝐴 ↦ (𝐹‘((𝑥𝑤)(+g𝑅)(𝑦𝑤)))) = ((𝐹𝑥) ∘f (+g𝑆)(𝐹𝑦)))
5631adantr 479 . . . . . . . 8 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑤𝐴) → 𝑅 ∈ Mnd)
5713, 39mndcl 18701 . . . . . . . 8 ((𝑅 ∈ Mnd ∧ (𝑥𝑤) ∈ (Base‘𝑅) ∧ (𝑦𝑤) ∈ (Base‘𝑅)) → ((𝑥𝑤)(+g𝑅)(𝑦𝑤)) ∈ (Base‘𝑅))
5856, 35, 38, 57syl3anc 1368 . . . . . . 7 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑤𝐴) → ((𝑥𝑤)(+g𝑅)(𝑦𝑤)) ∈ (Base‘𝑅))
59 eqid 2725 . . . . . . . . 9 (+g𝑌) = (+g𝑌)
605, 17, 31, 32, 33, 36, 39, 59pwsplusgval 17471 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝑌)𝑦) = (𝑥f (+g𝑅)𝑦))
61 fvexd 6907 . . . . . . . . 9 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑤𝐴) → (𝑥𝑤) ∈ V)
62 fvexd 6907 . . . . . . . . 9 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑤𝐴) → (𝑦𝑤) ∈ V)
6332, 61, 62, 46, 51offval2 7702 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥f (+g𝑅)𝑦) = (𝑤𝐴 ↦ ((𝑥𝑤)(+g𝑅)(𝑦𝑤))))
6460, 63eqtrd 2765 . . . . . . 7 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝑌)𝑦) = (𝑤𝐴 ↦ ((𝑥𝑤)(+g𝑅)(𝑦𝑤))))
65 fveq2 6892 . . . . . . 7 (𝑧 = ((𝑥𝑤)(+g𝑅)(𝑦𝑤)) → (𝐹𝑧) = (𝐹‘((𝑥𝑤)(+g𝑅)(𝑦𝑤))))
6658, 64, 48, 65fmptco 7134 . . . . . 6 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝐹 ∘ (𝑥(+g𝑌)𝑦)) = (𝑤𝐴 ↦ (𝐹‘((𝑥𝑤)(+g𝑅)(𝑦𝑤)))))
6729, 8syl 17 . . . . . . 7 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝑆 ∈ Mnd)
68 fco 6742 . . . . . . . . 9 ((𝐹:(Base‘𝑅)⟶(Base‘𝑆) ∧ 𝑥:𝐴⟶(Base‘𝑅)) → (𝐹𝑥):𝐴⟶(Base‘𝑆))
6947, 34, 68syl2anc 582 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝐹𝑥):𝐴⟶(Base‘𝑆))
7010, 14, 24pwselbasb 17469 . . . . . . . . 9 ((𝑆 ∈ Mnd ∧ 𝐴𝑉) → ((𝐹𝑥) ∈ (Base‘𝑍) ↔ (𝐹𝑥):𝐴⟶(Base‘𝑆)))
7167, 32, 70syl2anc 582 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ((𝐹𝑥) ∈ (Base‘𝑍) ↔ (𝐹𝑥):𝐴⟶(Base‘𝑆)))
7269, 71mpbird 256 . . . . . . 7 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝐹𝑥) ∈ (Base‘𝑍))
73 fco 6742 . . . . . . . . 9 ((𝐹:(Base‘𝑅)⟶(Base‘𝑆) ∧ 𝑦:𝐴⟶(Base‘𝑅)) → (𝐹𝑦):𝐴⟶(Base‘𝑆))
7447, 37, 73syl2anc 582 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝐹𝑦):𝐴⟶(Base‘𝑆))
7510, 14, 24pwselbasb 17469 . . . . . . . . 9 ((𝑆 ∈ Mnd ∧ 𝐴𝑉) → ((𝐹𝑦) ∈ (Base‘𝑍) ↔ (𝐹𝑦):𝐴⟶(Base‘𝑆)))
7667, 32, 75syl2anc 582 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ((𝐹𝑦) ∈ (Base‘𝑍) ↔ (𝐹𝑦):𝐴⟶(Base‘𝑆)))
7774, 76mpbird 256 . . . . . . 7 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝐹𝑦) ∈ (Base‘𝑍))
78 eqid 2725 . . . . . . 7 (+g𝑍) = (+g𝑍)
7910, 24, 67, 32, 72, 77, 40, 78pwsplusgval 17471 . . . . . 6 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ((𝐹𝑥)(+g𝑍)(𝐹𝑦)) = ((𝐹𝑥) ∘f (+g𝑆)(𝐹𝑦)))
8055, 66, 793eqtr4d 2775 . . . . 5 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝐹 ∘ (𝑥(+g𝑌)𝑦)) = ((𝐹𝑥)(+g𝑍)(𝐹𝑦)))
81 eqid 2725 . . . . . 6 (𝑔𝐵 ↦ (𝐹𝑔)) = (𝑔𝐵 ↦ (𝐹𝑔))
82 coeq2 5855 . . . . . 6 (𝑔 = (𝑥(+g𝑌)𝑦) → (𝐹𝑔) = (𝐹 ∘ (𝑥(+g𝑌)𝑦)))
8317, 59mndcl 18701 . . . . . . . 8 ((𝑌 ∈ Mnd ∧ 𝑥𝐵𝑦𝐵) → (𝑥(+g𝑌)𝑦) ∈ 𝐵)
84833expb 1117 . . . . . . 7 ((𝑌 ∈ Mnd ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝑌)𝑦) ∈ 𝐵)
857, 84sylan 578 . . . . . 6 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝑌)𝑦) ∈ 𝐵)
86 coexg 7935 . . . . . . 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 7023 . . . . 5 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ((𝑔𝐵 ↦ (𝐹𝑔))‘(𝑥(+g𝑌)𝑦)) = (𝐹 ∘ (𝑥(+g𝑌)𝑦)))
89 coeq2 5855 . . . . . . 7 (𝑔 = 𝑥 → (𝐹𝑔) = (𝐹𝑥))
9081, 89, 33, 72fvmptd3 7023 . . . . . 6 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ((𝑔𝐵 ↦ (𝐹𝑔))‘𝑥) = (𝐹𝑥))
91 coeq2 5855 . . . . . . 7 (𝑔 = 𝑦 → (𝐹𝑔) = (𝐹𝑦))
9281, 91, 36, 77fvmptd3 7023 . . . . . 6 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ((𝑔𝐵 ↦ (𝐹𝑔))‘𝑦) = (𝐹𝑦))
9390, 92oveq12d 7434 . . . . 5 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (((𝑔𝐵 ↦ (𝐹𝑔))‘𝑥)(+g𝑍)((𝑔𝐵 ↦ (𝐹𝑔))‘𝑦)) = ((𝐹𝑥)(+g𝑍)(𝐹𝑦)))
9480, 88, 933eqtr4d 2775 . . . 4 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ((𝑔𝐵 ↦ (𝐹𝑔))‘(𝑥(+g𝑌)𝑦)) = (((𝑔𝐵 ↦ (𝐹𝑔))‘𝑥)(+g𝑍)((𝑔𝐵 ↦ (𝐹𝑔))‘𝑦)))
9594ralrimivva 3191 . . 3 (𝜑 → ∀𝑥𝐵𝑦𝐵 ((𝑔𝐵 ↦ (𝐹𝑔))‘(𝑥(+g𝑌)𝑦)) = (((𝑔𝐵 ↦ (𝐹𝑔))‘𝑥)(+g𝑍)((𝑔𝐵 ↦ (𝐹𝑔))‘𝑦)))
96 coeq2 5855 . . . . 5 (𝑔 = (0g𝑌) → (𝐹𝑔) = (𝐹 ∘ (0g𝑌)))
97 eqid 2725 . . . . . . 7 (0g𝑌) = (0g𝑌)
9817, 97mndidcl 18708 . . . . . 6 (𝑌 ∈ Mnd → (0g𝑌) ∈ 𝐵)
997, 98syl 17 . . . . 5 (𝜑 → (0g𝑌) ∈ 𝐵)
100 coexg 7935 . . . . . 6 ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ (0g𝑌) ∈ 𝐵) → (𝐹 ∘ (0g𝑌)) ∈ V)
1011, 99, 100syl2anc 582 . . . . 5 (𝜑 → (𝐹 ∘ (0g𝑌)) ∈ V)
10281, 96, 99, 101fvmptd3 7023 . . . 4 (𝜑 → ((𝑔𝐵 ↦ (𝐹𝑔))‘(0g𝑌)) = (𝐹 ∘ (0g𝑌)))
10316ffnd 6718 . . . . . 6 (𝜑𝐹 Fn (Base‘𝑅))
104 eqid 2725 . . . . . . . 8 (0g𝑅) = (0g𝑅)
10513, 104mndidcl 18708 . . . . . . 7 (𝑅 ∈ Mnd → (0g𝑅) ∈ (Base‘𝑅))
1063, 105syl 17 . . . . . 6 (𝜑 → (0g𝑅) ∈ (Base‘𝑅))
107 fcoconst 7139 . . . . . 6 ((𝐹 Fn (Base‘𝑅) ∧ (0g𝑅) ∈ (Base‘𝑅)) → (𝐹 ∘ (𝐴 × {(0g𝑅)})) = (𝐴 × {(𝐹‘(0g𝑅))}))
108103, 106, 107syl2anc 582 . . . . 5 (𝜑 → (𝐹 ∘ (𝐴 × {(0g𝑅)})) = (𝐴 × {(𝐹‘(0g𝑅))}))
1095, 104pws0g 18729 . . . . . . 7 ((𝑅 ∈ Mnd ∧ 𝐴𝑉) → (𝐴 × {(0g𝑅)}) = (0g𝑌))
1103, 4, 109syl2anc 582 . . . . . 6 (𝜑 → (𝐴 × {(0g𝑅)}) = (0g𝑌))
111110coeq2d 5859 . . . . 5 (𝜑 → (𝐹 ∘ (𝐴 × {(0g𝑅)})) = (𝐹 ∘ (0g𝑌)))
112 eqid 2725 . . . . . . . . 9 (0g𝑆) = (0g𝑆)
113104, 112mhm0 18750 . . . . . . . 8 (𝐹 ∈ (𝑅 MndHom 𝑆) → (𝐹‘(0g𝑅)) = (0g𝑆))
1141, 113syl 17 . . . . . . 7 (𝜑 → (𝐹‘(0g𝑅)) = (0g𝑆))
115114sneqd 4636 . . . . . 6 (𝜑 → {(𝐹‘(0g𝑅))} = {(0g𝑆)})
116115xpeq2d 5702 . . . . 5 (𝜑 → (𝐴 × {(𝐹‘(0g𝑅))}) = (𝐴 × {(0g𝑆)}))
117108, 111, 1163eqtr3d 2773 . . . 4 (𝜑 → (𝐹 ∘ (0g𝑌)) = (𝐴 × {(0g𝑆)}))
11810, 112pws0g 18729 . . . . 5 ((𝑆 ∈ Mnd ∧ 𝐴𝑉) → (𝐴 × {(0g𝑆)}) = (0g𝑍))
1199, 4, 118syl2anc 582 . . . 4 (𝜑 → (𝐴 × {(0g𝑆)}) = (0g𝑍))
120102, 117, 1193eqtrd 2769 . . 3 (𝜑 → ((𝑔𝐵 ↦ (𝐹𝑔))‘(0g𝑌)) = (0g𝑍))
12128, 95, 1203jca 1125 . 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 2725 . . 3 (0g𝑍) = (0g𝑍)
12317, 24, 59, 78, 97, 122ismhm 18741 . 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 1341 1 (𝜑 → (𝑔𝐵 ↦ (𝐹𝑔)) ∈ (𝑌 MndHom 𝑍))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  w3a 1084   = wceq 1533  wcel 2098  wral 3051  Vcvv 3463  {csn 4624  cmpt 5226   × cxp 5670  ccom 5676   Fn wfn 6538  wf 6539  cfv 6543  (class class class)co 7416  f cof 7680  Basecbs 17179  +gcplusg 17232  0gc0g 17420  s cpws 17427  Mndcmnd 18693   MndHom cmhm 18737
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7738  ax-cnex 11194  ax-resscn 11195  ax-1cn 11196  ax-icn 11197  ax-addcl 11198  ax-addrcl 11199  ax-mulcl 11200  ax-mulrcl 11201  ax-mulcom 11202  ax-addass 11203  ax-mulass 11204  ax-distr 11205  ax-i2m1 11206  ax-1ne0 11207  ax-1rid 11208  ax-rnegex 11209  ax-rrecex 11210  ax-cnre 11211  ax-pre-lttri 11212  ax-pre-lttrn 11213  ax-pre-ltadd 11214  ax-pre-mulgt0 11215
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-pss 3959  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-tp 4629  df-op 4631  df-uni 4904  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-tr 5261  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7372  df-ov 7419  df-oprab 7420  df-mpo 7421  df-of 7682  df-om 7869  df-1st 7991  df-2nd 7992  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-1o 8485  df-er 8723  df-map 8845  df-ixp 8915  df-en 8963  df-dom 8964  df-sdom 8965  df-fin 8966  df-sup 9465  df-pnf 11280  df-mnf 11281  df-xr 11282  df-ltxr 11283  df-le 11284  df-sub 11476  df-neg 11477  df-nn 12243  df-2 12305  df-3 12306  df-4 12307  df-5 12308  df-6 12309  df-7 12310  df-8 12311  df-9 12312  df-n0 12503  df-z 12589  df-dec 12708  df-uz 12853  df-fz 13517  df-struct 17115  df-slot 17150  df-ndx 17162  df-base 17180  df-plusg 17245  df-mulr 17246  df-sca 17248  df-vsca 17249  df-ip 17250  df-tset 17251  df-ple 17252  df-ds 17254  df-hom 17256  df-cco 17257  df-0g 17422  df-prds 17428  df-pws 17430  df-mgm 18599  df-sgrp 18678  df-mnd 18694  df-mhm 18739
This theorem is referenced by:  pwsco2rhm  20446
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