Step | Hyp | Ref
| Expression |
1 | | simpr 486 |
. 2
β’ ((π β oMnd β§ (π βΎs π΄) β Mnd) β (π βΎs π΄) β Mnd) |
2 | | omndtos 31962 |
. . . 4
β’ (π β oMnd β π β Toset) |
3 | 2 | adantr 482 |
. . 3
β’ ((π β oMnd β§ (π βΎs π΄) β Mnd) β π β Toset) |
4 | | reldmress 17119 |
. . . . . . . 8
β’ Rel dom
βΎs |
5 | 4 | ovprc2 7398 |
. . . . . . 7
β’ (Β¬
π΄ β V β (π βΎs π΄) = β
) |
6 | 5 | fveq2d 6847 |
. . . . . 6
β’ (Β¬
π΄ β V β
(Baseβ(π
βΎs π΄)) =
(Baseββ
)) |
7 | 6 | adantl 483 |
. . . . 5
β’ (((π β oMnd β§ (π βΎs π΄) β Mnd) β§ Β¬ π΄ β V) β
(Baseβ(π
βΎs π΄)) =
(Baseββ
)) |
8 | | base0 17093 |
. . . . 5
β’ β
=
(Baseββ
) |
9 | 7, 8 | eqtr4di 2791 |
. . . 4
β’ (((π β oMnd β§ (π βΎs π΄) β Mnd) β§ Β¬ π΄ β V) β
(Baseβ(π
βΎs π΄)) =
β
) |
10 | | eqid 2733 |
. . . . . . . 8
β’
(Baseβ(π
βΎs π΄)) =
(Baseβ(π
βΎs π΄)) |
11 | | eqid 2733 |
. . . . . . . 8
β’
(0gβ(π βΎs π΄)) = (0gβ(π βΎs π΄)) |
12 | 10, 11 | mndidcl 18576 |
. . . . . . 7
β’ ((π βΎs π΄) β Mnd β
(0gβ(π
βΎs π΄))
β (Baseβ(π
βΎs π΄))) |
13 | 12 | ne0d 4296 |
. . . . . 6
β’ ((π βΎs π΄) β Mnd β
(Baseβ(π
βΎs π΄))
β β
) |
14 | 13 | ad2antlr 726 |
. . . . 5
β’ (((π β oMnd β§ (π βΎs π΄) β Mnd) β§ Β¬ π΄ β V) β
(Baseβ(π
βΎs π΄))
β β
) |
15 | 14 | neneqd 2945 |
. . . 4
β’ (((π β oMnd β§ (π βΎs π΄) β Mnd) β§ Β¬ π΄ β V) β Β¬
(Baseβ(π
βΎs π΄)) =
β
) |
16 | 9, 15 | condan 817 |
. . 3
β’ ((π β oMnd β§ (π βΎs π΄) β Mnd) β π΄ β V) |
17 | | resstos 31876 |
. . 3
β’ ((π β Toset β§ π΄ β V) β (π βΎs π΄) β Toset) |
18 | 3, 16, 17 | syl2anc 585 |
. 2
β’ ((π β oMnd β§ (π βΎs π΄) β Mnd) β (π βΎs π΄) β Toset) |
19 | | simplll 774 |
. . . . . 6
β’ ((((π β oMnd β§ (π βΎs π΄) β Mnd) β§ (π β (Baseβ(π βΎs π΄)) β§ π β (Baseβ(π βΎs π΄)) β§ π β (Baseβ(π βΎs π΄)))) β§ π(leβ(π βΎs π΄))π) β π β oMnd) |
20 | | eqid 2733 |
. . . . . . . . . . 11
β’ (π βΎs π΄) = (π βΎs π΄) |
21 | | eqid 2733 |
. . . . . . . . . . 11
β’
(Baseβπ) =
(Baseβπ) |
22 | 20, 21 | ressbas 17123 |
. . . . . . . . . 10
β’ (π΄ β V β (π΄ β© (Baseβπ)) = (Baseβ(π βΎs π΄))) |
23 | | inss2 4190 |
. . . . . . . . . 10
β’ (π΄ β© (Baseβπ)) β (Baseβπ) |
24 | 22, 23 | eqsstrrdi 4000 |
. . . . . . . . 9
β’ (π΄ β V β
(Baseβ(π
βΎs π΄))
β (Baseβπ)) |
25 | 16, 24 | syl 17 |
. . . . . . . 8
β’ ((π β oMnd β§ (π βΎs π΄) β Mnd) β
(Baseβ(π
βΎs π΄))
β (Baseβπ)) |
26 | 25 | ad2antrr 725 |
. . . . . . 7
β’ ((((π β oMnd β§ (π βΎs π΄) β Mnd) β§ (π β (Baseβ(π βΎs π΄)) β§ π β (Baseβ(π βΎs π΄)) β§ π β (Baseβ(π βΎs π΄)))) β§ π(leβ(π βΎs π΄))π) β (Baseβ(π βΎs π΄)) β (Baseβπ)) |
27 | | simplr1 1216 |
. . . . . . 7
β’ ((((π β oMnd β§ (π βΎs π΄) β Mnd) β§ (π β (Baseβ(π βΎs π΄)) β§ π β (Baseβ(π βΎs π΄)) β§ π β (Baseβ(π βΎs π΄)))) β§ π(leβ(π βΎs π΄))π) β π β (Baseβ(π βΎs π΄))) |
28 | 26, 27 | sseldd 3946 |
. . . . . 6
β’ ((((π β oMnd β§ (π βΎs π΄) β Mnd) β§ (π β (Baseβ(π βΎs π΄)) β§ π β (Baseβ(π βΎs π΄)) β§ π β (Baseβ(π βΎs π΄)))) β§ π(leβ(π βΎs π΄))π) β π β (Baseβπ)) |
29 | | simplr2 1217 |
. . . . . . 7
β’ ((((π β oMnd β§ (π βΎs π΄) β Mnd) β§ (π β (Baseβ(π βΎs π΄)) β§ π β (Baseβ(π βΎs π΄)) β§ π β (Baseβ(π βΎs π΄)))) β§ π(leβ(π βΎs π΄))π) β π β (Baseβ(π βΎs π΄))) |
30 | 26, 29 | sseldd 3946 |
. . . . . 6
β’ ((((π β oMnd β§ (π βΎs π΄) β Mnd) β§ (π β (Baseβ(π βΎs π΄)) β§ π β (Baseβ(π βΎs π΄)) β§ π β (Baseβ(π βΎs π΄)))) β§ π(leβ(π βΎs π΄))π) β π β (Baseβπ)) |
31 | | simplr3 1218 |
. . . . . . 7
β’ ((((π β oMnd β§ (π βΎs π΄) β Mnd) β§ (π β (Baseβ(π βΎs π΄)) β§ π β (Baseβ(π βΎs π΄)) β§ π β (Baseβ(π βΎs π΄)))) β§ π(leβ(π βΎs π΄))π) β π β (Baseβ(π βΎs π΄))) |
32 | 26, 31 | sseldd 3946 |
. . . . . 6
β’ ((((π β oMnd β§ (π βΎs π΄) β Mnd) β§ (π β (Baseβ(π βΎs π΄)) β§ π β (Baseβ(π βΎs π΄)) β§ π β (Baseβ(π βΎs π΄)))) β§ π(leβ(π βΎs π΄))π) β π β (Baseβπ)) |
33 | | eqid 2733 |
. . . . . . . . . . 11
β’
(leβπ) =
(leβπ) |
34 | 20, 33 | ressle 17266 |
. . . . . . . . . 10
β’ (π΄ β V β (leβπ) = (leβ(π βΎs π΄))) |
35 | 16, 34 | syl 17 |
. . . . . . . . 9
β’ ((π β oMnd β§ (π βΎs π΄) β Mnd) β
(leβπ) =
(leβ(π
βΎs π΄))) |
36 | 35 | adantr 482 |
. . . . . . . 8
β’ (((π β oMnd β§ (π βΎs π΄) β Mnd) β§ (π β (Baseβ(π βΎs π΄)) β§ π β (Baseβ(π βΎs π΄)) β§ π β (Baseβ(π βΎs π΄)))) β (leβπ) = (leβ(π βΎs π΄))) |
37 | 36 | breqd 5117 |
. . . . . . 7
β’ (((π β oMnd β§ (π βΎs π΄) β Mnd) β§ (π β (Baseβ(π βΎs π΄)) β§ π β (Baseβ(π βΎs π΄)) β§ π β (Baseβ(π βΎs π΄)))) β (π(leβπ)π β π(leβ(π βΎs π΄))π)) |
38 | 37 | biimpar 479 |
. . . . . 6
β’ ((((π β oMnd β§ (π βΎs π΄) β Mnd) β§ (π β (Baseβ(π βΎs π΄)) β§ π β (Baseβ(π βΎs π΄)) β§ π β (Baseβ(π βΎs π΄)))) β§ π(leβ(π βΎs π΄))π) β π(leβπ)π) |
39 | | eqid 2733 |
. . . . . . 7
β’
(+gβπ) = (+gβπ) |
40 | 21, 33, 39 | omndadd 31963 |
. . . . . 6
β’ ((π β oMnd β§ (π β (Baseβπ) β§ π β (Baseβπ) β§ π β (Baseβπ)) β§ π(leβπ)π) β (π(+gβπ)π)(leβπ)(π(+gβπ)π)) |
41 | 19, 28, 30, 32, 38, 40 | syl131anc 1384 |
. . . . 5
β’ ((((π β oMnd β§ (π βΎs π΄) β Mnd) β§ (π β (Baseβ(π βΎs π΄)) β§ π β (Baseβ(π βΎs π΄)) β§ π β (Baseβ(π βΎs π΄)))) β§ π(leβ(π βΎs π΄))π) β (π(+gβπ)π)(leβπ)(π(+gβπ)π)) |
42 | 16 | adantr 482 |
. . . . . . . . 9
β’ (((π β oMnd β§ (π βΎs π΄) β Mnd) β§ (π β (Baseβ(π βΎs π΄)) β§ π β (Baseβ(π βΎs π΄)) β§ π β (Baseβ(π βΎs π΄)))) β π΄ β V) |
43 | 20, 39 | ressplusg 17176 |
. . . . . . . . 9
β’ (π΄ β V β
(+gβπ) =
(+gβ(π
βΎs π΄))) |
44 | 42, 43 | syl 17 |
. . . . . . . 8
β’ (((π β oMnd β§ (π βΎs π΄) β Mnd) β§ (π β (Baseβ(π βΎs π΄)) β§ π β (Baseβ(π βΎs π΄)) β§ π β (Baseβ(π βΎs π΄)))) β (+gβπ) = (+gβ(π βΎs π΄))) |
45 | 44 | oveqd 7375 |
. . . . . . 7
β’ (((π β oMnd β§ (π βΎs π΄) β Mnd) β§ (π β (Baseβ(π βΎs π΄)) β§ π β (Baseβ(π βΎs π΄)) β§ π β (Baseβ(π βΎs π΄)))) β (π(+gβπ)π) = (π(+gβ(π βΎs π΄))π)) |
46 | 42, 34 | syl 17 |
. . . . . . 7
β’ (((π β oMnd β§ (π βΎs π΄) β Mnd) β§ (π β (Baseβ(π βΎs π΄)) β§ π β (Baseβ(π βΎs π΄)) β§ π β (Baseβ(π βΎs π΄)))) β (leβπ) = (leβ(π βΎs π΄))) |
47 | 44 | oveqd 7375 |
. . . . . . 7
β’ (((π β oMnd β§ (π βΎs π΄) β Mnd) β§ (π β (Baseβ(π βΎs π΄)) β§ π β (Baseβ(π βΎs π΄)) β§ π β (Baseβ(π βΎs π΄)))) β (π(+gβπ)π) = (π(+gβ(π βΎs π΄))π)) |
48 | 45, 46, 47 | breq123d 5120 |
. . . . . 6
β’ (((π β oMnd β§ (π βΎs π΄) β Mnd) β§ (π β (Baseβ(π βΎs π΄)) β§ π β (Baseβ(π βΎs π΄)) β§ π β (Baseβ(π βΎs π΄)))) β ((π(+gβπ)π)(leβπ)(π(+gβπ)π) β (π(+gβ(π βΎs π΄))π)(leβ(π βΎs π΄))(π(+gβ(π βΎs π΄))π))) |
49 | 48 | adantr 482 |
. . . . 5
β’ ((((π β oMnd β§ (π βΎs π΄) β Mnd) β§ (π β (Baseβ(π βΎs π΄)) β§ π β (Baseβ(π βΎs π΄)) β§ π β (Baseβ(π βΎs π΄)))) β§ π(leβ(π βΎs π΄))π) β ((π(+gβπ)π)(leβπ)(π(+gβπ)π) β (π(+gβ(π βΎs π΄))π)(leβ(π βΎs π΄))(π(+gβ(π βΎs π΄))π))) |
50 | 41, 49 | mpbid 231 |
. . . 4
β’ ((((π β oMnd β§ (π βΎs π΄) β Mnd) β§ (π β (Baseβ(π βΎs π΄)) β§ π β (Baseβ(π βΎs π΄)) β§ π β (Baseβ(π βΎs π΄)))) β§ π(leβ(π βΎs π΄))π) β (π(+gβ(π βΎs π΄))π)(leβ(π βΎs π΄))(π(+gβ(π βΎs π΄))π)) |
51 | 50 | ex 414 |
. . 3
β’ (((π β oMnd β§ (π βΎs π΄) β Mnd) β§ (π β (Baseβ(π βΎs π΄)) β§ π β (Baseβ(π βΎs π΄)) β§ π β (Baseβ(π βΎs π΄)))) β (π(leβ(π βΎs π΄))π β (π(+gβ(π βΎs π΄))π)(leβ(π βΎs π΄))(π(+gβ(π βΎs π΄))π))) |
52 | 51 | ralrimivvva 3197 |
. 2
β’ ((π β oMnd β§ (π βΎs π΄) β Mnd) β
βπ β
(Baseβ(π
βΎs π΄))βπ β (Baseβ(π βΎs π΄))βπ β (Baseβ(π βΎs π΄))(π(leβ(π βΎs π΄))π β (π(+gβ(π βΎs π΄))π)(leβ(π βΎs π΄))(π(+gβ(π βΎs π΄))π))) |
53 | | eqid 2733 |
. . 3
β’
(+gβ(π βΎs π΄)) = (+gβ(π βΎs π΄)) |
54 | | eqid 2733 |
. . 3
β’
(leβ(π
βΎs π΄)) =
(leβ(π
βΎs π΄)) |
55 | 10, 53, 54 | isomnd 31958 |
. 2
β’ ((π βΎs π΄) β oMnd β ((π βΎs π΄) β Mnd β§ (π βΎs π΄) β Toset β§
βπ β
(Baseβ(π
βΎs π΄))βπ β (Baseβ(π βΎs π΄))βπ β (Baseβ(π βΎs π΄))(π(leβ(π βΎs π΄))π β (π(+gβ(π βΎs π΄))π)(leβ(π βΎs π΄))(π(+gβ(π βΎs π΄))π)))) |
56 | 1, 18, 52, 55 | syl3anbrc 1344 |
1
β’ ((π β oMnd β§ (π βΎs π΄) β Mnd) β (π βΎs π΄) β oMnd) |