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Theorem pwsco1mhm 17542
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 . . . 4 (𝜑𝑅 ∈ Mnd)
2 pwsco1mhm.b . . . 4 (𝜑𝐵𝑊)
3 pwsco1mhm.z . . . . 5 𝑍 = (𝑅s 𝐵)
43pwsmnd 17497 . . . 4 ((𝑅 ∈ Mnd ∧ 𝐵𝑊) → 𝑍 ∈ Mnd)
51, 2, 4syl2anc 696 . . 3 (𝜑𝑍 ∈ Mnd)
6 pwsco1mhm.a . . . 4 (𝜑𝐴𝑉)
7 pwsco1mhm.y . . . . 5 𝑌 = (𝑅s 𝐴)
87pwsmnd 17497 . . . 4 ((𝑅 ∈ Mnd ∧ 𝐴𝑉) → 𝑌 ∈ Mnd)
91, 6, 8syl2anc 696 . . 3 (𝜑𝑌 ∈ Mnd)
105, 9jca 555 . 2 (𝜑 → (𝑍 ∈ Mnd ∧ 𝑌 ∈ Mnd))
11 eqid 2748 . . . . . . . . 9 (Base‘𝑅) = (Base‘𝑅)
12 pwsco1mhm.c . . . . . . . . 9 𝐶 = (Base‘𝑍)
133, 11, 12pwselbasb 16321 . . . . . . . 8 ((𝑅 ∈ Mnd ∧ 𝐵𝑊) → (𝑔𝐶𝑔:𝐵⟶(Base‘𝑅)))
141, 2, 13syl2anc 696 . . . . . . 7 (𝜑 → (𝑔𝐶𝑔:𝐵⟶(Base‘𝑅)))
1514biimpa 502 . . . . . 6 ((𝜑𝑔𝐶) → 𝑔:𝐵⟶(Base‘𝑅))
16 pwsco1mhm.f . . . . . . 7 (𝜑𝐹:𝐴𝐵)
1716adantr 472 . . . . . 6 ((𝜑𝑔𝐶) → 𝐹:𝐴𝐵)
18 fco 6207 . . . . . 6 ((𝑔:𝐵⟶(Base‘𝑅) ∧ 𝐹:𝐴𝐵) → (𝑔𝐹):𝐴⟶(Base‘𝑅))
1915, 17, 18syl2anc 696 . . . . 5 ((𝜑𝑔𝐶) → (𝑔𝐹):𝐴⟶(Base‘𝑅))
20 eqid 2748 . . . . . . . 8 (Base‘𝑌) = (Base‘𝑌)
217, 11, 20pwselbasb 16321 . . . . . . 7 ((𝑅 ∈ Mnd ∧ 𝐴𝑉) → ((𝑔𝐹) ∈ (Base‘𝑌) ↔ (𝑔𝐹):𝐴⟶(Base‘𝑅)))
221, 6, 21syl2anc 696 . . . . . 6 (𝜑 → ((𝑔𝐹) ∈ (Base‘𝑌) ↔ (𝑔𝐹):𝐴⟶(Base‘𝑅)))
2322adantr 472 . . . . 5 ((𝜑𝑔𝐶) → ((𝑔𝐹) ∈ (Base‘𝑌) ↔ (𝑔𝐹):𝐴⟶(Base‘𝑅)))
2419, 23mpbird 247 . . . 4 ((𝜑𝑔𝐶) → (𝑔𝐹) ∈ (Base‘𝑌))
25 eqid 2748 . . . 4 (𝑔𝐶 ↦ (𝑔𝐹)) = (𝑔𝐶 ↦ (𝑔𝐹))
2624, 25fmptd 6536 . . 3 (𝜑 → (𝑔𝐶 ↦ (𝑔𝐹)):𝐶⟶(Base‘𝑌))
276adantr 472 . . . . . . 7 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → 𝐴𝑉)
28 fvexd 6352 . . . . . . 7 (((𝜑 ∧ (𝑥𝐶𝑦𝐶)) ∧ 𝑧𝐴) → (𝑥‘(𝐹𝑧)) ∈ V)
29 fvexd 6352 . . . . . . 7 (((𝜑 ∧ (𝑥𝐶𝑦𝐶)) ∧ 𝑧𝐴) → (𝑦‘(𝐹𝑧)) ∈ V)
3016adantr 472 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → 𝐹:𝐴𝐵)
3130ffvelrnda 6510 . . . . . . . 8 (((𝜑 ∧ (𝑥𝐶𝑦𝐶)) ∧ 𝑧𝐴) → (𝐹𝑧) ∈ 𝐵)
3230feqmptd 6399 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → 𝐹 = (𝑧𝐴 ↦ (𝐹𝑧)))
331adantr 472 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → 𝑅 ∈ Mnd)
342adantr 472 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → 𝐵𝑊)
35 simprl 811 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → 𝑥𝐶)
363, 11, 12, 33, 34, 35pwselbas 16322 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → 𝑥:𝐵⟶(Base‘𝑅))
3736feqmptd 6399 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → 𝑥 = (𝑤𝐵 ↦ (𝑥𝑤)))
38 fveq2 6340 . . . . . . . 8 (𝑤 = (𝐹𝑧) → (𝑥𝑤) = (𝑥‘(𝐹𝑧)))
3931, 32, 37, 38fmptco 6547 . . . . . . 7 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → (𝑥𝐹) = (𝑧𝐴 ↦ (𝑥‘(𝐹𝑧))))
40 simprr 813 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → 𝑦𝐶)
413, 11, 12, 33, 34, 40pwselbas 16322 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → 𝑦:𝐵⟶(Base‘𝑅))
4241feqmptd 6399 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → 𝑦 = (𝑤𝐵 ↦ (𝑦𝑤)))
43 fveq2 6340 . . . . . . . 8 (𝑤 = (𝐹𝑧) → (𝑦𝑤) = (𝑦‘(𝐹𝑧)))
4431, 32, 42, 43fmptco 6547 . . . . . . 7 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → (𝑦𝐹) = (𝑧𝐴 ↦ (𝑦‘(𝐹𝑧))))
4527, 28, 29, 39, 44offval2 7067 . . . . . 6 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → ((𝑥𝐹) ∘𝑓 (+g𝑅)(𝑦𝐹)) = (𝑧𝐴 ↦ ((𝑥‘(𝐹𝑧))(+g𝑅)(𝑦‘(𝐹𝑧)))))
46 fco 6207 . . . . . . . . 9 ((𝑥:𝐵⟶(Base‘𝑅) ∧ 𝐹:𝐴𝐵) → (𝑥𝐹):𝐴⟶(Base‘𝑅))
4736, 30, 46syl2anc 696 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → (𝑥𝐹):𝐴⟶(Base‘𝑅))
487, 11, 20pwselbasb 16321 . . . . . . . . 9 ((𝑅 ∈ Mnd ∧ 𝐴𝑉) → ((𝑥𝐹) ∈ (Base‘𝑌) ↔ (𝑥𝐹):𝐴⟶(Base‘𝑅)))
4933, 27, 48syl2anc 696 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → ((𝑥𝐹) ∈ (Base‘𝑌) ↔ (𝑥𝐹):𝐴⟶(Base‘𝑅)))
5047, 49mpbird 247 . . . . . . 7 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → (𝑥𝐹) ∈ (Base‘𝑌))
51 fco 6207 . . . . . . . . 9 ((𝑦:𝐵⟶(Base‘𝑅) ∧ 𝐹:𝐴𝐵) → (𝑦𝐹):𝐴⟶(Base‘𝑅))
5241, 30, 51syl2anc 696 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → (𝑦𝐹):𝐴⟶(Base‘𝑅))
537, 11, 20pwselbasb 16321 . . . . . . . . 9 ((𝑅 ∈ Mnd ∧ 𝐴𝑉) → ((𝑦𝐹) ∈ (Base‘𝑌) ↔ (𝑦𝐹):𝐴⟶(Base‘𝑅)))
5433, 27, 53syl2anc 696 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → ((𝑦𝐹) ∈ (Base‘𝑌) ↔ (𝑦𝐹):𝐴⟶(Base‘𝑅)))
5552, 54mpbird 247 . . . . . . 7 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → (𝑦𝐹) ∈ (Base‘𝑌))
56 eqid 2748 . . . . . . 7 (+g𝑅) = (+g𝑅)
57 eqid 2748 . . . . . . 7 (+g𝑌) = (+g𝑌)
587, 20, 33, 27, 50, 55, 56, 57pwsplusgval 16323 . . . . . 6 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → ((𝑥𝐹)(+g𝑌)(𝑦𝐹)) = ((𝑥𝐹) ∘𝑓 (+g𝑅)(𝑦𝐹)))
59 eqid 2748 . . . . . . . . 9 (+g𝑍) = (+g𝑍)
603, 12, 33, 34, 35, 40, 56, 59pwsplusgval 16323 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → (𝑥(+g𝑍)𝑦) = (𝑥𝑓 (+g𝑅)𝑦))
61 fvexd 6352 . . . . . . . . 9 (((𝜑 ∧ (𝑥𝐶𝑦𝐶)) ∧ 𝑤𝐵) → (𝑥𝑤) ∈ V)
62 fvexd 6352 . . . . . . . . 9 (((𝜑 ∧ (𝑥𝐶𝑦𝐶)) ∧ 𝑤𝐵) → (𝑦𝑤) ∈ V)
6334, 61, 62, 37, 42offval2 7067 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → (𝑥𝑓 (+g𝑅)𝑦) = (𝑤𝐵 ↦ ((𝑥𝑤)(+g𝑅)(𝑦𝑤))))
6460, 63eqtrd 2782 . . . . . . 7 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → (𝑥(+g𝑍)𝑦) = (𝑤𝐵 ↦ ((𝑥𝑤)(+g𝑅)(𝑦𝑤))))
6538, 43oveq12d 6819 . . . . . . 7 (𝑤 = (𝐹𝑧) → ((𝑥𝑤)(+g𝑅)(𝑦𝑤)) = ((𝑥‘(𝐹𝑧))(+g𝑅)(𝑦‘(𝐹𝑧))))
6631, 32, 64, 65fmptco 6547 . . . . . 6 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → ((𝑥(+g𝑍)𝑦) ∘ 𝐹) = (𝑧𝐴 ↦ ((𝑥‘(𝐹𝑧))(+g𝑅)(𝑦‘(𝐹𝑧)))))
6745, 58, 663eqtr4rd 2793 . . . . 5 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → ((𝑥(+g𝑍)𝑦) ∘ 𝐹) = ((𝑥𝐹)(+g𝑌)(𝑦𝐹)))
6812, 59mndcl 17473 . . . . . . . 8 ((𝑍 ∈ Mnd ∧ 𝑥𝐶𝑦𝐶) → (𝑥(+g𝑍)𝑦) ∈ 𝐶)
69683expb 1113 . . . . . . 7 ((𝑍 ∈ Mnd ∧ (𝑥𝐶𝑦𝐶)) → (𝑥(+g𝑍)𝑦) ∈ 𝐶)
705, 69sylan 489 . . . . . 6 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → (𝑥(+g𝑍)𝑦) ∈ 𝐶)
71 ovex 6829 . . . . . . 7 (𝑥(+g𝑍)𝑦) ∈ V
72 fex 6641 . . . . . . . . 9 ((𝐹:𝐴𝐵𝐴𝑉) → 𝐹 ∈ V)
7316, 6, 72syl2anc 696 . . . . . . . 8 (𝜑𝐹 ∈ V)
7473adantr 472 . . . . . . 7 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → 𝐹 ∈ V)
75 coexg 7270 . . . . . . 7 (((𝑥(+g𝑍)𝑦) ∈ V ∧ 𝐹 ∈ V) → ((𝑥(+g𝑍)𝑦) ∘ 𝐹) ∈ V)
7671, 74, 75sylancr 698 . . . . . 6 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → ((𝑥(+g𝑍)𝑦) ∘ 𝐹) ∈ V)
77 coeq1 5423 . . . . . . 7 (𝑔 = (𝑥(+g𝑍)𝑦) → (𝑔𝐹) = ((𝑥(+g𝑍)𝑦) ∘ 𝐹))
7877, 25fvmptg 6430 . . . . . 6 (((𝑥(+g𝑍)𝑦) ∈ 𝐶 ∧ ((𝑥(+g𝑍)𝑦) ∘ 𝐹) ∈ V) → ((𝑔𝐶 ↦ (𝑔𝐹))‘(𝑥(+g𝑍)𝑦)) = ((𝑥(+g𝑍)𝑦) ∘ 𝐹))
7970, 76, 78syl2anc 696 . . . . 5 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → ((𝑔𝐶 ↦ (𝑔𝐹))‘(𝑥(+g𝑍)𝑦)) = ((𝑥(+g𝑍)𝑦) ∘ 𝐹))
80 coexg 7270 . . . . . . . 8 ((𝑥𝐶𝐹 ∈ V) → (𝑥𝐹) ∈ V)
8135, 74, 80syl2anc 696 . . . . . . 7 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → (𝑥𝐹) ∈ V)
82 coeq1 5423 . . . . . . . 8 (𝑔 = 𝑥 → (𝑔𝐹) = (𝑥𝐹))
8382, 25fvmptg 6430 . . . . . . 7 ((𝑥𝐶 ∧ (𝑥𝐹) ∈ V) → ((𝑔𝐶 ↦ (𝑔𝐹))‘𝑥) = (𝑥𝐹))
8435, 81, 83syl2anc 696 . . . . . 6 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → ((𝑔𝐶 ↦ (𝑔𝐹))‘𝑥) = (𝑥𝐹))
85 coexg 7270 . . . . . . . 8 ((𝑦𝐶𝐹 ∈ V) → (𝑦𝐹) ∈ V)
8640, 74, 85syl2anc 696 . . . . . . 7 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → (𝑦𝐹) ∈ V)
87 coeq1 5423 . . . . . . . 8 (𝑔 = 𝑦 → (𝑔𝐹) = (𝑦𝐹))
8887, 25fvmptg 6430 . . . . . . 7 ((𝑦𝐶 ∧ (𝑦𝐹) ∈ V) → ((𝑔𝐶 ↦ (𝑔𝐹))‘𝑦) = (𝑦𝐹))
8940, 86, 88syl2anc 696 . . . . . 6 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → ((𝑔𝐶 ↦ (𝑔𝐹))‘𝑦) = (𝑦𝐹))
9084, 89oveq12d 6819 . . . . 5 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → (((𝑔𝐶 ↦ (𝑔𝐹))‘𝑥)(+g𝑌)((𝑔𝐶 ↦ (𝑔𝐹))‘𝑦)) = ((𝑥𝐹)(+g𝑌)(𝑦𝐹)))
9167, 79, 903eqtr4d 2792 . . . 4 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → ((𝑔𝐶 ↦ (𝑔𝐹))‘(𝑥(+g𝑍)𝑦)) = (((𝑔𝐶 ↦ (𝑔𝐹))‘𝑥)(+g𝑌)((𝑔𝐶 ↦ (𝑔𝐹))‘𝑦)))
9291ralrimivva 3097 . . 3 (𝜑 → ∀𝑥𝐶𝑦𝐶 ((𝑔𝐶 ↦ (𝑔𝐹))‘(𝑥(+g𝑍)𝑦)) = (((𝑔𝐶 ↦ (𝑔𝐹))‘𝑥)(+g𝑌)((𝑔𝐶 ↦ (𝑔𝐹))‘𝑦)))
93 eqid 2748 . . . . . . 7 (0g𝑍) = (0g𝑍)
9412, 93mndidcl 17480 . . . . . 6 (𝑍 ∈ Mnd → (0g𝑍) ∈ 𝐶)
955, 94syl 17 . . . . 5 (𝜑 → (0g𝑍) ∈ 𝐶)
96 coexg 7270 . . . . . 6 (((0g𝑍) ∈ 𝐶𝐹 ∈ V) → ((0g𝑍) ∘ 𝐹) ∈ V)
9795, 73, 96syl2anc 696 . . . . 5 (𝜑 → ((0g𝑍) ∘ 𝐹) ∈ V)
98 coeq1 5423 . . . . . 6 (𝑔 = (0g𝑍) → (𝑔𝐹) = ((0g𝑍) ∘ 𝐹))
9998, 25fvmptg 6430 . . . . 5 (((0g𝑍) ∈ 𝐶 ∧ ((0g𝑍) ∘ 𝐹) ∈ V) → ((𝑔𝐶 ↦ (𝑔𝐹))‘(0g𝑍)) = ((0g𝑍) ∘ 𝐹))
10095, 97, 99syl2anc 696 . . . 4 (𝜑 → ((𝑔𝐶 ↦ (𝑔𝐹))‘(0g𝑍)) = ((0g𝑍) ∘ 𝐹))
1013, 11, 12, 1, 2, 95pwselbas 16322 . . . . . . 7 (𝜑 → (0g𝑍):𝐵⟶(Base‘𝑅))
102 fco 6207 . . . . . . 7 (((0g𝑍):𝐵⟶(Base‘𝑅) ∧ 𝐹:𝐴𝐵) → ((0g𝑍) ∘ 𝐹):𝐴⟶(Base‘𝑅))
103101, 16, 102syl2anc 696 . . . . . 6 (𝜑 → ((0g𝑍) ∘ 𝐹):𝐴⟶(Base‘𝑅))
104 ffn 6194 . . . . . 6 (((0g𝑍) ∘ 𝐹):𝐴⟶(Base‘𝑅) → ((0g𝑍) ∘ 𝐹) Fn 𝐴)
105103, 104syl 17 . . . . 5 (𝜑 → ((0g𝑍) ∘ 𝐹) Fn 𝐴)
106 fvexd 6352 . . . . . 6 (𝜑 → (0g𝑅) ∈ V)
107 fnconstg 6242 . . . . . 6 ((0g𝑅) ∈ V → (𝐴 × {(0g𝑅)}) Fn 𝐴)
108106, 107syl 17 . . . . 5 (𝜑 → (𝐴 × {(0g𝑅)}) Fn 𝐴)
109 eqid 2748 . . . . . . . . . . 11 (0g𝑅) = (0g𝑅)
1103, 109pws0g 17498 . . . . . . . . . 10 ((𝑅 ∈ Mnd ∧ 𝐵𝑊) → (𝐵 × {(0g𝑅)}) = (0g𝑍))
1111, 2, 110syl2anc 696 . . . . . . . . 9 (𝜑 → (𝐵 × {(0g𝑅)}) = (0g𝑍))
112111fveq1d 6342 . . . . . . . 8 (𝜑 → ((𝐵 × {(0g𝑅)})‘(𝐹𝑥)) = ((0g𝑍)‘(𝐹𝑥)))
113112adantr 472 . . . . . . 7 ((𝜑𝑥𝐴) → ((𝐵 × {(0g𝑅)})‘(𝐹𝑥)) = ((0g𝑍)‘(𝐹𝑥)))
114 fvex 6350 . . . . . . . 8 (0g𝑅) ∈ V
11516ffvelrnda 6510 . . . . . . . 8 ((𝜑𝑥𝐴) → (𝐹𝑥) ∈ 𝐵)
116 fvconst2g 6619 . . . . . . . 8 (((0g𝑅) ∈ V ∧ (𝐹𝑥) ∈ 𝐵) → ((𝐵 × {(0g𝑅)})‘(𝐹𝑥)) = (0g𝑅))
117114, 115, 116sylancr 698 . . . . . . 7 ((𝜑𝑥𝐴) → ((𝐵 × {(0g𝑅)})‘(𝐹𝑥)) = (0g𝑅))
118113, 117eqtr3d 2784 . . . . . 6 ((𝜑𝑥𝐴) → ((0g𝑍)‘(𝐹𝑥)) = (0g𝑅))
119 fvco3 6425 . . . . . . 7 ((𝐹:𝐴𝐵𝑥𝐴) → (((0g𝑍) ∘ 𝐹)‘𝑥) = ((0g𝑍)‘(𝐹𝑥)))
12016, 119sylan 489 . . . . . 6 ((𝜑𝑥𝐴) → (((0g𝑍) ∘ 𝐹)‘𝑥) = ((0g𝑍)‘(𝐹𝑥)))
121 fvconst2g 6619 . . . . . . 7 (((0g𝑅) ∈ V ∧ 𝑥𝐴) → ((𝐴 × {(0g𝑅)})‘𝑥) = (0g𝑅))
122106, 121sylan 489 . . . . . 6 ((𝜑𝑥𝐴) → ((𝐴 × {(0g𝑅)})‘𝑥) = (0g𝑅))
123118, 120, 1223eqtr4d 2792 . . . . 5 ((𝜑𝑥𝐴) → (((0g𝑍) ∘ 𝐹)‘𝑥) = ((𝐴 × {(0g𝑅)})‘𝑥))
124105, 108, 123eqfnfvd 6465 . . . 4 (𝜑 → ((0g𝑍) ∘ 𝐹) = (𝐴 × {(0g𝑅)}))
1257, 109pws0g 17498 . . . . 5 ((𝑅 ∈ Mnd ∧ 𝐴𝑉) → (𝐴 × {(0g𝑅)}) = (0g𝑌))
1261, 6, 125syl2anc 696 . . . 4 (𝜑 → (𝐴 × {(0g𝑅)}) = (0g𝑌))
127100, 124, 1263eqtrd 2786 . . 3 (𝜑 → ((𝑔𝐶 ↦ (𝑔𝐹))‘(0g𝑍)) = (0g𝑌))
12826, 92, 1273jca 1403 . 2 (𝜑 → ((𝑔𝐶 ↦ (𝑔𝐹)):𝐶⟶(Base‘𝑌) ∧ ∀𝑥𝐶𝑦𝐶 ((𝑔𝐶 ↦ (𝑔𝐹))‘(𝑥(+g𝑍)𝑦)) = (((𝑔𝐶 ↦ (𝑔𝐹))‘𝑥)(+g𝑌)((𝑔𝐶 ↦ (𝑔𝐹))‘𝑦)) ∧ ((𝑔𝐶 ↦ (𝑔𝐹))‘(0g𝑍)) = (0g𝑌)))
129 eqid 2748 . . 3 (0g𝑌) = (0g𝑌)
13012, 20, 59, 57, 93, 129ismhm 17509 . 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))))
13110, 128, 130sylanbrc 701 1 (𝜑 → (𝑔𝐶 ↦ (𝑔𝐹)) ∈ (𝑍 MndHom 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072   = wceq 1620  wcel 2127  wral 3038  Vcvv 3328  {csn 4309  cmpt 4869   × cxp 5252  ccom 5258   Fn wfn 6032  wf 6033  cfv 6037  (class class class)co 6801  𝑓 cof 7048  Basecbs 16030  +gcplusg 16114  0gc0g 16273  s cpws 16280  Mndcmnd 17466   MndHom cmhm 17505
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-8 2129  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-rep 4911  ax-sep 4921  ax-nul 4929  ax-pow 4980  ax-pr 5043  ax-un 7102  ax-cnex 10155  ax-resscn 10156  ax-1cn 10157  ax-icn 10158  ax-addcl 10159  ax-addrcl 10160  ax-mulcl 10161  ax-mulrcl 10162  ax-mulcom 10163  ax-addass 10164  ax-mulass 10165  ax-distr 10166  ax-i2m1 10167  ax-1ne0 10168  ax-1rid 10169  ax-rnegex 10170  ax-rrecex 10171  ax-cnre 10172  ax-pre-lttri 10173  ax-pre-lttrn 10174  ax-pre-ltadd 10175  ax-pre-mulgt0 10176
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ne 2921  df-nel 3024  df-ral 3043  df-rex 3044  df-reu 3045  df-rmo 3046  df-rab 3047  df-v 3330  df-sbc 3565  df-csb 3663  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-pss 3719  df-nul 4047  df-if 4219  df-pw 4292  df-sn 4310  df-pr 4312  df-tp 4314  df-op 4316  df-uni 4577  df-int 4616  df-iun 4662  df-br 4793  df-opab 4853  df-mpt 4870  df-tr 4893  df-id 5162  df-eprel 5167  df-po 5175  df-so 5176  df-fr 5213  df-we 5215  df-xp 5260  df-rel 5261  df-cnv 5262  df-co 5263  df-dm 5264  df-rn 5265  df-res 5266  df-ima 5267  df-pred 5829  df-ord 5875  df-on 5876  df-lim 5877  df-suc 5878  df-iota 6000  df-fun 6039  df-fn 6040  df-f 6041  df-f1 6042  df-fo 6043  df-f1o 6044  df-fv 6045  df-riota 6762  df-ov 6804  df-oprab 6805  df-mpt2 6806  df-of 7050  df-om 7219  df-1st 7321  df-2nd 7322  df-wrecs 7564  df-recs 7625  df-rdg 7663  df-1o 7717  df-oadd 7721  df-er 7899  df-map 8013  df-ixp 8063  df-en 8110  df-dom 8111  df-sdom 8112  df-fin 8113  df-sup 8501  df-pnf 10239  df-mnf 10240  df-xr 10241  df-ltxr 10242  df-le 10243  df-sub 10431  df-neg 10432  df-nn 11184  df-2 11242  df-3 11243  df-4 11244  df-5 11245  df-6 11246  df-7 11247  df-8 11248  df-9 11249  df-n0 11456  df-z 11541  df-dec 11657  df-uz 11851  df-fz 12491  df-struct 16032  df-ndx 16033  df-slot 16034  df-base 16036  df-plusg 16127  df-mulr 16128  df-sca 16130  df-vsca 16131  df-ip 16132  df-tset 16133  df-ple 16134  df-ds 16137  df-hom 16139  df-cco 16140  df-0g 16275  df-prds 16281  df-pws 16283  df-mgm 17414  df-sgrp 17456  df-mnd 17467  df-mhm 17507
This theorem is referenced by:  pwsco1rhm  18911
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