Metamath Proof Explorer


Theorem amgmlemALT

Description: Alternate proof of amgmlem using amgmwlem . (Contributed by Kunhao Zheng, 20-Jun-2021) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Hypotheses amgmlemALT.0 M = mulGrp fld
amgmlemALT.1 φ A Fin
amgmlemALT.2 φ A
amgmlemALT.3 φ F : A +
Assertion amgmlemALT φ M F 1 A fld F A

Proof

Step Hyp Ref Expression
1 amgmlemALT.0 M = mulGrp fld
2 amgmlemALT.1 φ A Fin
3 amgmlemALT.2 φ A
4 amgmlemALT.3 φ F : A +
5 hashnncl A Fin A A
6 2 5 syl φ A A
7 3 6 mpbird φ A
8 7 nnrpd φ A +
9 8 rpreccld φ 1 A +
10 fconst6g 1 A + A × 1 A : A +
11 9 10 syl φ A × 1 A : A +
12 fconstmpt A × 1 A = k A 1 A
13 12 a1i φ A × 1 A = k A 1 A
14 13 oveq2d φ fld A × 1 A = fld k A 1 A
15 7 nnrecred φ 1 A
16 15 recnd φ 1 A
17 simpl A Fin 1 A A Fin
18 simplr A Fin 1 A k A 1 A
19 17 18 gsumfsum A Fin 1 A fld k A 1 A = k A 1 A
20 2 16 19 syl2anc φ fld k A 1 A = k A 1 A
21 fsumconst A Fin 1 A k A 1 A = A 1 A
22 2 16 21 syl2anc φ k A 1 A = A 1 A
23 7 nncnd φ A
24 7 nnne0d φ A 0
25 23 24 recidd φ A 1 A = 1
26 22 25 eqtrd φ k A 1 A = 1
27 14 20 26 3eqtrd φ fld A × 1 A = 1
28 1 2 3 4 11 27 amgmwlem φ M F c f A × 1 A fld F × f A × 1 A
29 rpssre +
30 ax-resscn
31 29 30 sstri +
32 eqid M 𝑠 + = M 𝑠 +
33 cnfldbas = Base fld
34 1 33 mgpbas = Base M
35 32 34 ressbas2 + + = Base M 𝑠 +
36 31 35 ax-mp + = Base M 𝑠 +
37 cnfld1 1 = 1 fld
38 1 37 ringidval 1 = 0 M
39 1 oveq1i M 𝑠 0 = mulGrp fld 𝑠 0
40 39 rpmsubg + SubGrp M 𝑠 0
41 subgsubm + SubGrp M 𝑠 0 + SubMnd M 𝑠 0
42 40 41 ax-mp + SubMnd M 𝑠 0
43 cnring fld Ring
44 cnfld0 0 = 0 fld
45 cndrng fld DivRing
46 33 44 45 drngui 0 = Unit fld
47 46 1 unitsubm fld Ring 0 SubMnd M
48 43 47 ax-mp 0 SubMnd M
49 eqid M 𝑠 0 = M 𝑠 0
50 49 subsubm 0 SubMnd M + SubMnd M 𝑠 0 + SubMnd M + 0
51 48 50 ax-mp + SubMnd M 𝑠 0 + SubMnd M + 0
52 42 51 mpbi + SubMnd M + 0
53 52 simpli + SubMnd M
54 eqid 0 M = 0 M
55 32 54 subm0 + SubMnd M 0 M = 0 M 𝑠 +
56 53 55 ax-mp 0 M = 0 M 𝑠 +
57 38 56 eqtri 1 = 0 M 𝑠 +
58 cncrng fld CRing
59 1 crngmgp fld CRing M CMnd
60 58 59 ax-mp M CMnd
61 32 submmnd + SubMnd M M 𝑠 + Mnd
62 53 61 mp1i φ M 𝑠 + Mnd
63 32 subcmn M CMnd M 𝑠 + Mnd M 𝑠 + CMnd
64 60 62 63 sylancr φ M 𝑠 + CMnd
65 reex V
66 65 29 ssexi + V
67 cnfldmul × = fld
68 1 67 mgpplusg × = + M
69 32 68 ressplusg + V × = + M 𝑠 +
70 66 69 ax-mp × = + M 𝑠 +
71 eqid mulGrp fld 𝑠 0 = mulGrp fld 𝑠 0
72 71 rpmsubg + SubGrp mulGrp fld 𝑠 0
73 1 oveq1i M 𝑠 + = mulGrp fld 𝑠 +
74 cnex V
75 difss 0
76 74 75 ssexi 0 V
77 rpcndif0 w + w 0
78 77 ssriv + 0
79 ressabs 0 V + 0 mulGrp fld 𝑠 0 𝑠 + = mulGrp fld 𝑠 +
80 76 78 79 mp2an mulGrp fld 𝑠 0 𝑠 + = mulGrp fld 𝑠 +
81 73 80 eqtr4i M 𝑠 + = mulGrp fld 𝑠 0 𝑠 +
82 81 subggrp + SubGrp mulGrp fld 𝑠 0 M 𝑠 + Grp
83 72 82 mp1i φ M 𝑠 + Grp
84 simpr φ k + k +
85 15 adantr φ k + 1 A
86 84 85 rpcxpcld φ k + k 1 A +
87 eqid k + k 1 A = k + k 1 A
88 86 87 fmptd φ k + k 1 A : + +
89 simprl φ x + y + x +
90 89 rprege0d φ x + y + x 0 x
91 simprr φ x + y + y +
92 91 rprege0d φ x + y + y 0 y
93 16 adantr φ x + y + 1 A
94 mulcxp x 0 x y 0 y 1 A x y 1 A = x 1 A y 1 A
95 90 92 93 94 syl3anc φ x + y + x y 1 A = x 1 A y 1 A
96 rpmulcl x + y + x y +
97 96 adantl φ x + y + x y +
98 oveq1 k = x y k 1 A = x y 1 A
99 ovex k 1 A V
100 98 87 99 fvmpt3i x y + k + k 1 A x y = x y 1 A
101 97 100 syl φ x + y + k + k 1 A x y = x y 1 A
102 oveq1 k = x k 1 A = x 1 A
103 102 87 99 fvmpt3i x + k + k 1 A x = x 1 A
104 89 103 syl φ x + y + k + k 1 A x = x 1 A
105 oveq1 k = y k 1 A = y 1 A
106 105 87 99 fvmpt3i y + k + k 1 A y = y 1 A
107 91 106 syl φ x + y + k + k 1 A y = y 1 A
108 104 107 oveq12d φ x + y + k + k 1 A x k + k 1 A y = x 1 A y 1 A
109 95 101 108 3eqtr4d φ x + y + k + k 1 A x y = k + k 1 A x k + k 1 A y
110 36 36 70 70 83 83 88 109 isghmd φ k + k 1 A M 𝑠 + GrpHom M 𝑠 +
111 ghmmhm k + k 1 A M 𝑠 + GrpHom M 𝑠 + k + k 1 A M 𝑠 + MndHom M 𝑠 +
112 110 111 syl φ k + k 1 A M 𝑠 + MndHom M 𝑠 +
113 1red φ 1
114 4 2 113 fdmfifsupp φ finSupp 1 F
115 36 57 64 62 2 112 4 114 gsummhm φ M 𝑠 + k + k 1 A F = k + k 1 A M 𝑠 + F
116 53 a1i φ + SubMnd M
117 4 ffvelrnda φ k A F k +
118 15 adantr φ k A 1 A
119 117 118 rpcxpcld φ k A F k 1 A +
120 eqid k A F k 1 A = k A F k 1 A
121 119 120 fmptd φ k A F k 1 A : A +
122 2 116 121 32 gsumsubm φ M k A F k 1 A = M 𝑠 + k A F k 1 A
123 9 adantr φ k A 1 A +
124 4 feqmptd φ F = k A F k
125 2 117 123 124 13 offval2 φ F c f A × 1 A = k A F k 1 A
126 125 oveq2d φ M F c f A × 1 A = M k A F k 1 A
127 102 cbvmptv k + k 1 A = x + x 1 A
128 127 a1i φ k + k 1 A = x + x 1 A
129 oveq1 x = F k x 1 A = F k 1 A
130 117 124 128 129 fmptco φ k + k 1 A F = k A F k 1 A
131 130 oveq2d φ M 𝑠 + k + k 1 A F = M 𝑠 + k A F k 1 A
132 122 126 131 3eqtr4rd φ M 𝑠 + k + k 1 A F = M F c f A × 1 A
133 36 57 64 2 4 114 gsumcl φ M 𝑠 + F +
134 oveq1 k = M 𝑠 + F k 1 A = M 𝑠 + F 1 A
135 134 87 99 fvmpt3i M 𝑠 + F + k + k 1 A M 𝑠 + F = M 𝑠 + F 1 A
136 133 135 syl φ k + k 1 A M 𝑠 + F = M 𝑠 + F 1 A
137 2 116 4 32 gsumsubm φ M F = M 𝑠 + F
138 137 oveq1d φ M F 1 A = M 𝑠 + F 1 A
139 136 138 eqtr4d φ k + k 1 A M 𝑠 + F = M F 1 A
140 115 132 139 3eqtr3d φ M F c f A × 1 A = M F 1 A
141 117 rpcnd φ k A F k
142 2 141 fsumcl φ k A F k
143 142 23 24 divrecd φ k A F k A = k A F k 1 A
144 2 16 141 fsummulc1 φ k A F k 1 A = k A F k 1 A
145 143 144 eqtr2d φ k A F k 1 A = k A F k A
146 16 adantr φ k A 1 A
147 141 146 mulcld φ k A F k 1 A
148 2 147 gsumfsum φ fld k A F k 1 A = k A F k 1 A
149 2 141 gsumfsum φ fld k A F k = k A F k
150 149 oveq1d φ fld k A F k A = k A F k A
151 145 148 150 3eqtr4d φ fld k A F k 1 A = fld k A F k A
152 2 117 146 124 13 offval2 φ F × f A × 1 A = k A F k 1 A
153 152 oveq2d φ fld F × f A × 1 A = fld k A F k 1 A
154 124 oveq2d φ fld F = fld k A F k
155 154 oveq1d φ fld F A = fld k A F k A
156 151 153 155 3eqtr4d φ fld F × f A × 1 A = fld F A
157 28 140 156 3brtr3d φ M F 1 A fld F A