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Theorem pmatcollpwscmatlem1 20526
Description: Lemma 1 for pmatcollpwscmat 20528. (Contributed by AV, 2-Nov-2019.) (Revised by AV, 4-Dec-2019.)
Hypotheses
Ref Expression
pmatcollpwscmat.p 𝑃 = (Poly1𝑅)
pmatcollpwscmat.c 𝐶 = (𝑁 Mat 𝑃)
pmatcollpwscmat.b 𝐵 = (Base‘𝐶)
pmatcollpwscmat.m1 = ( ·𝑠𝐶)
pmatcollpwscmat.e1 = (.g‘(mulGrp‘𝑃))
pmatcollpwscmat.x 𝑋 = (var1𝑅)
pmatcollpwscmat.t 𝑇 = (𝑁 matToPolyMat 𝑅)
pmatcollpwscmat.a 𝐴 = (𝑁 Mat 𝑅)
pmatcollpwscmat.d 𝐷 = (Base‘𝐴)
pmatcollpwscmat.u 𝑈 = (algSc‘𝑃)
pmatcollpwscmat.k 𝐾 = (Base‘𝑅)
pmatcollpwscmat.e2 𝐸 = (Base‘𝑃)
pmatcollpwscmat.s 𝑆 = (algSc‘𝑃)
pmatcollpwscmat.1 1 = (1r𝐶)
pmatcollpwscmat.m2 𝑀 = (𝑄 1 )
Assertion
Ref Expression
pmatcollpwscmatlem1 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) ∧ (𝑎𝑁𝑏𝑁)) → (((coe1‘(𝑎𝑀𝑏))‘𝐿)( ·𝑠𝑃)(0(.g‘(mulGrp‘𝑃))(var1𝑅))) = if(𝑎 = 𝑏, (𝑈‘((coe1𝑄)‘𝐿)), (0g𝑃)))

Proof of Theorem pmatcollpwscmatlem1
StepHypRef Expression
1 pmatcollpwscmat.m2 . . . . . . . 8 𝑀 = (𝑄 1 )
21oveqi 6623 . . . . . . 7 (𝑎𝑀𝑏) = (𝑎(𝑄 1 )𝑏)
3 pmatcollpwscmat.p . . . . . . . . . . . 12 𝑃 = (Poly1𝑅)
43ply1ring 19550 . . . . . . . . . . 11 (𝑅 ∈ Ring → 𝑃 ∈ Ring)
54anim2i 592 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑁 ∈ Fin ∧ 𝑃 ∈ Ring))
6 simpr 477 . . . . . . . . . 10 ((𝐿 ∈ ℕ0𝑄𝐸) → 𝑄𝐸)
75, 6anim12i 589 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) → ((𝑁 ∈ Fin ∧ 𝑃 ∈ Ring) ∧ 𝑄𝐸))
8 df-3an 1038 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑃 ∈ Ring ∧ 𝑄𝐸) ↔ ((𝑁 ∈ Fin ∧ 𝑃 ∈ Ring) ∧ 𝑄𝐸))
97, 8sylibr 224 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) → (𝑁 ∈ Fin ∧ 𝑃 ∈ Ring ∧ 𝑄𝐸))
10 pmatcollpwscmat.c . . . . . . . . 9 𝐶 = (𝑁 Mat 𝑃)
11 pmatcollpwscmat.e2 . . . . . . . . 9 𝐸 = (Base‘𝑃)
12 eqid 2621 . . . . . . . . 9 (0g𝑃) = (0g𝑃)
13 pmatcollpwscmat.1 . . . . . . . . 9 1 = (1r𝐶)
14 pmatcollpwscmat.m1 . . . . . . . . 9 = ( ·𝑠𝐶)
1510, 11, 12, 13, 14scmatscmide 20245 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑃 ∈ Ring ∧ 𝑄𝐸) ∧ (𝑎𝑁𝑏𝑁)) → (𝑎(𝑄 1 )𝑏) = if(𝑎 = 𝑏, 𝑄, (0g𝑃)))
169, 15sylan 488 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) ∧ (𝑎𝑁𝑏𝑁)) → (𝑎(𝑄 1 )𝑏) = if(𝑎 = 𝑏, 𝑄, (0g𝑃)))
172, 16syl5eq 2667 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) ∧ (𝑎𝑁𝑏𝑁)) → (𝑎𝑀𝑏) = if(𝑎 = 𝑏, 𝑄, (0g𝑃)))
1817fveq2d 6157 . . . . 5 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) ∧ (𝑎𝑁𝑏𝑁)) → (coe1‘(𝑎𝑀𝑏)) = (coe1‘if(𝑎 = 𝑏, 𝑄, (0g𝑃))))
1918fveq1d 6155 . . . 4 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) ∧ (𝑎𝑁𝑏𝑁)) → ((coe1‘(𝑎𝑀𝑏))‘𝐿) = ((coe1‘if(𝑎 = 𝑏, 𝑄, (0g𝑃)))‘𝐿))
20 fvif 6166 . . . . . 6 (coe1‘if(𝑎 = 𝑏, 𝑄, (0g𝑃))) = if(𝑎 = 𝑏, (coe1𝑄), (coe1‘(0g𝑃)))
2120fveq1i 6154 . . . . 5 ((coe1‘if(𝑎 = 𝑏, 𝑄, (0g𝑃)))‘𝐿) = (if(𝑎 = 𝑏, (coe1𝑄), (coe1‘(0g𝑃)))‘𝐿)
22 iffv 6167 . . . . 5 (if(𝑎 = 𝑏, (coe1𝑄), (coe1‘(0g𝑃)))‘𝐿) = if(𝑎 = 𝑏, ((coe1𝑄)‘𝐿), ((coe1‘(0g𝑃))‘𝐿))
2321, 22eqtri 2643 . . . 4 ((coe1‘if(𝑎 = 𝑏, 𝑄, (0g𝑃)))‘𝐿) = if(𝑎 = 𝑏, ((coe1𝑄)‘𝐿), ((coe1‘(0g𝑃))‘𝐿))
2419, 23syl6eq 2671 . . 3 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) ∧ (𝑎𝑁𝑏𝑁)) → ((coe1‘(𝑎𝑀𝑏))‘𝐿) = if(𝑎 = 𝑏, ((coe1𝑄)‘𝐿), ((coe1‘(0g𝑃))‘𝐿)))
2524oveq1d 6625 . 2 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) ∧ (𝑎𝑁𝑏𝑁)) → (((coe1‘(𝑎𝑀𝑏))‘𝐿)( ·𝑠𝑃)(0(.g‘(mulGrp‘𝑃))(var1𝑅))) = (if(𝑎 = 𝑏, ((coe1𝑄)‘𝐿), ((coe1‘(0g𝑃))‘𝐿))( ·𝑠𝑃)(0(.g‘(mulGrp‘𝑃))(var1𝑅))))
26 ovif 6697 . . 3 (if(𝑎 = 𝑏, ((coe1𝑄)‘𝐿), ((coe1‘(0g𝑃))‘𝐿))( ·𝑠𝑃)(0(.g‘(mulGrp‘𝑃))(var1𝑅))) = if(𝑎 = 𝑏, (((coe1𝑄)‘𝐿)( ·𝑠𝑃)(0(.g‘(mulGrp‘𝑃))(var1𝑅))), (((coe1‘(0g𝑃))‘𝐿)( ·𝑠𝑃)(0(.g‘(mulGrp‘𝑃))(var1𝑅))))
27 eqid 2621 . . . . . . . . . . 11 (0g𝑅) = (0g𝑅)
283, 12, 27coe1z 19565 . . . . . . . . . 10 (𝑅 ∈ Ring → (coe1‘(0g𝑃)) = (ℕ0 × {(0g𝑅)}))
2928ad2antlr 762 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) → (coe1‘(0g𝑃)) = (ℕ0 × {(0g𝑅)}))
3029fveq1d 6155 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) → ((coe1‘(0g𝑃))‘𝐿) = ((ℕ0 × {(0g𝑅)})‘𝐿))
31 fvex 6163 . . . . . . . . . . 11 (0g𝑅) ∈ V
3231a1i 11 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (0g𝑅) ∈ V)
33 simpl 473 . . . . . . . . . 10 ((𝐿 ∈ ℕ0𝑄𝐸) → 𝐿 ∈ ℕ0)
3432, 33anim12i 589 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) → ((0g𝑅) ∈ V ∧ 𝐿 ∈ ℕ0))
35 fvconst2g 6427 . . . . . . . . 9 (((0g𝑅) ∈ V ∧ 𝐿 ∈ ℕ0) → ((ℕ0 × {(0g𝑅)})‘𝐿) = (0g𝑅))
3634, 35syl 17 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) → ((ℕ0 × {(0g𝑅)})‘𝐿) = (0g𝑅))
3730, 36eqtrd 2655 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) → ((coe1‘(0g𝑃))‘𝐿) = (0g𝑅))
3837oveq1d 6625 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) → (((coe1‘(0g𝑃))‘𝐿)( ·𝑠𝑃)(0(.g‘(mulGrp‘𝑃))(var1𝑅))) = ((0g𝑅)( ·𝑠𝑃)(0(.g‘(mulGrp‘𝑃))(var1𝑅))))
393ply1lmod 19554 . . . . . . . . 9 (𝑅 ∈ Ring → 𝑃 ∈ LMod)
4039ad2antlr 762 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) → 𝑃 ∈ LMod)
41 eqid 2621 . . . . . . . . . . . 12 (mulGrp‘𝑃) = (mulGrp‘𝑃)
4241ringmgp 18485 . . . . . . . . . . 11 (𝑃 ∈ Ring → (mulGrp‘𝑃) ∈ Mnd)
434, 42syl 17 . . . . . . . . . 10 (𝑅 ∈ Ring → (mulGrp‘𝑃) ∈ Mnd)
44 0nn0 11259 . . . . . . . . . . 11 0 ∈ ℕ0
4544a1i 11 . . . . . . . . . 10 (𝑅 ∈ Ring → 0 ∈ ℕ0)
46 eqid 2621 . . . . . . . . . . 11 (var1𝑅) = (var1𝑅)
4746, 3, 11vr1cl 19519 . . . . . . . . . 10 (𝑅 ∈ Ring → (var1𝑅) ∈ 𝐸)
4841, 11mgpbas 18427 . . . . . . . . . . 11 𝐸 = (Base‘(mulGrp‘𝑃))
49 eqid 2621 . . . . . . . . . . 11 (.g‘(mulGrp‘𝑃)) = (.g‘(mulGrp‘𝑃))
5048, 49mulgnn0cl 17490 . . . . . . . . . 10 (((mulGrp‘𝑃) ∈ Mnd ∧ 0 ∈ ℕ0 ∧ (var1𝑅) ∈ 𝐸) → (0(.g‘(mulGrp‘𝑃))(var1𝑅)) ∈ 𝐸)
5143, 45, 47, 50syl3anc 1323 . . . . . . . . 9 (𝑅 ∈ Ring → (0(.g‘(mulGrp‘𝑃))(var1𝑅)) ∈ 𝐸)
5251ad2antlr 762 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) → (0(.g‘(mulGrp‘𝑃))(var1𝑅)) ∈ 𝐸)
53 eqid 2621 . . . . . . . . 9 (Scalar‘𝑃) = (Scalar‘𝑃)
54 eqid 2621 . . . . . . . . 9 ( ·𝑠𝑃) = ( ·𝑠𝑃)
55 eqid 2621 . . . . . . . . 9 (0g‘(Scalar‘𝑃)) = (0g‘(Scalar‘𝑃))
5611, 53, 54, 55, 12lmod0vs 18828 . . . . . . . 8 ((𝑃 ∈ LMod ∧ (0(.g‘(mulGrp‘𝑃))(var1𝑅)) ∈ 𝐸) → ((0g‘(Scalar‘𝑃))( ·𝑠𝑃)(0(.g‘(mulGrp‘𝑃))(var1𝑅))) = (0g𝑃))
5740, 52, 56syl2anc 692 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) → ((0g‘(Scalar‘𝑃))( ·𝑠𝑃)(0(.g‘(mulGrp‘𝑃))(var1𝑅))) = (0g𝑃))
583ply1sca 19555 . . . . . . . . . . . 12 (𝑅 ∈ Ring → 𝑅 = (Scalar‘𝑃))
5958adantl 482 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑅 = (Scalar‘𝑃))
6059fveq2d 6157 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (0g𝑅) = (0g‘(Scalar‘𝑃)))
6160oveq1d 6625 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ((0g𝑅)( ·𝑠𝑃)(0(.g‘(mulGrp‘𝑃))(var1𝑅))) = ((0g‘(Scalar‘𝑃))( ·𝑠𝑃)(0(.g‘(mulGrp‘𝑃))(var1𝑅))))
6261eqeq1d 2623 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (((0g𝑅)( ·𝑠𝑃)(0(.g‘(mulGrp‘𝑃))(var1𝑅))) = (0g𝑃) ↔ ((0g‘(Scalar‘𝑃))( ·𝑠𝑃)(0(.g‘(mulGrp‘𝑃))(var1𝑅))) = (0g𝑃)))
6362adantr 481 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) → (((0g𝑅)( ·𝑠𝑃)(0(.g‘(mulGrp‘𝑃))(var1𝑅))) = (0g𝑃) ↔ ((0g‘(Scalar‘𝑃))( ·𝑠𝑃)(0(.g‘(mulGrp‘𝑃))(var1𝑅))) = (0g𝑃)))
6457, 63mpbird 247 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) → ((0g𝑅)( ·𝑠𝑃)(0(.g‘(mulGrp‘𝑃))(var1𝑅))) = (0g𝑃))
6538, 64eqtrd 2655 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) → (((coe1‘(0g𝑃))‘𝐿)( ·𝑠𝑃)(0(.g‘(mulGrp‘𝑃))(var1𝑅))) = (0g𝑃))
6665ifeq2d 4082 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) → if(𝑎 = 𝑏, (((coe1𝑄)‘𝐿)( ·𝑠𝑃)(0(.g‘(mulGrp‘𝑃))(var1𝑅))), (((coe1‘(0g𝑃))‘𝐿)( ·𝑠𝑃)(0(.g‘(mulGrp‘𝑃))(var1𝑅)))) = if(𝑎 = 𝑏, (((coe1𝑄)‘𝐿)( ·𝑠𝑃)(0(.g‘(mulGrp‘𝑃))(var1𝑅))), (0g𝑃)))
6766adantr 481 . . 3 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) ∧ (𝑎𝑁𝑏𝑁)) → if(𝑎 = 𝑏, (((coe1𝑄)‘𝐿)( ·𝑠𝑃)(0(.g‘(mulGrp‘𝑃))(var1𝑅))), (((coe1‘(0g𝑃))‘𝐿)( ·𝑠𝑃)(0(.g‘(mulGrp‘𝑃))(var1𝑅)))) = if(𝑎 = 𝑏, (((coe1𝑄)‘𝐿)( ·𝑠𝑃)(0(.g‘(mulGrp‘𝑃))(var1𝑅))), (0g𝑃)))
6826, 67syl5eq 2667 . 2 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) ∧ (𝑎𝑁𝑏𝑁)) → (if(𝑎 = 𝑏, ((coe1𝑄)‘𝐿), ((coe1‘(0g𝑃))‘𝐿))( ·𝑠𝑃)(0(.g‘(mulGrp‘𝑃))(var1𝑅))) = if(𝑎 = 𝑏, (((coe1𝑄)‘𝐿)( ·𝑠𝑃)(0(.g‘(mulGrp‘𝑃))(var1𝑅))), (0g𝑃)))
69 simpr 477 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) → (𝐿 ∈ ℕ0𝑄𝐸))
7069ancomd 467 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) → (𝑄𝐸𝐿 ∈ ℕ0))
71 eqid 2621 . . . . . . . . 9 (coe1𝑄) = (coe1𝑄)
72 pmatcollpwscmat.k . . . . . . . . 9 𝐾 = (Base‘𝑅)
7371, 11, 3, 72coe1fvalcl 19514 . . . . . . . 8 ((𝑄𝐸𝐿 ∈ ℕ0) → ((coe1𝑄)‘𝐿) ∈ 𝐾)
7470, 73syl 17 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) → ((coe1𝑄)‘𝐿) ∈ 𝐾)
7558eqcomd 2627 . . . . . . . . . . . 12 (𝑅 ∈ Ring → (Scalar‘𝑃) = 𝑅)
7675adantl 482 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (Scalar‘𝑃) = 𝑅)
7776fveq2d 6157 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (Base‘(Scalar‘𝑃)) = (Base‘𝑅))
7877, 72syl6eqr 2673 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (Base‘(Scalar‘𝑃)) = 𝐾)
7978eleq2d 2684 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (((coe1𝑄)‘𝐿) ∈ (Base‘(Scalar‘𝑃)) ↔ ((coe1𝑄)‘𝐿) ∈ 𝐾))
8079adantr 481 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) → (((coe1𝑄)‘𝐿) ∈ (Base‘(Scalar‘𝑃)) ↔ ((coe1𝑄)‘𝐿) ∈ 𝐾))
8174, 80mpbird 247 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) → ((coe1𝑄)‘𝐿) ∈ (Base‘(Scalar‘𝑃)))
82 pmatcollpwscmat.u . . . . . . 7 𝑈 = (algSc‘𝑃)
83 eqid 2621 . . . . . . 7 (Base‘(Scalar‘𝑃)) = (Base‘(Scalar‘𝑃))
84 eqid 2621 . . . . . . 7 (1r𝑃) = (1r𝑃)
8582, 53, 83, 54, 84asclval 19267 . . . . . 6 (((coe1𝑄)‘𝐿) ∈ (Base‘(Scalar‘𝑃)) → (𝑈‘((coe1𝑄)‘𝐿)) = (((coe1𝑄)‘𝐿)( ·𝑠𝑃)(1r𝑃)))
8681, 85syl 17 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) → (𝑈‘((coe1𝑄)‘𝐿)) = (((coe1𝑄)‘𝐿)( ·𝑠𝑃)(1r𝑃)))
873, 46, 41, 49ply1idvr1 19595 . . . . . . . 8 (𝑅 ∈ Ring → (0(.g‘(mulGrp‘𝑃))(var1𝑅)) = (1r𝑃))
8887eqcomd 2627 . . . . . . 7 (𝑅 ∈ Ring → (1r𝑃) = (0(.g‘(mulGrp‘𝑃))(var1𝑅)))
8988ad2antlr 762 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) → (1r𝑃) = (0(.g‘(mulGrp‘𝑃))(var1𝑅)))
9089oveq2d 6626 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) → (((coe1𝑄)‘𝐿)( ·𝑠𝑃)(1r𝑃)) = (((coe1𝑄)‘𝐿)( ·𝑠𝑃)(0(.g‘(mulGrp‘𝑃))(var1𝑅))))
9186, 90eqtr2d 2656 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) → (((coe1𝑄)‘𝐿)( ·𝑠𝑃)(0(.g‘(mulGrp‘𝑃))(var1𝑅))) = (𝑈‘((coe1𝑄)‘𝐿)))
9291ifeq1d 4081 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) → if(𝑎 = 𝑏, (((coe1𝑄)‘𝐿)( ·𝑠𝑃)(0(.g‘(mulGrp‘𝑃))(var1𝑅))), (0g𝑃)) = if(𝑎 = 𝑏, (𝑈‘((coe1𝑄)‘𝐿)), (0g𝑃)))
9392adantr 481 . 2 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) ∧ (𝑎𝑁𝑏𝑁)) → if(𝑎 = 𝑏, (((coe1𝑄)‘𝐿)( ·𝑠𝑃)(0(.g‘(mulGrp‘𝑃))(var1𝑅))), (0g𝑃)) = if(𝑎 = 𝑏, (𝑈‘((coe1𝑄)‘𝐿)), (0g𝑃)))
9425, 68, 933eqtrd 2659 1 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) ∧ (𝑎𝑁𝑏𝑁)) → (((coe1‘(𝑎𝑀𝑏))‘𝐿)( ·𝑠𝑃)(0(.g‘(mulGrp‘𝑃))(var1𝑅))) = if(𝑎 = 𝑏, (𝑈‘((coe1𝑄)‘𝐿)), (0g𝑃)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  Vcvv 3189  ifcif 4063  {csn 4153   × cxp 5077  cfv 5852  (class class class)co 6610  Fincfn 7907  0cc0 9888  0cn0 11244  Basecbs 15792  Scalarcsca 15876   ·𝑠 cvsca 15877  0gc0g 16032  Mndcmnd 17226  .gcmg 17472  mulGrpcmgp 18421  1rcur 18433  Ringcrg 18479  LModclmod 18795  algSccascl 19243  var1cv1 19478  Poly1cpl1 19479  coe1cco1 19480   Mat cmat 20145   matToPolyMat cmat2pmat 20441
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-inf2 8490  ax-cnex 9944  ax-resscn 9945  ax-1cn 9946  ax-icn 9947  ax-addcl 9948  ax-addrcl 9949  ax-mulcl 9950  ax-mulrcl 9951  ax-mulcom 9952  ax-addass 9953  ax-mulass 9954  ax-distr 9955  ax-i2m1 9956  ax-1ne0 9957  ax-1rid 9958  ax-rnegex 9959  ax-rrecex 9960  ax-cnre 9961  ax-pre-lttri 9962  ax-pre-lttrn 9963  ax-pre-ltadd 9964  ax-pre-mulgt0 9965
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-ot 4162  df-uni 4408  df-int 4446  df-iun 4492  df-iin 4493  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-se 5039  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-isom 5861  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-of 6857  df-ofr 6858  df-om 7020  df-1st 7120  df-2nd 7121  df-supp 7248  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-2o 7513  df-oadd 7516  df-er 7694  df-map 7811  df-pm 7812  df-ixp 7861  df-en 7908  df-dom 7909  df-sdom 7910  df-fin 7911  df-fsupp 8228  df-sup 8300  df-oi 8367  df-card 8717  df-pnf 10028  df-mnf 10029  df-xr 10030  df-ltxr 10031  df-le 10032  df-sub 10220  df-neg 10221  df-nn 10973  df-2 11031  df-3 11032  df-4 11033  df-5 11034  df-6 11035  df-7 11036  df-8 11037  df-9 11038  df-n0 11245  df-z 11330  df-dec 11446  df-uz 11640  df-fz 12277  df-fzo 12415  df-seq 12750  df-hash 13066  df-struct 15794  df-ndx 15795  df-slot 15796  df-base 15797  df-sets 15798  df-ress 15799  df-plusg 15886  df-mulr 15887  df-sca 15889  df-vsca 15890  df-ip 15891  df-tset 15892  df-ple 15893  df-ds 15896  df-hom 15898  df-cco 15899  df-0g 16034  df-gsum 16035  df-prds 16040  df-pws 16042  df-mre 16178  df-mrc 16179  df-acs 16181  df-mgm 17174  df-sgrp 17216  df-mnd 17227  df-mhm 17267  df-submnd 17268  df-grp 17357  df-minusg 17358  df-sbg 17359  df-mulg 17473  df-subg 17523  df-ghm 17590  df-cntz 17682  df-cmn 18127  df-abl 18128  df-mgp 18422  df-ur 18434  df-ring 18481  df-subrg 18710  df-lmod 18797  df-lss 18865  df-sra 19104  df-rgmod 19105  df-ascl 19246  df-psr 19288  df-mvr 19289  df-mpl 19290  df-opsr 19292  df-psr1 19482  df-vr1 19483  df-ply1 19484  df-coe1 19485  df-dsmm 20008  df-frlm 20023  df-mamu 20122  df-mat 20146
This theorem is referenced by:  pmatcollpwscmatlem2  20527
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