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Theorem pwsco1mhm 17990
 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 17940 . . 3 ((𝑅 ∈ Mnd ∧ 𝐵𝑊) → 𝑍 ∈ Mnd)
51, 2, 4syl2anc 586 . 2 (𝜑𝑍 ∈ Mnd)
6 pwsco1mhm.a . . 3 (𝜑𝐴𝑉)
7 pwsco1mhm.y . . . 4 𝑌 = (𝑅s 𝐴)
87pwsmnd 17940 . . 3 ((𝑅 ∈ Mnd ∧ 𝐴𝑉) → 𝑌 ∈ Mnd)
91, 6, 8syl2anc 586 . 2 (𝜑𝑌 ∈ Mnd)
10 eqid 2821 . . . . . . . . 9 (Base‘𝑅) = (Base‘𝑅)
11 pwsco1mhm.c . . . . . . . . 9 𝐶 = (Base‘𝑍)
123, 10, 11pwselbasb 16755 . . . . . . . 8 ((𝑅 ∈ Mnd ∧ 𝐵𝑊) → (𝑔𝐶𝑔:𝐵⟶(Base‘𝑅)))
131, 2, 12syl2anc 586 . . . . . . 7 (𝜑 → (𝑔𝐶𝑔:𝐵⟶(Base‘𝑅)))
1413biimpa 479 . . . . . 6 ((𝜑𝑔𝐶) → 𝑔:𝐵⟶(Base‘𝑅))
15 pwsco1mhm.f . . . . . . 7 (𝜑𝐹:𝐴𝐵)
1615adantr 483 . . . . . 6 ((𝜑𝑔𝐶) → 𝐹:𝐴𝐵)
17 fco 6525 . . . . . 6 ((𝑔:𝐵⟶(Base‘𝑅) ∧ 𝐹:𝐴𝐵) → (𝑔𝐹):𝐴⟶(Base‘𝑅))
1814, 16, 17syl2anc 586 . . . . 5 ((𝜑𝑔𝐶) → (𝑔𝐹):𝐴⟶(Base‘𝑅))
19 eqid 2821 . . . . . . . 8 (Base‘𝑌) = (Base‘𝑌)
207, 10, 19pwselbasb 16755 . . . . . . 7 ((𝑅 ∈ Mnd ∧ 𝐴𝑉) → ((𝑔𝐹) ∈ (Base‘𝑌) ↔ (𝑔𝐹):𝐴⟶(Base‘𝑅)))
211, 6, 20syl2anc 586 . . . . . 6 (𝜑 → ((𝑔𝐹) ∈ (Base‘𝑌) ↔ (𝑔𝐹):𝐴⟶(Base‘𝑅)))
2221adantr 483 . . . . 5 ((𝜑𝑔𝐶) → ((𝑔𝐹) ∈ (Base‘𝑌) ↔ (𝑔𝐹):𝐴⟶(Base‘𝑅)))
2318, 22mpbird 259 . . . 4 ((𝜑𝑔𝐶) → (𝑔𝐹) ∈ (Base‘𝑌))
2423fmpttd 6873 . . 3 (𝜑 → (𝑔𝐶 ↦ (𝑔𝐹)):𝐶⟶(Base‘𝑌))
256adantr 483 . . . . . . 7 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → 𝐴𝑉)
26 fvexd 6679 . . . . . . 7 (((𝜑 ∧ (𝑥𝐶𝑦𝐶)) ∧ 𝑧𝐴) → (𝑥‘(𝐹𝑧)) ∈ V)
27 fvexd 6679 . . . . . . 7 (((𝜑 ∧ (𝑥𝐶𝑦𝐶)) ∧ 𝑧𝐴) → (𝑦‘(𝐹𝑧)) ∈ V)
2815adantr 483 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → 𝐹:𝐴𝐵)
2928ffvelrnda 6845 . . . . . . . 8 (((𝜑 ∧ (𝑥𝐶𝑦𝐶)) ∧ 𝑧𝐴) → (𝐹𝑧) ∈ 𝐵)
3028feqmptd 6727 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → 𝐹 = (𝑧𝐴 ↦ (𝐹𝑧)))
311adantr 483 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → 𝑅 ∈ Mnd)
322adantr 483 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → 𝐵𝑊)
33 simprl 769 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → 𝑥𝐶)
343, 10, 11, 31, 32, 33pwselbas 16756 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → 𝑥:𝐵⟶(Base‘𝑅))
3534feqmptd 6727 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → 𝑥 = (𝑤𝐵 ↦ (𝑥𝑤)))
36 fveq2 6664 . . . . . . . 8 (𝑤 = (𝐹𝑧) → (𝑥𝑤) = (𝑥‘(𝐹𝑧)))
3729, 30, 35, 36fmptco 6885 . . . . . . 7 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → (𝑥𝐹) = (𝑧𝐴 ↦ (𝑥‘(𝐹𝑧))))
38 simprr 771 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → 𝑦𝐶)
393, 10, 11, 31, 32, 38pwselbas 16756 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → 𝑦:𝐵⟶(Base‘𝑅))
4039feqmptd 6727 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → 𝑦 = (𝑤𝐵 ↦ (𝑦𝑤)))
41 fveq2 6664 . . . . . . . 8 (𝑤 = (𝐹𝑧) → (𝑦𝑤) = (𝑦‘(𝐹𝑧)))
4229, 30, 40, 41fmptco 6885 . . . . . . 7 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → (𝑦𝐹) = (𝑧𝐴 ↦ (𝑦‘(𝐹𝑧))))
4325, 26, 27, 37, 42offval2 7420 . . . . . 6 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → ((𝑥𝐹) ∘f (+g𝑅)(𝑦𝐹)) = (𝑧𝐴 ↦ ((𝑥‘(𝐹𝑧))(+g𝑅)(𝑦‘(𝐹𝑧)))))
44 fco 6525 . . . . . . . . 9 ((𝑥:𝐵⟶(Base‘𝑅) ∧ 𝐹:𝐴𝐵) → (𝑥𝐹):𝐴⟶(Base‘𝑅))
4534, 28, 44syl2anc 586 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → (𝑥𝐹):𝐴⟶(Base‘𝑅))
467, 10, 19pwselbasb 16755 . . . . . . . . 9 ((𝑅 ∈ Mnd ∧ 𝐴𝑉) → ((𝑥𝐹) ∈ (Base‘𝑌) ↔ (𝑥𝐹):𝐴⟶(Base‘𝑅)))
4731, 25, 46syl2anc 586 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → ((𝑥𝐹) ∈ (Base‘𝑌) ↔ (𝑥𝐹):𝐴⟶(Base‘𝑅)))
4845, 47mpbird 259 . . . . . . 7 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → (𝑥𝐹) ∈ (Base‘𝑌))
49 fco 6525 . . . . . . . . 9 ((𝑦:𝐵⟶(Base‘𝑅) ∧ 𝐹:𝐴𝐵) → (𝑦𝐹):𝐴⟶(Base‘𝑅))
5039, 28, 49syl2anc 586 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → (𝑦𝐹):𝐴⟶(Base‘𝑅))
517, 10, 19pwselbasb 16755 . . . . . . . . 9 ((𝑅 ∈ Mnd ∧ 𝐴𝑉) → ((𝑦𝐹) ∈ (Base‘𝑌) ↔ (𝑦𝐹):𝐴⟶(Base‘𝑅)))
5231, 25, 51syl2anc 586 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → ((𝑦𝐹) ∈ (Base‘𝑌) ↔ (𝑦𝐹):𝐴⟶(Base‘𝑅)))
5350, 52mpbird 259 . . . . . . 7 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → (𝑦𝐹) ∈ (Base‘𝑌))
54 eqid 2821 . . . . . . 7 (+g𝑅) = (+g𝑅)
55 eqid 2821 . . . . . . 7 (+g𝑌) = (+g𝑌)
567, 19, 31, 25, 48, 53, 54, 55pwsplusgval 16757 . . . . . 6 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → ((𝑥𝐹)(+g𝑌)(𝑦𝐹)) = ((𝑥𝐹) ∘f (+g𝑅)(𝑦𝐹)))
57 eqid 2821 . . . . . . . . 9 (+g𝑍) = (+g𝑍)
583, 11, 31, 32, 33, 38, 54, 57pwsplusgval 16757 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → (𝑥(+g𝑍)𝑦) = (𝑥f (+g𝑅)𝑦))
59 fvexd 6679 . . . . . . . . 9 (((𝜑 ∧ (𝑥𝐶𝑦𝐶)) ∧ 𝑤𝐵) → (𝑥𝑤) ∈ V)
60 fvexd 6679 . . . . . . . . 9 (((𝜑 ∧ (𝑥𝐶𝑦𝐶)) ∧ 𝑤𝐵) → (𝑦𝑤) ∈ V)
6132, 59, 60, 35, 40offval2 7420 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → (𝑥f (+g𝑅)𝑦) = (𝑤𝐵 ↦ ((𝑥𝑤)(+g𝑅)(𝑦𝑤))))
6258, 61eqtrd 2856 . . . . . . 7 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → (𝑥(+g𝑍)𝑦) = (𝑤𝐵 ↦ ((𝑥𝑤)(+g𝑅)(𝑦𝑤))))
6336, 41oveq12d 7168 . . . . . . 7 (𝑤 = (𝐹𝑧) → ((𝑥𝑤)(+g𝑅)(𝑦𝑤)) = ((𝑥‘(𝐹𝑧))(+g𝑅)(𝑦‘(𝐹𝑧))))
6429, 30, 62, 63fmptco 6885 . . . . . 6 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → ((𝑥(+g𝑍)𝑦) ∘ 𝐹) = (𝑧𝐴 ↦ ((𝑥‘(𝐹𝑧))(+g𝑅)(𝑦‘(𝐹𝑧)))))
6543, 56, 643eqtr4rd 2867 . . . . 5 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → ((𝑥(+g𝑍)𝑦) ∘ 𝐹) = ((𝑥𝐹)(+g𝑌)(𝑦𝐹)))
66 eqid 2821 . . . . . 6 (𝑔𝐶 ↦ (𝑔𝐹)) = (𝑔𝐶 ↦ (𝑔𝐹))
67 coeq1 5722 . . . . . 6 (𝑔 = (𝑥(+g𝑍)𝑦) → (𝑔𝐹) = ((𝑥(+g𝑍)𝑦) ∘ 𝐹))
6811, 57mndcl 17913 . . . . . . . 8 ((𝑍 ∈ Mnd ∧ 𝑥𝐶𝑦𝐶) → (𝑥(+g𝑍)𝑦) ∈ 𝐶)
69683expb 1116 . . . . . . 7 ((𝑍 ∈ Mnd ∧ (𝑥𝐶𝑦𝐶)) → (𝑥(+g𝑍)𝑦) ∈ 𝐶)
705, 69sylan 582 . . . . . 6 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → (𝑥(+g𝑍)𝑦) ∈ 𝐶)
71 ovex 7183 . . . . . . 7 (𝑥(+g𝑍)𝑦) ∈ V
72 fex 6983 . . . . . . . . 9 ((𝐹:𝐴𝐵𝐴𝑉) → 𝐹 ∈ V)
7315, 6, 72syl2anc 586 . . . . . . . 8 (𝜑𝐹 ∈ V)
7473adantr 483 . . . . . . 7 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → 𝐹 ∈ V)
75 coexg 7628 . . . . . . 7 (((𝑥(+g𝑍)𝑦) ∈ V ∧ 𝐹 ∈ V) → ((𝑥(+g𝑍)𝑦) ∘ 𝐹) ∈ V)
7671, 74, 75sylancr 589 . . . . . 6 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → ((𝑥(+g𝑍)𝑦) ∘ 𝐹) ∈ V)
7766, 67, 70, 76fvmptd3 6785 . . . . 5 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → ((𝑔𝐶 ↦ (𝑔𝐹))‘(𝑥(+g𝑍)𝑦)) = ((𝑥(+g𝑍)𝑦) ∘ 𝐹))
78 coeq1 5722 . . . . . . 7 (𝑔 = 𝑥 → (𝑔𝐹) = (𝑥𝐹))
79 coexg 7628 . . . . . . . 8 ((𝑥𝐶𝐹 ∈ V) → (𝑥𝐹) ∈ V)
8033, 74, 79syl2anc 586 . . . . . . 7 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → (𝑥𝐹) ∈ V)
8166, 78, 33, 80fvmptd3 6785 . . . . . 6 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → ((𝑔𝐶 ↦ (𝑔𝐹))‘𝑥) = (𝑥𝐹))
82 coeq1 5722 . . . . . . 7 (𝑔 = 𝑦 → (𝑔𝐹) = (𝑦𝐹))
83 coexg 7628 . . . . . . . 8 ((𝑦𝐶𝐹 ∈ V) → (𝑦𝐹) ∈ V)
8438, 74, 83syl2anc 586 . . . . . . 7 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → (𝑦𝐹) ∈ V)
8566, 82, 38, 84fvmptd3 6785 . . . . . 6 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → ((𝑔𝐶 ↦ (𝑔𝐹))‘𝑦) = (𝑦𝐹))
8681, 85oveq12d 7168 . . . . 5 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → (((𝑔𝐶 ↦ (𝑔𝐹))‘𝑥)(+g𝑌)((𝑔𝐶 ↦ (𝑔𝐹))‘𝑦)) = ((𝑥𝐹)(+g𝑌)(𝑦𝐹)))
8765, 77, 863eqtr4d 2866 . . . 4 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → ((𝑔𝐶 ↦ (𝑔𝐹))‘(𝑥(+g𝑍)𝑦)) = (((𝑔𝐶 ↦ (𝑔𝐹))‘𝑥)(+g𝑌)((𝑔𝐶 ↦ (𝑔𝐹))‘𝑦)))
8887ralrimivva 3191 . . 3 (𝜑 → ∀𝑥𝐶𝑦𝐶 ((𝑔𝐶 ↦ (𝑔𝐹))‘(𝑥(+g𝑍)𝑦)) = (((𝑔𝐶 ↦ (𝑔𝐹))‘𝑥)(+g𝑌)((𝑔𝐶 ↦ (𝑔𝐹))‘𝑦)))
89 coeq1 5722 . . . . 5 (𝑔 = (0g𝑍) → (𝑔𝐹) = ((0g𝑍) ∘ 𝐹))
90 eqid 2821 . . . . . . 7 (0g𝑍) = (0g𝑍)
9111, 90mndidcl 17920 . . . . . 6 (𝑍 ∈ Mnd → (0g𝑍) ∈ 𝐶)
925, 91syl 17 . . . . 5 (𝜑 → (0g𝑍) ∈ 𝐶)
93 coexg 7628 . . . . . 6 (((0g𝑍) ∈ 𝐶𝐹 ∈ V) → ((0g𝑍) ∘ 𝐹) ∈ V)
9492, 73, 93syl2anc 586 . . . . 5 (𝜑 → ((0g𝑍) ∘ 𝐹) ∈ V)
9566, 89, 92, 94fvmptd3 6785 . . . 4 (𝜑 → ((𝑔𝐶 ↦ (𝑔𝐹))‘(0g𝑍)) = ((0g𝑍) ∘ 𝐹))
963, 10, 11, 1, 2, 92pwselbas 16756 . . . . . . 7 (𝜑 → (0g𝑍):𝐵⟶(Base‘𝑅))
97 fco 6525 . . . . . . 7 (((0g𝑍):𝐵⟶(Base‘𝑅) ∧ 𝐹:𝐴𝐵) → ((0g𝑍) ∘ 𝐹):𝐴⟶(Base‘𝑅))
9896, 15, 97syl2anc 586 . . . . . 6 (𝜑 → ((0g𝑍) ∘ 𝐹):𝐴⟶(Base‘𝑅))
9998ffnd 6509 . . . . 5 (𝜑 → ((0g𝑍) ∘ 𝐹) Fn 𝐴)
100 fvexd 6679 . . . . . 6 (𝜑 → (0g𝑅) ∈ V)
101 fnconstg 6561 . . . . . 6 ((0g𝑅) ∈ V → (𝐴 × {(0g𝑅)}) Fn 𝐴)
102100, 101syl 17 . . . . 5 (𝜑 → (𝐴 × {(0g𝑅)}) Fn 𝐴)
103 eqid 2821 . . . . . . . . . . 11 (0g𝑅) = (0g𝑅)
1043, 103pws0g 17941 . . . . . . . . . 10 ((𝑅 ∈ Mnd ∧ 𝐵𝑊) → (𝐵 × {(0g𝑅)}) = (0g𝑍))
1051, 2, 104syl2anc 586 . . . . . . . . 9 (𝜑 → (𝐵 × {(0g𝑅)}) = (0g𝑍))
106105fveq1d 6666 . . . . . . . 8 (𝜑 → ((𝐵 × {(0g𝑅)})‘(𝐹𝑥)) = ((0g𝑍)‘(𝐹𝑥)))
107106adantr 483 . . . . . . 7 ((𝜑𝑥𝐴) → ((𝐵 × {(0g𝑅)})‘(𝐹𝑥)) = ((0g𝑍)‘(𝐹𝑥)))
108 fvex 6677 . . . . . . . 8 (0g𝑅) ∈ V
10915ffvelrnda 6845 . . . . . . . 8 ((𝜑𝑥𝐴) → (𝐹𝑥) ∈ 𝐵)
110 fvconst2g 6958 . . . . . . . 8 (((0g𝑅) ∈ V ∧ (𝐹𝑥) ∈ 𝐵) → ((𝐵 × {(0g𝑅)})‘(𝐹𝑥)) = (0g𝑅))
111108, 109, 110sylancr 589 . . . . . . 7 ((𝜑𝑥𝐴) → ((𝐵 × {(0g𝑅)})‘(𝐹𝑥)) = (0g𝑅))
112107, 111eqtr3d 2858 . . . . . 6 ((𝜑𝑥𝐴) → ((0g𝑍)‘(𝐹𝑥)) = (0g𝑅))
113 fvco3 6754 . . . . . . 7 ((𝐹:𝐴𝐵𝑥𝐴) → (((0g𝑍) ∘ 𝐹)‘𝑥) = ((0g𝑍)‘(𝐹𝑥)))
11415, 113sylan 582 . . . . . 6 ((𝜑𝑥𝐴) → (((0g𝑍) ∘ 𝐹)‘𝑥) = ((0g𝑍)‘(𝐹𝑥)))
115 fvconst2g 6958 . . . . . . 7 (((0g𝑅) ∈ V ∧ 𝑥𝐴) → ((𝐴 × {(0g𝑅)})‘𝑥) = (0g𝑅))
116100, 115sylan 582 . . . . . 6 ((𝜑𝑥𝐴) → ((𝐴 × {(0g𝑅)})‘𝑥) = (0g𝑅))
117112, 114, 1163eqtr4d 2866 . . . . 5 ((𝜑𝑥𝐴) → (((0g𝑍) ∘ 𝐹)‘𝑥) = ((𝐴 × {(0g𝑅)})‘𝑥))
11899, 102, 117eqfnfvd 6799 . . . 4 (𝜑 → ((0g𝑍) ∘ 𝐹) = (𝐴 × {(0g𝑅)}))
1197, 103pws0g 17941 . . . . 5 ((𝑅 ∈ Mnd ∧ 𝐴𝑉) → (𝐴 × {(0g𝑅)}) = (0g𝑌))
1201, 6, 119syl2anc 586 . . . 4 (𝜑 → (𝐴 × {(0g𝑅)}) = (0g𝑌))
12195, 118, 1203eqtrd 2860 . . 3 (𝜑 → ((𝑔𝐶 ↦ (𝑔𝐹))‘(0g𝑍)) = (0g𝑌))
12224, 88, 1213jca 1124 . 2 (𝜑 → ((𝑔𝐶 ↦ (𝑔𝐹)):𝐶⟶(Base‘𝑌) ∧ ∀𝑥𝐶𝑦𝐶 ((𝑔𝐶 ↦ (𝑔𝐹))‘(𝑥(+g𝑍)𝑦)) = (((𝑔𝐶 ↦ (𝑔𝐹))‘𝑥)(+g𝑌)((𝑔𝐶 ↦ (𝑔𝐹))‘𝑦)) ∧ ((𝑔𝐶 ↦ (𝑔𝐹))‘(0g𝑍)) = (0g𝑌)))
123 eqid 2821 . . 3 (0g𝑌) = (0g𝑌)
12411, 19, 57, 55, 90, 123ismhm 17952 . 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))))
1255, 9, 122, 124syl21anbrc 1340 1 (𝜑 → (𝑔𝐶 ↦ (𝑔𝐹)) ∈ (𝑍 MndHom 𝑌))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1533   ∈ wcel 2110  ∀wral 3138  Vcvv 3494  {csn 4560   ↦ cmpt 5138   × cxp 5547   ∘ ccom 5553   Fn wfn 6344  ⟶wf 6345  ‘cfv 6349  (class class class)co 7150   ∘f cof 7401  Basecbs 16477  +gcplusg 16559  0gc0g 16707   ↑s cpws 16714  Mndcmnd 17905   MndHom cmhm 17948 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455  ax-cnex 10587  ax-resscn 10588  ax-1cn 10589  ax-icn 10590  ax-addcl 10591  ax-addrcl 10592  ax-mulcl 10593  ax-mulrcl 10594  ax-mulcom 10595  ax-addass 10596  ax-mulass 10597  ax-distr 10598  ax-i2m1 10599  ax-1ne0 10600  ax-1rid 10601  ax-rnegex 10602  ax-rrecex 10603  ax-cnre 10604  ax-pre-lttri 10605  ax-pre-lttrn 10606  ax-pre-ltadd 10607  ax-pre-mulgt0 10608 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-tp 4565  df-op 4567  df-uni 4832  df-int 4869  df-iun 4913  df-br 5059  df-opab 5121  df-mpt 5139  df-tr 5165  df-id 5454  df-eprel 5459  df-po 5468  df-so 5469  df-fr 5508  df-we 5510  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-pred 6142  df-ord 6188  df-on 6189  df-lim 6190  df-suc 6191  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-of 7403  df-om 7575  df-1st 7683  df-2nd 7684  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-1o 8096  df-oadd 8100  df-er 8283  df-map 8402  df-ixp 8456  df-en 8504  df-dom 8505  df-sdom 8506  df-fin 8507  df-sup 8900  df-pnf 10671  df-mnf 10672  df-xr 10673  df-ltxr 10674  df-le 10675  df-sub 10866  df-neg 10867  df-nn 11633  df-2 11694  df-3 11695  df-4 11696  df-5 11697  df-6 11698  df-7 11699  df-8 11700  df-9 11701  df-n0 11892  df-z 11976  df-dec 12093  df-uz 12238  df-fz 12887  df-struct 16479  df-ndx 16480  df-slot 16481  df-base 16483  df-plusg 16572  df-mulr 16573  df-sca 16575  df-vsca 16576  df-ip 16577  df-tset 16578  df-ple 16579  df-ds 16581  df-hom 16583  df-cco 16584  df-0g 16709  df-prds 16715  df-pws 16717  df-mgm 17846  df-sgrp 17895  df-mnd 17906  df-mhm 17950 This theorem is referenced by:  pwsco1rhm  19484
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