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Theorem pmatcollpwscmatlem1 22154
Description: Lemma 1 for pmatcollpwscmat 22156. (Contributed by AV, 2-Nov-2019.) (Revised by AV, 4-Dec-2019.)
Hypotheses
Ref Expression
pmatcollpwscmat.p 𝑃 = (Poly1β€˜π‘…)
pmatcollpwscmat.c 𝐢 = (𝑁 Mat 𝑃)
pmatcollpwscmat.b 𝐡 = (Baseβ€˜πΆ)
pmatcollpwscmat.m1 βˆ— = ( ·𝑠 β€˜πΆ)
pmatcollpwscmat.e1 ↑ = (.gβ€˜(mulGrpβ€˜π‘ƒ))
pmatcollpwscmat.x 𝑋 = (var1β€˜π‘…)
pmatcollpwscmat.t 𝑇 = (𝑁 matToPolyMat 𝑅)
pmatcollpwscmat.a 𝐴 = (𝑁 Mat 𝑅)
pmatcollpwscmat.d 𝐷 = (Baseβ€˜π΄)
pmatcollpwscmat.u π‘ˆ = (algScβ€˜π‘ƒ)
pmatcollpwscmat.k 𝐾 = (Baseβ€˜π‘…)
pmatcollpwscmat.e2 𝐸 = (Baseβ€˜π‘ƒ)
pmatcollpwscmat.s 𝑆 = (algScβ€˜π‘ƒ)
pmatcollpwscmat.1 1 = (1rβ€˜πΆ)
pmatcollpwscmat.m2 𝑀 = (𝑄 βˆ— 1 )
Assertion
Ref Expression
pmatcollpwscmatlem1 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ β„•0 ∧ 𝑄 ∈ 𝐸)) ∧ (π‘Ž ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) β†’ (((coe1β€˜(π‘Žπ‘€π‘))β€˜πΏ)( ·𝑠 β€˜π‘ƒ)(0(.gβ€˜(mulGrpβ€˜π‘ƒ))(var1β€˜π‘…))) = if(π‘Ž = 𝑏, (π‘ˆβ€˜((coe1β€˜π‘„)β€˜πΏ)), (0gβ€˜π‘ƒ)))

Proof of Theorem pmatcollpwscmatlem1
StepHypRef Expression
1 pmatcollpwscmat.m2 . . . . . . . 8 𝑀 = (𝑄 βˆ— 1 )
21oveqi 7371 . . . . . . 7 (π‘Žπ‘€π‘) = (π‘Ž(𝑄 βˆ— 1 )𝑏)
3 pmatcollpwscmat.p . . . . . . . . . . . 12 𝑃 = (Poly1β€˜π‘…)
43ply1ring 21635 . . . . . . . . . . 11 (𝑅 ∈ Ring β†’ 𝑃 ∈ Ring)
54anim2i 618 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) β†’ (𝑁 ∈ Fin ∧ 𝑃 ∈ Ring))
6 simpr 486 . . . . . . . . . 10 ((𝐿 ∈ β„•0 ∧ 𝑄 ∈ 𝐸) β†’ 𝑄 ∈ 𝐸)
75, 6anim12i 614 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ β„•0 ∧ 𝑄 ∈ 𝐸)) β†’ ((𝑁 ∈ Fin ∧ 𝑃 ∈ Ring) ∧ 𝑄 ∈ 𝐸))
8 df-3an 1090 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑃 ∈ Ring ∧ 𝑄 ∈ 𝐸) ↔ ((𝑁 ∈ Fin ∧ 𝑃 ∈ Ring) ∧ 𝑄 ∈ 𝐸))
97, 8sylibr 233 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ β„•0 ∧ 𝑄 ∈ 𝐸)) β†’ (𝑁 ∈ Fin ∧ 𝑃 ∈ Ring ∧ 𝑄 ∈ 𝐸))
10 pmatcollpwscmat.c . . . . . . . . 9 𝐢 = (𝑁 Mat 𝑃)
11 pmatcollpwscmat.e2 . . . . . . . . 9 𝐸 = (Baseβ€˜π‘ƒ)
12 eqid 2733 . . . . . . . . 9 (0gβ€˜π‘ƒ) = (0gβ€˜π‘ƒ)
13 pmatcollpwscmat.1 . . . . . . . . 9 1 = (1rβ€˜πΆ)
14 pmatcollpwscmat.m1 . . . . . . . . 9 βˆ— = ( ·𝑠 β€˜πΆ)
1510, 11, 12, 13, 14scmatscmide 21872 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑃 ∈ Ring ∧ 𝑄 ∈ 𝐸) ∧ (π‘Ž ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) β†’ (π‘Ž(𝑄 βˆ— 1 )𝑏) = if(π‘Ž = 𝑏, 𝑄, (0gβ€˜π‘ƒ)))
169, 15sylan 581 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ β„•0 ∧ 𝑄 ∈ 𝐸)) ∧ (π‘Ž ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) β†’ (π‘Ž(𝑄 βˆ— 1 )𝑏) = if(π‘Ž = 𝑏, 𝑄, (0gβ€˜π‘ƒ)))
172, 16eqtrid 2785 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ β„•0 ∧ 𝑄 ∈ 𝐸)) ∧ (π‘Ž ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) β†’ (π‘Žπ‘€π‘) = if(π‘Ž = 𝑏, 𝑄, (0gβ€˜π‘ƒ)))
1817fveq2d 6847 . . . . 5 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ β„•0 ∧ 𝑄 ∈ 𝐸)) ∧ (π‘Ž ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) β†’ (coe1β€˜(π‘Žπ‘€π‘)) = (coe1β€˜if(π‘Ž = 𝑏, 𝑄, (0gβ€˜π‘ƒ))))
1918fveq1d 6845 . . . 4 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ β„•0 ∧ 𝑄 ∈ 𝐸)) ∧ (π‘Ž ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) β†’ ((coe1β€˜(π‘Žπ‘€π‘))β€˜πΏ) = ((coe1β€˜if(π‘Ž = 𝑏, 𝑄, (0gβ€˜π‘ƒ)))β€˜πΏ))
20 fvif 6859 . . . . . 6 (coe1β€˜if(π‘Ž = 𝑏, 𝑄, (0gβ€˜π‘ƒ))) = if(π‘Ž = 𝑏, (coe1β€˜π‘„), (coe1β€˜(0gβ€˜π‘ƒ)))
2120fveq1i 6844 . . . . 5 ((coe1β€˜if(π‘Ž = 𝑏, 𝑄, (0gβ€˜π‘ƒ)))β€˜πΏ) = (if(π‘Ž = 𝑏, (coe1β€˜π‘„), (coe1β€˜(0gβ€˜π‘ƒ)))β€˜πΏ)
22 iffv 6860 . . . . 5 (if(π‘Ž = 𝑏, (coe1β€˜π‘„), (coe1β€˜(0gβ€˜π‘ƒ)))β€˜πΏ) = if(π‘Ž = 𝑏, ((coe1β€˜π‘„)β€˜πΏ), ((coe1β€˜(0gβ€˜π‘ƒ))β€˜πΏ))
2321, 22eqtri 2761 . . . 4 ((coe1β€˜if(π‘Ž = 𝑏, 𝑄, (0gβ€˜π‘ƒ)))β€˜πΏ) = if(π‘Ž = 𝑏, ((coe1β€˜π‘„)β€˜πΏ), ((coe1β€˜(0gβ€˜π‘ƒ))β€˜πΏ))
2419, 23eqtrdi 2789 . . 3 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ β„•0 ∧ 𝑄 ∈ 𝐸)) ∧ (π‘Ž ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) β†’ ((coe1β€˜(π‘Žπ‘€π‘))β€˜πΏ) = if(π‘Ž = 𝑏, ((coe1β€˜π‘„)β€˜πΏ), ((coe1β€˜(0gβ€˜π‘ƒ))β€˜πΏ)))
2524oveq1d 7373 . 2 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ β„•0 ∧ 𝑄 ∈ 𝐸)) ∧ (π‘Ž ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) β†’ (((coe1β€˜(π‘Žπ‘€π‘))β€˜πΏ)( ·𝑠 β€˜π‘ƒ)(0(.gβ€˜(mulGrpβ€˜π‘ƒ))(var1β€˜π‘…))) = (if(π‘Ž = 𝑏, ((coe1β€˜π‘„)β€˜πΏ), ((coe1β€˜(0gβ€˜π‘ƒ))β€˜πΏ))( ·𝑠 β€˜π‘ƒ)(0(.gβ€˜(mulGrpβ€˜π‘ƒ))(var1β€˜π‘…))))
26 ovif 7455 . . 3 (if(π‘Ž = 𝑏, ((coe1β€˜π‘„)β€˜πΏ), ((coe1β€˜(0gβ€˜π‘ƒ))β€˜πΏ))( ·𝑠 β€˜π‘ƒ)(0(.gβ€˜(mulGrpβ€˜π‘ƒ))(var1β€˜π‘…))) = if(π‘Ž = 𝑏, (((coe1β€˜π‘„)β€˜πΏ)( ·𝑠 β€˜π‘ƒ)(0(.gβ€˜(mulGrpβ€˜π‘ƒ))(var1β€˜π‘…))), (((coe1β€˜(0gβ€˜π‘ƒ))β€˜πΏ)( ·𝑠 β€˜π‘ƒ)(0(.gβ€˜(mulGrpβ€˜π‘ƒ))(var1β€˜π‘…))))
27 eqid 2733 . . . . . . . . . . 11 (0gβ€˜π‘…) = (0gβ€˜π‘…)
283, 12, 27coe1z 21650 . . . . . . . . . 10 (𝑅 ∈ Ring β†’ (coe1β€˜(0gβ€˜π‘ƒ)) = (β„•0 Γ— {(0gβ€˜π‘…)}))
2928ad2antlr 726 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ β„•0 ∧ 𝑄 ∈ 𝐸)) β†’ (coe1β€˜(0gβ€˜π‘ƒ)) = (β„•0 Γ— {(0gβ€˜π‘…)}))
3029fveq1d 6845 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ β„•0 ∧ 𝑄 ∈ 𝐸)) β†’ ((coe1β€˜(0gβ€˜π‘ƒ))β€˜πΏ) = ((β„•0 Γ— {(0gβ€˜π‘…)})β€˜πΏ))
31 fvexd 6858 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) β†’ (0gβ€˜π‘…) ∈ V)
32 simpl 484 . . . . . . . . . 10 ((𝐿 ∈ β„•0 ∧ 𝑄 ∈ 𝐸) β†’ 𝐿 ∈ β„•0)
3331, 32anim12i 614 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ β„•0 ∧ 𝑄 ∈ 𝐸)) β†’ ((0gβ€˜π‘…) ∈ V ∧ 𝐿 ∈ β„•0))
34 fvconst2g 7152 . . . . . . . . 9 (((0gβ€˜π‘…) ∈ V ∧ 𝐿 ∈ β„•0) β†’ ((β„•0 Γ— {(0gβ€˜π‘…)})β€˜πΏ) = (0gβ€˜π‘…))
3533, 34syl 17 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ β„•0 ∧ 𝑄 ∈ 𝐸)) β†’ ((β„•0 Γ— {(0gβ€˜π‘…)})β€˜πΏ) = (0gβ€˜π‘…))
3630, 35eqtrd 2773 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ β„•0 ∧ 𝑄 ∈ 𝐸)) β†’ ((coe1β€˜(0gβ€˜π‘ƒ))β€˜πΏ) = (0gβ€˜π‘…))
3736oveq1d 7373 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ β„•0 ∧ 𝑄 ∈ 𝐸)) β†’ (((coe1β€˜(0gβ€˜π‘ƒ))β€˜πΏ)( ·𝑠 β€˜π‘ƒ)(0(.gβ€˜(mulGrpβ€˜π‘ƒ))(var1β€˜π‘…))) = ((0gβ€˜π‘…)( ·𝑠 β€˜π‘ƒ)(0(.gβ€˜(mulGrpβ€˜π‘ƒ))(var1β€˜π‘…))))
383ply1lmod 21639 . . . . . . . . 9 (𝑅 ∈ Ring β†’ 𝑃 ∈ LMod)
3938ad2antlr 726 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ β„•0 ∧ 𝑄 ∈ 𝐸)) β†’ 𝑃 ∈ LMod)
40 eqid 2733 . . . . . . . . . . 11 (mulGrpβ€˜π‘ƒ) = (mulGrpβ€˜π‘ƒ)
4140, 11mgpbas 19907 . . . . . . . . . 10 𝐸 = (Baseβ€˜(mulGrpβ€˜π‘ƒ))
42 eqid 2733 . . . . . . . . . 10 (.gβ€˜(mulGrpβ€˜π‘ƒ)) = (.gβ€˜(mulGrpβ€˜π‘ƒ))
4340ringmgp 19975 . . . . . . . . . . 11 (𝑃 ∈ Ring β†’ (mulGrpβ€˜π‘ƒ) ∈ Mnd)
444, 43syl 17 . . . . . . . . . 10 (𝑅 ∈ Ring β†’ (mulGrpβ€˜π‘ƒ) ∈ Mnd)
45 0nn0 12433 . . . . . . . . . . 11 0 ∈ β„•0
4645a1i 11 . . . . . . . . . 10 (𝑅 ∈ Ring β†’ 0 ∈ β„•0)
47 eqid 2733 . . . . . . . . . . 11 (var1β€˜π‘…) = (var1β€˜π‘…)
4847, 3, 11vr1cl 21604 . . . . . . . . . 10 (𝑅 ∈ Ring β†’ (var1β€˜π‘…) ∈ 𝐸)
4941, 42, 44, 46, 48mulgnn0cld 18902 . . . . . . . . 9 (𝑅 ∈ Ring β†’ (0(.gβ€˜(mulGrpβ€˜π‘ƒ))(var1β€˜π‘…)) ∈ 𝐸)
5049ad2antlr 726 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ β„•0 ∧ 𝑄 ∈ 𝐸)) β†’ (0(.gβ€˜(mulGrpβ€˜π‘ƒ))(var1β€˜π‘…)) ∈ 𝐸)
51 eqid 2733 . . . . . . . . 9 (Scalarβ€˜π‘ƒ) = (Scalarβ€˜π‘ƒ)
52 eqid 2733 . . . . . . . . 9 ( ·𝑠 β€˜π‘ƒ) = ( ·𝑠 β€˜π‘ƒ)
53 eqid 2733 . . . . . . . . 9 (0gβ€˜(Scalarβ€˜π‘ƒ)) = (0gβ€˜(Scalarβ€˜π‘ƒ))
5411, 51, 52, 53, 12lmod0vs 20370 . . . . . . . 8 ((𝑃 ∈ LMod ∧ (0(.gβ€˜(mulGrpβ€˜π‘ƒ))(var1β€˜π‘…)) ∈ 𝐸) β†’ ((0gβ€˜(Scalarβ€˜π‘ƒ))( ·𝑠 β€˜π‘ƒ)(0(.gβ€˜(mulGrpβ€˜π‘ƒ))(var1β€˜π‘…))) = (0gβ€˜π‘ƒ))
5539, 50, 54syl2anc 585 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ β„•0 ∧ 𝑄 ∈ 𝐸)) β†’ ((0gβ€˜(Scalarβ€˜π‘ƒ))( ·𝑠 β€˜π‘ƒ)(0(.gβ€˜(mulGrpβ€˜π‘ƒ))(var1β€˜π‘…))) = (0gβ€˜π‘ƒ))
563ply1sca 21640 . . . . . . . . . . . 12 (𝑅 ∈ Ring β†’ 𝑅 = (Scalarβ€˜π‘ƒ))
5756adantl 483 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) β†’ 𝑅 = (Scalarβ€˜π‘ƒ))
5857fveq2d 6847 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) β†’ (0gβ€˜π‘…) = (0gβ€˜(Scalarβ€˜π‘ƒ)))
5958oveq1d 7373 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) β†’ ((0gβ€˜π‘…)( ·𝑠 β€˜π‘ƒ)(0(.gβ€˜(mulGrpβ€˜π‘ƒ))(var1β€˜π‘…))) = ((0gβ€˜(Scalarβ€˜π‘ƒ))( ·𝑠 β€˜π‘ƒ)(0(.gβ€˜(mulGrpβ€˜π‘ƒ))(var1β€˜π‘…))))
6059eqeq1d 2735 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) β†’ (((0gβ€˜π‘…)( ·𝑠 β€˜π‘ƒ)(0(.gβ€˜(mulGrpβ€˜π‘ƒ))(var1β€˜π‘…))) = (0gβ€˜π‘ƒ) ↔ ((0gβ€˜(Scalarβ€˜π‘ƒ))( ·𝑠 β€˜π‘ƒ)(0(.gβ€˜(mulGrpβ€˜π‘ƒ))(var1β€˜π‘…))) = (0gβ€˜π‘ƒ)))
6160adantr 482 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ β„•0 ∧ 𝑄 ∈ 𝐸)) β†’ (((0gβ€˜π‘…)( ·𝑠 β€˜π‘ƒ)(0(.gβ€˜(mulGrpβ€˜π‘ƒ))(var1β€˜π‘…))) = (0gβ€˜π‘ƒ) ↔ ((0gβ€˜(Scalarβ€˜π‘ƒ))( ·𝑠 β€˜π‘ƒ)(0(.gβ€˜(mulGrpβ€˜π‘ƒ))(var1β€˜π‘…))) = (0gβ€˜π‘ƒ)))
6255, 61mpbird 257 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ β„•0 ∧ 𝑄 ∈ 𝐸)) β†’ ((0gβ€˜π‘…)( ·𝑠 β€˜π‘ƒ)(0(.gβ€˜(mulGrpβ€˜π‘ƒ))(var1β€˜π‘…))) = (0gβ€˜π‘ƒ))
6337, 62eqtrd 2773 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ β„•0 ∧ 𝑄 ∈ 𝐸)) β†’ (((coe1β€˜(0gβ€˜π‘ƒ))β€˜πΏ)( ·𝑠 β€˜π‘ƒ)(0(.gβ€˜(mulGrpβ€˜π‘ƒ))(var1β€˜π‘…))) = (0gβ€˜π‘ƒ))
6463ifeq2d 4507 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ β„•0 ∧ 𝑄 ∈ 𝐸)) β†’ if(π‘Ž = 𝑏, (((coe1β€˜π‘„)β€˜πΏ)( ·𝑠 β€˜π‘ƒ)(0(.gβ€˜(mulGrpβ€˜π‘ƒ))(var1β€˜π‘…))), (((coe1β€˜(0gβ€˜π‘ƒ))β€˜πΏ)( ·𝑠 β€˜π‘ƒ)(0(.gβ€˜(mulGrpβ€˜π‘ƒ))(var1β€˜π‘…)))) = if(π‘Ž = 𝑏, (((coe1β€˜π‘„)β€˜πΏ)( ·𝑠 β€˜π‘ƒ)(0(.gβ€˜(mulGrpβ€˜π‘ƒ))(var1β€˜π‘…))), (0gβ€˜π‘ƒ)))
6564adantr 482 . . 3 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ β„•0 ∧ 𝑄 ∈ 𝐸)) ∧ (π‘Ž ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) β†’ if(π‘Ž = 𝑏, (((coe1β€˜π‘„)β€˜πΏ)( ·𝑠 β€˜π‘ƒ)(0(.gβ€˜(mulGrpβ€˜π‘ƒ))(var1β€˜π‘…))), (((coe1β€˜(0gβ€˜π‘ƒ))β€˜πΏ)( ·𝑠 β€˜π‘ƒ)(0(.gβ€˜(mulGrpβ€˜π‘ƒ))(var1β€˜π‘…)))) = if(π‘Ž = 𝑏, (((coe1β€˜π‘„)β€˜πΏ)( ·𝑠 β€˜π‘ƒ)(0(.gβ€˜(mulGrpβ€˜π‘ƒ))(var1β€˜π‘…))), (0gβ€˜π‘ƒ)))
6626, 65eqtrid 2785 . 2 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ β„•0 ∧ 𝑄 ∈ 𝐸)) ∧ (π‘Ž ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) β†’ (if(π‘Ž = 𝑏, ((coe1β€˜π‘„)β€˜πΏ), ((coe1β€˜(0gβ€˜π‘ƒ))β€˜πΏ))( ·𝑠 β€˜π‘ƒ)(0(.gβ€˜(mulGrpβ€˜π‘ƒ))(var1β€˜π‘…))) = if(π‘Ž = 𝑏, (((coe1β€˜π‘„)β€˜πΏ)( ·𝑠 β€˜π‘ƒ)(0(.gβ€˜(mulGrpβ€˜π‘ƒ))(var1β€˜π‘…))), (0gβ€˜π‘ƒ)))
67 simpr 486 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ β„•0 ∧ 𝑄 ∈ 𝐸)) β†’ (𝐿 ∈ β„•0 ∧ 𝑄 ∈ 𝐸))
6867ancomd 463 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ β„•0 ∧ 𝑄 ∈ 𝐸)) β†’ (𝑄 ∈ 𝐸 ∧ 𝐿 ∈ β„•0))
69 eqid 2733 . . . . . . . . 9 (coe1β€˜π‘„) = (coe1β€˜π‘„)
70 pmatcollpwscmat.k . . . . . . . . 9 𝐾 = (Baseβ€˜π‘…)
7169, 11, 3, 70coe1fvalcl 21599 . . . . . . . 8 ((𝑄 ∈ 𝐸 ∧ 𝐿 ∈ β„•0) β†’ ((coe1β€˜π‘„)β€˜πΏ) ∈ 𝐾)
7268, 71syl 17 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ β„•0 ∧ 𝑄 ∈ 𝐸)) β†’ ((coe1β€˜π‘„)β€˜πΏ) ∈ 𝐾)
7356eqcomd 2739 . . . . . . . . . . . 12 (𝑅 ∈ Ring β†’ (Scalarβ€˜π‘ƒ) = 𝑅)
7473adantl 483 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) β†’ (Scalarβ€˜π‘ƒ) = 𝑅)
7574fveq2d 6847 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) β†’ (Baseβ€˜(Scalarβ€˜π‘ƒ)) = (Baseβ€˜π‘…))
7675, 70eqtr4di 2791 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) β†’ (Baseβ€˜(Scalarβ€˜π‘ƒ)) = 𝐾)
7776eleq2d 2820 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) β†’ (((coe1β€˜π‘„)β€˜πΏ) ∈ (Baseβ€˜(Scalarβ€˜π‘ƒ)) ↔ ((coe1β€˜π‘„)β€˜πΏ) ∈ 𝐾))
7877adantr 482 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ β„•0 ∧ 𝑄 ∈ 𝐸)) β†’ (((coe1β€˜π‘„)β€˜πΏ) ∈ (Baseβ€˜(Scalarβ€˜π‘ƒ)) ↔ ((coe1β€˜π‘„)β€˜πΏ) ∈ 𝐾))
7972, 78mpbird 257 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ β„•0 ∧ 𝑄 ∈ 𝐸)) β†’ ((coe1β€˜π‘„)β€˜πΏ) ∈ (Baseβ€˜(Scalarβ€˜π‘ƒ)))
80 pmatcollpwscmat.u . . . . . . 7 π‘ˆ = (algScβ€˜π‘ƒ)
81 eqid 2733 . . . . . . 7 (Baseβ€˜(Scalarβ€˜π‘ƒ)) = (Baseβ€˜(Scalarβ€˜π‘ƒ))
82 eqid 2733 . . . . . . 7 (1rβ€˜π‘ƒ) = (1rβ€˜π‘ƒ)
8380, 51, 81, 52, 82asclval 21299 . . . . . 6 (((coe1β€˜π‘„)β€˜πΏ) ∈ (Baseβ€˜(Scalarβ€˜π‘ƒ)) β†’ (π‘ˆβ€˜((coe1β€˜π‘„)β€˜πΏ)) = (((coe1β€˜π‘„)β€˜πΏ)( ·𝑠 β€˜π‘ƒ)(1rβ€˜π‘ƒ)))
8479, 83syl 17 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ β„•0 ∧ 𝑄 ∈ 𝐸)) β†’ (π‘ˆβ€˜((coe1β€˜π‘„)β€˜πΏ)) = (((coe1β€˜π‘„)β€˜πΏ)( ·𝑠 β€˜π‘ƒ)(1rβ€˜π‘ƒ)))
853, 47, 40, 42ply1idvr1 21680 . . . . . . . 8 (𝑅 ∈ Ring β†’ (0(.gβ€˜(mulGrpβ€˜π‘ƒ))(var1β€˜π‘…)) = (1rβ€˜π‘ƒ))
8685eqcomd 2739 . . . . . . 7 (𝑅 ∈ Ring β†’ (1rβ€˜π‘ƒ) = (0(.gβ€˜(mulGrpβ€˜π‘ƒ))(var1β€˜π‘…)))
8786ad2antlr 726 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ β„•0 ∧ 𝑄 ∈ 𝐸)) β†’ (1rβ€˜π‘ƒ) = (0(.gβ€˜(mulGrpβ€˜π‘ƒ))(var1β€˜π‘…)))
8887oveq2d 7374 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ β„•0 ∧ 𝑄 ∈ 𝐸)) β†’ (((coe1β€˜π‘„)β€˜πΏ)( ·𝑠 β€˜π‘ƒ)(1rβ€˜π‘ƒ)) = (((coe1β€˜π‘„)β€˜πΏ)( ·𝑠 β€˜π‘ƒ)(0(.gβ€˜(mulGrpβ€˜π‘ƒ))(var1β€˜π‘…))))
8984, 88eqtr2d 2774 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ β„•0 ∧ 𝑄 ∈ 𝐸)) β†’ (((coe1β€˜π‘„)β€˜πΏ)( ·𝑠 β€˜π‘ƒ)(0(.gβ€˜(mulGrpβ€˜π‘ƒ))(var1β€˜π‘…))) = (π‘ˆβ€˜((coe1β€˜π‘„)β€˜πΏ)))
9089ifeq1d 4506 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ β„•0 ∧ 𝑄 ∈ 𝐸)) β†’ if(π‘Ž = 𝑏, (((coe1β€˜π‘„)β€˜πΏ)( ·𝑠 β€˜π‘ƒ)(0(.gβ€˜(mulGrpβ€˜π‘ƒ))(var1β€˜π‘…))), (0gβ€˜π‘ƒ)) = if(π‘Ž = 𝑏, (π‘ˆβ€˜((coe1β€˜π‘„)β€˜πΏ)), (0gβ€˜π‘ƒ)))
9190adantr 482 . 2 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ β„•0 ∧ 𝑄 ∈ 𝐸)) ∧ (π‘Ž ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) β†’ if(π‘Ž = 𝑏, (((coe1β€˜π‘„)β€˜πΏ)( ·𝑠 β€˜π‘ƒ)(0(.gβ€˜(mulGrpβ€˜π‘ƒ))(var1β€˜π‘…))), (0gβ€˜π‘ƒ)) = if(π‘Ž = 𝑏, (π‘ˆβ€˜((coe1β€˜π‘„)β€˜πΏ)), (0gβ€˜π‘ƒ)))
9225, 66, 913eqtrd 2777 1 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ β„•0 ∧ 𝑄 ∈ 𝐸)) ∧ (π‘Ž ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) β†’ (((coe1β€˜(π‘Žπ‘€π‘))β€˜πΏ)( ·𝑠 β€˜π‘ƒ)(0(.gβ€˜(mulGrpβ€˜π‘ƒ))(var1β€˜π‘…))) = if(π‘Ž = 𝑏, (π‘ˆβ€˜((coe1β€˜π‘„)β€˜πΏ)), (0gβ€˜π‘ƒ)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  Vcvv 3444  ifcif 4487  {csn 4587   Γ— cxp 5632  β€˜cfv 6497  (class class class)co 7358  Fincfn 8886  0cc0 11056  β„•0cn0 12418  Basecbs 17088  Scalarcsca 17141   ·𝑠 cvsca 17142  0gc0g 17326  Mndcmnd 18561  .gcmg 18877  mulGrpcmgp 19901  1rcur 19918  Ringcrg 19969  LModclmod 20336  algSccascl 21274  var1cv1 21563  Poly1cpl1 21564  coe1cco1 21565   Mat cmat 21770   matToPolyMat cmat2pmat 22069
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-cnex 11112  ax-resscn 11113  ax-1cn 11114  ax-icn 11115  ax-addcl 11116  ax-addrcl 11117  ax-mulcl 11118  ax-mulrcl 11119  ax-mulcom 11120  ax-addass 11121  ax-mulass 11122  ax-distr 11123  ax-i2m1 11124  ax-1ne0 11125  ax-1rid 11126  ax-rnegex 11127  ax-rrecex 11128  ax-cnre 11129  ax-pre-lttri 11130  ax-pre-lttrn 11131  ax-pre-ltadd 11132  ax-pre-mulgt0 11133
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-ot 4596  df-uni 4867  df-int 4909  df-iun 4957  df-iin 4958  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-se 5590  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-isom 6506  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-of 7618  df-ofr 7619  df-om 7804  df-1st 7922  df-2nd 7923  df-supp 8094  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-1o 8413  df-er 8651  df-map 8770  df-pm 8771  df-ixp 8839  df-en 8887  df-dom 8888  df-sdom 8889  df-fin 8890  df-fsupp 9309  df-sup 9383  df-oi 9451  df-card 9880  df-pnf 11196  df-mnf 11197  df-xr 11198  df-ltxr 11199  df-le 11200  df-sub 11392  df-neg 11393  df-nn 12159  df-2 12221  df-3 12222  df-4 12223  df-5 12224  df-6 12225  df-7 12226  df-8 12227  df-9 12228  df-n0 12419  df-z 12505  df-dec 12624  df-uz 12769  df-fz 13431  df-fzo 13574  df-seq 13913  df-hash 14237  df-struct 17024  df-sets 17041  df-slot 17059  df-ndx 17071  df-base 17089  df-ress 17118  df-plusg 17151  df-mulr 17152  df-sca 17154  df-vsca 17155  df-ip 17156  df-tset 17157  df-ple 17158  df-ds 17160  df-hom 17162  df-cco 17163  df-0g 17328  df-gsum 17329  df-prds 17334  df-pws 17336  df-mre 17471  df-mrc 17472  df-acs 17474  df-mgm 18502  df-sgrp 18551  df-mnd 18562  df-mhm 18606  df-submnd 18607  df-grp 18756  df-minusg 18757  df-sbg 18758  df-mulg 18878  df-subg 18930  df-ghm 19011  df-cntz 19102  df-cmn 19569  df-abl 19570  df-mgp 19902  df-ur 19919  df-ring 19971  df-subrg 20234  df-lmod 20338  df-lss 20408  df-sra 20649  df-rgmod 20650  df-dsmm 21154  df-frlm 21169  df-ascl 21277  df-psr 21327  df-mvr 21328  df-mpl 21329  df-opsr 21331  df-psr1 21567  df-vr1 21568  df-ply1 21569  df-coe1 21570  df-mamu 21749  df-mat 21771
This theorem is referenced by:  pmatcollpwscmatlem2  22155
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