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Theorem pwsco2mhm 17311
 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 . . . . 5 (𝜑𝐹 ∈ (𝑅 MndHom 𝑆))
2 mhmrcl1 17278 . . . . 5 (𝐹 ∈ (𝑅 MndHom 𝑆) → 𝑅 ∈ Mnd)
31, 2syl 17 . . . 4 (𝜑𝑅 ∈ Mnd)
4 pwsco2mhm.a . . . 4 (𝜑𝐴𝑉)
5 pwsco2mhm.y . . . . 5 𝑌 = (𝑅s 𝐴)
65pwsmnd 17265 . . . 4 ((𝑅 ∈ Mnd ∧ 𝐴𝑉) → 𝑌 ∈ Mnd)
73, 4, 6syl2anc 692 . . 3 (𝜑𝑌 ∈ Mnd)
8 mhmrcl2 17279 . . . . 5 (𝐹 ∈ (𝑅 MndHom 𝑆) → 𝑆 ∈ Mnd)
91, 8syl 17 . . . 4 (𝜑𝑆 ∈ Mnd)
10 pwsco2mhm.z . . . . 5 𝑍 = (𝑆s 𝐴)
1110pwsmnd 17265 . . . 4 ((𝑆 ∈ Mnd ∧ 𝐴𝑉) → 𝑍 ∈ Mnd)
129, 4, 11syl2anc 692 . . 3 (𝜑𝑍 ∈ Mnd)
137, 12jca 554 . 2 (𝜑 → (𝑌 ∈ Mnd ∧ 𝑍 ∈ Mnd))
14 eqid 2621 . . . . . . . . 9 (Base‘𝑅) = (Base‘𝑅)
15 eqid 2621 . . . . . . . . 9 (Base‘𝑆) = (Base‘𝑆)
1614, 15mhmf 17280 . . . . . . . 8 (𝐹 ∈ (𝑅 MndHom 𝑆) → 𝐹:(Base‘𝑅)⟶(Base‘𝑆))
171, 16syl 17 . . . . . . 7 (𝜑𝐹:(Base‘𝑅)⟶(Base‘𝑆))
1817adantr 481 . . . . . 6 ((𝜑𝑔𝐵) → 𝐹:(Base‘𝑅)⟶(Base‘𝑆))
19 pwsco2mhm.b . . . . . . 7 𝐵 = (Base‘𝑌)
203adantr 481 . . . . . . 7 ((𝜑𝑔𝐵) → 𝑅 ∈ Mnd)
214adantr 481 . . . . . . 7 ((𝜑𝑔𝐵) → 𝐴𝑉)
22 simpr 477 . . . . . . 7 ((𝜑𝑔𝐵) → 𝑔𝐵)
235, 14, 19, 20, 21, 22pwselbas 16089 . . . . . 6 ((𝜑𝑔𝐵) → 𝑔:𝐴⟶(Base‘𝑅))
24 fco 6025 . . . . . 6 ((𝐹:(Base‘𝑅)⟶(Base‘𝑆) ∧ 𝑔:𝐴⟶(Base‘𝑅)) → (𝐹𝑔):𝐴⟶(Base‘𝑆))
2518, 23, 24syl2anc 692 . . . . 5 ((𝜑𝑔𝐵) → (𝐹𝑔):𝐴⟶(Base‘𝑆))
269adantr 481 . . . . . 6 ((𝜑𝑔𝐵) → 𝑆 ∈ Mnd)
27 eqid 2621 . . . . . . 7 (Base‘𝑍) = (Base‘𝑍)
2810, 15, 27pwselbasb 16088 . . . . . 6 ((𝑆 ∈ Mnd ∧ 𝐴𝑉) → ((𝐹𝑔) ∈ (Base‘𝑍) ↔ (𝐹𝑔):𝐴⟶(Base‘𝑆)))
2926, 21, 28syl2anc 692 . . . . 5 ((𝜑𝑔𝐵) → ((𝐹𝑔) ∈ (Base‘𝑍) ↔ (𝐹𝑔):𝐴⟶(Base‘𝑆)))
3025, 29mpbird 247 . . . 4 ((𝜑𝑔𝐵) → (𝐹𝑔) ∈ (Base‘𝑍))
31 eqid 2621 . . . 4 (𝑔𝐵 ↦ (𝐹𝑔)) = (𝑔𝐵 ↦ (𝐹𝑔))
3230, 31fmptd 6351 . . 3 (𝜑 → (𝑔𝐵 ↦ (𝐹𝑔)):𝐵⟶(Base‘𝑍))
331adantr 481 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝐹 ∈ (𝑅 MndHom 𝑆))
3433adantr 481 . . . . . . . . 9 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑤𝐴) → 𝐹 ∈ (𝑅 MndHom 𝑆))
3533, 2syl 17 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝑅 ∈ Mnd)
364adantr 481 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝐴𝑉)
37 simprl 793 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝑥𝐵)
385, 14, 19, 35, 36, 37pwselbas 16089 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝑥:𝐴⟶(Base‘𝑅))
3938ffvelrnda 6325 . . . . . . . . 9 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑤𝐴) → (𝑥𝑤) ∈ (Base‘𝑅))
40 simprr 795 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝑦𝐵)
415, 14, 19, 35, 36, 40pwselbas 16089 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝑦:𝐴⟶(Base‘𝑅))
4241ffvelrnda 6325 . . . . . . . . 9 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑤𝐴) → (𝑦𝑤) ∈ (Base‘𝑅))
43 eqid 2621 . . . . . . . . . 10 (+g𝑅) = (+g𝑅)
44 eqid 2621 . . . . . . . . . 10 (+g𝑆) = (+g𝑆)
4514, 43, 44mhmlin 17282 . . . . . . . . 9 ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ (𝑥𝑤) ∈ (Base‘𝑅) ∧ (𝑦𝑤) ∈ (Base‘𝑅)) → (𝐹‘((𝑥𝑤)(+g𝑅)(𝑦𝑤))) = ((𝐹‘(𝑥𝑤))(+g𝑆)(𝐹‘(𝑦𝑤))))
4634, 39, 42, 45syl3anc 1323 . . . . . . . 8 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑤𝐴) → (𝐹‘((𝑥𝑤)(+g𝑅)(𝑦𝑤))) = ((𝐹‘(𝑥𝑤))(+g𝑆)(𝐹‘(𝑦𝑤))))
4746mpteq2dva 4714 . . . . . . 7 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑤𝐴 ↦ (𝐹‘((𝑥𝑤)(+g𝑅)(𝑦𝑤)))) = (𝑤𝐴 ↦ ((𝐹‘(𝑥𝑤))(+g𝑆)(𝐹‘(𝑦𝑤)))))
48 fvexd 6170 . . . . . . . 8 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑤𝐴) → (𝐹‘(𝑥𝑤)) ∈ V)
49 fvexd 6170 . . . . . . . 8 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑤𝐴) → (𝐹‘(𝑦𝑤)) ∈ V)
5038feqmptd 6216 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝑥 = (𝑤𝐴 ↦ (𝑥𝑤)))
5133, 16syl 17 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝐹:(Base‘𝑅)⟶(Base‘𝑆))
5251feqmptd 6216 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝐹 = (𝑧 ∈ (Base‘𝑅) ↦ (𝐹𝑧)))
53 fveq2 6158 . . . . . . . . 9 (𝑧 = (𝑥𝑤) → (𝐹𝑧) = (𝐹‘(𝑥𝑤)))
5439, 50, 52, 53fmptco 6362 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝐹𝑥) = (𝑤𝐴 ↦ (𝐹‘(𝑥𝑤))))
5541feqmptd 6216 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝑦 = (𝑤𝐴 ↦ (𝑦𝑤)))
56 fveq2 6158 . . . . . . . . 9 (𝑧 = (𝑦𝑤) → (𝐹𝑧) = (𝐹‘(𝑦𝑤)))
5742, 55, 52, 56fmptco 6362 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝐹𝑦) = (𝑤𝐴 ↦ (𝐹‘(𝑦𝑤))))
5836, 48, 49, 54, 57offval2 6879 . . . . . . 7 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ((𝐹𝑥) ∘𝑓 (+g𝑆)(𝐹𝑦)) = (𝑤𝐴 ↦ ((𝐹‘(𝑥𝑤))(+g𝑆)(𝐹‘(𝑦𝑤)))))
5947, 58eqtr4d 2658 . . . . . 6 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑤𝐴 ↦ (𝐹‘((𝑥𝑤)(+g𝑅)(𝑦𝑤)))) = ((𝐹𝑥) ∘𝑓 (+g𝑆)(𝐹𝑦)))
6035adantr 481 . . . . . . . 8 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑤𝐴) → 𝑅 ∈ Mnd)
6114, 43mndcl 17241 . . . . . . . 8 ((𝑅 ∈ Mnd ∧ (𝑥𝑤) ∈ (Base‘𝑅) ∧ (𝑦𝑤) ∈ (Base‘𝑅)) → ((𝑥𝑤)(+g𝑅)(𝑦𝑤)) ∈ (Base‘𝑅))
6260, 39, 42, 61syl3anc 1323 . . . . . . 7 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑤𝐴) → ((𝑥𝑤)(+g𝑅)(𝑦𝑤)) ∈ (Base‘𝑅))
63 eqid 2621 . . . . . . . . 9 (+g𝑌) = (+g𝑌)
645, 19, 35, 36, 37, 40, 43, 63pwsplusgval 16090 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝑌)𝑦) = (𝑥𝑓 (+g𝑅)𝑦))
65 fvexd 6170 . . . . . . . . 9 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑤𝐴) → (𝑥𝑤) ∈ V)
66 fvexd 6170 . . . . . . . . 9 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑤𝐴) → (𝑦𝑤) ∈ V)
6736, 65, 66, 50, 55offval2 6879 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥𝑓 (+g𝑅)𝑦) = (𝑤𝐴 ↦ ((𝑥𝑤)(+g𝑅)(𝑦𝑤))))
6864, 67eqtrd 2655 . . . . . . 7 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝑌)𝑦) = (𝑤𝐴 ↦ ((𝑥𝑤)(+g𝑅)(𝑦𝑤))))
69 fveq2 6158 . . . . . . 7 (𝑧 = ((𝑥𝑤)(+g𝑅)(𝑦𝑤)) → (𝐹𝑧) = (𝐹‘((𝑥𝑤)(+g𝑅)(𝑦𝑤))))
7062, 68, 52, 69fmptco 6362 . . . . . 6 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝐹 ∘ (𝑥(+g𝑌)𝑦)) = (𝑤𝐴 ↦ (𝐹‘((𝑥𝑤)(+g𝑅)(𝑦𝑤)))))
7133, 8syl 17 . . . . . . 7 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝑆 ∈ Mnd)
72 fco 6025 . . . . . . . . 9 ((𝐹:(Base‘𝑅)⟶(Base‘𝑆) ∧ 𝑥:𝐴⟶(Base‘𝑅)) → (𝐹𝑥):𝐴⟶(Base‘𝑆))
7351, 38, 72syl2anc 692 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝐹𝑥):𝐴⟶(Base‘𝑆))
7410, 15, 27pwselbasb 16088 . . . . . . . . 9 ((𝑆 ∈ Mnd ∧ 𝐴𝑉) → ((𝐹𝑥) ∈ (Base‘𝑍) ↔ (𝐹𝑥):𝐴⟶(Base‘𝑆)))
7571, 36, 74syl2anc 692 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ((𝐹𝑥) ∈ (Base‘𝑍) ↔ (𝐹𝑥):𝐴⟶(Base‘𝑆)))
7673, 75mpbird 247 . . . . . . 7 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝐹𝑥) ∈ (Base‘𝑍))
77 fco 6025 . . . . . . . . 9 ((𝐹:(Base‘𝑅)⟶(Base‘𝑆) ∧ 𝑦:𝐴⟶(Base‘𝑅)) → (𝐹𝑦):𝐴⟶(Base‘𝑆))
7851, 41, 77syl2anc 692 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝐹𝑦):𝐴⟶(Base‘𝑆))
7910, 15, 27pwselbasb 16088 . . . . . . . . 9 ((𝑆 ∈ Mnd ∧ 𝐴𝑉) → ((𝐹𝑦) ∈ (Base‘𝑍) ↔ (𝐹𝑦):𝐴⟶(Base‘𝑆)))
8071, 36, 79syl2anc 692 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ((𝐹𝑦) ∈ (Base‘𝑍) ↔ (𝐹𝑦):𝐴⟶(Base‘𝑆)))
8178, 80mpbird 247 . . . . . . 7 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝐹𝑦) ∈ (Base‘𝑍))
82 eqid 2621 . . . . . . 7 (+g𝑍) = (+g𝑍)
8310, 27, 71, 36, 76, 81, 44, 82pwsplusgval 16090 . . . . . 6 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ((𝐹𝑥)(+g𝑍)(𝐹𝑦)) = ((𝐹𝑥) ∘𝑓 (+g𝑆)(𝐹𝑦)))
8459, 70, 833eqtr4d 2665 . . . . 5 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝐹 ∘ (𝑥(+g𝑌)𝑦)) = ((𝐹𝑥)(+g𝑍)(𝐹𝑦)))
8519, 63mndcl 17241 . . . . . . . 8 ((𝑌 ∈ Mnd ∧ 𝑥𝐵𝑦𝐵) → (𝑥(+g𝑌)𝑦) ∈ 𝐵)
86853expb 1263 . . . . . . 7 ((𝑌 ∈ Mnd ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝑌)𝑦) ∈ 𝐵)
877, 86sylan 488 . . . . . 6 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝑌)𝑦) ∈ 𝐵)
88 coexg 7079 . . . . . . 7 ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ (𝑥(+g𝑌)𝑦) ∈ 𝐵) → (𝐹 ∘ (𝑥(+g𝑌)𝑦)) ∈ V)
8933, 87, 88syl2anc 692 . . . . . 6 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝐹 ∘ (𝑥(+g𝑌)𝑦)) ∈ V)
90 coeq2 5250 . . . . . . 7 (𝑔 = (𝑥(+g𝑌)𝑦) → (𝐹𝑔) = (𝐹 ∘ (𝑥(+g𝑌)𝑦)))
9190, 31fvmptg 6247 . . . . . 6 (((𝑥(+g𝑌)𝑦) ∈ 𝐵 ∧ (𝐹 ∘ (𝑥(+g𝑌)𝑦)) ∈ V) → ((𝑔𝐵 ↦ (𝐹𝑔))‘(𝑥(+g𝑌)𝑦)) = (𝐹 ∘ (𝑥(+g𝑌)𝑦)))
9287, 89, 91syl2anc 692 . . . . 5 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ((𝑔𝐵 ↦ (𝐹𝑔))‘(𝑥(+g𝑌)𝑦)) = (𝐹 ∘ (𝑥(+g𝑌)𝑦)))
93 coeq2 5250 . . . . . . . 8 (𝑔 = 𝑥 → (𝐹𝑔) = (𝐹𝑥))
9493, 31fvmptg 6247 . . . . . . 7 ((𝑥𝐵 ∧ (𝐹𝑥) ∈ (Base‘𝑍)) → ((𝑔𝐵 ↦ (𝐹𝑔))‘𝑥) = (𝐹𝑥))
9537, 76, 94syl2anc 692 . . . . . 6 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ((𝑔𝐵 ↦ (𝐹𝑔))‘𝑥) = (𝐹𝑥))
96 coeq2 5250 . . . . . . . 8 (𝑔 = 𝑦 → (𝐹𝑔) = (𝐹𝑦))
9796, 31fvmptg 6247 . . . . . . 7 ((𝑦𝐵 ∧ (𝐹𝑦) ∈ (Base‘𝑍)) → ((𝑔𝐵 ↦ (𝐹𝑔))‘𝑦) = (𝐹𝑦))
9840, 81, 97syl2anc 692 . . . . . 6 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ((𝑔𝐵 ↦ (𝐹𝑔))‘𝑦) = (𝐹𝑦))
9995, 98oveq12d 6633 . . . . 5 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (((𝑔𝐵 ↦ (𝐹𝑔))‘𝑥)(+g𝑍)((𝑔𝐵 ↦ (𝐹𝑔))‘𝑦)) = ((𝐹𝑥)(+g𝑍)(𝐹𝑦)))
10084, 92, 993eqtr4d 2665 . . . 4 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ((𝑔𝐵 ↦ (𝐹𝑔))‘(𝑥(+g𝑌)𝑦)) = (((𝑔𝐵 ↦ (𝐹𝑔))‘𝑥)(+g𝑍)((𝑔𝐵 ↦ (𝐹𝑔))‘𝑦)))
101100ralrimivva 2967 . . 3 (𝜑 → ∀𝑥𝐵𝑦𝐵 ((𝑔𝐵 ↦ (𝐹𝑔))‘(𝑥(+g𝑌)𝑦)) = (((𝑔𝐵 ↦ (𝐹𝑔))‘𝑥)(+g𝑍)((𝑔𝐵 ↦ (𝐹𝑔))‘𝑦)))
102 eqid 2621 . . . . . . 7 (0g𝑌) = (0g𝑌)
10319, 102mndidcl 17248 . . . . . 6 (𝑌 ∈ Mnd → (0g𝑌) ∈ 𝐵)
1047, 103syl 17 . . . . 5 (𝜑 → (0g𝑌) ∈ 𝐵)
105 coexg 7079 . . . . . 6 ((𝐹 ∈ (𝑅 MndHom 𝑆) ∧ (0g𝑌) ∈ 𝐵) → (𝐹 ∘ (0g𝑌)) ∈ V)
1061, 104, 105syl2anc 692 . . . . 5 (𝜑 → (𝐹 ∘ (0g𝑌)) ∈ V)
107 coeq2 5250 . . . . . 6 (𝑔 = (0g𝑌) → (𝐹𝑔) = (𝐹 ∘ (0g𝑌)))
108107, 31fvmptg 6247 . . . . 5 (((0g𝑌) ∈ 𝐵 ∧ (𝐹 ∘ (0g𝑌)) ∈ V) → ((𝑔𝐵 ↦ (𝐹𝑔))‘(0g𝑌)) = (𝐹 ∘ (0g𝑌)))
109104, 106, 108syl2anc 692 . . . 4 (𝜑 → ((𝑔𝐵 ↦ (𝐹𝑔))‘(0g𝑌)) = (𝐹 ∘ (0g𝑌)))
110 ffn 6012 . . . . . . 7 (𝐹:(Base‘𝑅)⟶(Base‘𝑆) → 𝐹 Fn (Base‘𝑅))
11117, 110syl 17 . . . . . 6 (𝜑𝐹 Fn (Base‘𝑅))
112 eqid 2621 . . . . . . . 8 (0g𝑅) = (0g𝑅)
11314, 112mndidcl 17248 . . . . . . 7 (𝑅 ∈ Mnd → (0g𝑅) ∈ (Base‘𝑅))
1143, 113syl 17 . . . . . 6 (𝜑 → (0g𝑅) ∈ (Base‘𝑅))
115 fcoconst 6366 . . . . . 6 ((𝐹 Fn (Base‘𝑅) ∧ (0g𝑅) ∈ (Base‘𝑅)) → (𝐹 ∘ (𝐴 × {(0g𝑅)})) = (𝐴 × {(𝐹‘(0g𝑅))}))
116111, 114, 115syl2anc 692 . . . . 5 (𝜑 → (𝐹 ∘ (𝐴 × {(0g𝑅)})) = (𝐴 × {(𝐹‘(0g𝑅))}))
1175, 112pws0g 17266 . . . . . . 7 ((𝑅 ∈ Mnd ∧ 𝐴𝑉) → (𝐴 × {(0g𝑅)}) = (0g𝑌))
1183, 4, 117syl2anc 692 . . . . . 6 (𝜑 → (𝐴 × {(0g𝑅)}) = (0g𝑌))
119118coeq2d 5254 . . . . 5 (𝜑 → (𝐹 ∘ (𝐴 × {(0g𝑅)})) = (𝐹 ∘ (0g𝑌)))
120 eqid 2621 . . . . . . . . 9 (0g𝑆) = (0g𝑆)
121112, 120mhm0 17283 . . . . . . . 8 (𝐹 ∈ (𝑅 MndHom 𝑆) → (𝐹‘(0g𝑅)) = (0g𝑆))
1221, 121syl 17 . . . . . . 7 (𝜑 → (𝐹‘(0g𝑅)) = (0g𝑆))
123122sneqd 4167 . . . . . 6 (𝜑 → {(𝐹‘(0g𝑅))} = {(0g𝑆)})
124123xpeq2d 5109 . . . . 5 (𝜑 → (𝐴 × {(𝐹‘(0g𝑅))}) = (𝐴 × {(0g𝑆)}))
125116, 119, 1243eqtr3d 2663 . . . 4 (𝜑 → (𝐹 ∘ (0g𝑌)) = (𝐴 × {(0g𝑆)}))
12610, 120pws0g 17266 . . . . 5 ((𝑆 ∈ Mnd ∧ 𝐴𝑉) → (𝐴 × {(0g𝑆)}) = (0g𝑍))
1279, 4, 126syl2anc 692 . . . 4 (𝜑 → (𝐴 × {(0g𝑆)}) = (0g𝑍))
128109, 125, 1273eqtrd 2659 . . 3 (𝜑 → ((𝑔𝐵 ↦ (𝐹𝑔))‘(0g𝑌)) = (0g𝑍))
12932, 101, 1283jca 1240 . 2 (𝜑 → ((𝑔𝐵 ↦ (𝐹𝑔)):𝐵⟶(Base‘𝑍) ∧ ∀𝑥𝐵𝑦𝐵 ((𝑔𝐵 ↦ (𝐹𝑔))‘(𝑥(+g𝑌)𝑦)) = (((𝑔𝐵 ↦ (𝐹𝑔))‘𝑥)(+g𝑍)((𝑔𝐵 ↦ (𝐹𝑔))‘𝑦)) ∧ ((𝑔𝐵 ↦ (𝐹𝑔))‘(0g𝑌)) = (0g𝑍)))
130 eqid 2621 . . 3 (0g𝑍) = (0g𝑍)
13119, 27, 63, 82, 102, 130ismhm 17277 . 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))))
13213, 129, 131sylanbrc 697 1 (𝜑 → (𝑔𝐵 ↦ (𝐹𝑔)) ∈ (𝑌 MndHom 𝑍))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1036   = wceq 1480   ∈ wcel 1987  ∀wral 2908  Vcvv 3190  {csn 4155   ↦ cmpt 4683   × cxp 5082   ∘ ccom 5088   Fn wfn 5852  ⟶wf 5853  ‘cfv 5857  (class class class)co 6615   ∘𝑓 cof 6860  Basecbs 15800  +gcplusg 15881  0gc0g 16040   ↑s cpws 16047  Mndcmnd 17234   MndHom cmhm 17273 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-cnex 9952  ax-resscn 9953  ax-1cn 9954  ax-icn 9955  ax-addcl 9956  ax-addrcl 9957  ax-mulcl 9958  ax-mulrcl 9959  ax-mulcom 9960  ax-addass 9961  ax-mulass 9962  ax-distr 9963  ax-i2m1 9964  ax-1ne0 9965  ax-1rid 9966  ax-rnegex 9967  ax-rrecex 9968  ax-cnre 9969  ax-pre-lttri 9970  ax-pre-lttrn 9971  ax-pre-ltadd 9972  ax-pre-mulgt0 9973 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-int 4448  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-of 6862  df-om 7028  df-1st 7128  df-2nd 7129  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-1o 7520  df-oadd 7524  df-er 7702  df-map 7819  df-ixp 7869  df-en 7916  df-dom 7917  df-sdom 7918  df-fin 7919  df-sup 8308  df-pnf 10036  df-mnf 10037  df-xr 10038  df-ltxr 10039  df-le 10040  df-sub 10228  df-neg 10229  df-nn 10981  df-2 11039  df-3 11040  df-4 11041  df-5 11042  df-6 11043  df-7 11044  df-8 11045  df-9 11046  df-n0 11253  df-z 11338  df-dec 11454  df-uz 11648  df-fz 12285  df-struct 15802  df-ndx 15803  df-slot 15804  df-base 15805  df-plusg 15894  df-mulr 15895  df-sca 15897  df-vsca 15898  df-ip 15899  df-tset 15900  df-ple 15901  df-ds 15904  df-hom 15906  df-cco 15907  df-0g 16042  df-prds 16048  df-pws 16050  df-mgm 17182  df-sgrp 17224  df-mnd 17235  df-mhm 17275 This theorem is referenced by:  pwsco2rhm  18679
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