MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ply1coe Structured version   Visualization version   GIF version

Theorem ply1coe 21683
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 2733 . . 3 (1o mPoly 𝑅) = (1o mPoly 𝑅)
2 psr1baslem 21572 . . 3 (β„•0 ↑m 1o) = {𝑑 ∈ (β„•0 ↑m 1o) ∣ (◑𝑑 β€œ β„•) ∈ Fin}
3 eqid 2733 . . 3 (0gβ€˜π‘…) = (0gβ€˜π‘…)
4 eqid 2733 . . 3 (1rβ€˜π‘…) = (1rβ€˜π‘…)
5 1onn 8587 . . . 4 1o ∈ Ο‰
65a1i 11 . . 3 ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡) β†’ 1o ∈ Ο‰)
7 ply1coe.p . . . 4 𝑃 = (Poly1β€˜π‘…)
8 eqid 2733 . . . 4 (PwSer1β€˜π‘…) = (PwSer1β€˜π‘…)
9 ply1coe.b . . . 4 𝐡 = (Baseβ€˜π‘ƒ)
107, 8, 9ply1bas 21582 . . 3 𝐡 = (Baseβ€˜(1o mPoly 𝑅))
11 ply1coe.n . . . 4 Β· = ( ·𝑠 β€˜π‘ƒ)
127, 1, 11ply1vsca 21613 . . 3 Β· = ( ·𝑠 β€˜(1o mPoly 𝑅))
13 simpl 484 . . 3 ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡) β†’ 𝑅 ∈ Ring)
14 simpr 486 . . 3 ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡) β†’ 𝐾 ∈ 𝐡)
151, 2, 3, 4, 6, 10, 12, 13, 14mplcoe1 21454 . 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 21594 . . . . . 6 ((𝐾 ∈ 𝐡 ∧ π‘Ž ∈ (β„•0 ↑m 1o)) β†’ (πΎβ€˜π‘Ž) = (π΄β€˜(π‘Žβ€˜βˆ…)))
1817adantll 713 . . . . 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 2733 . . . . . . 7 (mulGrpβ€˜(1o mPoly 𝑅)) = (mulGrpβ€˜(1o mPoly 𝑅))
21 eqid 2733 . . . . . . 7 (.gβ€˜(mulGrpβ€˜(1o mPoly 𝑅))) = (.gβ€˜(mulGrpβ€˜(1o mPoly 𝑅)))
22 eqid 2733 . . . . . . 7 (1o mVar 𝑅) = (1o mVar 𝑅)
23 simpll 766 . . . . . . 7 (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡) ∧ π‘Ž ∈ (β„•0 ↑m 1o)) β†’ 𝑅 ∈ Ring)
24 simpr 486 . . . . . . 7 (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡) ∧ π‘Ž ∈ (β„•0 ↑m 1o)) β†’ π‘Ž ∈ (β„•0 ↑m 1o))
25 eqidd 2734 . . . . . . . . . 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 5265 . . . . . . . . . . 11 βˆ… ∈ V
27 fveq2 6843 . . . . . . . . . . . . 13 (𝑏 = βˆ… β†’ ((1o mVar 𝑅)β€˜π‘) = ((1o mVar 𝑅)β€˜βˆ…))
2827oveq1d 7373 . . . . . . . . . . . 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 7374 . . . . . . . . . . . 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 2749 . . . . . . . . . . 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 4643 . . . . . . . . . 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 6843 . . . . . . . . . . . . 13 (π‘₯ = βˆ… β†’ ((1o mVar 𝑅)β€˜π‘₯) = ((1o mVar 𝑅)β€˜βˆ…))
3433oveq2d 7374 . . . . . . . . . . . 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 7373 . . . . . . . . . . . 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 2749 . . . . . . . . . . 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 3171 . . . . . . . . . 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 4643 . . . . . . . . 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 8420 . . . . . . . . 9 1o = {βˆ…}
4140raleqi 3310 . . . . . . . . 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 3314 . . . . . . . 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 21457 . . . . . 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 5202 . . . . . . . 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 7369 . . . . . . 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 21440 . . . . . . . . . . 11 ((1o ∈ Ο‰ ∧ 𝑅 ∈ Ring) β†’ (1o mPoly 𝑅) ∈ Ring)
485, 47mpan 689 . . . . . . . . . 10 (𝑅 ∈ Ring β†’ (1o mPoly 𝑅) ∈ Ring)
4920ringmgp 19975 . . . . . . . . . 10 ((1o mPoly 𝑅) ∈ Ring β†’ (mulGrpβ€˜(1o mPoly 𝑅)) ∈ Mnd)
5048, 49syl 17 . . . . . . . . 9 (𝑅 ∈ Ring β†’ (mulGrpβ€˜(1o mPoly 𝑅)) ∈ Mnd)
5150ad2antrr 725 . . . . . . . 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 19907 . . . . . . . . . . . . 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 19907 . . . . . . . . . . . . 13 𝐡 = (Baseβ€˜π‘€)
5857a1i 11 . . . . . . . . . . . 12 ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡) β†’ 𝐡 = (Baseβ€˜π‘€))
59 ssv 3969 . . . . . . . . . . . . 13 𝐡 βŠ† V
6059a1i 11 . . . . . . . . . . . 12 ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡) β†’ 𝐡 βŠ† V)
61 ovexd 7393 . . . . . . . . . . . 12 (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡) ∧ (π‘Ž ∈ V ∧ 𝑏 ∈ V)) β†’ (π‘Ž(+gβ€˜(mulGrpβ€˜(1o mPoly 𝑅)))𝑏) ∈ V)
62 eqid 2733 . . . . . . . . . . . . . . . . 17 (.rβ€˜π‘ƒ) = (.rβ€˜π‘ƒ)
637, 1, 62ply1mulr 21614 . . . . . . . . . . . . . . . 16 (.rβ€˜π‘ƒ) = (.rβ€˜(1o mPoly 𝑅))
6420, 63mgpplusg 19905 . . . . . . . . . . . . . . 15 (.rβ€˜π‘ƒ) = (+gβ€˜(mulGrpβ€˜(1o mPoly 𝑅)))
6556, 62mgpplusg 19905 . . . . . . . . . . . . . . 15 (.rβ€˜π‘ƒ) = (+gβ€˜π‘€)
6664, 65eqtr3i 2763 . . . . . . . . . . . . . 14 (+gβ€˜(mulGrpβ€˜(1o mPoly 𝑅))) = (+gβ€˜π‘€)
6766oveqi 7371 . . . . . . . . . . . . 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 18923 . . . . . . . . . . 11 ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡) β†’ (.gβ€˜(mulGrpβ€˜(1o mPoly 𝑅))) = ↑ )
7069oveqd 7375 . . . . . . . . . 10 ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡) β†’ ((π‘Žβ€˜βˆ…)(.gβ€˜(mulGrpβ€˜(1o mPoly 𝑅)))𝑋) = ((π‘Žβ€˜βˆ…) ↑ 𝑋))
7170adantr 482 . . . . . . . . 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 21635 . . . . . . . . . . . 12 (𝑅 ∈ Ring β†’ 𝑃 ∈ Ring)
7356ringmgp 19975 . . . . . . . . . . . 12 (𝑃 ∈ Ring β†’ 𝑀 ∈ Mnd)
7472, 73syl 17 . . . . . . . . . . 11 (𝑅 ∈ Ring β†’ 𝑀 ∈ Mnd)
7574ad2antrr 725 . . . . . . . . . 10 (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡) ∧ π‘Ž ∈ (β„•0 ↑m 1o)) β†’ 𝑀 ∈ Mnd)
76 elmapi 8790 . . . . . . . . . . . 12 (π‘Ž ∈ (β„•0 ↑m 1o) β†’ π‘Ž:1oβŸΆβ„•0)
77 0lt1o 8451 . . . . . . . . . . . 12 βˆ… ∈ 1o
78 ffvelcdm 7033 . . . . . . . . . . . 12 ((π‘Ž:1oβŸΆβ„•0 ∧ βˆ… ∈ 1o) β†’ (π‘Žβ€˜βˆ…) ∈ β„•0)
7976, 77, 78sylancl 587 . . . . . . . . . . 11 (π‘Ž ∈ (β„•0 ↑m 1o) β†’ (π‘Žβ€˜βˆ…) ∈ β„•0)
8079adantl 483 . . . . . . . . . 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 21604 . . . . . . . . . . 11 (𝑅 ∈ Ring β†’ 𝑋 ∈ 𝐡)
8382ad2antrr 725 . . . . . . . . . 10 (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡) ∧ π‘Ž ∈ (β„•0 ↑m 1o)) β†’ 𝑋 ∈ 𝐡)
8457, 53, 75, 80, 83mulgnn0cld 18902 . . . . . . . . 9 (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡) ∧ π‘Ž ∈ (β„•0 ↑m 1o)) β†’ ((π‘Žβ€˜βˆ…) ↑ 𝑋) ∈ 𝐡)
8571, 84eqeltrd 2834 . . . . . . . 8 (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡) ∧ π‘Ž ∈ (β„•0 ↑m 1o)) β†’ ((π‘Žβ€˜βˆ…)(.gβ€˜(mulGrpβ€˜(1o mPoly 𝑅)))𝑋) ∈ 𝐡)
86 fveq2 6843 . . . . . . . . . 10 (𝑐 = βˆ… β†’ (π‘Žβ€˜π‘) = (π‘Žβ€˜βˆ…))
87 fveq2 6843 . . . . . . . . . . 11 (𝑐 = βˆ… β†’ ((1o mVar 𝑅)β€˜π‘) = ((1o mVar 𝑅)β€˜βˆ…))
8881vr1val 21579 . . . . . . . . . . 11 𝑋 = ((1o mVar 𝑅)β€˜βˆ…)
8987, 88eqtr4di 2791 . . . . . . . . . 10 (𝑐 = βˆ… β†’ ((1o mVar 𝑅)β€˜π‘) = 𝑋)
9086, 89oveq12d 7376 . . . . . . . . 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 19736 . . . . . . . 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 1372 . . . . . . 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 2785 . . . . . 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 2777 . . . . 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 7376 . . . 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 5206 . . 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 7374 . 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 12424 . . . . . 6 β„•0 ∈ V
9998mptex 7174 . . . . 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 6857 . . . . 5 𝑃 ∈ V
102101a1i 11 . . . 4 ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡) β†’ 𝑃 ∈ V)
103 ovexd 7393 . . . 4 ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡) β†’ (1o mPoly 𝑅) ∈ V)
1049, 10eqtr3i 2763 . . . . 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 2733 . . . . . 6 (+gβ€˜π‘ƒ) = (+gβ€˜π‘ƒ)
1077, 1, 106ply1plusg 21612 . . . . 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 18538 . . 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 2733 . . . . 5 (0gβ€˜π‘ƒ) = (0gβ€˜π‘ƒ)
1111, 7, 110ply1mpl0 21642 . . . 4 (0gβ€˜π‘ƒ) = (0gβ€˜(1o mPoly 𝑅))
1121mpllmod 21439 . . . . . 6 ((1o ∈ Ο‰ ∧ 𝑅 ∈ Ring) β†’ (1o mPoly 𝑅) ∈ LMod)
1135, 13, 112sylancr 588 . . . . 5 ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡) β†’ (1o mPoly 𝑅) ∈ LMod)
114 lmodcmn 20385 . . . . 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 21639 . . . . . . 7 (𝑅 ∈ Ring β†’ 𝑃 ∈ LMod)
118117ad2antrr 725 . . . . . 6 (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡) ∧ π‘˜ ∈ β„•0) β†’ 𝑃 ∈ LMod)
119 eqid 2733 . . . . . . . . . 10 (Baseβ€˜π‘…) = (Baseβ€˜π‘…)
12016, 9, 7, 119coe1f 21598 . . . . . . . . 9 (𝐾 ∈ 𝐡 β†’ 𝐴:β„•0⟢(Baseβ€˜π‘…))
121120adantl 483 . . . . . . . 8 ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡) β†’ 𝐴:β„•0⟢(Baseβ€˜π‘…))
122121ffvelcdmda 7036 . . . . . . 7 (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡) ∧ π‘˜ ∈ β„•0) β†’ (π΄β€˜π‘˜) ∈ (Baseβ€˜π‘…))
1237ply1sca 21640 . . . . . . . . . 10 (𝑅 ∈ Ring β†’ 𝑅 = (Scalarβ€˜π‘ƒ))
124123eqcomd 2739 . . . . . . . . 9 (𝑅 ∈ Ring β†’ (Scalarβ€˜π‘ƒ) = 𝑅)
125124ad2antrr 725 . . . . . . . 8 (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡) ∧ π‘˜ ∈ β„•0) β†’ (Scalarβ€˜π‘ƒ) = 𝑅)
126125fveq2d 6847 . . . . . . 7 (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡) ∧ π‘˜ ∈ β„•0) β†’ (Baseβ€˜(Scalarβ€˜π‘ƒ)) = (Baseβ€˜π‘…))
127122, 126eleqtrrd 2837 . . . . . 6 (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡) ∧ π‘˜ ∈ β„•0) β†’ (π΄β€˜π‘˜) ∈ (Baseβ€˜(Scalarβ€˜π‘ƒ)))
12874ad2antrr 725 . . . . . . 7 (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡) ∧ π‘˜ ∈ β„•0) β†’ 𝑀 ∈ Mnd)
129 simpr 486 . . . . . . 7 (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡) ∧ π‘˜ ∈ β„•0) β†’ π‘˜ ∈ β„•0)
13082ad2antrr 725 . . . . . . 7 (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡) ∧ π‘˜ ∈ β„•0) β†’ 𝑋 ∈ 𝐡)
13157, 53, 128, 129, 130mulgnn0cld 18902 . . . . . 6 (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡) ∧ π‘˜ ∈ β„•0) β†’ (π‘˜ ↑ 𝑋) ∈ 𝐡)
132 eqid 2733 . . . . . . 7 (Scalarβ€˜π‘ƒ) = (Scalarβ€˜π‘ƒ)
133 eqid 2733 . . . . . . 7 (Baseβ€˜(Scalarβ€˜π‘ƒ)) = (Baseβ€˜(Scalarβ€˜π‘ƒ))
1349, 132, 11, 133lmodvscl 20354 . . . . . 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 1372 . . . . 5 (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡) ∧ π‘˜ ∈ β„•0) β†’ ((π΄β€˜π‘˜) Β· (π‘˜ ↑ 𝑋)) ∈ 𝐡)
136135fmpttd 7064 . . . 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 21682 . . . 4 ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡) β†’ (π‘˜ ∈ β„•0 ↦ ((π΄β€˜π‘˜) Β· (π‘˜ ↑ 𝑋))) finSupp (0gβ€˜π‘ƒ))
138 eqid 2733 . . . . . 6 (π‘Ž ∈ (β„•0 ↑m 1o) ↦ (π‘Žβ€˜βˆ…)) = (π‘Ž ∈ (β„•0 ↑m 1o) ↦ (π‘Žβ€˜βˆ…))
13940, 98, 26, 138mapsnf1o2 8835 . . . . 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 19698 . . 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 2734 . . . . 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 2734 . . . . 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 6843 . . . . . 6 (π‘˜ = (π‘Žβ€˜βˆ…) β†’ (π΄β€˜π‘˜) = (π΄β€˜(π‘Žβ€˜βˆ…)))
145 oveq1 7365 . . . . . 6 (π‘˜ = (π‘Žβ€˜βˆ…) β†’ (π‘˜ ↑ 𝑋) = ((π‘Žβ€˜βˆ…) ↑ 𝑋))
146144, 145oveq12d 7376 . . . . 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 7076 . . . 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 7374 . . 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 2778 . 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 2777 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 397   = wceq 1542   ∈ wcel 2107  βˆ€wral 3061  Vcvv 3444   βŠ† wss 3911  βˆ…c0 4283  ifcif 4487  {csn 4587   ↦ cmpt 5189   ∘ ccom 5638  βŸΆwf 6493  β€“1-1-ontoβ†’wf1o 6496  β€˜cfv 6497  (class class class)co 7358  Ο‰com 7803  1oc1o 8406   ↑m cmap 8768  β„•0cn0 12418  Basecbs 17088  +gcplusg 17138  .rcmulr 17139  Scalarcsca 17141   ·𝑠 cvsca 17142  0gc0g 17326   Ξ£g cgsu 17327  Mndcmnd 18561  .gcmg 18877  CMndccmn 19567  mulGrpcmgp 19901  1rcur 19918  Ringcrg 19969  LModclmod 20336   mVar cmvr 21323   mPoly cmpl 21324  PwSer1cps1 21562  var1cv1 21563  Poly1cpl1 21564  coe1cco1 21565
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-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-srg 19923  df-ring 19971  df-subrg 20234  df-lmod 20338  df-lss 20408  df-psr 21327  df-mvr 21328  df-mpl 21329  df-opsr 21331  df-psr1 21567  df-vr1 21568  df-ply1 21569  df-coe1 21570
This theorem is referenced by:  eqcoe1ply1eq  21684  pmatcollpw1lem2  22140  mp2pm2mp  22176  plypf1  25589  evls1fpws  32320  ply1degltdimlem  32374
  Copyright terms: Public domain W3C validator