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Theorem pwsco1mhm 17193
 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 17148 . . . 4 ((𝑅 ∈ Mnd ∧ 𝐵𝑊) → 𝑍 ∈ Mnd)
51, 2, 4syl2anc 691 . . 3 (𝜑𝑍 ∈ Mnd)
6 pwsco1mhm.a . . . 4 (𝜑𝐴𝑉)
7 pwsco1mhm.y . . . . 5 𝑌 = (𝑅s 𝐴)
87pwsmnd 17148 . . . 4 ((𝑅 ∈ Mnd ∧ 𝐴𝑉) → 𝑌 ∈ Mnd)
91, 6, 8syl2anc 691 . . 3 (𝜑𝑌 ∈ Mnd)
105, 9jca 553 . 2 (𝜑 → (𝑍 ∈ Mnd ∧ 𝑌 ∈ Mnd))
11 eqid 2610 . . . . . . . . 9 (Base‘𝑅) = (Base‘𝑅)
12 pwsco1mhm.c . . . . . . . . 9 𝐶 = (Base‘𝑍)
133, 11, 12pwselbasb 15971 . . . . . . . 8 ((𝑅 ∈ Mnd ∧ 𝐵𝑊) → (𝑔𝐶𝑔:𝐵⟶(Base‘𝑅)))
141, 2, 13syl2anc 691 . . . . . . 7 (𝜑 → (𝑔𝐶𝑔:𝐵⟶(Base‘𝑅)))
1514biimpa 500 . . . . . 6 ((𝜑𝑔𝐶) → 𝑔:𝐵⟶(Base‘𝑅))
16 pwsco1mhm.f . . . . . . 7 (𝜑𝐹:𝐴𝐵)
1716adantr 480 . . . . . 6 ((𝜑𝑔𝐶) → 𝐹:𝐴𝐵)
18 fco 5971 . . . . . 6 ((𝑔:𝐵⟶(Base‘𝑅) ∧ 𝐹:𝐴𝐵) → (𝑔𝐹):𝐴⟶(Base‘𝑅))
1915, 17, 18syl2anc 691 . . . . 5 ((𝜑𝑔𝐶) → (𝑔𝐹):𝐴⟶(Base‘𝑅))
20 eqid 2610 . . . . . . . 8 (Base‘𝑌) = (Base‘𝑌)
217, 11, 20pwselbasb 15971 . . . . . . 7 ((𝑅 ∈ Mnd ∧ 𝐴𝑉) → ((𝑔𝐹) ∈ (Base‘𝑌) ↔ (𝑔𝐹):𝐴⟶(Base‘𝑅)))
221, 6, 21syl2anc 691 . . . . . 6 (𝜑 → ((𝑔𝐹) ∈ (Base‘𝑌) ↔ (𝑔𝐹):𝐴⟶(Base‘𝑅)))
2322adantr 480 . . . . 5 ((𝜑𝑔𝐶) → ((𝑔𝐹) ∈ (Base‘𝑌) ↔ (𝑔𝐹):𝐴⟶(Base‘𝑅)))
2419, 23mpbird 246 . . . 4 ((𝜑𝑔𝐶) → (𝑔𝐹) ∈ (Base‘𝑌))
25 eqid 2610 . . . 4 (𝑔𝐶 ↦ (𝑔𝐹)) = (𝑔𝐶 ↦ (𝑔𝐹))
2624, 25fmptd 6292 . . 3 (𝜑 → (𝑔𝐶 ↦ (𝑔𝐹)):𝐶⟶(Base‘𝑌))
276adantr 480 . . . . . . 7 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → 𝐴𝑉)
28 fvex 6113 . . . . . . . 8 (𝑥‘(𝐹𝑧)) ∈ V
2928a1i 11 . . . . . . 7 (((𝜑 ∧ (𝑥𝐶𝑦𝐶)) ∧ 𝑧𝐴) → (𝑥‘(𝐹𝑧)) ∈ V)
30 fvex 6113 . . . . . . . 8 (𝑦‘(𝐹𝑧)) ∈ V
3130a1i 11 . . . . . . 7 (((𝜑 ∧ (𝑥𝐶𝑦𝐶)) ∧ 𝑧𝐴) → (𝑦‘(𝐹𝑧)) ∈ V)
3216adantr 480 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → 𝐹:𝐴𝐵)
3332ffvelrnda 6267 . . . . . . . 8 (((𝜑 ∧ (𝑥𝐶𝑦𝐶)) ∧ 𝑧𝐴) → (𝐹𝑧) ∈ 𝐵)
3432feqmptd 6159 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → 𝐹 = (𝑧𝐴 ↦ (𝐹𝑧)))
351adantr 480 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → 𝑅 ∈ Mnd)
362adantr 480 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → 𝐵𝑊)
37 simprl 790 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → 𝑥𝐶)
383, 11, 12, 35, 36, 37pwselbas 15972 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → 𝑥:𝐵⟶(Base‘𝑅))
3938feqmptd 6159 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → 𝑥 = (𝑤𝐵 ↦ (𝑥𝑤)))
40 fveq2 6103 . . . . . . . 8 (𝑤 = (𝐹𝑧) → (𝑥𝑤) = (𝑥‘(𝐹𝑧)))
4133, 34, 39, 40fmptco 6303 . . . . . . 7 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → (𝑥𝐹) = (𝑧𝐴 ↦ (𝑥‘(𝐹𝑧))))
42 simprr 792 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → 𝑦𝐶)
433, 11, 12, 35, 36, 42pwselbas 15972 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → 𝑦:𝐵⟶(Base‘𝑅))
4443feqmptd 6159 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → 𝑦 = (𝑤𝐵 ↦ (𝑦𝑤)))
45 fveq2 6103 . . . . . . . 8 (𝑤 = (𝐹𝑧) → (𝑦𝑤) = (𝑦‘(𝐹𝑧)))
4633, 34, 44, 45fmptco 6303 . . . . . . 7 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → (𝑦𝐹) = (𝑧𝐴 ↦ (𝑦‘(𝐹𝑧))))
4727, 29, 31, 41, 46offval2 6812 . . . . . 6 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → ((𝑥𝐹) ∘𝑓 (+g𝑅)(𝑦𝐹)) = (𝑧𝐴 ↦ ((𝑥‘(𝐹𝑧))(+g𝑅)(𝑦‘(𝐹𝑧)))))
48 fco 5971 . . . . . . . . 9 ((𝑥:𝐵⟶(Base‘𝑅) ∧ 𝐹:𝐴𝐵) → (𝑥𝐹):𝐴⟶(Base‘𝑅))
4938, 32, 48syl2anc 691 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → (𝑥𝐹):𝐴⟶(Base‘𝑅))
507, 11, 20pwselbasb 15971 . . . . . . . . 9 ((𝑅 ∈ Mnd ∧ 𝐴𝑉) → ((𝑥𝐹) ∈ (Base‘𝑌) ↔ (𝑥𝐹):𝐴⟶(Base‘𝑅)))
5135, 27, 50syl2anc 691 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → ((𝑥𝐹) ∈ (Base‘𝑌) ↔ (𝑥𝐹):𝐴⟶(Base‘𝑅)))
5249, 51mpbird 246 . . . . . . 7 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → (𝑥𝐹) ∈ (Base‘𝑌))
53 fco 5971 . . . . . . . . 9 ((𝑦:𝐵⟶(Base‘𝑅) ∧ 𝐹:𝐴𝐵) → (𝑦𝐹):𝐴⟶(Base‘𝑅))
5443, 32, 53syl2anc 691 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → (𝑦𝐹):𝐴⟶(Base‘𝑅))
557, 11, 20pwselbasb 15971 . . . . . . . . 9 ((𝑅 ∈ Mnd ∧ 𝐴𝑉) → ((𝑦𝐹) ∈ (Base‘𝑌) ↔ (𝑦𝐹):𝐴⟶(Base‘𝑅)))
5635, 27, 55syl2anc 691 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → ((𝑦𝐹) ∈ (Base‘𝑌) ↔ (𝑦𝐹):𝐴⟶(Base‘𝑅)))
5754, 56mpbird 246 . . . . . . 7 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → (𝑦𝐹) ∈ (Base‘𝑌))
58 eqid 2610 . . . . . . 7 (+g𝑅) = (+g𝑅)
59 eqid 2610 . . . . . . 7 (+g𝑌) = (+g𝑌)
607, 20, 35, 27, 52, 57, 58, 59pwsplusgval 15973 . . . . . 6 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → ((𝑥𝐹)(+g𝑌)(𝑦𝐹)) = ((𝑥𝐹) ∘𝑓 (+g𝑅)(𝑦𝐹)))
61 eqid 2610 . . . . . . . . 9 (+g𝑍) = (+g𝑍)
623, 12, 35, 36, 37, 42, 58, 61pwsplusgval 15973 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → (𝑥(+g𝑍)𝑦) = (𝑥𝑓 (+g𝑅)𝑦))
63 fvex 6113 . . . . . . . . . 10 (𝑥𝑤) ∈ V
6463a1i 11 . . . . . . . . 9 (((𝜑 ∧ (𝑥𝐶𝑦𝐶)) ∧ 𝑤𝐵) → (𝑥𝑤) ∈ V)
65 fvex 6113 . . . . . . . . . 10 (𝑦𝑤) ∈ V
6665a1i 11 . . . . . . . . 9 (((𝜑 ∧ (𝑥𝐶𝑦𝐶)) ∧ 𝑤𝐵) → (𝑦𝑤) ∈ V)
6736, 64, 66, 39, 44offval2 6812 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → (𝑥𝑓 (+g𝑅)𝑦) = (𝑤𝐵 ↦ ((𝑥𝑤)(+g𝑅)(𝑦𝑤))))
6862, 67eqtrd 2644 . . . . . . 7 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → (𝑥(+g𝑍)𝑦) = (𝑤𝐵 ↦ ((𝑥𝑤)(+g𝑅)(𝑦𝑤))))
6940, 45oveq12d 6567 . . . . . . 7 (𝑤 = (𝐹𝑧) → ((𝑥𝑤)(+g𝑅)(𝑦𝑤)) = ((𝑥‘(𝐹𝑧))(+g𝑅)(𝑦‘(𝐹𝑧))))
7033, 34, 68, 69fmptco 6303 . . . . . 6 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → ((𝑥(+g𝑍)𝑦) ∘ 𝐹) = (𝑧𝐴 ↦ ((𝑥‘(𝐹𝑧))(+g𝑅)(𝑦‘(𝐹𝑧)))))
7147, 60, 703eqtr4rd 2655 . . . . 5 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → ((𝑥(+g𝑍)𝑦) ∘ 𝐹) = ((𝑥𝐹)(+g𝑌)(𝑦𝐹)))
7212, 61mndcl 17124 . . . . . . . 8 ((𝑍 ∈ Mnd ∧ 𝑥𝐶𝑦𝐶) → (𝑥(+g𝑍)𝑦) ∈ 𝐶)
73723expb 1258 . . . . . . 7 ((𝑍 ∈ Mnd ∧ (𝑥𝐶𝑦𝐶)) → (𝑥(+g𝑍)𝑦) ∈ 𝐶)
745, 73sylan 487 . . . . . 6 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → (𝑥(+g𝑍)𝑦) ∈ 𝐶)
75 ovex 6577 . . . . . . 7 (𝑥(+g𝑍)𝑦) ∈ V
76 fex 6394 . . . . . . . . 9 ((𝐹:𝐴𝐵𝐴𝑉) → 𝐹 ∈ V)
7716, 6, 76syl2anc 691 . . . . . . . 8 (𝜑𝐹 ∈ V)
7877adantr 480 . . . . . . 7 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → 𝐹 ∈ V)
79 coexg 7010 . . . . . . 7 (((𝑥(+g𝑍)𝑦) ∈ V ∧ 𝐹 ∈ V) → ((𝑥(+g𝑍)𝑦) ∘ 𝐹) ∈ V)
8075, 78, 79sylancr 694 . . . . . 6 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → ((𝑥(+g𝑍)𝑦) ∘ 𝐹) ∈ V)
81 coeq1 5201 . . . . . . 7 (𝑔 = (𝑥(+g𝑍)𝑦) → (𝑔𝐹) = ((𝑥(+g𝑍)𝑦) ∘ 𝐹))
8281, 25fvmptg 6189 . . . . . 6 (((𝑥(+g𝑍)𝑦) ∈ 𝐶 ∧ ((𝑥(+g𝑍)𝑦) ∘ 𝐹) ∈ V) → ((𝑔𝐶 ↦ (𝑔𝐹))‘(𝑥(+g𝑍)𝑦)) = ((𝑥(+g𝑍)𝑦) ∘ 𝐹))
8374, 80, 82syl2anc 691 . . . . 5 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → ((𝑔𝐶 ↦ (𝑔𝐹))‘(𝑥(+g𝑍)𝑦)) = ((𝑥(+g𝑍)𝑦) ∘ 𝐹))
84 coexg 7010 . . . . . . . 8 ((𝑥𝐶𝐹 ∈ V) → (𝑥𝐹) ∈ V)
8537, 78, 84syl2anc 691 . . . . . . 7 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → (𝑥𝐹) ∈ V)
86 coeq1 5201 . . . . . . . 8 (𝑔 = 𝑥 → (𝑔𝐹) = (𝑥𝐹))
8786, 25fvmptg 6189 . . . . . . 7 ((𝑥𝐶 ∧ (𝑥𝐹) ∈ V) → ((𝑔𝐶 ↦ (𝑔𝐹))‘𝑥) = (𝑥𝐹))
8837, 85, 87syl2anc 691 . . . . . 6 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → ((𝑔𝐶 ↦ (𝑔𝐹))‘𝑥) = (𝑥𝐹))
89 coexg 7010 . . . . . . . 8 ((𝑦𝐶𝐹 ∈ V) → (𝑦𝐹) ∈ V)
9042, 78, 89syl2anc 691 . . . . . . 7 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → (𝑦𝐹) ∈ V)
91 coeq1 5201 . . . . . . . 8 (𝑔 = 𝑦 → (𝑔𝐹) = (𝑦𝐹))
9291, 25fvmptg 6189 . . . . . . 7 ((𝑦𝐶 ∧ (𝑦𝐹) ∈ V) → ((𝑔𝐶 ↦ (𝑔𝐹))‘𝑦) = (𝑦𝐹))
9342, 90, 92syl2anc 691 . . . . . 6 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → ((𝑔𝐶 ↦ (𝑔𝐹))‘𝑦) = (𝑦𝐹))
9488, 93oveq12d 6567 . . . . 5 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → (((𝑔𝐶 ↦ (𝑔𝐹))‘𝑥)(+g𝑌)((𝑔𝐶 ↦ (𝑔𝐹))‘𝑦)) = ((𝑥𝐹)(+g𝑌)(𝑦𝐹)))
9571, 83, 943eqtr4d 2654 . . . 4 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → ((𝑔𝐶 ↦ (𝑔𝐹))‘(𝑥(+g𝑍)𝑦)) = (((𝑔𝐶 ↦ (𝑔𝐹))‘𝑥)(+g𝑌)((𝑔𝐶 ↦ (𝑔𝐹))‘𝑦)))
9695ralrimivva 2954 . . 3 (𝜑 → ∀𝑥𝐶𝑦𝐶 ((𝑔𝐶 ↦ (𝑔𝐹))‘(𝑥(+g𝑍)𝑦)) = (((𝑔𝐶 ↦ (𝑔𝐹))‘𝑥)(+g𝑌)((𝑔𝐶 ↦ (𝑔𝐹))‘𝑦)))
97 eqid 2610 . . . . . . 7 (0g𝑍) = (0g𝑍)
9812, 97mndidcl 17131 . . . . . 6 (𝑍 ∈ Mnd → (0g𝑍) ∈ 𝐶)
995, 98syl 17 . . . . 5 (𝜑 → (0g𝑍) ∈ 𝐶)
100 coexg 7010 . . . . . 6 (((0g𝑍) ∈ 𝐶𝐹 ∈ V) → ((0g𝑍) ∘ 𝐹) ∈ V)
10199, 77, 100syl2anc 691 . . . . 5 (𝜑 → ((0g𝑍) ∘ 𝐹) ∈ V)
102 coeq1 5201 . . . . . 6 (𝑔 = (0g𝑍) → (𝑔𝐹) = ((0g𝑍) ∘ 𝐹))
103102, 25fvmptg 6189 . . . . 5 (((0g𝑍) ∈ 𝐶 ∧ ((0g𝑍) ∘ 𝐹) ∈ V) → ((𝑔𝐶 ↦ (𝑔𝐹))‘(0g𝑍)) = ((0g𝑍) ∘ 𝐹))
10499, 101, 103syl2anc 691 . . . 4 (𝜑 → ((𝑔𝐶 ↦ (𝑔𝐹))‘(0g𝑍)) = ((0g𝑍) ∘ 𝐹))
1053, 11, 12, 1, 2, 99pwselbas 15972 . . . . . . 7 (𝜑 → (0g𝑍):𝐵⟶(Base‘𝑅))
106 fco 5971 . . . . . . 7 (((0g𝑍):𝐵⟶(Base‘𝑅) ∧ 𝐹:𝐴𝐵) → ((0g𝑍) ∘ 𝐹):𝐴⟶(Base‘𝑅))
107105, 16, 106syl2anc 691 . . . . . 6 (𝜑 → ((0g𝑍) ∘ 𝐹):𝐴⟶(Base‘𝑅))
108 ffn 5958 . . . . . 6 (((0g𝑍) ∘ 𝐹):𝐴⟶(Base‘𝑅) → ((0g𝑍) ∘ 𝐹) Fn 𝐴)
109107, 108syl 17 . . . . 5 (𝜑 → ((0g𝑍) ∘ 𝐹) Fn 𝐴)
110 fvex 6113 . . . . . . 7 (0g𝑅) ∈ V
111110a1i 11 . . . . . 6 (𝜑 → (0g𝑅) ∈ V)
112 fnconstg 6006 . . . . . 6 ((0g𝑅) ∈ V → (𝐴 × {(0g𝑅)}) Fn 𝐴)
113111, 112syl 17 . . . . 5 (𝜑 → (𝐴 × {(0g𝑅)}) Fn 𝐴)
114 eqid 2610 . . . . . . . . . . 11 (0g𝑅) = (0g𝑅)
1153, 114pws0g 17149 . . . . . . . . . 10 ((𝑅 ∈ Mnd ∧ 𝐵𝑊) → (𝐵 × {(0g𝑅)}) = (0g𝑍))
1161, 2, 115syl2anc 691 . . . . . . . . 9 (𝜑 → (𝐵 × {(0g𝑅)}) = (0g𝑍))
117116fveq1d 6105 . . . . . . . 8 (𝜑 → ((𝐵 × {(0g𝑅)})‘(𝐹𝑥)) = ((0g𝑍)‘(𝐹𝑥)))
118117adantr 480 . . . . . . 7 ((𝜑𝑥𝐴) → ((𝐵 × {(0g𝑅)})‘(𝐹𝑥)) = ((0g𝑍)‘(𝐹𝑥)))
11916ffvelrnda 6267 . . . . . . . 8 ((𝜑𝑥𝐴) → (𝐹𝑥) ∈ 𝐵)
120 fvconst2g 6372 . . . . . . . 8 (((0g𝑅) ∈ V ∧ (𝐹𝑥) ∈ 𝐵) → ((𝐵 × {(0g𝑅)})‘(𝐹𝑥)) = (0g𝑅))
121110, 119, 120sylancr 694 . . . . . . 7 ((𝜑𝑥𝐴) → ((𝐵 × {(0g𝑅)})‘(𝐹𝑥)) = (0g𝑅))
122118, 121eqtr3d 2646 . . . . . 6 ((𝜑𝑥𝐴) → ((0g𝑍)‘(𝐹𝑥)) = (0g𝑅))
123 fvco3 6185 . . . . . . 7 ((𝐹:𝐴𝐵𝑥𝐴) → (((0g𝑍) ∘ 𝐹)‘𝑥) = ((0g𝑍)‘(𝐹𝑥)))
12416, 123sylan 487 . . . . . 6 ((𝜑𝑥𝐴) → (((0g𝑍) ∘ 𝐹)‘𝑥) = ((0g𝑍)‘(𝐹𝑥)))
125 fvconst2g 6372 . . . . . . 7 (((0g𝑅) ∈ V ∧ 𝑥𝐴) → ((𝐴 × {(0g𝑅)})‘𝑥) = (0g𝑅))
126111, 125sylan 487 . . . . . 6 ((𝜑𝑥𝐴) → ((𝐴 × {(0g𝑅)})‘𝑥) = (0g𝑅))
127122, 124, 1263eqtr4d 2654 . . . . 5 ((𝜑𝑥𝐴) → (((0g𝑍) ∘ 𝐹)‘𝑥) = ((𝐴 × {(0g𝑅)})‘𝑥))
128109, 113, 127eqfnfvd 6222 . . . 4 (𝜑 → ((0g𝑍) ∘ 𝐹) = (𝐴 × {(0g𝑅)}))
1297, 114pws0g 17149 . . . . 5 ((𝑅 ∈ Mnd ∧ 𝐴𝑉) → (𝐴 × {(0g𝑅)}) = (0g𝑌))
1301, 6, 129syl2anc 691 . . . 4 (𝜑 → (𝐴 × {(0g𝑅)}) = (0g𝑌))
131104, 128, 1303eqtrd 2648 . . 3 (𝜑 → ((𝑔𝐶 ↦ (𝑔𝐹))‘(0g𝑍)) = (0g𝑌))
13226, 96, 1313jca 1235 . 2 (𝜑 → ((𝑔𝐶 ↦ (𝑔𝐹)):𝐶⟶(Base‘𝑌) ∧ ∀𝑥𝐶𝑦𝐶 ((𝑔𝐶 ↦ (𝑔𝐹))‘(𝑥(+g𝑍)𝑦)) = (((𝑔𝐶 ↦ (𝑔𝐹))‘𝑥)(+g𝑌)((𝑔𝐶 ↦ (𝑔𝐹))‘𝑦)) ∧ ((𝑔𝐶 ↦ (𝑔𝐹))‘(0g𝑍)) = (0g𝑌)))
133 eqid 2610 . . 3 (0g𝑌) = (0g𝑌)
13412, 20, 61, 59, 97, 133ismhm 17160 . 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))))
13510, 132, 134sylanbrc 695 1 (𝜑 → (𝑔𝐶 ↦ (𝑔𝐹)) ∈ (𝑍 MndHom 𝑌))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  Vcvv 3173  {csn 4125   ↦ cmpt 4643   × cxp 5036   ∘ ccom 5042   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ∘𝑓 cof 6793  Basecbs 15695  +gcplusg 15768  0gc0g 15923   ↑s cpws 15930  Mndcmnd 17117   MndHom cmhm 17156 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-ixp 7795  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-sup 8231  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-uz 11564  df-fz 12198  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-plusg 15781  df-mulr 15782  df-sca 15784  df-vsca 15785  df-ip 15786  df-tset 15787  df-ple 15788  df-ds 15791  df-hom 15793  df-cco 15794  df-0g 15925  df-prds 15931  df-pws 15933  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-mhm 17158 This theorem is referenced by:  pwsco1rhm  18561
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