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Theorem pwsco1mhm 18845
Description: Right composition with a function on the index sets yields a monoid homomorphism of structure powers. (Contributed by Mario Carneiro, 12-Jun-2015.)
Hypotheses
Ref Expression
pwsco1mhm.y 𝑌 = (𝑅s 𝐴)
pwsco1mhm.z 𝑍 = (𝑅s 𝐵)
pwsco1mhm.c 𝐶 = (Base‘𝑍)
pwsco1mhm.r (𝜑𝑅 ∈ Mnd)
pwsco1mhm.a (𝜑𝐴𝑉)
pwsco1mhm.b (𝜑𝐵𝑊)
pwsco1mhm.f (𝜑𝐹:𝐴𝐵)
Assertion
Ref Expression
pwsco1mhm (𝜑 → (𝑔𝐶 ↦ (𝑔𝐹)) ∈ (𝑍 MndHom 𝑌))
Distinct variable groups:   𝐶,𝑔   𝑔,𝑌   𝑔,𝑍   𝑔,𝐹   𝜑,𝑔
Allowed substitution hints:   𝐴(𝑔)   𝐵(𝑔)   𝑅(𝑔)   𝑉(𝑔)   𝑊(𝑔)

Proof of Theorem pwsco1mhm
Dummy variables 𝑥 𝑧 𝑤 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pwsco1mhm.r . . 3 (𝜑𝑅 ∈ Mnd)
2 pwsco1mhm.b . . 3 (𝜑𝐵𝑊)
3 pwsco1mhm.z . . . 4 𝑍 = (𝑅s 𝐵)
43pwsmnd 18785 . . 3 ((𝑅 ∈ Mnd ∧ 𝐵𝑊) → 𝑍 ∈ Mnd)
51, 2, 4syl2anc 584 . 2 (𝜑𝑍 ∈ Mnd)
6 pwsco1mhm.a . . 3 (𝜑𝐴𝑉)
7 pwsco1mhm.y . . . 4 𝑌 = (𝑅s 𝐴)
87pwsmnd 18785 . . 3 ((𝑅 ∈ Mnd ∧ 𝐴𝑉) → 𝑌 ∈ Mnd)
91, 6, 8syl2anc 584 . 2 (𝜑𝑌 ∈ Mnd)
10 eqid 2737 . . . . . . . . 9 (Base‘𝑅) = (Base‘𝑅)
11 pwsco1mhm.c . . . . . . . . 9 𝐶 = (Base‘𝑍)
123, 10, 11pwselbasb 17533 . . . . . . . 8 ((𝑅 ∈ Mnd ∧ 𝐵𝑊) → (𝑔𝐶𝑔:𝐵⟶(Base‘𝑅)))
131, 2, 12syl2anc 584 . . . . . . 7 (𝜑 → (𝑔𝐶𝑔:𝐵⟶(Base‘𝑅)))
1413biimpa 476 . . . . . 6 ((𝜑𝑔𝐶) → 𝑔:𝐵⟶(Base‘𝑅))
15 pwsco1mhm.f . . . . . . 7 (𝜑𝐹:𝐴𝐵)
1615adantr 480 . . . . . 6 ((𝜑𝑔𝐶) → 𝐹:𝐴𝐵)
17 fco 6760 . . . . . 6 ((𝑔:𝐵⟶(Base‘𝑅) ∧ 𝐹:𝐴𝐵) → (𝑔𝐹):𝐴⟶(Base‘𝑅))
1814, 16, 17syl2anc 584 . . . . 5 ((𝜑𝑔𝐶) → (𝑔𝐹):𝐴⟶(Base‘𝑅))
19 eqid 2737 . . . . . . . 8 (Base‘𝑌) = (Base‘𝑌)
207, 10, 19pwselbasb 17533 . . . . . . 7 ((𝑅 ∈ Mnd ∧ 𝐴𝑉) → ((𝑔𝐹) ∈ (Base‘𝑌) ↔ (𝑔𝐹):𝐴⟶(Base‘𝑅)))
211, 6, 20syl2anc 584 . . . . . 6 (𝜑 → ((𝑔𝐹) ∈ (Base‘𝑌) ↔ (𝑔𝐹):𝐴⟶(Base‘𝑅)))
2221adantr 480 . . . . 5 ((𝜑𝑔𝐶) → ((𝑔𝐹) ∈ (Base‘𝑌) ↔ (𝑔𝐹):𝐴⟶(Base‘𝑅)))
2318, 22mpbird 257 . . . 4 ((𝜑𝑔𝐶) → (𝑔𝐹) ∈ (Base‘𝑌))
2423fmpttd 7135 . . 3 (𝜑 → (𝑔𝐶 ↦ (𝑔𝐹)):𝐶⟶(Base‘𝑌))
256adantr 480 . . . . . . 7 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → 𝐴𝑉)
26 fvexd 6921 . . . . . . 7 (((𝜑 ∧ (𝑥𝐶𝑦𝐶)) ∧ 𝑧𝐴) → (𝑥‘(𝐹𝑧)) ∈ V)
27 fvexd 6921 . . . . . . 7 (((𝜑 ∧ (𝑥𝐶𝑦𝐶)) ∧ 𝑧𝐴) → (𝑦‘(𝐹𝑧)) ∈ V)
2815adantr 480 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → 𝐹:𝐴𝐵)
2928ffvelcdmda 7104 . . . . . . . 8 (((𝜑 ∧ (𝑥𝐶𝑦𝐶)) ∧ 𝑧𝐴) → (𝐹𝑧) ∈ 𝐵)
3028feqmptd 6977 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → 𝐹 = (𝑧𝐴 ↦ (𝐹𝑧)))
311adantr 480 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → 𝑅 ∈ Mnd)
322adantr 480 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → 𝐵𝑊)
33 simprl 771 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → 𝑥𝐶)
343, 10, 11, 31, 32, 33pwselbas 17534 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → 𝑥:𝐵⟶(Base‘𝑅))
3534feqmptd 6977 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → 𝑥 = (𝑤𝐵 ↦ (𝑥𝑤)))
36 fveq2 6906 . . . . . . . 8 (𝑤 = (𝐹𝑧) → (𝑥𝑤) = (𝑥‘(𝐹𝑧)))
3729, 30, 35, 36fmptco 7149 . . . . . . 7 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → (𝑥𝐹) = (𝑧𝐴 ↦ (𝑥‘(𝐹𝑧))))
38 simprr 773 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → 𝑦𝐶)
393, 10, 11, 31, 32, 38pwselbas 17534 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → 𝑦:𝐵⟶(Base‘𝑅))
4039feqmptd 6977 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → 𝑦 = (𝑤𝐵 ↦ (𝑦𝑤)))
41 fveq2 6906 . . . . . . . 8 (𝑤 = (𝐹𝑧) → (𝑦𝑤) = (𝑦‘(𝐹𝑧)))
4229, 30, 40, 41fmptco 7149 . . . . . . 7 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → (𝑦𝐹) = (𝑧𝐴 ↦ (𝑦‘(𝐹𝑧))))
4325, 26, 27, 37, 42offval2 7717 . . . . . 6 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → ((𝑥𝐹) ∘f (+g𝑅)(𝑦𝐹)) = (𝑧𝐴 ↦ ((𝑥‘(𝐹𝑧))(+g𝑅)(𝑦‘(𝐹𝑧)))))
44 fco 6760 . . . . . . . . 9 ((𝑥:𝐵⟶(Base‘𝑅) ∧ 𝐹:𝐴𝐵) → (𝑥𝐹):𝐴⟶(Base‘𝑅))
4534, 28, 44syl2anc 584 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → (𝑥𝐹):𝐴⟶(Base‘𝑅))
467, 10, 19pwselbasb 17533 . . . . . . . . 9 ((𝑅 ∈ Mnd ∧ 𝐴𝑉) → ((𝑥𝐹) ∈ (Base‘𝑌) ↔ (𝑥𝐹):𝐴⟶(Base‘𝑅)))
4731, 25, 46syl2anc 584 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → ((𝑥𝐹) ∈ (Base‘𝑌) ↔ (𝑥𝐹):𝐴⟶(Base‘𝑅)))
4845, 47mpbird 257 . . . . . . 7 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → (𝑥𝐹) ∈ (Base‘𝑌))
49 fco 6760 . . . . . . . . 9 ((𝑦:𝐵⟶(Base‘𝑅) ∧ 𝐹:𝐴𝐵) → (𝑦𝐹):𝐴⟶(Base‘𝑅))
5039, 28, 49syl2anc 584 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → (𝑦𝐹):𝐴⟶(Base‘𝑅))
517, 10, 19pwselbasb 17533 . . . . . . . . 9 ((𝑅 ∈ Mnd ∧ 𝐴𝑉) → ((𝑦𝐹) ∈ (Base‘𝑌) ↔ (𝑦𝐹):𝐴⟶(Base‘𝑅)))
5231, 25, 51syl2anc 584 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → ((𝑦𝐹) ∈ (Base‘𝑌) ↔ (𝑦𝐹):𝐴⟶(Base‘𝑅)))
5350, 52mpbird 257 . . . . . . 7 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → (𝑦𝐹) ∈ (Base‘𝑌))
54 eqid 2737 . . . . . . 7 (+g𝑅) = (+g𝑅)
55 eqid 2737 . . . . . . 7 (+g𝑌) = (+g𝑌)
567, 19, 31, 25, 48, 53, 54, 55pwsplusgval 17535 . . . . . 6 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → ((𝑥𝐹)(+g𝑌)(𝑦𝐹)) = ((𝑥𝐹) ∘f (+g𝑅)(𝑦𝐹)))
57 eqid 2737 . . . . . . . . 9 (+g𝑍) = (+g𝑍)
583, 11, 31, 32, 33, 38, 54, 57pwsplusgval 17535 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → (𝑥(+g𝑍)𝑦) = (𝑥f (+g𝑅)𝑦))
59 fvexd 6921 . . . . . . . . 9 (((𝜑 ∧ (𝑥𝐶𝑦𝐶)) ∧ 𝑤𝐵) → (𝑥𝑤) ∈ V)
60 fvexd 6921 . . . . . . . . 9 (((𝜑 ∧ (𝑥𝐶𝑦𝐶)) ∧ 𝑤𝐵) → (𝑦𝑤) ∈ V)
6132, 59, 60, 35, 40offval2 7717 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → (𝑥f (+g𝑅)𝑦) = (𝑤𝐵 ↦ ((𝑥𝑤)(+g𝑅)(𝑦𝑤))))
6258, 61eqtrd 2777 . . . . . . 7 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → (𝑥(+g𝑍)𝑦) = (𝑤𝐵 ↦ ((𝑥𝑤)(+g𝑅)(𝑦𝑤))))
6336, 41oveq12d 7449 . . . . . . 7 (𝑤 = (𝐹𝑧) → ((𝑥𝑤)(+g𝑅)(𝑦𝑤)) = ((𝑥‘(𝐹𝑧))(+g𝑅)(𝑦‘(𝐹𝑧))))
6429, 30, 62, 63fmptco 7149 . . . . . 6 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → ((𝑥(+g𝑍)𝑦) ∘ 𝐹) = (𝑧𝐴 ↦ ((𝑥‘(𝐹𝑧))(+g𝑅)(𝑦‘(𝐹𝑧)))))
6543, 56, 643eqtr4rd 2788 . . . . 5 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → ((𝑥(+g𝑍)𝑦) ∘ 𝐹) = ((𝑥𝐹)(+g𝑌)(𝑦𝐹)))
66 eqid 2737 . . . . . 6 (𝑔𝐶 ↦ (𝑔𝐹)) = (𝑔𝐶 ↦ (𝑔𝐹))
67 coeq1 5868 . . . . . 6 (𝑔 = (𝑥(+g𝑍)𝑦) → (𝑔𝐹) = ((𝑥(+g𝑍)𝑦) ∘ 𝐹))
6811, 57mndcl 18755 . . . . . . . 8 ((𝑍 ∈ Mnd ∧ 𝑥𝐶𝑦𝐶) → (𝑥(+g𝑍)𝑦) ∈ 𝐶)
69683expb 1121 . . . . . . 7 ((𝑍 ∈ Mnd ∧ (𝑥𝐶𝑦𝐶)) → (𝑥(+g𝑍)𝑦) ∈ 𝐶)
705, 69sylan 580 . . . . . 6 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → (𝑥(+g𝑍)𝑦) ∈ 𝐶)
71 ovex 7464 . . . . . . 7 (𝑥(+g𝑍)𝑦) ∈ V
7215, 6fexd 7247 . . . . . . . 8 (𝜑𝐹 ∈ V)
7372adantr 480 . . . . . . 7 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → 𝐹 ∈ V)
74 coexg 7951 . . . . . . 7 (((𝑥(+g𝑍)𝑦) ∈ V ∧ 𝐹 ∈ V) → ((𝑥(+g𝑍)𝑦) ∘ 𝐹) ∈ V)
7571, 73, 74sylancr 587 . . . . . 6 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → ((𝑥(+g𝑍)𝑦) ∘ 𝐹) ∈ V)
7666, 67, 70, 75fvmptd3 7039 . . . . 5 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → ((𝑔𝐶 ↦ (𝑔𝐹))‘(𝑥(+g𝑍)𝑦)) = ((𝑥(+g𝑍)𝑦) ∘ 𝐹))
77 coeq1 5868 . . . . . . 7 (𝑔 = 𝑥 → (𝑔𝐹) = (𝑥𝐹))
78 coexg 7951 . . . . . . . 8 ((𝑥𝐶𝐹 ∈ V) → (𝑥𝐹) ∈ V)
7933, 73, 78syl2anc 584 . . . . . . 7 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → (𝑥𝐹) ∈ V)
8066, 77, 33, 79fvmptd3 7039 . . . . . 6 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → ((𝑔𝐶 ↦ (𝑔𝐹))‘𝑥) = (𝑥𝐹))
81 coeq1 5868 . . . . . . 7 (𝑔 = 𝑦 → (𝑔𝐹) = (𝑦𝐹))
82 coexg 7951 . . . . . . . 8 ((𝑦𝐶𝐹 ∈ V) → (𝑦𝐹) ∈ V)
8338, 73, 82syl2anc 584 . . . . . . 7 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → (𝑦𝐹) ∈ V)
8466, 81, 38, 83fvmptd3 7039 . . . . . 6 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → ((𝑔𝐶 ↦ (𝑔𝐹))‘𝑦) = (𝑦𝐹))
8580, 84oveq12d 7449 . . . . 5 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → (((𝑔𝐶 ↦ (𝑔𝐹))‘𝑥)(+g𝑌)((𝑔𝐶 ↦ (𝑔𝐹))‘𝑦)) = ((𝑥𝐹)(+g𝑌)(𝑦𝐹)))
8665, 76, 853eqtr4d 2787 . . . 4 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → ((𝑔𝐶 ↦ (𝑔𝐹))‘(𝑥(+g𝑍)𝑦)) = (((𝑔𝐶 ↦ (𝑔𝐹))‘𝑥)(+g𝑌)((𝑔𝐶 ↦ (𝑔𝐹))‘𝑦)))
8786ralrimivva 3202 . . 3 (𝜑 → ∀𝑥𝐶𝑦𝐶 ((𝑔𝐶 ↦ (𝑔𝐹))‘(𝑥(+g𝑍)𝑦)) = (((𝑔𝐶 ↦ (𝑔𝐹))‘𝑥)(+g𝑌)((𝑔𝐶 ↦ (𝑔𝐹))‘𝑦)))
88 coeq1 5868 . . . . 5 (𝑔 = (0g𝑍) → (𝑔𝐹) = ((0g𝑍) ∘ 𝐹))
89 eqid 2737 . . . . . . 7 (0g𝑍) = (0g𝑍)
9011, 89mndidcl 18762 . . . . . 6 (𝑍 ∈ Mnd → (0g𝑍) ∈ 𝐶)
915, 90syl 17 . . . . 5 (𝜑 → (0g𝑍) ∈ 𝐶)
92 coexg 7951 . . . . . 6 (((0g𝑍) ∈ 𝐶𝐹 ∈ V) → ((0g𝑍) ∘ 𝐹) ∈ V)
9391, 72, 92syl2anc 584 . . . . 5 (𝜑 → ((0g𝑍) ∘ 𝐹) ∈ V)
9466, 88, 91, 93fvmptd3 7039 . . . 4 (𝜑 → ((𝑔𝐶 ↦ (𝑔𝐹))‘(0g𝑍)) = ((0g𝑍) ∘ 𝐹))
953, 10, 11, 1, 2, 91pwselbas 17534 . . . . . . 7 (𝜑 → (0g𝑍):𝐵⟶(Base‘𝑅))
96 fco 6760 . . . . . . 7 (((0g𝑍):𝐵⟶(Base‘𝑅) ∧ 𝐹:𝐴𝐵) → ((0g𝑍) ∘ 𝐹):𝐴⟶(Base‘𝑅))
9795, 15, 96syl2anc 584 . . . . . 6 (𝜑 → ((0g𝑍) ∘ 𝐹):𝐴⟶(Base‘𝑅))
9897ffnd 6737 . . . . 5 (𝜑 → ((0g𝑍) ∘ 𝐹) Fn 𝐴)
99 fvexd 6921 . . . . . 6 (𝜑 → (0g𝑅) ∈ V)
100 fnconstg 6796 . . . . . 6 ((0g𝑅) ∈ V → (𝐴 × {(0g𝑅)}) Fn 𝐴)
10199, 100syl 17 . . . . 5 (𝜑 → (𝐴 × {(0g𝑅)}) Fn 𝐴)
102 eqid 2737 . . . . . . . . . . 11 (0g𝑅) = (0g𝑅)
1033, 102pws0g 18786 . . . . . . . . . 10 ((𝑅 ∈ Mnd ∧ 𝐵𝑊) → (𝐵 × {(0g𝑅)}) = (0g𝑍))
1041, 2, 103syl2anc 584 . . . . . . . . 9 (𝜑 → (𝐵 × {(0g𝑅)}) = (0g𝑍))
105104fveq1d 6908 . . . . . . . 8 (𝜑 → ((𝐵 × {(0g𝑅)})‘(𝐹𝑥)) = ((0g𝑍)‘(𝐹𝑥)))
106105adantr 480 . . . . . . 7 ((𝜑𝑥𝐴) → ((𝐵 × {(0g𝑅)})‘(𝐹𝑥)) = ((0g𝑍)‘(𝐹𝑥)))
107 fvex 6919 . . . . . . . 8 (0g𝑅) ∈ V
10815ffvelcdmda 7104 . . . . . . . 8 ((𝜑𝑥𝐴) → (𝐹𝑥) ∈ 𝐵)
109 fvconst2g 7222 . . . . . . . 8 (((0g𝑅) ∈ V ∧ (𝐹𝑥) ∈ 𝐵) → ((𝐵 × {(0g𝑅)})‘(𝐹𝑥)) = (0g𝑅))
110107, 108, 109sylancr 587 . . . . . . 7 ((𝜑𝑥𝐴) → ((𝐵 × {(0g𝑅)})‘(𝐹𝑥)) = (0g𝑅))
111106, 110eqtr3d 2779 . . . . . 6 ((𝜑𝑥𝐴) → ((0g𝑍)‘(𝐹𝑥)) = (0g𝑅))
112 fvco3 7008 . . . . . . 7 ((𝐹:𝐴𝐵𝑥𝐴) → (((0g𝑍) ∘ 𝐹)‘𝑥) = ((0g𝑍)‘(𝐹𝑥)))
11315, 112sylan 580 . . . . . 6 ((𝜑𝑥𝐴) → (((0g𝑍) ∘ 𝐹)‘𝑥) = ((0g𝑍)‘(𝐹𝑥)))
114 fvconst2g 7222 . . . . . . 7 (((0g𝑅) ∈ V ∧ 𝑥𝐴) → ((𝐴 × {(0g𝑅)})‘𝑥) = (0g𝑅))
11599, 114sylan 580 . . . . . 6 ((𝜑𝑥𝐴) → ((𝐴 × {(0g𝑅)})‘𝑥) = (0g𝑅))
116111, 113, 1153eqtr4d 2787 . . . . 5 ((𝜑𝑥𝐴) → (((0g𝑍) ∘ 𝐹)‘𝑥) = ((𝐴 × {(0g𝑅)})‘𝑥))
11798, 101, 116eqfnfvd 7054 . . . 4 (𝜑 → ((0g𝑍) ∘ 𝐹) = (𝐴 × {(0g𝑅)}))
1187, 102pws0g 18786 . . . . 5 ((𝑅 ∈ Mnd ∧ 𝐴𝑉) → (𝐴 × {(0g𝑅)}) = (0g𝑌))
1191, 6, 118syl2anc 584 . . . 4 (𝜑 → (𝐴 × {(0g𝑅)}) = (0g𝑌))
12094, 117, 1193eqtrd 2781 . . 3 (𝜑 → ((𝑔𝐶 ↦ (𝑔𝐹))‘(0g𝑍)) = (0g𝑌))
12124, 87, 1203jca 1129 . 2 (𝜑 → ((𝑔𝐶 ↦ (𝑔𝐹)):𝐶⟶(Base‘𝑌) ∧ ∀𝑥𝐶𝑦𝐶 ((𝑔𝐶 ↦ (𝑔𝐹))‘(𝑥(+g𝑍)𝑦)) = (((𝑔𝐶 ↦ (𝑔𝐹))‘𝑥)(+g𝑌)((𝑔𝐶 ↦ (𝑔𝐹))‘𝑦)) ∧ ((𝑔𝐶 ↦ (𝑔𝐹))‘(0g𝑍)) = (0g𝑌)))
122 eqid 2737 . . 3 (0g𝑌) = (0g𝑌)
12311, 19, 57, 55, 89, 122ismhm 18798 . 2 ((𝑔𝐶 ↦ (𝑔𝐹)) ∈ (𝑍 MndHom 𝑌) ↔ ((𝑍 ∈ Mnd ∧ 𝑌 ∈ Mnd) ∧ ((𝑔𝐶 ↦ (𝑔𝐹)):𝐶⟶(Base‘𝑌) ∧ ∀𝑥𝐶𝑦𝐶 ((𝑔𝐶 ↦ (𝑔𝐹))‘(𝑥(+g𝑍)𝑦)) = (((𝑔𝐶 ↦ (𝑔𝐹))‘𝑥)(+g𝑌)((𝑔𝐶 ↦ (𝑔𝐹))‘𝑦)) ∧ ((𝑔𝐶 ↦ (𝑔𝐹))‘(0g𝑍)) = (0g𝑌))))
1245, 9, 121, 123syl21anbrc 1345 1 (𝜑 → (𝑔𝐶 ↦ (𝑔𝐹)) ∈ (𝑍 MndHom 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  wral 3061  Vcvv 3480  {csn 4626  cmpt 5225   × cxp 5683  ccom 5689   Fn wfn 6556  wf 6557  cfv 6561  (class class class)co 7431  f cof 7695  Basecbs 17247  +gcplusg 17297  0gc0g 17484  s cpws 17491  Mndcmnd 18747   MndHom cmhm 18794
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-map 8868  df-ixp 8938  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-sup 9482  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-z 12614  df-dec 12734  df-uz 12879  df-fz 13548  df-struct 17184  df-slot 17219  df-ndx 17231  df-base 17248  df-plusg 17310  df-mulr 17311  df-sca 17313  df-vsca 17314  df-ip 17315  df-tset 17316  df-ple 17317  df-ds 17319  df-hom 17321  df-cco 17322  df-0g 17486  df-prds 17492  df-pws 17494  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-mhm 18796
This theorem is referenced by:  pwsco1rhm  20502
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