Step | Hyp | Ref
| Expression |
1 | | vex 3451 |
. . . . . 6
β’ π₯ β V |
2 | | eqid 2733 |
. . . . . . 7
β’ (π β π β¦ ((πβπ) β (πβπ))) = (π β π β¦ ((πβπ) β (πβπ))) |
3 | 2 | elrnmpt 5915 |
. . . . . 6
β’ (π₯ β V β (π₯ β ran (π β π β¦ ((πβπ) β (πβπ))) β βπ β π π₯ = ((πβπ) β (πβπ)))) |
4 | 1, 3 | mp1i 13 |
. . . . 5
β’ (π β (π₯ β ran (π β π β¦ ((πβπ) β (πβπ))) β βπ β π π₯ = ((πβπ) β (πβπ)))) |
5 | | vex 3451 |
. . . . . . . 8
β’ π¦ β V |
6 | 2 | elrnmpt 5915 |
. . . . . . . 8
β’ (π¦ β V β (π¦ β ran (π β π β¦ ((πβπ) β (πβπ))) β βπ β π π¦ = ((πβπ) β (πβπ)))) |
7 | 5, 6 | mp1i 13 |
. . . . . . 7
β’ (π β (π¦ β ran (π β π β¦ ((πβπ) β (πβπ))) β βπ β π π¦ = ((πβπ) β (πβπ)))) |
8 | | fveq2 6846 |
. . . . . . . . . . 11
β’ (π = π β (πβπ) = (πβπ)) |
9 | | fveq2 6846 |
. . . . . . . . . . 11
β’ (π = π β (πβπ) = (πβπ)) |
10 | 8, 9 | oveq12d 7379 |
. . . . . . . . . 10
β’ (π = π β ((πβπ) β (πβπ)) = ((πβπ) β (πβπ))) |
11 | 10 | eqeq2d 2744 |
. . . . . . . . 9
β’ (π = π β (π¦ = ((πβπ) β (πβπ)) β π¦ = ((πβπ) β (πβπ)))) |
12 | 11 | cbvrexvw 3225 |
. . . . . . . 8
β’
(βπ β
π π¦ = ((πβπ) β (πβπ)) β βπ β π π¦ = ((πβπ) β (πβπ))) |
13 | | eqid 2733 |
. . . . . . . . . . . . . 14
β’
(Baseβπ) =
(Baseβπ) |
14 | | mplcoe2.g |
. . . . . . . . . . . . . . . 16
β’ πΊ = (mulGrpβπ) |
15 | | eqid 2733 |
. . . . . . . . . . . . . . . 16
β’
(.rβπ) = (.rβπ) |
16 | 14, 15 | mgpplusg 19908 |
. . . . . . . . . . . . . . 15
β’
(.rβπ) = (+gβπΊ) |
17 | 16 | eqcomi 2742 |
. . . . . . . . . . . . . 14
β’
(+gβπΊ) = (.rβπ) |
18 | | mplcoe2.m |
. . . . . . . . . . . . . 14
β’ β =
(.gβπΊ) |
19 | | mplcoe1.i |
. . . . . . . . . . . . . . . . . 18
β’ (π β πΌ β π) |
20 | | mplcoe5.r |
. . . . . . . . . . . . . . . . . 18
β’ (π β π
β Ring) |
21 | | mplcoe1.p |
. . . . . . . . . . . . . . . . . . 19
β’ π = (πΌ mPoly π
) |
22 | 21 | mplring 21447 |
. . . . . . . . . . . . . . . . . 18
β’ ((πΌ β π β§ π
β Ring) β π β Ring) |
23 | 19, 20, 22 | syl2anc 585 |
. . . . . . . . . . . . . . . . 17
β’ (π β π β Ring) |
24 | | ringsrg 20021 |
. . . . . . . . . . . . . . . . 17
β’ (π β Ring β π β SRing) |
25 | 23, 24 | syl 17 |
. . . . . . . . . . . . . . . 16
β’ (π β π β SRing) |
26 | 25 | adantr 482 |
. . . . . . . . . . . . . . 15
β’ ((π β§ π β π) β π β SRing) |
27 | 26 | adantr 482 |
. . . . . . . . . . . . . 14
β’ (((π β§ π β π) β§ π β π) β π β SRing) |
28 | 14, 13 | mgpbas 19910 |
. . . . . . . . . . . . . . . 16
β’
(Baseβπ) =
(BaseβπΊ) |
29 | 14 | ringmgp 19978 |
. . . . . . . . . . . . . . . . . 18
β’ (π β Ring β πΊ β Mnd) |
30 | 23, 29 | syl 17 |
. . . . . . . . . . . . . . . . 17
β’ (π β πΊ β Mnd) |
31 | 30 | adantr 482 |
. . . . . . . . . . . . . . . 16
β’ ((π β§ π β π) β πΊ β Mnd) |
32 | | mplcoe5.s |
. . . . . . . . . . . . . . . . . . 19
β’ (π β π β πΌ) |
33 | 32 | sseld 3947 |
. . . . . . . . . . . . . . . . . 18
β’ (π β (π β π β π β πΌ)) |
34 | 33 | imdistani 570 |
. . . . . . . . . . . . . . . . 17
β’ ((π β§ π β π) β (π β§ π β πΌ)) |
35 | | mplcoe5.y |
. . . . . . . . . . . . . . . . . . . 20
β’ (π β π β π·) |
36 | | mplcoe1.d |
. . . . . . . . . . . . . . . . . . . . . 22
β’ π· = {π β (β0
βm πΌ)
β£ (β‘π β β) β
Fin} |
37 | 36 | psrbag 21342 |
. . . . . . . . . . . . . . . . . . . . 21
β’ (πΌ β π β (π β π· β (π:πΌβΆβ0 β§ (β‘π β β) β
Fin))) |
38 | 19, 37 | syl 17 |
. . . . . . . . . . . . . . . . . . . 20
β’ (π β (π β π· β (π:πΌβΆβ0 β§ (β‘π β β) β
Fin))) |
39 | 35, 38 | mpbid 231 |
. . . . . . . . . . . . . . . . . . 19
β’ (π β (π:πΌβΆβ0 β§ (β‘π β β) β
Fin)) |
40 | 39 | simpld 496 |
. . . . . . . . . . . . . . . . . 18
β’ (π β π:πΌβΆβ0) |
41 | 40 | ffvelcdmda 7039 |
. . . . . . . . . . . . . . . . 17
β’ ((π β§ π β πΌ) β (πβπ) β
β0) |
42 | 34, 41 | syl 17 |
. . . . . . . . . . . . . . . 16
β’ ((π β§ π β π) β (πβπ) β
β0) |
43 | | mplcoe2.v |
. . . . . . . . . . . . . . . . 17
β’ π = (πΌ mVar π
) |
44 | 19 | adantr 482 |
. . . . . . . . . . . . . . . . 17
β’ ((π β§ π β π) β πΌ β π) |
45 | 20 | adantr 482 |
. . . . . . . . . . . . . . . . 17
β’ ((π β§ π β π) β π
β Ring) |
46 | 32 | sselda 3948 |
. . . . . . . . . . . . . . . . 17
β’ ((π β§ π β π) β π β πΌ) |
47 | 21, 43, 13, 44, 45, 46 | mvrcl 21444 |
. . . . . . . . . . . . . . . 16
β’ ((π β§ π β π) β (πβπ) β (Baseβπ)) |
48 | 28, 18, 31, 42, 47 | mulgnn0cld 18905 |
. . . . . . . . . . . . . . 15
β’ ((π β§ π β π) β ((πβπ) β (πβπ)) β (Baseβπ)) |
49 | 48 | adantr 482 |
. . . . . . . . . . . . . 14
β’ (((π β§ π β π) β§ π β π) β ((πβπ) β (πβπ)) β (Baseβπ)) |
50 | 19 | adantr 482 |
. . . . . . . . . . . . . . . 16
β’ ((π β§ π β π) β πΌ β π) |
51 | 20 | adantr 482 |
. . . . . . . . . . . . . . . 16
β’ ((π β§ π β π) β π
β Ring) |
52 | 32 | sselda 3948 |
. . . . . . . . . . . . . . . 16
β’ ((π β§ π β π) β π β πΌ) |
53 | 21, 43, 13, 50, 51, 52 | mvrcl 21444 |
. . . . . . . . . . . . . . 15
β’ ((π β§ π β π) β (πβπ) β (Baseβπ)) |
54 | 53 | adantlr 714 |
. . . . . . . . . . . . . 14
β’ (((π β§ π β π) β§ π β π) β (πβπ) β (Baseβπ)) |
55 | 40 | ffvelcdmda 7039 |
. . . . . . . . . . . . . . . 16
β’ ((π β§ π β πΌ) β (πβπ) β
β0) |
56 | 52, 55 | syldan 592 |
. . . . . . . . . . . . . . 15
β’ ((π β§ π β π) β (πβπ) β
β0) |
57 | 56 | adantlr 714 |
. . . . . . . . . . . . . 14
β’ (((π β§ π β π) β§ π β π) β (πβπ) β
β0) |
58 | 47 | adantr 482 |
. . . . . . . . . . . . . . 15
β’ (((π β§ π β π) β§ π β π) β (πβπ) β (Baseβπ)) |
59 | 42 | adantr 482 |
. . . . . . . . . . . . . . 15
β’ (((π β§ π β π) β§ π β π) β (πβπ) β
β0) |
60 | | mplcoe5.c |
. . . . . . . . . . . . . . . . 17
β’ (π β βπ₯ β πΌ βπ¦ β πΌ ((πβπ¦)(+gβπΊ)(πβπ₯)) = ((πβπ₯)(+gβπΊ)(πβπ¦))) |
61 | | fveq2 6846 |
. . . . . . . . . . . . . . . . . . . . . 22
β’ (π₯ = π β (πβπ₯) = (πβπ)) |
62 | 61 | oveq2d 7377 |
. . . . . . . . . . . . . . . . . . . . 21
β’ (π₯ = π β ((πβπ¦)(+gβπΊ)(πβπ₯)) = ((πβπ¦)(+gβπΊ)(πβπ))) |
63 | 61 | oveq1d 7376 |
. . . . . . . . . . . . . . . . . . . . 21
β’ (π₯ = π β ((πβπ₯)(+gβπΊ)(πβπ¦)) = ((πβπ)(+gβπΊ)(πβπ¦))) |
64 | 62, 63 | eqeq12d 2749 |
. . . . . . . . . . . . . . . . . . . 20
β’ (π₯ = π β (((πβπ¦)(+gβπΊ)(πβπ₯)) = ((πβπ₯)(+gβπΊ)(πβπ¦)) β ((πβπ¦)(+gβπΊ)(πβπ)) = ((πβπ)(+gβπΊ)(πβπ¦)))) |
65 | | fveq2 6846 |
. . . . . . . . . . . . . . . . . . . . . 22
β’ (π¦ = π β (πβπ¦) = (πβπ)) |
66 | 65 | oveq1d 7376 |
. . . . . . . . . . . . . . . . . . . . 21
β’ (π¦ = π β ((πβπ¦)(+gβπΊ)(πβπ)) = ((πβπ)(+gβπΊ)(πβπ))) |
67 | 65 | oveq2d 7377 |
. . . . . . . . . . . . . . . . . . . . 21
β’ (π¦ = π β ((πβπ)(+gβπΊ)(πβπ¦)) = ((πβπ)(+gβπΊ)(πβπ))) |
68 | 66, 67 | eqeq12d 2749 |
. . . . . . . . . . . . . . . . . . . 20
β’ (π¦ = π β (((πβπ¦)(+gβπΊ)(πβπ)) = ((πβπ)(+gβπΊ)(πβπ¦)) β ((πβπ)(+gβπΊ)(πβπ)) = ((πβπ)(+gβπΊ)(πβπ)))) |
69 | 64, 68 | rspc2v 3592 |
. . . . . . . . . . . . . . . . . . 19
β’ ((π β πΌ β§ π β πΌ) β (βπ₯ β πΌ βπ¦ β πΌ ((πβπ¦)(+gβπΊ)(πβπ₯)) = ((πβπ₯)(+gβπΊ)(πβπ¦)) β ((πβπ)(+gβπΊ)(πβπ)) = ((πβπ)(+gβπΊ)(πβπ)))) |
70 | 46, 52 | anim12dan 620 |
. . . . . . . . . . . . . . . . . . 19
β’ ((π β§ (π β π β§ π β π)) β (π β πΌ β§ π β πΌ)) |
71 | 69, 70 | syl11 33 |
. . . . . . . . . . . . . . . . . 18
β’
(βπ₯ β
πΌ βπ¦ β πΌ ((πβπ¦)(+gβπΊ)(πβπ₯)) = ((πβπ₯)(+gβπΊ)(πβπ¦)) β ((π β§ (π β π β§ π β π)) β ((πβπ)(+gβπΊ)(πβπ)) = ((πβπ)(+gβπΊ)(πβπ)))) |
72 | 71 | expd 417 |
. . . . . . . . . . . . . . . . 17
β’
(βπ₯ β
πΌ βπ¦ β πΌ ((πβπ¦)(+gβπΊ)(πβπ₯)) = ((πβπ₯)(+gβπΊ)(πβπ¦)) β (π β ((π β π β§ π β π) β ((πβπ)(+gβπΊ)(πβπ)) = ((πβπ)(+gβπΊ)(πβπ))))) |
73 | 60, 72 | mpcom 38 |
. . . . . . . . . . . . . . . 16
β’ (π β ((π β π β§ π β π) β ((πβπ)(+gβπΊ)(πβπ)) = ((πβπ)(+gβπΊ)(πβπ)))) |
74 | 73 | impl 457 |
. . . . . . . . . . . . . . 15
β’ (((π β§ π β π) β§ π β π) β ((πβπ)(+gβπΊ)(πβπ)) = ((πβπ)(+gβπΊ)(πβπ))) |
75 | 13, 17, 14, 18, 27, 54, 58, 59, 74 | srgpcomp 19957 |
. . . . . . . . . . . . . 14
β’ (((π β§ π β π) β§ π β π) β (((πβπ) β (πβπ))(+gβπΊ)(πβπ)) = ((πβπ)(+gβπΊ)((πβπ) β (πβπ)))) |
76 | 13, 17, 14, 18, 27, 49, 54, 57, 75 | srgpcomp 19957 |
. . . . . . . . . . . . 13
β’ (((π β§ π β π) β§ π β π) β (((πβπ) β (πβπ))(+gβπΊ)((πβπ) β (πβπ))) = (((πβπ) β (πβπ))(+gβπΊ)((πβπ) β (πβπ)))) |
77 | | oveq12 7370 |
. . . . . . . . . . . . . 14
β’ ((π₯ = ((πβπ) β (πβπ)) β§ π¦ = ((πβπ) β (πβπ))) β (π₯(+gβπΊ)π¦) = (((πβπ) β (πβπ))(+gβπΊ)((πβπ) β (πβπ)))) |
78 | | oveq12 7370 |
. . . . . . . . . . . . . . 15
β’ ((π¦ = ((πβπ) β (πβπ)) β§ π₯ = ((πβπ) β (πβπ))) β (π¦(+gβπΊ)π₯) = (((πβπ) β (πβπ))(+gβπΊ)((πβπ) β (πβπ)))) |
79 | 78 | ancoms 460 |
. . . . . . . . . . . . . 14
β’ ((π₯ = ((πβπ) β (πβπ)) β§ π¦ = ((πβπ) β (πβπ))) β (π¦(+gβπΊ)π₯) = (((πβπ) β (πβπ))(+gβπΊ)((πβπ) β (πβπ)))) |
80 | 77, 79 | eqeq12d 2749 |
. . . . . . . . . . . . 13
β’ ((π₯ = ((πβπ) β (πβπ)) β§ π¦ = ((πβπ) β (πβπ))) β ((π₯(+gβπΊ)π¦) = (π¦(+gβπΊ)π₯) β (((πβπ) β (πβπ))(+gβπΊ)((πβπ) β (πβπ))) = (((πβπ) β (πβπ))(+gβπΊ)((πβπ) β (πβπ))))) |
81 | 76, 80 | syl5ibrcom 247 |
. . . . . . . . . . . 12
β’ (((π β§ π β π) β§ π β π) β ((π₯ = ((πβπ) β (πβπ)) β§ π¦ = ((πβπ) β (πβπ))) β (π₯(+gβπΊ)π¦) = (π¦(+gβπΊ)π₯))) |
82 | 81 | expd 417 |
. . . . . . . . . . 11
β’ (((π β§ π β π) β§ π β π) β (π₯ = ((πβπ) β (πβπ)) β (π¦ = ((πβπ) β (πβπ)) β (π₯(+gβπΊ)π¦) = (π¦(+gβπΊ)π₯)))) |
83 | 82 | rexlimdva 3149 |
. . . . . . . . . 10
β’ ((π β§ π β π) β (βπ β π π₯ = ((πβπ) β (πβπ)) β (π¦ = ((πβπ) β (πβπ)) β (π₯(+gβπΊ)π¦) = (π¦(+gβπΊ)π₯)))) |
84 | 83 | com23 86 |
. . . . . . . . 9
β’ ((π β§ π β π) β (π¦ = ((πβπ) β (πβπ)) β (βπ β π π₯ = ((πβπ) β (πβπ)) β (π₯(+gβπΊ)π¦) = (π¦(+gβπΊ)π₯)))) |
85 | 84 | rexlimdva 3149 |
. . . . . . . 8
β’ (π β (βπ β π π¦ = ((πβπ) β (πβπ)) β (βπ β π π₯ = ((πβπ) β (πβπ)) β (π₯(+gβπΊ)π¦) = (π¦(+gβπΊ)π₯)))) |
86 | 12, 85 | biimtrid 241 |
. . . . . . 7
β’ (π β (βπ β π π¦ = ((πβπ) β (πβπ)) β (βπ β π π₯ = ((πβπ) β (πβπ)) β (π₯(+gβπΊ)π¦) = (π¦(+gβπΊ)π₯)))) |
87 | 7, 86 | sylbid 239 |
. . . . . 6
β’ (π β (π¦ β ran (π β π β¦ ((πβπ) β (πβπ))) β (βπ β π π₯ = ((πβπ) β (πβπ)) β (π₯(+gβπΊ)π¦) = (π¦(+gβπΊ)π₯)))) |
88 | 87 | com23 86 |
. . . . 5
β’ (π β (βπ β π π₯ = ((πβπ) β (πβπ)) β (π¦ β ran (π β π β¦ ((πβπ) β (πβπ))) β (π₯(+gβπΊ)π¦) = (π¦(+gβπΊ)π₯)))) |
89 | 4, 88 | sylbid 239 |
. . . 4
β’ (π β (π₯ β ran (π β π β¦ ((πβπ) β (πβπ))) β (π¦ β ran (π β π β¦ ((πβπ) β (πβπ))) β (π₯(+gβπΊ)π¦) = (π¦(+gβπΊ)π₯)))) |
90 | 89 | imp32 420 |
. . 3
β’ ((π β§ (π₯ β ran (π β π β¦ ((πβπ) β (πβπ))) β§ π¦ β ran (π β π β¦ ((πβπ) β (πβπ))))) β (π₯(+gβπΊ)π¦) = (π¦(+gβπΊ)π₯)) |
91 | 90 | ralrimivva 3194 |
. 2
β’ (π β βπ₯ β ran (π β π β¦ ((πβπ) β (πβπ)))βπ¦ β ran (π β π β¦ ((πβπ) β (πβπ)))(π₯(+gβπΊ)π¦) = (π¦(+gβπΊ)π₯)) |
92 | | eqid 2733 |
. . . . . 6
β’
(BaseβπΊ) =
(BaseβπΊ) |
93 | 30 | adantr 482 |
. . . . . 6
β’ ((π β§ π β π) β πΊ β Mnd) |
94 | 32 | sseld 3947 |
. . . . . . . 8
β’ (π β (π β π β π β πΌ)) |
95 | 94 | imdistani 570 |
. . . . . . 7
β’ ((π β§ π β π) β (π β§ π β πΌ)) |
96 | 95, 55 | syl 17 |
. . . . . 6
β’ ((π β§ π β π) β (πβπ) β
β0) |
97 | 53, 28 | eleqtrdi 2844 |
. . . . . 6
β’ ((π β§ π β π) β (πβπ) β (BaseβπΊ)) |
98 | 92, 18, 93, 96, 97 | mulgnn0cld 18905 |
. . . . 5
β’ ((π β§ π β π) β ((πβπ) β (πβπ)) β (BaseβπΊ)) |
99 | 98 | fmpttd 7067 |
. . . 4
β’ (π β (π β π β¦ ((πβπ) β (πβπ))):πβΆ(BaseβπΊ)) |
100 | 99 | frnd 6680 |
. . 3
β’ (π β ran (π β π β¦ ((πβπ) β (πβπ))) β (BaseβπΊ)) |
101 | | eqid 2733 |
. . . 4
β’
(+gβπΊ) = (+gβπΊ) |
102 | | eqid 2733 |
. . . 4
β’
(CntzβπΊ) =
(CntzβπΊ) |
103 | 92, 101, 102 | sscntz 19114 |
. . 3
β’ ((ran
(π β π β¦ ((πβπ) β (πβπ))) β (BaseβπΊ) β§ ran (π β π β¦ ((πβπ) β (πβπ))) β (BaseβπΊ)) β (ran (π β π β¦ ((πβπ) β (πβπ))) β ((CntzβπΊ)βran (π β π β¦ ((πβπ) β (πβπ)))) β βπ₯ β ran (π β π β¦ ((πβπ) β (πβπ)))βπ¦ β ran (π β π β¦ ((πβπ) β (πβπ)))(π₯(+gβπΊ)π¦) = (π¦(+gβπΊ)π₯))) |
104 | 100, 100,
103 | syl2anc 585 |
. 2
β’ (π β (ran (π β π β¦ ((πβπ) β (πβπ))) β ((CntzβπΊ)βran (π β π β¦ ((πβπ) β (πβπ)))) β βπ₯ β ran (π β π β¦ ((πβπ) β (πβπ)))βπ¦ β ran (π β π β¦ ((πβπ) β (πβπ)))(π₯(+gβπΊ)π¦) = (π¦(+gβπΊ)π₯))) |
105 | 91, 104 | mpbird 257 |
1
β’ (π β ran (π β π β¦ ((πβπ) β (πβπ))) β ((CntzβπΊ)βran (π β π β¦ ((πβπ) β (πβπ))))) |