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Theorem pwsco1mhm 18470
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 18420 . . 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 18420 . . 3 ((𝑅 ∈ Mnd ∧ 𝐴𝑉) → 𝑌 ∈ Mnd)
91, 6, 8syl2anc 584 . 2 (𝜑𝑌 ∈ Mnd)
10 eqid 2738 . . . . . . . . 9 (Base‘𝑅) = (Base‘𝑅)
11 pwsco1mhm.c . . . . . . . . 9 𝐶 = (Base‘𝑍)
123, 10, 11pwselbasb 17199 . . . . . . . 8 ((𝑅 ∈ Mnd ∧ 𝐵𝑊) → (𝑔𝐶𝑔:𝐵⟶(Base‘𝑅)))
131, 2, 12syl2anc 584 . . . . . . 7 (𝜑 → (𝑔𝐶𝑔:𝐵⟶(Base‘𝑅)))
1413biimpa 477 . . . . . 6 ((𝜑𝑔𝐶) → 𝑔:𝐵⟶(Base‘𝑅))
15 pwsco1mhm.f . . . . . . 7 (𝜑𝐹:𝐴𝐵)
1615adantr 481 . . . . . 6 ((𝜑𝑔𝐶) → 𝐹:𝐴𝐵)
17 fco 6624 . . . . . 6 ((𝑔:𝐵⟶(Base‘𝑅) ∧ 𝐹:𝐴𝐵) → (𝑔𝐹):𝐴⟶(Base‘𝑅))
1814, 16, 17syl2anc 584 . . . . 5 ((𝜑𝑔𝐶) → (𝑔𝐹):𝐴⟶(Base‘𝑅))
19 eqid 2738 . . . . . . . 8 (Base‘𝑌) = (Base‘𝑌)
207, 10, 19pwselbasb 17199 . . . . . . 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 481 . . . . 5 ((𝜑𝑔𝐶) → ((𝑔𝐹) ∈ (Base‘𝑌) ↔ (𝑔𝐹):𝐴⟶(Base‘𝑅)))
2318, 22mpbird 256 . . . 4 ((𝜑𝑔𝐶) → (𝑔𝐹) ∈ (Base‘𝑌))
2423fmpttd 6989 . . 3 (𝜑 → (𝑔𝐶 ↦ (𝑔𝐹)):𝐶⟶(Base‘𝑌))
256adantr 481 . . . . . . 7 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → 𝐴𝑉)
26 fvexd 6789 . . . . . . 7 (((𝜑 ∧ (𝑥𝐶𝑦𝐶)) ∧ 𝑧𝐴) → (𝑥‘(𝐹𝑧)) ∈ V)
27 fvexd 6789 . . . . . . 7 (((𝜑 ∧ (𝑥𝐶𝑦𝐶)) ∧ 𝑧𝐴) → (𝑦‘(𝐹𝑧)) ∈ V)
2815adantr 481 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → 𝐹:𝐴𝐵)
2928ffvelrnda 6961 . . . . . . . 8 (((𝜑 ∧ (𝑥𝐶𝑦𝐶)) ∧ 𝑧𝐴) → (𝐹𝑧) ∈ 𝐵)
3028feqmptd 6837 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → 𝐹 = (𝑧𝐴 ↦ (𝐹𝑧)))
311adantr 481 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → 𝑅 ∈ Mnd)
322adantr 481 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → 𝐵𝑊)
33 simprl 768 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → 𝑥𝐶)
343, 10, 11, 31, 32, 33pwselbas 17200 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → 𝑥:𝐵⟶(Base‘𝑅))
3534feqmptd 6837 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → 𝑥 = (𝑤𝐵 ↦ (𝑥𝑤)))
36 fveq2 6774 . . . . . . . 8 (𝑤 = (𝐹𝑧) → (𝑥𝑤) = (𝑥‘(𝐹𝑧)))
3729, 30, 35, 36fmptco 7001 . . . . . . 7 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → (𝑥𝐹) = (𝑧𝐴 ↦ (𝑥‘(𝐹𝑧))))
38 simprr 770 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → 𝑦𝐶)
393, 10, 11, 31, 32, 38pwselbas 17200 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → 𝑦:𝐵⟶(Base‘𝑅))
4039feqmptd 6837 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → 𝑦 = (𝑤𝐵 ↦ (𝑦𝑤)))
41 fveq2 6774 . . . . . . . 8 (𝑤 = (𝐹𝑧) → (𝑦𝑤) = (𝑦‘(𝐹𝑧)))
4229, 30, 40, 41fmptco 7001 . . . . . . 7 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → (𝑦𝐹) = (𝑧𝐴 ↦ (𝑦‘(𝐹𝑧))))
4325, 26, 27, 37, 42offval2 7553 . . . . . 6 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → ((𝑥𝐹) ∘f (+g𝑅)(𝑦𝐹)) = (𝑧𝐴 ↦ ((𝑥‘(𝐹𝑧))(+g𝑅)(𝑦‘(𝐹𝑧)))))
44 fco 6624 . . . . . . . . 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 17199 . . . . . . . . 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 256 . . . . . . 7 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → (𝑥𝐹) ∈ (Base‘𝑌))
49 fco 6624 . . . . . . . . 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 17199 . . . . . . . . 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 256 . . . . . . 7 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → (𝑦𝐹) ∈ (Base‘𝑌))
54 eqid 2738 . . . . . . 7 (+g𝑅) = (+g𝑅)
55 eqid 2738 . . . . . . 7 (+g𝑌) = (+g𝑌)
567, 19, 31, 25, 48, 53, 54, 55pwsplusgval 17201 . . . . . 6 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → ((𝑥𝐹)(+g𝑌)(𝑦𝐹)) = ((𝑥𝐹) ∘f (+g𝑅)(𝑦𝐹)))
57 eqid 2738 . . . . . . . . 9 (+g𝑍) = (+g𝑍)
583, 11, 31, 32, 33, 38, 54, 57pwsplusgval 17201 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → (𝑥(+g𝑍)𝑦) = (𝑥f (+g𝑅)𝑦))
59 fvexd 6789 . . . . . . . . 9 (((𝜑 ∧ (𝑥𝐶𝑦𝐶)) ∧ 𝑤𝐵) → (𝑥𝑤) ∈ V)
60 fvexd 6789 . . . . . . . . 9 (((𝜑 ∧ (𝑥𝐶𝑦𝐶)) ∧ 𝑤𝐵) → (𝑦𝑤) ∈ V)
6132, 59, 60, 35, 40offval2 7553 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → (𝑥f (+g𝑅)𝑦) = (𝑤𝐵 ↦ ((𝑥𝑤)(+g𝑅)(𝑦𝑤))))
6258, 61eqtrd 2778 . . . . . . 7 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → (𝑥(+g𝑍)𝑦) = (𝑤𝐵 ↦ ((𝑥𝑤)(+g𝑅)(𝑦𝑤))))
6336, 41oveq12d 7293 . . . . . . 7 (𝑤 = (𝐹𝑧) → ((𝑥𝑤)(+g𝑅)(𝑦𝑤)) = ((𝑥‘(𝐹𝑧))(+g𝑅)(𝑦‘(𝐹𝑧))))
6429, 30, 62, 63fmptco 7001 . . . . . 6 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → ((𝑥(+g𝑍)𝑦) ∘ 𝐹) = (𝑧𝐴 ↦ ((𝑥‘(𝐹𝑧))(+g𝑅)(𝑦‘(𝐹𝑧)))))
6543, 56, 643eqtr4rd 2789 . . . . 5 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → ((𝑥(+g𝑍)𝑦) ∘ 𝐹) = ((𝑥𝐹)(+g𝑌)(𝑦𝐹)))
66 eqid 2738 . . . . . 6 (𝑔𝐶 ↦ (𝑔𝐹)) = (𝑔𝐶 ↦ (𝑔𝐹))
67 coeq1 5766 . . . . . 6 (𝑔 = (𝑥(+g𝑍)𝑦) → (𝑔𝐹) = ((𝑥(+g𝑍)𝑦) ∘ 𝐹))
6811, 57mndcl 18393 . . . . . . . 8 ((𝑍 ∈ Mnd ∧ 𝑥𝐶𝑦𝐶) → (𝑥(+g𝑍)𝑦) ∈ 𝐶)
69683expb 1119 . . . . . . 7 ((𝑍 ∈ Mnd ∧ (𝑥𝐶𝑦𝐶)) → (𝑥(+g𝑍)𝑦) ∈ 𝐶)
705, 69sylan 580 . . . . . 6 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → (𝑥(+g𝑍)𝑦) ∈ 𝐶)
71 ovex 7308 . . . . . . 7 (𝑥(+g𝑍)𝑦) ∈ V
7215, 6fexd 7103 . . . . . . . 8 (𝜑𝐹 ∈ V)
7372adantr 481 . . . . . . 7 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → 𝐹 ∈ V)
74 coexg 7776 . . . . . . 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 6898 . . . . 5 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → ((𝑔𝐶 ↦ (𝑔𝐹))‘(𝑥(+g𝑍)𝑦)) = ((𝑥(+g𝑍)𝑦) ∘ 𝐹))
77 coeq1 5766 . . . . . . 7 (𝑔 = 𝑥 → (𝑔𝐹) = (𝑥𝐹))
78 coexg 7776 . . . . . . . 8 ((𝑥𝐶𝐹 ∈ V) → (𝑥𝐹) ∈ V)
7933, 73, 78syl2anc 584 . . . . . . 7 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → (𝑥𝐹) ∈ V)
8066, 77, 33, 79fvmptd3 6898 . . . . . 6 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → ((𝑔𝐶 ↦ (𝑔𝐹))‘𝑥) = (𝑥𝐹))
81 coeq1 5766 . . . . . . 7 (𝑔 = 𝑦 → (𝑔𝐹) = (𝑦𝐹))
82 coexg 7776 . . . . . . . 8 ((𝑦𝐶𝐹 ∈ V) → (𝑦𝐹) ∈ V)
8338, 73, 82syl2anc 584 . . . . . . 7 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → (𝑦𝐹) ∈ V)
8466, 81, 38, 83fvmptd3 6898 . . . . . 6 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → ((𝑔𝐶 ↦ (𝑔𝐹))‘𝑦) = (𝑦𝐹))
8580, 84oveq12d 7293 . . . . 5 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → (((𝑔𝐶 ↦ (𝑔𝐹))‘𝑥)(+g𝑌)((𝑔𝐶 ↦ (𝑔𝐹))‘𝑦)) = ((𝑥𝐹)(+g𝑌)(𝑦𝐹)))
8665, 76, 853eqtr4d 2788 . . . 4 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → ((𝑔𝐶 ↦ (𝑔𝐹))‘(𝑥(+g𝑍)𝑦)) = (((𝑔𝐶 ↦ (𝑔𝐹))‘𝑥)(+g𝑌)((𝑔𝐶 ↦ (𝑔𝐹))‘𝑦)))
8786ralrimivva 3123 . . 3 (𝜑 → ∀𝑥𝐶𝑦𝐶 ((𝑔𝐶 ↦ (𝑔𝐹))‘(𝑥(+g𝑍)𝑦)) = (((𝑔𝐶 ↦ (𝑔𝐹))‘𝑥)(+g𝑌)((𝑔𝐶 ↦ (𝑔𝐹))‘𝑦)))
88 coeq1 5766 . . . . 5 (𝑔 = (0g𝑍) → (𝑔𝐹) = ((0g𝑍) ∘ 𝐹))
89 eqid 2738 . . . . . . 7 (0g𝑍) = (0g𝑍)
9011, 89mndidcl 18400 . . . . . 6 (𝑍 ∈ Mnd → (0g𝑍) ∈ 𝐶)
915, 90syl 17 . . . . 5 (𝜑 → (0g𝑍) ∈ 𝐶)
92 coexg 7776 . . . . . 6 (((0g𝑍) ∈ 𝐶𝐹 ∈ V) → ((0g𝑍) ∘ 𝐹) ∈ V)
9391, 72, 92syl2anc 584 . . . . 5 (𝜑 → ((0g𝑍) ∘ 𝐹) ∈ V)
9466, 88, 91, 93fvmptd3 6898 . . . 4 (𝜑 → ((𝑔𝐶 ↦ (𝑔𝐹))‘(0g𝑍)) = ((0g𝑍) ∘ 𝐹))
953, 10, 11, 1, 2, 91pwselbas 17200 . . . . . . 7 (𝜑 → (0g𝑍):𝐵⟶(Base‘𝑅))
96 fco 6624 . . . . . . 7 (((0g𝑍):𝐵⟶(Base‘𝑅) ∧ 𝐹:𝐴𝐵) → ((0g𝑍) ∘ 𝐹):𝐴⟶(Base‘𝑅))
9795, 15, 96syl2anc 584 . . . . . 6 (𝜑 → ((0g𝑍) ∘ 𝐹):𝐴⟶(Base‘𝑅))
9897ffnd 6601 . . . . 5 (𝜑 → ((0g𝑍) ∘ 𝐹) Fn 𝐴)
99 fvexd 6789 . . . . . 6 (𝜑 → (0g𝑅) ∈ V)
100 fnconstg 6662 . . . . . 6 ((0g𝑅) ∈ V → (𝐴 × {(0g𝑅)}) Fn 𝐴)
10199, 100syl 17 . . . . 5 (𝜑 → (𝐴 × {(0g𝑅)}) Fn 𝐴)
102 eqid 2738 . . . . . . . . . . 11 (0g𝑅) = (0g𝑅)
1033, 102pws0g 18421 . . . . . . . . . 10 ((𝑅 ∈ Mnd ∧ 𝐵𝑊) → (𝐵 × {(0g𝑅)}) = (0g𝑍))
1041, 2, 103syl2anc 584 . . . . . . . . 9 (𝜑 → (𝐵 × {(0g𝑅)}) = (0g𝑍))
105104fveq1d 6776 . . . . . . . 8 (𝜑 → ((𝐵 × {(0g𝑅)})‘(𝐹𝑥)) = ((0g𝑍)‘(𝐹𝑥)))
106105adantr 481 . . . . . . 7 ((𝜑𝑥𝐴) → ((𝐵 × {(0g𝑅)})‘(𝐹𝑥)) = ((0g𝑍)‘(𝐹𝑥)))
107 fvex 6787 . . . . . . . 8 (0g𝑅) ∈ V
10815ffvelrnda 6961 . . . . . . . 8 ((𝜑𝑥𝐴) → (𝐹𝑥) ∈ 𝐵)
109 fvconst2g 7077 . . . . . . . 8 (((0g𝑅) ∈ V ∧ (𝐹𝑥) ∈ 𝐵) → ((𝐵 × {(0g𝑅)})‘(𝐹𝑥)) = (0g𝑅))
110107, 108, 109sylancr 587 . . . . . . 7 ((𝜑𝑥𝐴) → ((𝐵 × {(0g𝑅)})‘(𝐹𝑥)) = (0g𝑅))
111106, 110eqtr3d 2780 . . . . . 6 ((𝜑𝑥𝐴) → ((0g𝑍)‘(𝐹𝑥)) = (0g𝑅))
112 fvco3 6867 . . . . . . 7 ((𝐹:𝐴𝐵𝑥𝐴) → (((0g𝑍) ∘ 𝐹)‘𝑥) = ((0g𝑍)‘(𝐹𝑥)))
11315, 112sylan 580 . . . . . 6 ((𝜑𝑥𝐴) → (((0g𝑍) ∘ 𝐹)‘𝑥) = ((0g𝑍)‘(𝐹𝑥)))
114 fvconst2g 7077 . . . . . . 7 (((0g𝑅) ∈ V ∧ 𝑥𝐴) → ((𝐴 × {(0g𝑅)})‘𝑥) = (0g𝑅))
11599, 114sylan 580 . . . . . 6 ((𝜑𝑥𝐴) → ((𝐴 × {(0g𝑅)})‘𝑥) = (0g𝑅))
116111, 113, 1153eqtr4d 2788 . . . . 5 ((𝜑𝑥𝐴) → (((0g𝑍) ∘ 𝐹)‘𝑥) = ((𝐴 × {(0g𝑅)})‘𝑥))
11798, 101, 116eqfnfvd 6912 . . . 4 (𝜑 → ((0g𝑍) ∘ 𝐹) = (𝐴 × {(0g𝑅)}))
1187, 102pws0g 18421 . . . . 5 ((𝑅 ∈ Mnd ∧ 𝐴𝑉) → (𝐴 × {(0g𝑅)}) = (0g𝑌))
1191, 6, 118syl2anc 584 . . . 4 (𝜑 → (𝐴 × {(0g𝑅)}) = (0g𝑌))
12094, 117, 1193eqtrd 2782 . . 3 (𝜑 → ((𝑔𝐶 ↦ (𝑔𝐹))‘(0g𝑍)) = (0g𝑌))
12124, 87, 1203jca 1127 . 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 2738 . . 3 (0g𝑌) = (0g𝑌)
12311, 19, 57, 55, 89, 122ismhm 18432 . 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 1343 1 (𝜑 → (𝑔𝐶 ↦ (𝑔𝐹)) ∈ (𝑍 MndHom 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1086   = wceq 1539  wcel 2106  wral 3064  Vcvv 3432  {csn 4561  cmpt 5157   × cxp 5587  ccom 5593   Fn wfn 6428  wf 6429  cfv 6433  (class class class)co 7275  f cof 7531  Basecbs 16912  +gcplusg 16962  0gc0g 17150  s cpws 17157  Mndcmnd 18385   MndHom cmhm 18428
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-of 7533  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-er 8498  df-map 8617  df-ixp 8686  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-sup 9201  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-nn 11974  df-2 12036  df-3 12037  df-4 12038  df-5 12039  df-6 12040  df-7 12041  df-8 12042  df-9 12043  df-n0 12234  df-z 12320  df-dec 12438  df-uz 12583  df-fz 13240  df-struct 16848  df-slot 16883  df-ndx 16895  df-base 16913  df-plusg 16975  df-mulr 16976  df-sca 16978  df-vsca 16979  df-ip 16980  df-tset 16981  df-ple 16982  df-ds 16984  df-hom 16986  df-cco 16987  df-0g 17152  df-prds 17158  df-pws 17160  df-mgm 18326  df-sgrp 18375  df-mnd 18386  df-mhm 18430
This theorem is referenced by:  pwsco1rhm  19982
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