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Theorem ply1coe 21811
Description: Decompose a univariate polynomial as a sum of powers. (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by AV, 7-Oct-2019.)
Hypotheses
Ref Expression
ply1coe.p 𝑃 = (Poly1β€˜π‘…)
ply1coe.x 𝑋 = (var1β€˜π‘…)
ply1coe.b 𝐡 = (Baseβ€˜π‘ƒ)
ply1coe.n Β· = ( ·𝑠 β€˜π‘ƒ)
ply1coe.m 𝑀 = (mulGrpβ€˜π‘ƒ)
ply1coe.e ↑ = (.gβ€˜π‘€)
ply1coe.a 𝐴 = (coe1β€˜πΎ)
Assertion
Ref Expression
ply1coe ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡) β†’ 𝐾 = (𝑃 Ξ£g (π‘˜ ∈ β„•0 ↦ ((π΄β€˜π‘˜) Β· (π‘˜ ↑ 𝑋)))))
Distinct variable groups:   𝐴,π‘˜   𝐡,π‘˜   π‘˜,𝐾   π‘˜,𝑋   ↑ ,π‘˜   𝑅,π‘˜   Β· ,π‘˜   𝑃,π‘˜
Allowed substitution hint:   𝑀(π‘˜)

Proof of Theorem ply1coe
Dummy variables π‘Ž 𝑏 𝑐 π‘₯ 𝑑 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2732 . . 3 (1o mPoly 𝑅) = (1o mPoly 𝑅)
2 psr1baslem 21700 . . 3 (β„•0 ↑m 1o) = {𝑑 ∈ (β„•0 ↑m 1o) ∣ (◑𝑑 β€œ β„•) ∈ Fin}
3 eqid 2732 . . 3 (0gβ€˜π‘…) = (0gβ€˜π‘…)
4 eqid 2732 . . 3 (1rβ€˜π‘…) = (1rβ€˜π‘…)
5 1onn 8635 . . . 4 1o ∈ Ο‰
65a1i 11 . . 3 ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡) β†’ 1o ∈ Ο‰)
7 ply1coe.p . . . 4 𝑃 = (Poly1β€˜π‘…)
8 eqid 2732 . . . 4 (PwSer1β€˜π‘…) = (PwSer1β€˜π‘…)
9 ply1coe.b . . . 4 𝐡 = (Baseβ€˜π‘ƒ)
107, 8, 9ply1bas 21710 . . 3 𝐡 = (Baseβ€˜(1o mPoly 𝑅))
11 ply1coe.n . . . 4 Β· = ( ·𝑠 β€˜π‘ƒ)
127, 1, 11ply1vsca 21739 . . 3 Β· = ( ·𝑠 β€˜(1o mPoly 𝑅))
13 simpl 483 . . 3 ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡) β†’ 𝑅 ∈ Ring)
14 simpr 485 . . 3 ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡) β†’ 𝐾 ∈ 𝐡)
151, 2, 3, 4, 6, 10, 12, 13, 14mplcoe1 21583 . 2 ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡) β†’ 𝐾 = ((1o mPoly 𝑅) Ξ£g (π‘Ž ∈ (β„•0 ↑m 1o) ↦ ((πΎβ€˜π‘Ž) Β· (𝑏 ∈ (β„•0 ↑m 1o) ↦ if(𝑏 = π‘Ž, (1rβ€˜π‘…), (0gβ€˜π‘…)))))))
16 ply1coe.a . . . . . . 7 𝐴 = (coe1β€˜πΎ)
1716fvcoe1 21722 . . . . . 6 ((𝐾 ∈ 𝐡 ∧ π‘Ž ∈ (β„•0 ↑m 1o)) β†’ (πΎβ€˜π‘Ž) = (π΄β€˜(π‘Žβ€˜βˆ…)))
1817adantll 712 . . . . 5 (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡) ∧ π‘Ž ∈ (β„•0 ↑m 1o)) β†’ (πΎβ€˜π‘Ž) = (π΄β€˜(π‘Žβ€˜βˆ…)))
195a1i 11 . . . . . . 7 (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡) ∧ π‘Ž ∈ (β„•0 ↑m 1o)) β†’ 1o ∈ Ο‰)
20 eqid 2732 . . . . . . 7 (mulGrpβ€˜(1o mPoly 𝑅)) = (mulGrpβ€˜(1o mPoly 𝑅))
21 eqid 2732 . . . . . . 7 (.gβ€˜(mulGrpβ€˜(1o mPoly 𝑅))) = (.gβ€˜(mulGrpβ€˜(1o mPoly 𝑅)))
22 eqid 2732 . . . . . . 7 (1o mVar 𝑅) = (1o mVar 𝑅)
23 simpll 765 . . . . . . 7 (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡) ∧ π‘Ž ∈ (β„•0 ↑m 1o)) β†’ 𝑅 ∈ Ring)
24 simpr 485 . . . . . . 7 (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡) ∧ π‘Ž ∈ (β„•0 ↑m 1o)) β†’ π‘Ž ∈ (β„•0 ↑m 1o))
25 eqidd 2733 . . . . . . . . . 10 (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡) ∧ π‘Ž ∈ (β„•0 ↑m 1o)) β†’ (((1o mVar 𝑅)β€˜βˆ…)(+gβ€˜(mulGrpβ€˜(1o mPoly 𝑅)))((1o mVar 𝑅)β€˜βˆ…)) = (((1o mVar 𝑅)β€˜βˆ…)(+gβ€˜(mulGrpβ€˜(1o mPoly 𝑅)))((1o mVar 𝑅)β€˜βˆ…)))
26 0ex 5306 . . . . . . . . . . 11 βˆ… ∈ V
27 fveq2 6888 . . . . . . . . . . . . 13 (𝑏 = βˆ… β†’ ((1o mVar 𝑅)β€˜π‘) = ((1o mVar 𝑅)β€˜βˆ…))
2827oveq1d 7420 . . . . . . . . . . . 12 (𝑏 = βˆ… β†’ (((1o mVar 𝑅)β€˜π‘)(+gβ€˜(mulGrpβ€˜(1o mPoly 𝑅)))((1o mVar 𝑅)β€˜βˆ…)) = (((1o mVar 𝑅)β€˜βˆ…)(+gβ€˜(mulGrpβ€˜(1o mPoly 𝑅)))((1o mVar 𝑅)β€˜βˆ…)))
2927oveq2d 7421 . . . . . . . . . . . 12 (𝑏 = βˆ… β†’ (((1o mVar 𝑅)β€˜βˆ…)(+gβ€˜(mulGrpβ€˜(1o mPoly 𝑅)))((1o mVar 𝑅)β€˜π‘)) = (((1o mVar 𝑅)β€˜βˆ…)(+gβ€˜(mulGrpβ€˜(1o mPoly 𝑅)))((1o mVar 𝑅)β€˜βˆ…)))
3028, 29eqeq12d 2748 . . . . . . . . . . 11 (𝑏 = βˆ… β†’ ((((1o mVar 𝑅)β€˜π‘)(+gβ€˜(mulGrpβ€˜(1o mPoly 𝑅)))((1o mVar 𝑅)β€˜βˆ…)) = (((1o mVar 𝑅)β€˜βˆ…)(+gβ€˜(mulGrpβ€˜(1o mPoly 𝑅)))((1o mVar 𝑅)β€˜π‘)) ↔ (((1o mVar 𝑅)β€˜βˆ…)(+gβ€˜(mulGrpβ€˜(1o mPoly 𝑅)))((1o mVar 𝑅)β€˜βˆ…)) = (((1o mVar 𝑅)β€˜βˆ…)(+gβ€˜(mulGrpβ€˜(1o mPoly 𝑅)))((1o mVar 𝑅)β€˜βˆ…))))
3126, 30ralsn 4684 . . . . . . . . . 10 (βˆ€π‘ ∈ {βˆ…} (((1o mVar 𝑅)β€˜π‘)(+gβ€˜(mulGrpβ€˜(1o mPoly 𝑅)))((1o mVar 𝑅)β€˜βˆ…)) = (((1o mVar 𝑅)β€˜βˆ…)(+gβ€˜(mulGrpβ€˜(1o mPoly 𝑅)))((1o mVar 𝑅)β€˜π‘)) ↔ (((1o mVar 𝑅)β€˜βˆ…)(+gβ€˜(mulGrpβ€˜(1o mPoly 𝑅)))((1o mVar 𝑅)β€˜βˆ…)) = (((1o mVar 𝑅)β€˜βˆ…)(+gβ€˜(mulGrpβ€˜(1o mPoly 𝑅)))((1o mVar 𝑅)β€˜βˆ…)))
3225, 31sylibr 233 . . . . . . . . 9 (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡) ∧ π‘Ž ∈ (β„•0 ↑m 1o)) β†’ βˆ€π‘ ∈ {βˆ…} (((1o mVar 𝑅)β€˜π‘)(+gβ€˜(mulGrpβ€˜(1o mPoly 𝑅)))((1o mVar 𝑅)β€˜βˆ…)) = (((1o mVar 𝑅)β€˜βˆ…)(+gβ€˜(mulGrpβ€˜(1o mPoly 𝑅)))((1o mVar 𝑅)β€˜π‘)))
33 fveq2 6888 . . . . . . . . . . . . 13 (π‘₯ = βˆ… β†’ ((1o mVar 𝑅)β€˜π‘₯) = ((1o mVar 𝑅)β€˜βˆ…))
3433oveq2d 7421 . . . . . . . . . . . 12 (π‘₯ = βˆ… β†’ (((1o mVar 𝑅)β€˜π‘)(+gβ€˜(mulGrpβ€˜(1o mPoly 𝑅)))((1o mVar 𝑅)β€˜π‘₯)) = (((1o mVar 𝑅)β€˜π‘)(+gβ€˜(mulGrpβ€˜(1o mPoly 𝑅)))((1o mVar 𝑅)β€˜βˆ…)))
3533oveq1d 7420 . . . . . . . . . . . 12 (π‘₯ = βˆ… β†’ (((1o mVar 𝑅)β€˜π‘₯)(+gβ€˜(mulGrpβ€˜(1o mPoly 𝑅)))((1o mVar 𝑅)β€˜π‘)) = (((1o mVar 𝑅)β€˜βˆ…)(+gβ€˜(mulGrpβ€˜(1o mPoly 𝑅)))((1o mVar 𝑅)β€˜π‘)))
3634, 35eqeq12d 2748 . . . . . . . . . . 11 (π‘₯ = βˆ… β†’ ((((1o mVar 𝑅)β€˜π‘)(+gβ€˜(mulGrpβ€˜(1o mPoly 𝑅)))((1o mVar 𝑅)β€˜π‘₯)) = (((1o mVar 𝑅)β€˜π‘₯)(+gβ€˜(mulGrpβ€˜(1o mPoly 𝑅)))((1o mVar 𝑅)β€˜π‘)) ↔ (((1o mVar 𝑅)β€˜π‘)(+gβ€˜(mulGrpβ€˜(1o mPoly 𝑅)))((1o mVar 𝑅)β€˜βˆ…)) = (((1o mVar 𝑅)β€˜βˆ…)(+gβ€˜(mulGrpβ€˜(1o mPoly 𝑅)))((1o mVar 𝑅)β€˜π‘))))
3736ralbidv 3177 . . . . . . . . . 10 (π‘₯ = βˆ… β†’ (βˆ€π‘ ∈ {βˆ…} (((1o mVar 𝑅)β€˜π‘)(+gβ€˜(mulGrpβ€˜(1o mPoly 𝑅)))((1o mVar 𝑅)β€˜π‘₯)) = (((1o mVar 𝑅)β€˜π‘₯)(+gβ€˜(mulGrpβ€˜(1o mPoly 𝑅)))((1o mVar 𝑅)β€˜π‘)) ↔ βˆ€π‘ ∈ {βˆ…} (((1o mVar 𝑅)β€˜π‘)(+gβ€˜(mulGrpβ€˜(1o mPoly 𝑅)))((1o mVar 𝑅)β€˜βˆ…)) = (((1o mVar 𝑅)β€˜βˆ…)(+gβ€˜(mulGrpβ€˜(1o mPoly 𝑅)))((1o mVar 𝑅)β€˜π‘))))
3826, 37ralsn 4684 . . . . . . . . 9 (βˆ€π‘₯ ∈ {βˆ…}βˆ€π‘ ∈ {βˆ…} (((1o mVar 𝑅)β€˜π‘)(+gβ€˜(mulGrpβ€˜(1o mPoly 𝑅)))((1o mVar 𝑅)β€˜π‘₯)) = (((1o mVar 𝑅)β€˜π‘₯)(+gβ€˜(mulGrpβ€˜(1o mPoly 𝑅)))((1o mVar 𝑅)β€˜π‘)) ↔ βˆ€π‘ ∈ {βˆ…} (((1o mVar 𝑅)β€˜π‘)(+gβ€˜(mulGrpβ€˜(1o mPoly 𝑅)))((1o mVar 𝑅)β€˜βˆ…)) = (((1o mVar 𝑅)β€˜βˆ…)(+gβ€˜(mulGrpβ€˜(1o mPoly 𝑅)))((1o mVar 𝑅)β€˜π‘)))
3932, 38sylibr 233 . . . . . . . 8 (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡) ∧ π‘Ž ∈ (β„•0 ↑m 1o)) β†’ βˆ€π‘₯ ∈ {βˆ…}βˆ€π‘ ∈ {βˆ…} (((1o mVar 𝑅)β€˜π‘)(+gβ€˜(mulGrpβ€˜(1o mPoly 𝑅)))((1o mVar 𝑅)β€˜π‘₯)) = (((1o mVar 𝑅)β€˜π‘₯)(+gβ€˜(mulGrpβ€˜(1o mPoly 𝑅)))((1o mVar 𝑅)β€˜π‘)))
40 df1o2 8469 . . . . . . . . 9 1o = {βˆ…}
4140raleqi 3323 . . . . . . . . 9 (βˆ€π‘ ∈ 1o (((1o mVar 𝑅)β€˜π‘)(+gβ€˜(mulGrpβ€˜(1o mPoly 𝑅)))((1o mVar 𝑅)β€˜π‘₯)) = (((1o mVar 𝑅)β€˜π‘₯)(+gβ€˜(mulGrpβ€˜(1o mPoly 𝑅)))((1o mVar 𝑅)β€˜π‘)) ↔ βˆ€π‘ ∈ {βˆ…} (((1o mVar 𝑅)β€˜π‘)(+gβ€˜(mulGrpβ€˜(1o mPoly 𝑅)))((1o mVar 𝑅)β€˜π‘₯)) = (((1o mVar 𝑅)β€˜π‘₯)(+gβ€˜(mulGrpβ€˜(1o mPoly 𝑅)))((1o mVar 𝑅)β€˜π‘)))
4240, 41raleqbii 3338 . . . . . . . 8 (βˆ€π‘₯ ∈ 1o βˆ€π‘ ∈ 1o (((1o mVar 𝑅)β€˜π‘)(+gβ€˜(mulGrpβ€˜(1o mPoly 𝑅)))((1o mVar 𝑅)β€˜π‘₯)) = (((1o mVar 𝑅)β€˜π‘₯)(+gβ€˜(mulGrpβ€˜(1o mPoly 𝑅)))((1o mVar 𝑅)β€˜π‘)) ↔ βˆ€π‘₯ ∈ {βˆ…}βˆ€π‘ ∈ {βˆ…} (((1o mVar 𝑅)β€˜π‘)(+gβ€˜(mulGrpβ€˜(1o mPoly 𝑅)))((1o mVar 𝑅)β€˜π‘₯)) = (((1o mVar 𝑅)β€˜π‘₯)(+gβ€˜(mulGrpβ€˜(1o mPoly 𝑅)))((1o mVar 𝑅)β€˜π‘)))
4339, 42sylibr 233 . . . . . . 7 (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡) ∧ π‘Ž ∈ (β„•0 ↑m 1o)) β†’ βˆ€π‘₯ ∈ 1o βˆ€π‘ ∈ 1o (((1o mVar 𝑅)β€˜π‘)(+gβ€˜(mulGrpβ€˜(1o mPoly 𝑅)))((1o mVar 𝑅)β€˜π‘₯)) = (((1o mVar 𝑅)β€˜π‘₯)(+gβ€˜(mulGrpβ€˜(1o mPoly 𝑅)))((1o mVar 𝑅)β€˜π‘)))
441, 2, 3, 4, 19, 20, 21, 22, 23, 24, 43mplcoe5 21586 . . . . . 6 (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡) ∧ π‘Ž ∈ (β„•0 ↑m 1o)) β†’ (𝑏 ∈ (β„•0 ↑m 1o) ↦ if(𝑏 = π‘Ž, (1rβ€˜π‘…), (0gβ€˜π‘…))) = ((mulGrpβ€˜(1o mPoly 𝑅)) Ξ£g (𝑐 ∈ 1o ↦ ((π‘Žβ€˜π‘)(.gβ€˜(mulGrpβ€˜(1o mPoly 𝑅)))((1o mVar 𝑅)β€˜π‘)))))
4540mpteq1i 5243 . . . . . . . 8 (𝑐 ∈ 1o ↦ ((π‘Žβ€˜π‘)(.gβ€˜(mulGrpβ€˜(1o mPoly 𝑅)))((1o mVar 𝑅)β€˜π‘))) = (𝑐 ∈ {βˆ…} ↦ ((π‘Žβ€˜π‘)(.gβ€˜(mulGrpβ€˜(1o mPoly 𝑅)))((1o mVar 𝑅)β€˜π‘)))
4645oveq2i 7416 . . . . . . 7 ((mulGrpβ€˜(1o mPoly 𝑅)) Ξ£g (𝑐 ∈ 1o ↦ ((π‘Žβ€˜π‘)(.gβ€˜(mulGrpβ€˜(1o mPoly 𝑅)))((1o mVar 𝑅)β€˜π‘)))) = ((mulGrpβ€˜(1o mPoly 𝑅)) Ξ£g (𝑐 ∈ {βˆ…} ↦ ((π‘Žβ€˜π‘)(.gβ€˜(mulGrpβ€˜(1o mPoly 𝑅)))((1o mVar 𝑅)β€˜π‘))))
471mplring 21569 . . . . . . . . . . 11 ((1o ∈ Ο‰ ∧ 𝑅 ∈ Ring) β†’ (1o mPoly 𝑅) ∈ Ring)
485, 47mpan 688 . . . . . . . . . 10 (𝑅 ∈ Ring β†’ (1o mPoly 𝑅) ∈ Ring)
4920ringmgp 20055 . . . . . . . . . 10 ((1o mPoly 𝑅) ∈ Ring β†’ (mulGrpβ€˜(1o mPoly 𝑅)) ∈ Mnd)
5048, 49syl 17 . . . . . . . . 9 (𝑅 ∈ Ring β†’ (mulGrpβ€˜(1o mPoly 𝑅)) ∈ Mnd)
5150ad2antrr 724 . . . . . . . 8 (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡) ∧ π‘Ž ∈ (β„•0 ↑m 1o)) β†’ (mulGrpβ€˜(1o mPoly 𝑅)) ∈ Mnd)
5226a1i 11 . . . . . . . 8 (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡) ∧ π‘Ž ∈ (β„•0 ↑m 1o)) β†’ βˆ… ∈ V)
53 ply1coe.e . . . . . . . . . . . 12 ↑ = (.gβ€˜π‘€)
5420, 10mgpbas 19987 . . . . . . . . . . . . 13 𝐡 = (Baseβ€˜(mulGrpβ€˜(1o mPoly 𝑅)))
5554a1i 11 . . . . . . . . . . . 12 ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡) β†’ 𝐡 = (Baseβ€˜(mulGrpβ€˜(1o mPoly 𝑅))))
56 ply1coe.m . . . . . . . . . . . . . 14 𝑀 = (mulGrpβ€˜π‘ƒ)
5756, 9mgpbas 19987 . . . . . . . . . . . . 13 𝐡 = (Baseβ€˜π‘€)
5857a1i 11 . . . . . . . . . . . 12 ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡) β†’ 𝐡 = (Baseβ€˜π‘€))
59 ssv 4005 . . . . . . . . . . . . 13 𝐡 βŠ† V
6059a1i 11 . . . . . . . . . . . 12 ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡) β†’ 𝐡 βŠ† V)
61 ovexd 7440 . . . . . . . . . . . 12 (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡) ∧ (π‘Ž ∈ V ∧ 𝑏 ∈ V)) β†’ (π‘Ž(+gβ€˜(mulGrpβ€˜(1o mPoly 𝑅)))𝑏) ∈ V)
62 eqid 2732 . . . . . . . . . . . . . . . . 17 (.rβ€˜π‘ƒ) = (.rβ€˜π‘ƒ)
637, 1, 62ply1mulr 21740 . . . . . . . . . . . . . . . 16 (.rβ€˜π‘ƒ) = (.rβ€˜(1o mPoly 𝑅))
6420, 63mgpplusg 19985 . . . . . . . . . . . . . . 15 (.rβ€˜π‘ƒ) = (+gβ€˜(mulGrpβ€˜(1o mPoly 𝑅)))
6556, 62mgpplusg 19985 . . . . . . . . . . . . . . 15 (.rβ€˜π‘ƒ) = (+gβ€˜π‘€)
6664, 65eqtr3i 2762 . . . . . . . . . . . . . 14 (+gβ€˜(mulGrpβ€˜(1o mPoly 𝑅))) = (+gβ€˜π‘€)
6766oveqi 7418 . . . . . . . . . . . . 13 (π‘Ž(+gβ€˜(mulGrpβ€˜(1o mPoly 𝑅)))𝑏) = (π‘Ž(+gβ€˜π‘€)𝑏)
6867a1i 11 . . . . . . . . . . . 12 (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡) ∧ (π‘Ž ∈ V ∧ 𝑏 ∈ V)) β†’ (π‘Ž(+gβ€˜(mulGrpβ€˜(1o mPoly 𝑅)))𝑏) = (π‘Ž(+gβ€˜π‘€)𝑏))
6921, 53, 55, 58, 60, 61, 68mulgpropd 18990 . . . . . . . . . . 11 ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡) β†’ (.gβ€˜(mulGrpβ€˜(1o mPoly 𝑅))) = ↑ )
7069oveqd 7422 . . . . . . . . . 10 ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡) β†’ ((π‘Žβ€˜βˆ…)(.gβ€˜(mulGrpβ€˜(1o mPoly 𝑅)))𝑋) = ((π‘Žβ€˜βˆ…) ↑ 𝑋))
7170adantr 481 . . . . . . . . 9 (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡) ∧ π‘Ž ∈ (β„•0 ↑m 1o)) β†’ ((π‘Žβ€˜βˆ…)(.gβ€˜(mulGrpβ€˜(1o mPoly 𝑅)))𝑋) = ((π‘Žβ€˜βˆ…) ↑ 𝑋))
727ply1ring 21761 . . . . . . . . . . . 12 (𝑅 ∈ Ring β†’ 𝑃 ∈ Ring)
7356ringmgp 20055 . . . . . . . . . . . 12 (𝑃 ∈ Ring β†’ 𝑀 ∈ Mnd)
7472, 73syl 17 . . . . . . . . . . 11 (𝑅 ∈ Ring β†’ 𝑀 ∈ Mnd)
7574ad2antrr 724 . . . . . . . . . 10 (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡) ∧ π‘Ž ∈ (β„•0 ↑m 1o)) β†’ 𝑀 ∈ Mnd)
76 elmapi 8839 . . . . . . . . . . . 12 (π‘Ž ∈ (β„•0 ↑m 1o) β†’ π‘Ž:1oβŸΆβ„•0)
77 0lt1o 8500 . . . . . . . . . . . 12 βˆ… ∈ 1o
78 ffvelcdm 7080 . . . . . . . . . . . 12 ((π‘Ž:1oβŸΆβ„•0 ∧ βˆ… ∈ 1o) β†’ (π‘Žβ€˜βˆ…) ∈ β„•0)
7976, 77, 78sylancl 586 . . . . . . . . . . 11 (π‘Ž ∈ (β„•0 ↑m 1o) β†’ (π‘Žβ€˜βˆ…) ∈ β„•0)
8079adantl 482 . . . . . . . . . 10 (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡) ∧ π‘Ž ∈ (β„•0 ↑m 1o)) β†’ (π‘Žβ€˜βˆ…) ∈ β„•0)
81 ply1coe.x . . . . . . . . . . . 12 𝑋 = (var1β€˜π‘…)
8281, 7, 9vr1cl 21732 . . . . . . . . . . 11 (𝑅 ∈ Ring β†’ 𝑋 ∈ 𝐡)
8382ad2antrr 724 . . . . . . . . . 10 (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡) ∧ π‘Ž ∈ (β„•0 ↑m 1o)) β†’ 𝑋 ∈ 𝐡)
8457, 53, 75, 80, 83mulgnn0cld 18969 . . . . . . . . 9 (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡) ∧ π‘Ž ∈ (β„•0 ↑m 1o)) β†’ ((π‘Žβ€˜βˆ…) ↑ 𝑋) ∈ 𝐡)
8571, 84eqeltrd 2833 . . . . . . . 8 (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡) ∧ π‘Ž ∈ (β„•0 ↑m 1o)) β†’ ((π‘Žβ€˜βˆ…)(.gβ€˜(mulGrpβ€˜(1o mPoly 𝑅)))𝑋) ∈ 𝐡)
86 fveq2 6888 . . . . . . . . . 10 (𝑐 = βˆ… β†’ (π‘Žβ€˜π‘) = (π‘Žβ€˜βˆ…))
87 fveq2 6888 . . . . . . . . . . 11 (𝑐 = βˆ… β†’ ((1o mVar 𝑅)β€˜π‘) = ((1o mVar 𝑅)β€˜βˆ…))
8881vr1val 21707 . . . . . . . . . . 11 𝑋 = ((1o mVar 𝑅)β€˜βˆ…)
8987, 88eqtr4di 2790 . . . . . . . . . 10 (𝑐 = βˆ… β†’ ((1o mVar 𝑅)β€˜π‘) = 𝑋)
9086, 89oveq12d 7423 . . . . . . . . 9 (𝑐 = βˆ… β†’ ((π‘Žβ€˜π‘)(.gβ€˜(mulGrpβ€˜(1o mPoly 𝑅)))((1o mVar 𝑅)β€˜π‘)) = ((π‘Žβ€˜βˆ…)(.gβ€˜(mulGrpβ€˜(1o mPoly 𝑅)))𝑋))
9154, 90gsumsn 19816 . . . . . . . 8 (((mulGrpβ€˜(1o mPoly 𝑅)) ∈ Mnd ∧ βˆ… ∈ V ∧ ((π‘Žβ€˜βˆ…)(.gβ€˜(mulGrpβ€˜(1o mPoly 𝑅)))𝑋) ∈ 𝐡) β†’ ((mulGrpβ€˜(1o mPoly 𝑅)) Ξ£g (𝑐 ∈ {βˆ…} ↦ ((π‘Žβ€˜π‘)(.gβ€˜(mulGrpβ€˜(1o mPoly 𝑅)))((1o mVar 𝑅)β€˜π‘)))) = ((π‘Žβ€˜βˆ…)(.gβ€˜(mulGrpβ€˜(1o mPoly 𝑅)))𝑋))
9251, 52, 85, 91syl3anc 1371 . . . . . . 7 (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡) ∧ π‘Ž ∈ (β„•0 ↑m 1o)) β†’ ((mulGrpβ€˜(1o mPoly 𝑅)) Ξ£g (𝑐 ∈ {βˆ…} ↦ ((π‘Žβ€˜π‘)(.gβ€˜(mulGrpβ€˜(1o mPoly 𝑅)))((1o mVar 𝑅)β€˜π‘)))) = ((π‘Žβ€˜βˆ…)(.gβ€˜(mulGrpβ€˜(1o mPoly 𝑅)))𝑋))
9346, 92eqtrid 2784 . . . . . 6 (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡) ∧ π‘Ž ∈ (β„•0 ↑m 1o)) β†’ ((mulGrpβ€˜(1o mPoly 𝑅)) Ξ£g (𝑐 ∈ 1o ↦ ((π‘Žβ€˜π‘)(.gβ€˜(mulGrpβ€˜(1o mPoly 𝑅)))((1o mVar 𝑅)β€˜π‘)))) = ((π‘Žβ€˜βˆ…)(.gβ€˜(mulGrpβ€˜(1o mPoly 𝑅)))𝑋))
9444, 93, 713eqtrd 2776 . . . . 5 (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡) ∧ π‘Ž ∈ (β„•0 ↑m 1o)) β†’ (𝑏 ∈ (β„•0 ↑m 1o) ↦ if(𝑏 = π‘Ž, (1rβ€˜π‘…), (0gβ€˜π‘…))) = ((π‘Žβ€˜βˆ…) ↑ 𝑋))
9518, 94oveq12d 7423 . . . 4 (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡) ∧ π‘Ž ∈ (β„•0 ↑m 1o)) β†’ ((πΎβ€˜π‘Ž) Β· (𝑏 ∈ (β„•0 ↑m 1o) ↦ if(𝑏 = π‘Ž, (1rβ€˜π‘…), (0gβ€˜π‘…)))) = ((π΄β€˜(π‘Žβ€˜βˆ…)) Β· ((π‘Žβ€˜βˆ…) ↑ 𝑋)))
9695mpteq2dva 5247 . . 3 ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡) β†’ (π‘Ž ∈ (β„•0 ↑m 1o) ↦ ((πΎβ€˜π‘Ž) Β· (𝑏 ∈ (β„•0 ↑m 1o) ↦ if(𝑏 = π‘Ž, (1rβ€˜π‘…), (0gβ€˜π‘…))))) = (π‘Ž ∈ (β„•0 ↑m 1o) ↦ ((π΄β€˜(π‘Žβ€˜βˆ…)) Β· ((π‘Žβ€˜βˆ…) ↑ 𝑋))))
9796oveq2d 7421 . 2 ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡) β†’ ((1o mPoly 𝑅) Ξ£g (π‘Ž ∈ (β„•0 ↑m 1o) ↦ ((πΎβ€˜π‘Ž) Β· (𝑏 ∈ (β„•0 ↑m 1o) ↦ if(𝑏 = π‘Ž, (1rβ€˜π‘…), (0gβ€˜π‘…)))))) = ((1o mPoly 𝑅) Ξ£g (π‘Ž ∈ (β„•0 ↑m 1o) ↦ ((π΄β€˜(π‘Žβ€˜βˆ…)) Β· ((π‘Žβ€˜βˆ…) ↑ 𝑋)))))
98 nn0ex 12474 . . . . . 6 β„•0 ∈ V
9998mptex 7221 . . . . 5 (π‘˜ ∈ β„•0 ↦ ((π΄β€˜π‘˜) Β· (π‘˜ ↑ 𝑋))) ∈ V
10099a1i 11 . . . 4 ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡) β†’ (π‘˜ ∈ β„•0 ↦ ((π΄β€˜π‘˜) Β· (π‘˜ ↑ 𝑋))) ∈ V)
1017fvexi 6902 . . . . 5 𝑃 ∈ V
102101a1i 11 . . . 4 ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡) β†’ 𝑃 ∈ V)
103 ovexd 7440 . . . 4 ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡) β†’ (1o mPoly 𝑅) ∈ V)
1049, 10eqtr3i 2762 . . . . 5 (Baseβ€˜π‘ƒ) = (Baseβ€˜(1o mPoly 𝑅))
105104a1i 11 . . . 4 ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡) β†’ (Baseβ€˜π‘ƒ) = (Baseβ€˜(1o mPoly 𝑅)))
106 eqid 2732 . . . . . 6 (+gβ€˜π‘ƒ) = (+gβ€˜π‘ƒ)
1077, 1, 106ply1plusg 21738 . . . . 5 (+gβ€˜π‘ƒ) = (+gβ€˜(1o mPoly 𝑅))
108107a1i 11 . . . 4 ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡) β†’ (+gβ€˜π‘ƒ) = (+gβ€˜(1o mPoly 𝑅)))
109100, 102, 103, 105, 108gsumpropd 18593 . . 3 ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡) β†’ (𝑃 Ξ£g (π‘˜ ∈ β„•0 ↦ ((π΄β€˜π‘˜) Β· (π‘˜ ↑ 𝑋)))) = ((1o mPoly 𝑅) Ξ£g (π‘˜ ∈ β„•0 ↦ ((π΄β€˜π‘˜) Β· (π‘˜ ↑ 𝑋)))))
110 eqid 2732 . . . . 5 (0gβ€˜π‘ƒ) = (0gβ€˜π‘ƒ)
1111, 7, 110ply1mpl0 21768 . . . 4 (0gβ€˜π‘ƒ) = (0gβ€˜(1o mPoly 𝑅))
1121mpllmod 21568 . . . . . 6 ((1o ∈ Ο‰ ∧ 𝑅 ∈ Ring) β†’ (1o mPoly 𝑅) ∈ LMod)
1135, 13, 112sylancr 587 . . . . 5 ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡) β†’ (1o mPoly 𝑅) ∈ LMod)
114 lmodcmn 20512 . . . . 5 ((1o mPoly 𝑅) ∈ LMod β†’ (1o mPoly 𝑅) ∈ CMnd)
115113, 114syl 17 . . . 4 ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡) β†’ (1o mPoly 𝑅) ∈ CMnd)
11698a1i 11 . . . 4 ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡) β†’ β„•0 ∈ V)
1177ply1lmod 21765 . . . . . . 7 (𝑅 ∈ Ring β†’ 𝑃 ∈ LMod)
118117ad2antrr 724 . . . . . 6 (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡) ∧ π‘˜ ∈ β„•0) β†’ 𝑃 ∈ LMod)
119 eqid 2732 . . . . . . . . . 10 (Baseβ€˜π‘…) = (Baseβ€˜π‘…)
12016, 9, 7, 119coe1f 21726 . . . . . . . . 9 (𝐾 ∈ 𝐡 β†’ 𝐴:β„•0⟢(Baseβ€˜π‘…))
121120adantl 482 . . . . . . . 8 ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡) β†’ 𝐴:β„•0⟢(Baseβ€˜π‘…))
122121ffvelcdmda 7083 . . . . . . 7 (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡) ∧ π‘˜ ∈ β„•0) β†’ (π΄β€˜π‘˜) ∈ (Baseβ€˜π‘…))
1237ply1sca 21766 . . . . . . . . . 10 (𝑅 ∈ Ring β†’ 𝑅 = (Scalarβ€˜π‘ƒ))
124123eqcomd 2738 . . . . . . . . 9 (𝑅 ∈ Ring β†’ (Scalarβ€˜π‘ƒ) = 𝑅)
125124ad2antrr 724 . . . . . . . 8 (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡) ∧ π‘˜ ∈ β„•0) β†’ (Scalarβ€˜π‘ƒ) = 𝑅)
126125fveq2d 6892 . . . . . . 7 (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡) ∧ π‘˜ ∈ β„•0) β†’ (Baseβ€˜(Scalarβ€˜π‘ƒ)) = (Baseβ€˜π‘…))
127122, 126eleqtrrd 2836 . . . . . 6 (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡) ∧ π‘˜ ∈ β„•0) β†’ (π΄β€˜π‘˜) ∈ (Baseβ€˜(Scalarβ€˜π‘ƒ)))
12874ad2antrr 724 . . . . . . 7 (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡) ∧ π‘˜ ∈ β„•0) β†’ 𝑀 ∈ Mnd)
129 simpr 485 . . . . . . 7 (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡) ∧ π‘˜ ∈ β„•0) β†’ π‘˜ ∈ β„•0)
13082ad2antrr 724 . . . . . . 7 (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡) ∧ π‘˜ ∈ β„•0) β†’ 𝑋 ∈ 𝐡)
13157, 53, 128, 129, 130mulgnn0cld 18969 . . . . . 6 (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡) ∧ π‘˜ ∈ β„•0) β†’ (π‘˜ ↑ 𝑋) ∈ 𝐡)
132 eqid 2732 . . . . . . 7 (Scalarβ€˜π‘ƒ) = (Scalarβ€˜π‘ƒ)
133 eqid 2732 . . . . . . 7 (Baseβ€˜(Scalarβ€˜π‘ƒ)) = (Baseβ€˜(Scalarβ€˜π‘ƒ))
1349, 132, 11, 133lmodvscl 20481 . . . . . 6 ((𝑃 ∈ LMod ∧ (π΄β€˜π‘˜) ∈ (Baseβ€˜(Scalarβ€˜π‘ƒ)) ∧ (π‘˜ ↑ 𝑋) ∈ 𝐡) β†’ ((π΄β€˜π‘˜) Β· (π‘˜ ↑ 𝑋)) ∈ 𝐡)
135118, 127, 131, 134syl3anc 1371 . . . . 5 (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡) ∧ π‘˜ ∈ β„•0) β†’ ((π΄β€˜π‘˜) Β· (π‘˜ ↑ 𝑋)) ∈ 𝐡)
136135fmpttd 7111 . . . 4 ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡) β†’ (π‘˜ ∈ β„•0 ↦ ((π΄β€˜π‘˜) Β· (π‘˜ ↑ 𝑋))):β„•0⟢𝐡)
1377, 81, 9, 11, 56, 53, 16ply1coefsupp 21810 . . . 4 ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡) β†’ (π‘˜ ∈ β„•0 ↦ ((π΄β€˜π‘˜) Β· (π‘˜ ↑ 𝑋))) finSupp (0gβ€˜π‘ƒ))
138 eqid 2732 . . . . . 6 (π‘Ž ∈ (β„•0 ↑m 1o) ↦ (π‘Žβ€˜βˆ…)) = (π‘Ž ∈ (β„•0 ↑m 1o) ↦ (π‘Žβ€˜βˆ…))
13940, 98, 26, 138mapsnf1o2 8884 . . . . 5 (π‘Ž ∈ (β„•0 ↑m 1o) ↦ (π‘Žβ€˜βˆ…)):(β„•0 ↑m 1o)–1-1-ontoβ†’β„•0
140139a1i 11 . . . 4 ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡) β†’ (π‘Ž ∈ (β„•0 ↑m 1o) ↦ (π‘Žβ€˜βˆ…)):(β„•0 ↑m 1o)–1-1-ontoβ†’β„•0)
14110, 111, 115, 116, 136, 137, 140gsumf1o 19778 . . 3 ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡) β†’ ((1o mPoly 𝑅) Ξ£g (π‘˜ ∈ β„•0 ↦ ((π΄β€˜π‘˜) Β· (π‘˜ ↑ 𝑋)))) = ((1o mPoly 𝑅) Ξ£g ((π‘˜ ∈ β„•0 ↦ ((π΄β€˜π‘˜) Β· (π‘˜ ↑ 𝑋))) ∘ (π‘Ž ∈ (β„•0 ↑m 1o) ↦ (π‘Žβ€˜βˆ…)))))
142 eqidd 2733 . . . . 5 ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡) β†’ (π‘Ž ∈ (β„•0 ↑m 1o) ↦ (π‘Žβ€˜βˆ…)) = (π‘Ž ∈ (β„•0 ↑m 1o) ↦ (π‘Žβ€˜βˆ…)))
143 eqidd 2733 . . . . 5 ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡) β†’ (π‘˜ ∈ β„•0 ↦ ((π΄β€˜π‘˜) Β· (π‘˜ ↑ 𝑋))) = (π‘˜ ∈ β„•0 ↦ ((π΄β€˜π‘˜) Β· (π‘˜ ↑ 𝑋))))
144 fveq2 6888 . . . . . 6 (π‘˜ = (π‘Žβ€˜βˆ…) β†’ (π΄β€˜π‘˜) = (π΄β€˜(π‘Žβ€˜βˆ…)))
145 oveq1 7412 . . . . . 6 (π‘˜ = (π‘Žβ€˜βˆ…) β†’ (π‘˜ ↑ 𝑋) = ((π‘Žβ€˜βˆ…) ↑ 𝑋))
146144, 145oveq12d 7423 . . . . 5 (π‘˜ = (π‘Žβ€˜βˆ…) β†’ ((π΄β€˜π‘˜) Β· (π‘˜ ↑ 𝑋)) = ((π΄β€˜(π‘Žβ€˜βˆ…)) Β· ((π‘Žβ€˜βˆ…) ↑ 𝑋)))
14780, 142, 143, 146fmptco 7123 . . . 4 ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡) β†’ ((π‘˜ ∈ β„•0 ↦ ((π΄β€˜π‘˜) Β· (π‘˜ ↑ 𝑋))) ∘ (π‘Ž ∈ (β„•0 ↑m 1o) ↦ (π‘Žβ€˜βˆ…))) = (π‘Ž ∈ (β„•0 ↑m 1o) ↦ ((π΄β€˜(π‘Žβ€˜βˆ…)) Β· ((π‘Žβ€˜βˆ…) ↑ 𝑋))))
148147oveq2d 7421 . . 3 ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡) β†’ ((1o mPoly 𝑅) Ξ£g ((π‘˜ ∈ β„•0 ↦ ((π΄β€˜π‘˜) Β· (π‘˜ ↑ 𝑋))) ∘ (π‘Ž ∈ (β„•0 ↑m 1o) ↦ (π‘Žβ€˜βˆ…)))) = ((1o mPoly 𝑅) Ξ£g (π‘Ž ∈ (β„•0 ↑m 1o) ↦ ((π΄β€˜(π‘Žβ€˜βˆ…)) Β· ((π‘Žβ€˜βˆ…) ↑ 𝑋)))))
149109, 141, 1483eqtrrd 2777 . 2 ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡) β†’ ((1o mPoly 𝑅) Ξ£g (π‘Ž ∈ (β„•0 ↑m 1o) ↦ ((π΄β€˜(π‘Žβ€˜βˆ…)) Β· ((π‘Žβ€˜βˆ…) ↑ 𝑋)))) = (𝑃 Ξ£g (π‘˜ ∈ β„•0 ↦ ((π΄β€˜π‘˜) Β· (π‘˜ ↑ 𝑋)))))
15015, 97, 1493eqtrd 2776 1 ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡) β†’ 𝐾 = (𝑃 Ξ£g (π‘˜ ∈ β„•0 ↦ ((π΄β€˜π‘˜) Β· (π‘˜ ↑ 𝑋)))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061  Vcvv 3474   βŠ† wss 3947  βˆ…c0 4321  ifcif 4527  {csn 4627   ↦ cmpt 5230   ∘ ccom 5679  βŸΆwf 6536  β€“1-1-ontoβ†’wf1o 6539  β€˜cfv 6540  (class class class)co 7405  Ο‰com 7851  1oc1o 8455   ↑m cmap 8816  β„•0cn0 12468  Basecbs 17140  +gcplusg 17193  .rcmulr 17194  Scalarcsca 17196   ·𝑠 cvsca 17197  0gc0g 17381   Ξ£g cgsu 17382  Mndcmnd 18621  .gcmg 18944  CMndccmn 19642  mulGrpcmgp 19981  1rcur 19998  Ringcrg 20049  LModclmod 20463   mVar cmvr 21449   mPoly cmpl 21450  PwSer1cps1 21690  var1cv1 21691  Poly1cpl1 21692  coe1cco1 21693
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-tp 4632  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-iin 4999  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-se 5631  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-isom 6549  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-of 7666  df-ofr 7667  df-om 7852  df-1st 7971  df-2nd 7972  df-supp 8143  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-1o 8462  df-er 8699  df-map 8818  df-pm 8819  df-ixp 8888  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-fsupp 9358  df-sup 9433  df-oi 9501  df-card 9930  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-nn 12209  df-2 12271  df-3 12272  df-4 12273  df-5 12274  df-6 12275  df-7 12276  df-8 12277  df-9 12278  df-n0 12469  df-z 12555  df-dec 12674  df-uz 12819  df-fz 13481  df-fzo 13624  df-seq 13963  df-hash 14287  df-struct 17076  df-sets 17093  df-slot 17111  df-ndx 17123  df-base 17141  df-ress 17170  df-plusg 17206  df-mulr 17207  df-sca 17209  df-vsca 17210  df-ip 17211  df-tset 17212  df-ple 17213  df-ds 17215  df-hom 17217  df-cco 17218  df-0g 17383  df-gsum 17384  df-prds 17389  df-pws 17391  df-mre 17526  df-mrc 17527  df-acs 17529  df-mgm 18557  df-sgrp 18606  df-mnd 18622  df-mhm 18667  df-submnd 18668  df-grp 18818  df-minusg 18819  df-sbg 18820  df-mulg 18945  df-subg 18997  df-ghm 19084  df-cntz 19175  df-cmn 19644  df-abl 19645  df-mgp 19982  df-ur 19999  df-srg 20003  df-ring 20051  df-subrg 20353  df-lmod 20465  df-lss 20535  df-psr 21453  df-mvr 21454  df-mpl 21455  df-opsr 21457  df-psr1 21695  df-vr1 21696  df-ply1 21697  df-coe1 21698
This theorem is referenced by:  eqcoe1ply1eq  21812  pmatcollpw1lem2  22268  mp2pm2mp  22304  plypf1  25717  evls1fpws  32634  ply1degltdimlem  32695
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