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Theorem ply1coe 22224
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 2728 . . 3 (1o mPoly 𝑅) = (1o mPoly 𝑅)
2 psr1baslem 22111 . . 3 (β„•0 ↑m 1o) = {𝑑 ∈ (β„•0 ↑m 1o) ∣ (◑𝑑 β€œ β„•) ∈ Fin}
3 eqid 2728 . . 3 (0gβ€˜π‘…) = (0gβ€˜π‘…)
4 eqid 2728 . . 3 (1rβ€˜π‘…) = (1rβ€˜π‘…)
5 1onn 8667 . . . 4 1o ∈ Ο‰
65a1i 11 . . 3 ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡) β†’ 1o ∈ Ο‰)
7 ply1coe.p . . . 4 𝑃 = (Poly1β€˜π‘…)
8 eqid 2728 . . . 4 (PwSer1β€˜π‘…) = (PwSer1β€˜π‘…)
9 ply1coe.b . . . 4 𝐡 = (Baseβ€˜π‘ƒ)
107, 8, 9ply1bas 22121 . . 3 𝐡 = (Baseβ€˜(1o mPoly 𝑅))
11 ply1coe.n . . . 4 Β· = ( ·𝑠 β€˜π‘ƒ)
127, 1, 11ply1vsca 22150 . . 3 Β· = ( ·𝑠 β€˜(1o mPoly 𝑅))
13 simpl 481 . . 3 ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡) β†’ 𝑅 ∈ Ring)
14 simpr 483 . . 3 ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡) β†’ 𝐾 ∈ 𝐡)
151, 2, 3, 4, 6, 10, 12, 13, 14mplcoe1 21982 . 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 22133 . . . . . 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 2728 . . . . . . 7 (mulGrpβ€˜(1o mPoly 𝑅)) = (mulGrpβ€˜(1o mPoly 𝑅))
21 eqid 2728 . . . . . . 7 (.gβ€˜(mulGrpβ€˜(1o mPoly 𝑅))) = (.gβ€˜(mulGrpβ€˜(1o mPoly 𝑅)))
22 eqid 2728 . . . . . . 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 483 . . . . . . 7 (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡) ∧ π‘Ž ∈ (β„•0 ↑m 1o)) β†’ π‘Ž ∈ (β„•0 ↑m 1o))
25 eqidd 2729 . . . . . . . . . 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 5311 . . . . . . . . . . 11 βˆ… ∈ V
27 fveq2 6902 . . . . . . . . . . . . 13 (𝑏 = βˆ… β†’ ((1o mVar 𝑅)β€˜π‘) = ((1o mVar 𝑅)β€˜βˆ…))
2827oveq1d 7441 . . . . . . . . . . . 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 7442 . . . . . . . . . . . 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 2744 . . . . . . . . . . 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 4690 . . . . . . . . . 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 6902 . . . . . . . . . . . . 13 (π‘₯ = βˆ… β†’ ((1o mVar 𝑅)β€˜π‘₯) = ((1o mVar 𝑅)β€˜βˆ…))
3433oveq2d 7442 . . . . . . . . . . . 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 7441 . . . . . . . . . . . 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 2744 . . . . . . . . . . 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 3175 . . . . . . . . . 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 4690 . . . . . . . . 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 8500 . . . . . . . . 9 1o = {βˆ…}
4140raleqi 3321 . . . . . . . . 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 3336 . . . . . . . 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 21985 . . . . . 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 5248 . . . . . . . 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 7437 . . . . . . 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 21968 . . . . . . . . . . 11 ((1o ∈ Ο‰ ∧ 𝑅 ∈ Ring) β†’ (1o mPoly 𝑅) ∈ Ring)
485, 47mpan 688 . . . . . . . . . 10 (𝑅 ∈ Ring β†’ (1o mPoly 𝑅) ∈ Ring)
4920ringmgp 20186 . . . . . . . . . 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 20087 . . . . . . . . . . . . 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 20087 . . . . . . . . . . . . 13 𝐡 = (Baseβ€˜π‘€)
5857a1i 11 . . . . . . . . . . . 12 ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡) β†’ 𝐡 = (Baseβ€˜π‘€))
59 ssv 4006 . . . . . . . . . . . . 13 𝐡 βŠ† V
6059a1i 11 . . . . . . . . . . . 12 ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡) β†’ 𝐡 βŠ† V)
61 ovexd 7461 . . . . . . . . . . . 12 (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡) ∧ (π‘Ž ∈ V ∧ 𝑏 ∈ V)) β†’ (π‘Ž(+gβ€˜(mulGrpβ€˜(1o mPoly 𝑅)))𝑏) ∈ V)
62 eqid 2728 . . . . . . . . . . . . . . . . 17 (.rβ€˜π‘ƒ) = (.rβ€˜π‘ƒ)
637, 1, 62ply1mulr 22151 . . . . . . . . . . . . . . . 16 (.rβ€˜π‘ƒ) = (.rβ€˜(1o mPoly 𝑅))
6420, 63mgpplusg 20085 . . . . . . . . . . . . . . 15 (.rβ€˜π‘ƒ) = (+gβ€˜(mulGrpβ€˜(1o mPoly 𝑅)))
6556, 62mgpplusg 20085 . . . . . . . . . . . . . . 15 (.rβ€˜π‘ƒ) = (+gβ€˜π‘€)
6664, 65eqtr3i 2758 . . . . . . . . . . . . . 14 (+gβ€˜(mulGrpβ€˜(1o mPoly 𝑅))) = (+gβ€˜π‘€)
6766oveqi 7439 . . . . . . . . . . . . 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 19078 . . . . . . . . . . 11 ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡) β†’ (.gβ€˜(mulGrpβ€˜(1o mPoly 𝑅))) = ↑ )
7069oveqd 7443 . . . . . . . . . 10 ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡) β†’ ((π‘Žβ€˜βˆ…)(.gβ€˜(mulGrpβ€˜(1o mPoly 𝑅)))𝑋) = ((π‘Žβ€˜βˆ…) ↑ 𝑋))
7170adantr 479 . . . . . . . . 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 22173 . . . . . . . . . . . 12 (𝑅 ∈ Ring β†’ 𝑃 ∈ Ring)
7356ringmgp 20186 . . . . . . . . . . . 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 8874 . . . . . . . . . . . 12 (π‘Ž ∈ (β„•0 ↑m 1o) β†’ π‘Ž:1oβŸΆβ„•0)
77 0lt1o 8531 . . . . . . . . . . . 12 βˆ… ∈ 1o
78 ffvelcdm 7096 . . . . . . . . . . . 12 ((π‘Ž:1oβŸΆβ„•0 ∧ βˆ… ∈ 1o) β†’ (π‘Žβ€˜βˆ…) ∈ β„•0)
7976, 77, 78sylancl 584 . . . . . . . . . . 11 (π‘Ž ∈ (β„•0 ↑m 1o) β†’ (π‘Žβ€˜βˆ…) ∈ β„•0)
8079adantl 480 . . . . . . . . . 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 22143 . . . . . . . . . . 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 19057 . . . . . . . . 9 (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡) ∧ π‘Ž ∈ (β„•0 ↑m 1o)) β†’ ((π‘Žβ€˜βˆ…) ↑ 𝑋) ∈ 𝐡)
8571, 84eqeltrd 2829 . . . . . . . 8 (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡) ∧ π‘Ž ∈ (β„•0 ↑m 1o)) β†’ ((π‘Žβ€˜βˆ…)(.gβ€˜(mulGrpβ€˜(1o mPoly 𝑅)))𝑋) ∈ 𝐡)
86 fveq2 6902 . . . . . . . . . 10 (𝑐 = βˆ… β†’ (π‘Žβ€˜π‘) = (π‘Žβ€˜βˆ…))
87 fveq2 6902 . . . . . . . . . . 11 (𝑐 = βˆ… β†’ ((1o mVar 𝑅)β€˜π‘) = ((1o mVar 𝑅)β€˜βˆ…))
8881vr1val 22118 . . . . . . . . . . 11 𝑋 = ((1o mVar 𝑅)β€˜βˆ…)
8987, 88eqtr4di 2786 . . . . . . . . . 10 (𝑐 = βˆ… β†’ ((1o mVar 𝑅)β€˜π‘) = 𝑋)
9086, 89oveq12d 7444 . . . . . . . . 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 19916 . . . . . . . 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 1368 . . . . . . 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 2780 . . . . . 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 2772 . . . . 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 7444 . . . 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 5252 . . 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 7442 . 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 12516 . . . . . 6 β„•0 ∈ V
9998mptex 7241 . . . . 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 6916 . . . . 5 𝑃 ∈ V
102101a1i 11 . . . 4 ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡) β†’ 𝑃 ∈ V)
103 ovexd 7461 . . . 4 ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡) β†’ (1o mPoly 𝑅) ∈ V)
1049, 10eqtr3i 2758 . . . . 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 2728 . . . . . 6 (+gβ€˜π‘ƒ) = (+gβ€˜π‘ƒ)
1077, 1, 106ply1plusg 22149 . . . . 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 18645 . . 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 2728 . . . . 5 (0gβ€˜π‘ƒ) = (0gβ€˜π‘ƒ)
1111, 7, 110ply1mpl0 22181 . . . 4 (0gβ€˜π‘ƒ) = (0gβ€˜(1o mPoly 𝑅))
1121mpllmod 21967 . . . . . 6 ((1o ∈ Ο‰ ∧ 𝑅 ∈ Ring) β†’ (1o mPoly 𝑅) ∈ LMod)
1135, 13, 112sylancr 585 . . . . 5 ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡) β†’ (1o mPoly 𝑅) ∈ LMod)
114 lmodcmn 20800 . . . . 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 22177 . . . . . . 7 (𝑅 ∈ Ring β†’ 𝑃 ∈ LMod)
118117ad2antrr 724 . . . . . 6 (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡) ∧ π‘˜ ∈ β„•0) β†’ 𝑃 ∈ LMod)
119 eqid 2728 . . . . . . . . . 10 (Baseβ€˜π‘…) = (Baseβ€˜π‘…)
12016, 9, 7, 119coe1f 22137 . . . . . . . . 9 (𝐾 ∈ 𝐡 β†’ 𝐴:β„•0⟢(Baseβ€˜π‘…))
121120adantl 480 . . . . . . . 8 ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡) β†’ 𝐴:β„•0⟢(Baseβ€˜π‘…))
122121ffvelcdmda 7099 . . . . . . 7 (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡) ∧ π‘˜ ∈ β„•0) β†’ (π΄β€˜π‘˜) ∈ (Baseβ€˜π‘…))
1237ply1sca 22178 . . . . . . . . . 10 (𝑅 ∈ Ring β†’ 𝑅 = (Scalarβ€˜π‘ƒ))
124123eqcomd 2734 . . . . . . . . 9 (𝑅 ∈ Ring β†’ (Scalarβ€˜π‘ƒ) = 𝑅)
125124ad2antrr 724 . . . . . . . 8 (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡) ∧ π‘˜ ∈ β„•0) β†’ (Scalarβ€˜π‘ƒ) = 𝑅)
126125fveq2d 6906 . . . . . . 7 (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡) ∧ π‘˜ ∈ β„•0) β†’ (Baseβ€˜(Scalarβ€˜π‘ƒ)) = (Baseβ€˜π‘…))
127122, 126eleqtrrd 2832 . . . . . 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 483 . . . . . . 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 19057 . . . . . 6 (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡) ∧ π‘˜ ∈ β„•0) β†’ (π‘˜ ↑ 𝑋) ∈ 𝐡)
132 eqid 2728 . . . . . . 7 (Scalarβ€˜π‘ƒ) = (Scalarβ€˜π‘ƒ)
133 eqid 2728 . . . . . . 7 (Baseβ€˜(Scalarβ€˜π‘ƒ)) = (Baseβ€˜(Scalarβ€˜π‘ƒ))
1349, 132, 11, 133lmodvscl 20768 . . . . . 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 1368 . . . . 5 (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡) ∧ π‘˜ ∈ β„•0) β†’ ((π΄β€˜π‘˜) Β· (π‘˜ ↑ 𝑋)) ∈ 𝐡)
136135fmpttd 7130 . . . 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 22223 . . . 4 ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡) β†’ (π‘˜ ∈ β„•0 ↦ ((π΄β€˜π‘˜) Β· (π‘˜ ↑ 𝑋))) finSupp (0gβ€˜π‘ƒ))
138 eqid 2728 . . . . . 6 (π‘Ž ∈ (β„•0 ↑m 1o) ↦ (π‘Žβ€˜βˆ…)) = (π‘Ž ∈ (β„•0 ↑m 1o) ↦ (π‘Žβ€˜βˆ…))
13940, 98, 26, 138mapsnf1o2 8919 . . . . 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 19878 . . 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 2729 . . . . 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 2729 . . . . 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 6902 . . . . . 6 (π‘˜ = (π‘Žβ€˜βˆ…) β†’ (π΄β€˜π‘˜) = (π΄β€˜(π‘Žβ€˜βˆ…)))
145 oveq1 7433 . . . . . 6 (π‘˜ = (π‘Žβ€˜βˆ…) β†’ (π‘˜ ↑ 𝑋) = ((π‘Žβ€˜βˆ…) ↑ 𝑋))
146144, 145oveq12d 7444 . . . . 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 7144 . . . 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 7442 . . 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 2773 . 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 2772 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 394   = wceq 1533   ∈ wcel 2098  βˆ€wral 3058  Vcvv 3473   βŠ† wss 3949  βˆ…c0 4326  ifcif 4532  {csn 4632   ↦ cmpt 5235   ∘ ccom 5686  βŸΆwf 6549  β€“1-1-ontoβ†’wf1o 6552  β€˜cfv 6553  (class class class)co 7426  Ο‰com 7876  1oc1o 8486   ↑m cmap 8851  β„•0cn0 12510  Basecbs 17187  +gcplusg 17240  .rcmulr 17241  Scalarcsca 17243   ·𝑠 cvsca 17244  0gc0g 17428   Ξ£g cgsu 17429  Mndcmnd 18701  .gcmg 19030  CMndccmn 19742  mulGrpcmgp 20081  1rcur 20128  Ringcrg 20180  LModclmod 20750   mVar cmvr 21845   mPoly cmpl 21846  PwSer1cps1 22101  var1cv1 22102  Poly1cpl1 22103  coe1cco1 22104
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-cnex 11202  ax-resscn 11203  ax-1cn 11204  ax-icn 11205  ax-addcl 11206  ax-addrcl 11207  ax-mulcl 11208  ax-mulrcl 11209  ax-mulcom 11210  ax-addass 11211  ax-mulass 11212  ax-distr 11213  ax-i2m1 11214  ax-1ne0 11215  ax-1rid 11216  ax-rnegex 11217  ax-rrecex 11218  ax-cnre 11219  ax-pre-lttri 11220  ax-pre-lttrn 11221  ax-pre-ltadd 11222  ax-pre-mulgt0 11223
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-tp 4637  df-op 4639  df-uni 4913  df-int 4954  df-iun 5002  df-iin 5003  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-se 5638  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-isom 6562  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-of 7691  df-ofr 7692  df-om 7877  df-1st 7999  df-2nd 8000  df-supp 8172  df-frecs 8293  df-wrecs 8324  df-recs 8398  df-rdg 8437  df-1o 8493  df-er 8731  df-map 8853  df-pm 8854  df-ixp 8923  df-en 8971  df-dom 8972  df-sdom 8973  df-fin 8974  df-fsupp 9394  df-sup 9473  df-oi 9541  df-card 9970  df-pnf 11288  df-mnf 11289  df-xr 11290  df-ltxr 11291  df-le 11292  df-sub 11484  df-neg 11485  df-nn 12251  df-2 12313  df-3 12314  df-4 12315  df-5 12316  df-6 12317  df-7 12318  df-8 12319  df-9 12320  df-n0 12511  df-z 12597  df-dec 12716  df-uz 12861  df-fz 13525  df-fzo 13668  df-seq 14007  df-hash 14330  df-struct 17123  df-sets 17140  df-slot 17158  df-ndx 17170  df-base 17188  df-ress 17217  df-plusg 17253  df-mulr 17254  df-sca 17256  df-vsca 17257  df-ip 17258  df-tset 17259  df-ple 17260  df-ds 17262  df-hom 17264  df-cco 17265  df-0g 17430  df-gsum 17431  df-prds 17436  df-pws 17438  df-mre 17573  df-mrc 17574  df-acs 17576  df-mgm 18607  df-sgrp 18686  df-mnd 18702  df-mhm 18747  df-submnd 18748  df-grp 18900  df-minusg 18901  df-sbg 18902  df-mulg 19031  df-subg 19085  df-ghm 19175  df-cntz 19275  df-cmn 19744  df-abl 19745  df-mgp 20082  df-rng 20100  df-ur 20129  df-srg 20134  df-ring 20182  df-subrng 20490  df-subrg 20515  df-lmod 20752  df-lss 20823  df-psr 21849  df-mvr 21850  df-mpl 21851  df-opsr 21853  df-psr1 22106  df-vr1 22107  df-ply1 22108  df-coe1 22109
This theorem is referenced by:  eqcoe1ply1eq  22225  evls1fpws  22295  pmatcollpw1lem2  22697  mp2pm2mp  22733  plypf1  26166  ply1degltdimlem  33353
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