Step | Hyp | Ref
| Expression |
1 | | crngorngo 36462 |
. . . 4
β’ (π
β CRingOps β π
β
RingOps) |
2 | | ispridlc.1 |
. . . . 5
β’ πΊ = (1st βπ
) |
3 | | ispridlc.2 |
. . . . 5
β’ π» = (2nd βπ
) |
4 | | ispridlc.3 |
. . . . 5
β’ π = ran πΊ |
5 | 2, 3, 4 | ispridl 36496 |
. . . 4
β’ (π
β RingOps β (π β (PrIdlβπ
) β (π β (Idlβπ
) β§ π β π β§ βπ β (Idlβπ
)βπ β (Idlβπ
)(βπ₯ β π βπ¦ β π (π₯π»π¦) β π β (π β π β¨ π β π))))) |
6 | 1, 5 | syl 17 |
. . 3
β’ (π
β CRingOps β (π β (PrIdlβπ
) β (π β (Idlβπ
) β§ π β π β§ βπ β (Idlβπ
)βπ β (Idlβπ
)(βπ₯ β π βπ¦ β π (π₯π»π¦) β π β (π β π β¨ π β π))))) |
7 | | snssi 4769 |
. . . . . . . . . . . . 13
β’ (π β π β {π} β π) |
8 | 2, 4 | igenidl 36525 |
. . . . . . . . . . . . 13
β’ ((π
β RingOps β§ {π} β π) β (π
IdlGen {π}) β (Idlβπ
)) |
9 | 1, 7, 8 | syl2an 597 |
. . . . . . . . . . . 12
β’ ((π
β CRingOps β§ π β π) β (π
IdlGen {π}) β (Idlβπ
)) |
10 | 9 | adantrr 716 |
. . . . . . . . . . 11
β’ ((π
β CRingOps β§ (π β π β§ π β π)) β (π
IdlGen {π}) β (Idlβπ
)) |
11 | | snssi 4769 |
. . . . . . . . . . . . 13
β’ (π β π β {π} β π) |
12 | 2, 4 | igenidl 36525 |
. . . . . . . . . . . . 13
β’ ((π
β RingOps β§ {π} β π) β (π
IdlGen {π}) β (Idlβπ
)) |
13 | 1, 11, 12 | syl2an 597 |
. . . . . . . . . . . 12
β’ ((π
β CRingOps β§ π β π) β (π
IdlGen {π}) β (Idlβπ
)) |
14 | 13 | adantrl 715 |
. . . . . . . . . . 11
β’ ((π
β CRingOps β§ (π β π β§ π β π)) β (π
IdlGen {π}) β (Idlβπ
)) |
15 | | raleq 3310 |
. . . . . . . . . . . . 13
β’ (π = (π
IdlGen {π}) β (βπ₯ β π βπ¦ β π (π₯π»π¦) β π β βπ₯ β (π
IdlGen {π})βπ¦ β π (π₯π»π¦) β π)) |
16 | | sseq1 3970 |
. . . . . . . . . . . . . 14
β’ (π = (π
IdlGen {π}) β (π β π β (π
IdlGen {π}) β π)) |
17 | 16 | orbi1d 916 |
. . . . . . . . . . . . 13
β’ (π = (π
IdlGen {π}) β ((π β π β¨ π β π) β ((π
IdlGen {π}) β π β¨ π β π))) |
18 | 15, 17 | imbi12d 345 |
. . . . . . . . . . . 12
β’ (π = (π
IdlGen {π}) β ((βπ₯ β π βπ¦ β π (π₯π»π¦) β π β (π β π β¨ π β π)) β (βπ₯ β (π
IdlGen {π})βπ¦ β π (π₯π»π¦) β π β ((π
IdlGen {π}) β π β¨ π β π)))) |
19 | | raleq 3310 |
. . . . . . . . . . . . . 14
β’ (π = (π
IdlGen {π}) β (βπ¦ β π (π₯π»π¦) β π β βπ¦ β (π
IdlGen {π})(π₯π»π¦) β π)) |
20 | 19 | ralbidv 3175 |
. . . . . . . . . . . . 13
β’ (π = (π
IdlGen {π}) β (βπ₯ β (π
IdlGen {π})βπ¦ β π (π₯π»π¦) β π β βπ₯ β (π
IdlGen {π})βπ¦ β (π
IdlGen {π})(π₯π»π¦) β π)) |
21 | | sseq1 3970 |
. . . . . . . . . . . . . 14
β’ (π = (π
IdlGen {π}) β (π β π β (π
IdlGen {π}) β π)) |
22 | 21 | orbi2d 915 |
. . . . . . . . . . . . 13
β’ (π = (π
IdlGen {π}) β (((π
IdlGen {π}) β π β¨ π β π) β ((π
IdlGen {π}) β π β¨ (π
IdlGen {π}) β π))) |
23 | 20, 22 | imbi12d 345 |
. . . . . . . . . . . 12
β’ (π = (π
IdlGen {π}) β ((βπ₯ β (π
IdlGen {π})βπ¦ β π (π₯π»π¦) β π β ((π
IdlGen {π}) β π β¨ π β π)) β (βπ₯ β (π
IdlGen {π})βπ¦ β (π
IdlGen {π})(π₯π»π¦) β π β ((π
IdlGen {π}) β π β¨ (π
IdlGen {π}) β π)))) |
24 | 18, 23 | rspc2v 3591 |
. . . . . . . . . . 11
β’ (((π
IdlGen {π}) β (Idlβπ
) β§ (π
IdlGen {π}) β (Idlβπ
)) β (βπ β (Idlβπ
)βπ β (Idlβπ
)(βπ₯ β π βπ¦ β π (π₯π»π¦) β π β (π β π β¨ π β π)) β (βπ₯ β (π
IdlGen {π})βπ¦ β (π
IdlGen {π})(π₯π»π¦) β π β ((π
IdlGen {π}) β π β¨ (π
IdlGen {π}) β π)))) |
25 | 10, 14, 24 | syl2anc 585 |
. . . . . . . . . 10
β’ ((π
β CRingOps β§ (π β π β§ π β π)) β (βπ β (Idlβπ
)βπ β (Idlβπ
)(βπ₯ β π βπ¦ β π (π₯π»π¦) β π β (π β π β¨ π β π)) β (βπ₯ β (π
IdlGen {π})βπ¦ β (π
IdlGen {π})(π₯π»π¦) β π β ((π
IdlGen {π}) β π β¨ (π
IdlGen {π}) β π)))) |
26 | 25 | adantlr 714 |
. . . . . . . . 9
β’ (((π
β CRingOps β§ π β (Idlβπ
)) β§ (π β π β§ π β π)) β (βπ β (Idlβπ
)βπ β (Idlβπ
)(βπ₯ β π βπ¦ β π (π₯π»π¦) β π β (π β π β¨ π β π)) β (βπ₯ β (π
IdlGen {π})βπ¦ β (π
IdlGen {π})(π₯π»π¦) β π β ((π
IdlGen {π}) β π β¨ (π
IdlGen {π}) β π)))) |
27 | 2, 3, 4 | prnc 36529 |
. . . . . . . . . . . . . . . . . . 19
β’ ((π
β CRingOps β§ π β π) β (π
IdlGen {π}) = {π₯ β π β£ βπ β π π₯ = (ππ»π)}) |
28 | | df-rab 3409 |
. . . . . . . . . . . . . . . . . . 19
β’ {π₯ β π β£ βπ β π π₯ = (ππ»π)} = {π₯ β£ (π₯ β π β§ βπ β π π₯ = (ππ»π))} |
29 | 27, 28 | eqtrdi 2793 |
. . . . . . . . . . . . . . . . . 18
β’ ((π
β CRingOps β§ π β π) β (π
IdlGen {π}) = {π₯ β£ (π₯ β π β§ βπ β π π₯ = (ππ»π))}) |
30 | 29 | eqabd 2881 |
. . . . . . . . . . . . . . . . 17
β’ ((π
β CRingOps β§ π β π) β (π₯ β (π
IdlGen {π}) β (π₯ β π β§ βπ β π π₯ = (ππ»π)))) |
31 | 30 | adantrr 716 |
. . . . . . . . . . . . . . . 16
β’ ((π
β CRingOps β§ (π β π β§ π β π)) β (π₯ β (π
IdlGen {π}) β (π₯ β π β§ βπ β π π₯ = (ππ»π)))) |
32 | 2, 3, 4 | prnc 36529 |
. . . . . . . . . . . . . . . . . . 19
β’ ((π
β CRingOps β§ π β π) β (π
IdlGen {π}) = {π¦ β π β£ βπ β π π¦ = (π π»π)}) |
33 | | df-rab 3409 |
. . . . . . . . . . . . . . . . . . 19
β’ {π¦ β π β£ βπ β π π¦ = (π π»π)} = {π¦ β£ (π¦ β π β§ βπ β π π¦ = (π π»π))} |
34 | 32, 33 | eqtrdi 2793 |
. . . . . . . . . . . . . . . . . 18
β’ ((π
β CRingOps β§ π β π) β (π
IdlGen {π}) = {π¦ β£ (π¦ β π β§ βπ β π π¦ = (π π»π))}) |
35 | 34 | eqabd 2881 |
. . . . . . . . . . . . . . . . 17
β’ ((π
β CRingOps β§ π β π) β (π¦ β (π
IdlGen {π}) β (π¦ β π β§ βπ β π π¦ = (π π»π)))) |
36 | 35 | adantrl 715 |
. . . . . . . . . . . . . . . 16
β’ ((π
β CRingOps β§ (π β π β§ π β π)) β (π¦ β (π
IdlGen {π}) β (π¦ β π β§ βπ β π π¦ = (π π»π)))) |
37 | 31, 36 | anbi12d 632 |
. . . . . . . . . . . . . . 15
β’ ((π
β CRingOps β§ (π β π β§ π β π)) β ((π₯ β (π
IdlGen {π}) β§ π¦ β (π
IdlGen {π})) β ((π₯ β π β§ βπ β π π₯ = (ππ»π)) β§ (π¦ β π β§ βπ β π π¦ = (π π»π))))) |
38 | 37 | adantlr 714 |
. . . . . . . . . . . . . 14
β’ (((π
β CRingOps β§ π β (Idlβπ
)) β§ (π β π β§ π β π)) β ((π₯ β (π
IdlGen {π}) β§ π¦ β (π
IdlGen {π})) β ((π₯ β π β§ βπ β π π₯ = (ππ»π)) β§ (π¦ β π β§ βπ β π π¦ = (π π»π))))) |
39 | 38 | adantr 482 |
. . . . . . . . . . . . 13
β’ ((((π
β CRingOps β§ π β (Idlβπ
)) β§ (π β π β§ π β π)) β§ (ππ»π) β π) β ((π₯ β (π
IdlGen {π}) β§ π¦ β (π
IdlGen {π})) β ((π₯ β π β§ βπ β π π₯ = (ππ»π)) β§ (π¦ β π β§ βπ β π π¦ = (π π»π))))) |
40 | | reeanv 3218 |
. . . . . . . . . . . . . . . 16
β’
(βπ β
π βπ β π (π₯ = (ππ»π) β§ π¦ = (π π»π)) β (βπ β π π₯ = (ππ»π) β§ βπ β π π¦ = (π π»π))) |
41 | 40 | anbi2i 624 |
. . . . . . . . . . . . . . 15
β’ (((π₯ β π β§ π¦ β π) β§ βπ β π βπ β π (π₯ = (ππ»π) β§ π¦ = (π π»π))) β ((π₯ β π β§ π¦ β π) β§ (βπ β π π₯ = (ππ»π) β§ βπ β π π¦ = (π π»π)))) |
42 | | an4 655 |
. . . . . . . . . . . . . . 15
β’ (((π₯ β π β§ π¦ β π) β§ (βπ β π π₯ = (ππ»π) β§ βπ β π π¦ = (π π»π))) β ((π₯ β π β§ βπ β π π₯ = (ππ»π)) β§ (π¦ β π β§ βπ β π π¦ = (π π»π)))) |
43 | 41, 42 | bitri 275 |
. . . . . . . . . . . . . 14
β’ (((π₯ β π β§ π¦ β π) β§ βπ β π βπ β π (π₯ = (ππ»π) β§ π¦ = (π π»π))) β ((π₯ β π β§ βπ β π π₯ = (ππ»π)) β§ (π¦ β π β§ βπ β π π¦ = (π π»π)))) |
44 | 2, 3, 4 | crngm4 36465 |
. . . . . . . . . . . . . . . . . . . . . 22
β’ ((π
β CRingOps β§ (π β π β§ π β π) β§ (π β π β§ π β π)) β ((ππ»π )π»(ππ»π)) = ((ππ»π)π»(π π»π))) |
45 | 44 | 3com23 1127 |
. . . . . . . . . . . . . . . . . . . . 21
β’ ((π
β CRingOps β§ (π β π β§ π β π) β§ (π β π β§ π β π)) β ((ππ»π )π»(ππ»π)) = ((ππ»π)π»(π π»π))) |
46 | 45 | 3expa 1119 |
. . . . . . . . . . . . . . . . . . . 20
β’ (((π
β CRingOps β§ (π β π β§ π β π)) β§ (π β π β§ π β π)) β ((ππ»π )π»(ππ»π)) = ((ππ»π)π»(π π»π))) |
47 | 46 | adantllr 718 |
. . . . . . . . . . . . . . . . . . 19
β’ ((((π
β CRingOps β§ π β (Idlβπ
)) β§ (π β π β§ π β π)) β§ (π β π β§ π β π)) β ((ππ»π )π»(ππ»π)) = ((ππ»π)π»(π π»π))) |
48 | 47 | adantlr 714 |
. . . . . . . . . . . . . . . . . 18
β’
(((((π
β
CRingOps β§ π β
(Idlβπ
)) β§ (π β π β§ π β π)) β§ (ππ»π) β π) β§ (π β π β§ π β π)) β ((ππ»π )π»(ππ»π)) = ((ππ»π)π»(π π»π))) |
49 | 2, 3, 4 | rngocl 36363 |
. . . . . . . . . . . . . . . . . . . . . . . 24
β’ ((π
β RingOps β§ π β π β§ π β π) β (ππ»π ) β π) |
50 | 1, 49 | syl3an1 1164 |
. . . . . . . . . . . . . . . . . . . . . . 23
β’ ((π
β CRingOps β§ π β π β§ π β π) β (ππ»π ) β π) |
51 | 50 | 3expb 1121 |
. . . . . . . . . . . . . . . . . . . . . 22
β’ ((π
β CRingOps β§ (π β π β§ π β π)) β (ππ»π ) β π) |
52 | 51 | adantlr 714 |
. . . . . . . . . . . . . . . . . . . . 21
β’ (((π
β CRingOps β§ π β (Idlβπ
)) β§ (π β π β§ π β π)) β (ππ»π ) β π) |
53 | 52 | adantlr 714 |
. . . . . . . . . . . . . . . . . . . 20
β’ ((((π
β CRingOps β§ π β (Idlβπ
)) β§ (ππ»π) β π) β§ (π β π β§ π β π)) β (ππ»π ) β π) |
54 | 2, 3, 4 | idllmulcl 36482 |
. . . . . . . . . . . . . . . . . . . . . 22
β’ (((π
β RingOps β§ π β (Idlβπ
)) β§ ((ππ»π) β π β§ (ππ»π ) β π)) β ((ππ»π )π»(ππ»π)) β π) |
55 | 1, 54 | sylanl1 679 |
. . . . . . . . . . . . . . . . . . . . 21
β’ (((π
β CRingOps β§ π β (Idlβπ
)) β§ ((ππ»π) β π β§ (ππ»π ) β π)) β ((ππ»π )π»(ππ»π)) β π) |
56 | 55 | anassrs 469 |
. . . . . . . . . . . . . . . . . . . 20
β’ ((((π
β CRingOps β§ π β (Idlβπ
)) β§ (ππ»π) β π) β§ (ππ»π ) β π) β ((ππ»π )π»(ππ»π)) β π) |
57 | 53, 56 | syldan 592 |
. . . . . . . . . . . . . . . . . . 19
β’ ((((π
β CRingOps β§ π β (Idlβπ
)) β§ (ππ»π) β π) β§ (π β π β§ π β π)) β ((ππ»π )π»(ππ»π)) β π) |
58 | 57 | adantllr 718 |
. . . . . . . . . . . . . . . . . 18
β’
(((((π
β
CRingOps β§ π β
(Idlβπ
)) β§ (π β π β§ π β π)) β§ (ππ»π) β π) β§ (π β π β§ π β π)) β ((ππ»π )π»(ππ»π)) β π) |
59 | 48, 58 | eqeltrrd 2839 |
. . . . . . . . . . . . . . . . 17
β’
(((((π
β
CRingOps β§ π β
(Idlβπ
)) β§ (π β π β§ π β π)) β§ (ππ»π) β π) β§ (π β π β§ π β π)) β ((ππ»π)π»(π π»π)) β π) |
60 | | oveq12 7367 |
. . . . . . . . . . . . . . . . . 18
β’ ((π₯ = (ππ»π) β§ π¦ = (π π»π)) β (π₯π»π¦) = ((ππ»π)π»(π π»π))) |
61 | 60 | eleq1d 2823 |
. . . . . . . . . . . . . . . . 17
β’ ((π₯ = (ππ»π) β§ π¦ = (π π»π)) β ((π₯π»π¦) β π β ((ππ»π)π»(π π»π)) β π)) |
62 | 59, 61 | syl5ibrcom 247 |
. . . . . . . . . . . . . . . 16
β’
(((((π
β
CRingOps β§ π β
(Idlβπ
)) β§ (π β π β§ π β π)) β§ (ππ»π) β π) β§ (π β π β§ π β π)) β ((π₯ = (ππ»π) β§ π¦ = (π π»π)) β (π₯π»π¦) β π)) |
63 | 62 | rexlimdvva 3206 |
. . . . . . . . . . . . . . 15
β’ ((((π
β CRingOps β§ π β (Idlβπ
)) β§ (π β π β§ π β π)) β§ (ππ»π) β π) β (βπ β π βπ β π (π₯ = (ππ»π) β§ π¦ = (π π»π)) β (π₯π»π¦) β π)) |
64 | 63 | adantld 492 |
. . . . . . . . . . . . . 14
β’ ((((π
β CRingOps β§ π β (Idlβπ
)) β§ (π β π β§ π β π)) β§ (ππ»π) β π) β (((π₯ β π β§ π¦ β π) β§ βπ β π βπ β π (π₯ = (ππ»π) β§ π¦ = (π π»π))) β (π₯π»π¦) β π)) |
65 | 43, 64 | biimtrrid 242 |
. . . . . . . . . . . . 13
β’ ((((π
β CRingOps β§ π β (Idlβπ
)) β§ (π β π β§ π β π)) β§ (ππ»π) β π) β (((π₯ β π β§ βπ β π π₯ = (ππ»π)) β§ (π¦ β π β§ βπ β π π¦ = (π π»π))) β (π₯π»π¦) β π)) |
66 | 39, 65 | sylbid 239 |
. . . . . . . . . . . 12
β’ ((((π
β CRingOps β§ π β (Idlβπ
)) β§ (π β π β§ π β π)) β§ (ππ»π) β π) β ((π₯ β (π
IdlGen {π}) β§ π¦ β (π
IdlGen {π})) β (π₯π»π¦) β π)) |
67 | 66 | ralrimivv 3196 |
. . . . . . . . . . 11
β’ ((((π
β CRingOps β§ π β (Idlβπ
)) β§ (π β π β§ π β π)) β§ (ππ»π) β π) β βπ₯ β (π
IdlGen {π})βπ¦ β (π
IdlGen {π})(π₯π»π¦) β π) |
68 | 67 | ex 414 |
. . . . . . . . . 10
β’ (((π
β CRingOps β§ π β (Idlβπ
)) β§ (π β π β§ π β π)) β ((ππ»π) β π β βπ₯ β (π
IdlGen {π})βπ¦ β (π
IdlGen {π})(π₯π»π¦) β π)) |
69 | 2, 4 | igenss 36524 |
. . . . . . . . . . . . . . . 16
β’ ((π
β RingOps β§ {π} β π) β {π} β (π
IdlGen {π})) |
70 | 1, 7, 69 | syl2an 597 |
. . . . . . . . . . . . . . 15
β’ ((π
β CRingOps β§ π β π) β {π} β (π
IdlGen {π})) |
71 | | vex 3450 |
. . . . . . . . . . . . . . . 16
β’ π β V |
72 | 71 | snss 4747 |
. . . . . . . . . . . . . . 15
β’ (π β (π
IdlGen {π}) β {π} β (π
IdlGen {π})) |
73 | 70, 72 | sylibr 233 |
. . . . . . . . . . . . . 14
β’ ((π
β CRingOps β§ π β π) β π β (π
IdlGen {π})) |
74 | 73 | adantrr 716 |
. . . . . . . . . . . . 13
β’ ((π
β CRingOps β§ (π β π β§ π β π)) β π β (π
IdlGen {π})) |
75 | | ssel 3938 |
. . . . . . . . . . . . 13
β’ ((π
IdlGen {π}) β π β (π β (π
IdlGen {π}) β π β π)) |
76 | 74, 75 | syl5com 31 |
. . . . . . . . . . . 12
β’ ((π
β CRingOps β§ (π β π β§ π β π)) β ((π
IdlGen {π}) β π β π β π)) |
77 | 2, 4 | igenss 36524 |
. . . . . . . . . . . . . . . 16
β’ ((π
β RingOps β§ {π} β π) β {π} β (π
IdlGen {π})) |
78 | 1, 11, 77 | syl2an 597 |
. . . . . . . . . . . . . . 15
β’ ((π
β CRingOps β§ π β π) β {π} β (π
IdlGen {π})) |
79 | | vex 3450 |
. . . . . . . . . . . . . . . 16
β’ π β V |
80 | 79 | snss 4747 |
. . . . . . . . . . . . . . 15
β’ (π β (π
IdlGen {π}) β {π} β (π
IdlGen {π})) |
81 | 78, 80 | sylibr 233 |
. . . . . . . . . . . . . 14
β’ ((π
β CRingOps β§ π β π) β π β (π
IdlGen {π})) |
82 | 81 | adantrl 715 |
. . . . . . . . . . . . 13
β’ ((π
β CRingOps β§ (π β π β§ π β π)) β π β (π
IdlGen {π})) |
83 | | ssel 3938 |
. . . . . . . . . . . . 13
β’ ((π
IdlGen {π}) β π β (π β (π
IdlGen {π}) β π β π)) |
84 | 82, 83 | syl5com 31 |
. . . . . . . . . . . 12
β’ ((π
β CRingOps β§ (π β π β§ π β π)) β ((π
IdlGen {π}) β π β π β π)) |
85 | 76, 84 | orim12d 964 |
. . . . . . . . . . 11
β’ ((π
β CRingOps β§ (π β π β§ π β π)) β (((π
IdlGen {π}) β π β¨ (π
IdlGen {π}) β π) β (π β π β¨ π β π))) |
86 | 85 | adantlr 714 |
. . . . . . . . . 10
β’ (((π
β CRingOps β§ π β (Idlβπ
)) β§ (π β π β§ π β π)) β (((π
IdlGen {π}) β π β¨ (π
IdlGen {π}) β π) β (π β π β¨ π β π))) |
87 | 68, 86 | imim12d 81 |
. . . . . . . . 9
β’ (((π
β CRingOps β§ π β (Idlβπ
)) β§ (π β π β§ π β π)) β ((βπ₯ β (π
IdlGen {π})βπ¦ β (π
IdlGen {π})(π₯π»π¦) β π β ((π
IdlGen {π}) β π β¨ (π
IdlGen {π}) β π)) β ((ππ»π) β π β (π β π β¨ π β π)))) |
88 | 26, 87 | syld 47 |
. . . . . . . 8
β’ (((π
β CRingOps β§ π β (Idlβπ
)) β§ (π β π β§ π β π)) β (βπ β (Idlβπ
)βπ β (Idlβπ
)(βπ₯ β π βπ¦ β π (π₯π»π¦) β π β (π β π β¨ π β π)) β ((ππ»π) β π β (π β π β¨ π β π)))) |
89 | 88 | ralrimdvva 3204 |
. . . . . . 7
β’ ((π
β CRingOps β§ π β (Idlβπ
)) β (βπ β (Idlβπ
)βπ β (Idlβπ
)(βπ₯ β π βπ¦ β π (π₯π»π¦) β π β (π β π β¨ π β π)) β βπ β π βπ β π ((ππ»π) β π β (π β π β¨ π β π)))) |
90 | 89 | ex 414 |
. . . . . 6
β’ (π
β CRingOps β (π β (Idlβπ
) β (βπ β (Idlβπ
)βπ β (Idlβπ
)(βπ₯ β π βπ¦ β π (π₯π»π¦) β π β (π β π β¨ π β π)) β βπ β π βπ β π ((ππ»π) β π β (π β π β¨ π β π))))) |
91 | 90 | adantrd 493 |
. . . . 5
β’ (π
β CRingOps β ((π β (Idlβπ
) β§ π β π) β (βπ β (Idlβπ
)βπ β (Idlβπ
)(βπ₯ β π βπ¦ β π (π₯π»π¦) β π β (π β π β¨ π β π)) β βπ β π βπ β π ((ππ»π) β π β (π β π β¨ π β π))))) |
92 | 91 | imdistand 572 |
. . . 4
β’ (π
β CRingOps β (((π β (Idlβπ
) β§ π β π) β§ βπ β (Idlβπ
)βπ β (Idlβπ
)(βπ₯ β π βπ¦ β π (π₯π»π¦) β π β (π β π β¨ π β π))) β ((π β (Idlβπ
) β§ π β π) β§ βπ β π βπ β π ((ππ»π) β π β (π β π β¨ π β π))))) |
93 | | df-3an 1090 |
. . . 4
β’ ((π β (Idlβπ
) β§ π β π β§ βπ β (Idlβπ
)βπ β (Idlβπ
)(βπ₯ β π βπ¦ β π (π₯π»π¦) β π β (π β π β¨ π β π))) β ((π β (Idlβπ
) β§ π β π) β§ βπ β (Idlβπ
)βπ β (Idlβπ
)(βπ₯ β π βπ¦ β π (π₯π»π¦) β π β (π β π β¨ π β π)))) |
94 | | df-3an 1090 |
. . . 4
β’ ((π β (Idlβπ
) β§ π β π β§ βπ β π βπ β π ((ππ»π) β π β (π β π β¨ π β π))) β ((π β (Idlβπ
) β§ π β π) β§ βπ β π βπ β π ((ππ»π) β π β (π β π β¨ π β π)))) |
95 | 92, 93, 94 | 3imtr4g 296 |
. . 3
β’ (π
β CRingOps β ((π β (Idlβπ
) β§ π β π β§ βπ β (Idlβπ
)βπ β (Idlβπ
)(βπ₯ β π βπ¦ β π (π₯π»π¦) β π β (π β π β¨ π β π))) β (π β (Idlβπ
) β§ π β π β§ βπ β π βπ β π ((ππ»π) β π β (π β π β¨ π β π))))) |
96 | 6, 95 | sylbid 239 |
. 2
β’ (π
β CRingOps β (π β (PrIdlβπ
) β (π β (Idlβπ
) β§ π β π β§ βπ β π βπ β π ((ππ»π) β π β (π β π β¨ π β π))))) |
97 | 2, 3, 4 | ispridl2 36500 |
. . . 4
β’ ((π
β RingOps β§ (π β (Idlβπ
) β§ π β π β§ βπ β π βπ β π ((ππ»π) β π β (π β π β¨ π β π)))) β π β (PrIdlβπ
)) |
98 | 97 | ex 414 |
. . 3
β’ (π
β RingOps β ((π β (Idlβπ
) β§ π β π β§ βπ β π βπ β π ((ππ»π) β π β (π β π β¨ π β π))) β π β (PrIdlβπ
))) |
99 | 1, 98 | syl 17 |
. 2
β’ (π
β CRingOps β ((π β (Idlβπ
) β§ π β π β§ βπ β π βπ β π ((ππ»π) β π β (π β π β¨ π β π))) β π β (PrIdlβπ
))) |
100 | 96, 99 | impbid 211 |
1
β’ (π
β CRingOps β (π β (PrIdlβπ
) β (π β (Idlβπ
) β§ π β π β§ βπ β π βπ β π ((ππ»π) β π β (π β π β¨ π β π))))) |