Step | Hyp | Ref
| Expression |
1 | | nns.s |
. 2
β’ π = (π β ββ β¦
(π β Ο β¦
if(π = β
,
1o, (πββͺ π)))) |
2 | | 1lt2o 6442 |
. . . . . . 7
β’
1o β 2o |
3 | 2 | a1i 9 |
. . . . . 6
β’ ((π β
ββ β§ π β Ο) β 1o β
2o) |
4 | | nninff 7120 |
. . . . . . . 8
β’ (π β
ββ β π:ΟβΆ2o) |
5 | 4 | adantr 276 |
. . . . . . 7
β’ ((π β
ββ β§ π β Ο) β π:ΟβΆ2o) |
6 | | nnpredcl 4622 |
. . . . . . . 8
β’ (π β Ο β βͺ π
β Ο) |
7 | 6 | adantl 277 |
. . . . . . 7
β’ ((π β
ββ β§ π β Ο) β βͺ π
β Ο) |
8 | 5, 7 | ffvelcdmd 5652 |
. . . . . 6
β’ ((π β
ββ β§ π β Ο) β (πββͺ π) β
2o) |
9 | | nndceq0 4617 |
. . . . . . 7
β’ (π β Ο β
DECID π =
β
) |
10 | 9 | adantl 277 |
. . . . . 6
β’ ((π β
ββ β§ π β Ο) β DECID
π =
β
) |
11 | 3, 8, 10 | ifcldcd 3570 |
. . . . 5
β’ ((π β
ββ β§ π β Ο) β if(π = β
, 1o, (πββͺ π))
β 2o) |
12 | | eqid 2177 |
. . . . 5
β’ (π β Ο β¦ if(π = β
, 1o,
(πββͺ π)))
= (π β Ο β¦
if(π = β
,
1o, (πββͺ π))) |
13 | 11, 12 | fmptd 5670 |
. . . 4
β’ (π β
ββ β (π β Ο β¦ if(π = β
, 1o, (πββͺ π))):ΟβΆ2o) |
14 | | 2onn 6521 |
. . . . 5
β’
2o β Ο |
15 | | omex 4592 |
. . . . 5
β’ Ο
β V |
16 | | elmapg 6660 |
. . . . 5
β’
((2o β Ο β§ Ο β V) β ((π β Ο β¦ if(π = β
, 1o,
(πββͺ π)))
β (2o βπ Ο) β (π β Ο β¦ if(π = β
, 1o,
(πββͺ π))):ΟβΆ2o)) |
17 | 14, 15, 16 | mp2an 426 |
. . . 4
β’ ((π β Ο β¦ if(π = β
, 1o,
(πββͺ π)))
β (2o βπ Ο) β (π β Ο β¦ if(π = β
, 1o,
(πββͺ π))):ΟβΆ2o) |
18 | 13, 17 | sylibr 134 |
. . 3
β’ (π β
ββ β (π β Ο β¦ if(π = β
, 1o, (πββͺ π)))
β (2o βπ Ο)) |
19 | | 1on 6423 |
. . . . . . . . 9
β’
1o β On |
20 | 19 | ontrci 4427 |
. . . . . . . 8
β’ Tr
1o |
21 | 2 | a1i 9 |
. . . . . . . . . . 11
β’ ((π β
ββ β§ π β Ο) β 1o β
2o) |
22 | 4 | adantr 276 |
. . . . . . . . . . . 12
β’ ((π β
ββ β§ π β Ο) β π:ΟβΆ2o) |
23 | | peano2 4594 |
. . . . . . . . . . . . . 14
β’ (π β Ο β suc π β
Ο) |
24 | 23 | adantl 277 |
. . . . . . . . . . . . 13
β’ ((π β
ββ β§ π β Ο) β suc π β
Ο) |
25 | | nnpredcl 4622 |
. . . . . . . . . . . . 13
β’ (suc
π β Ο β
βͺ suc π β Ο) |
26 | 24, 25 | syl 14 |
. . . . . . . . . . . 12
β’ ((π β
ββ β§ π β Ο) β βͺ suc π β Ο) |
27 | 22, 26 | ffvelcdmd 5652 |
. . . . . . . . . . 11
β’ ((π β
ββ β§ π β Ο) β (πββͺ suc π) β
2o) |
28 | | nndceq0 4617 |
. . . . . . . . . . . 12
β’ (suc
π β Ο β
DECID suc π
= β
) |
29 | 24, 28 | syl 14 |
. . . . . . . . . . 11
β’ ((π β
ββ β§ π β Ο) β DECID
suc π =
β
) |
30 | 21, 27, 29 | ifcldcd 3570 |
. . . . . . . . . 10
β’ ((π β
ββ β§ π β Ο) β if(suc π = β
, 1o,
(πββͺ suc π)) β 2o) |
31 | 30 | adantr 276 |
. . . . . . . . 9
β’ (((π β
ββ β§ π β Ο) β§ π = β
) β if(suc π = β
, 1o, (πββͺ suc π)) β 2o) |
32 | | df-2o 6417 |
. . . . . . . . 9
β’
2o = suc 1o |
33 | 31, 32 | eleqtrdi 2270 |
. . . . . . . 8
β’ (((π β
ββ β§ π β Ο) β§ π = β
) β if(suc π = β
, 1o, (πββͺ suc π)) β suc 1o) |
34 | | trsucss 4423 |
. . . . . . . 8
β’ (Tr
1o β (if(suc π = β
, 1o, (πββͺ suc π)) β suc 1o β if(suc
π = β
, 1o,
(πββͺ suc π)) β 1o)) |
35 | 20, 33, 34 | mpsyl 65 |
. . . . . . 7
β’ (((π β
ββ β§ π β Ο) β§ π = β
) β if(suc π = β
, 1o, (πββͺ suc π)) β 1o) |
36 | | iftrue 3539 |
. . . . . . . 8
β’ (π = β
β if(π = β
, 1o,
(πββͺ π))
= 1o) |
37 | 36 | adantl 277 |
. . . . . . 7
β’ (((π β
ββ β§ π β Ο) β§ π = β
) β if(π = β
, 1o, (πββͺ π))
= 1o) |
38 | 35, 37 | sseqtrrd 3194 |
. . . . . 6
β’ (((π β
ββ β§ π β Ο) β§ π = β
) β if(suc π = β
, 1o, (πββͺ suc π)) β if(π = β
, 1o, (πββͺ π))) |
39 | | simpr 110 |
. . . . . . . . . . . 12
β’ ((π β
ββ β§ π β Ο) β π β Ο) |
40 | 39 | adantr 276 |
. . . . . . . . . . 11
β’ (((π β
ββ β§ π β Ο) β§ Β¬ π = β
) β π β
Ο) |
41 | | nnord 4611 |
. . . . . . . . . . 11
β’ (π β Ο β Ord π) |
42 | | ordtr 4378 |
. . . . . . . . . . 11
β’ (Ord
π β Tr π) |
43 | 40, 41, 42 | 3syl 17 |
. . . . . . . . . 10
β’ (((π β
ββ β§ π β Ο) β§ Β¬ π = β
) β Tr π) |
44 | | unisucg 4414 |
. . . . . . . . . . 11
β’ (π β Ο β (Tr π β βͺ suc π = π)) |
45 | 40, 44 | syl 14 |
. . . . . . . . . 10
β’ (((π β
ββ β§ π β Ο) β§ Β¬ π = β
) β (Tr π β βͺ suc π = π)) |
46 | 43, 45 | mpbid 147 |
. . . . . . . . 9
β’ (((π β
ββ β§ π β Ο) β§ Β¬ π = β
) β βͺ suc π = π) |
47 | 46 | fveq2d 5519 |
. . . . . . . 8
β’ (((π β
ββ β§ π β Ο) β§ Β¬ π = β
) β (πββͺ suc π) = (πβπ)) |
48 | | simpr 110 |
. . . . . . . . . . . 12
β’ (((π β
ββ β§ π β Ο) β§ Β¬ π = β
) β Β¬ π = β
) |
49 | 48 | neqned 2354 |
. . . . . . . . . . 11
β’ (((π β
ββ β§ π β Ο) β§ Β¬ π = β
) β π β β
) |
50 | | nnsucpred 4616 |
. . . . . . . . . . 11
β’ ((π β Ο β§ π β β
) β suc βͺ π =
π) |
51 | 40, 49, 50 | syl2anc 411 |
. . . . . . . . . 10
β’ (((π β
ββ β§ π β Ο) β§ Β¬ π = β
) β suc βͺ π =
π) |
52 | 51 | fveq2d 5519 |
. . . . . . . . 9
β’ (((π β
ββ β§ π β Ο) β§ Β¬ π = β
) β (πβsuc βͺ π) =
(πβπ)) |
53 | | suceq 4402 |
. . . . . . . . . . . 12
β’ (π = βͺ
π β suc π = suc βͺ π) |
54 | 53 | fveq2d 5519 |
. . . . . . . . . . 11
β’ (π = βͺ
π β (πβsuc π) = (πβsuc βͺ π)) |
55 | | fveq2 5515 |
. . . . . . . . . . 11
β’ (π = βͺ
π β (πβπ) = (πββͺ π)) |
56 | 54, 55 | sseq12d 3186 |
. . . . . . . . . 10
β’ (π = βͺ
π β ((πβsuc π) β (πβπ) β (πβsuc βͺ π) β (πββͺ π))) |
57 | | fveq1 5514 |
. . . . . . . . . . . . . . . 16
β’ (π = π β (πβsuc π) = (πβsuc π)) |
58 | | fveq1 5514 |
. . . . . . . . . . . . . . . 16
β’ (π = π β (πβπ) = (πβπ)) |
59 | 57, 58 | sseq12d 3186 |
. . . . . . . . . . . . . . 15
β’ (π = π β ((πβsuc π) β (πβπ) β (πβsuc π) β (πβπ))) |
60 | 59 | ralbidv 2477 |
. . . . . . . . . . . . . 14
β’ (π = π β (βπ β Ο (πβsuc π) β (πβπ) β βπ β Ο (πβsuc π) β (πβπ))) |
61 | | df-nninf 7118 |
. . . . . . . . . . . . . 14
β’
ββ = {π β (2o
βπ Ο) β£ βπ β Ο (πβsuc π) β (πβπ)} |
62 | 60, 61 | elrab2 2896 |
. . . . . . . . . . . . 13
β’ (π β
ββ β (π β (2o
βπ Ο) β§ βπ β Ο (πβsuc π) β (πβπ))) |
63 | 62 | simprbi 275 |
. . . . . . . . . . . 12
β’ (π β
ββ β βπ β Ο (πβsuc π) β (πβπ)) |
64 | | suceq 4402 |
. . . . . . . . . . . . . . 15
β’ (π = π β suc π = suc π) |
65 | 64 | fveq2d 5519 |
. . . . . . . . . . . . . 14
β’ (π = π β (πβsuc π) = (πβsuc π)) |
66 | | fveq2 5515 |
. . . . . . . . . . . . . 14
β’ (π = π β (πβπ) = (πβπ)) |
67 | 65, 66 | sseq12d 3186 |
. . . . . . . . . . . . 13
β’ (π = π β ((πβsuc π) β (πβπ) β (πβsuc π) β (πβπ))) |
68 | 67 | cbvralv 2703 |
. . . . . . . . . . . 12
β’
(βπ β
Ο (πβsuc π) β (πβπ) β βπ β Ο (πβsuc π) β (πβπ)) |
69 | 63, 68 | sylib 122 |
. . . . . . . . . . 11
β’ (π β
ββ β βπ β Ο (πβsuc π) β (πβπ)) |
70 | 69 | ad2antrr 488 |
. . . . . . . . . 10
β’ (((π β
ββ β§ π β Ο) β§ Β¬ π = β
) β βπ β Ο (πβsuc π) β (πβπ)) |
71 | | nnpredcl 4622 |
. . . . . . . . . . . 12
β’ (π β Ο β βͺ π
β Ο) |
72 | 71 | adantl 277 |
. . . . . . . . . . 11
β’ ((π β
ββ β§ π β Ο) β βͺ π
β Ο) |
73 | 72 | adantr 276 |
. . . . . . . . . 10
β’ (((π β
ββ β§ π β Ο) β§ Β¬ π = β
) β βͺ π
β Ο) |
74 | 56, 70, 73 | rspcdva 2846 |
. . . . . . . . 9
β’ (((π β
ββ β§ π β Ο) β§ Β¬ π = β
) β (πβsuc βͺ π)
β (πββͺ π)) |
75 | 52, 74 | eqsstrrd 3192 |
. . . . . . . 8
β’ (((π β
ββ β§ π β Ο) β§ Β¬ π = β
) β (πβπ) β (πββͺ π)) |
76 | 47, 75 | eqsstrd 3191 |
. . . . . . 7
β’ (((π β
ββ β§ π β Ο) β§ Β¬ π = β
) β (πββͺ suc π) β (πββͺ π)) |
77 | | peano3 4595 |
. . . . . . . . . 10
β’ (π β Ο β suc π β β
) |
78 | 77 | neneqd 2368 |
. . . . . . . . 9
β’ (π β Ο β Β¬ suc
π =
β
) |
79 | 78 | ad2antlr 489 |
. . . . . . . 8
β’ (((π β
ββ β§ π β Ο) β§ Β¬ π = β
) β Β¬ suc
π =
β
) |
80 | 79 | iffalsed 3544 |
. . . . . . 7
β’ (((π β
ββ β§ π β Ο) β§ Β¬ π = β
) β if(suc π = β
, 1o,
(πββͺ suc π)) = (πββͺ suc π)) |
81 | 48 | iffalsed 3544 |
. . . . . . 7
β’ (((π β
ββ β§ π β Ο) β§ Β¬ π = β
) β if(π = β
, 1o,
(πββͺ π))
= (πββͺ π)) |
82 | 76, 80, 81 | 3sstr4d 3200 |
. . . . . 6
β’ (((π β
ββ β§ π β Ο) β§ Β¬ π = β
) β if(suc π = β
, 1o,
(πββͺ suc π)) β if(π = β
, 1o, (πββͺ π))) |
83 | | nndceq0 4617 |
. . . . . . . 8
β’ (π β Ο β
DECID π =
β
) |
84 | 83 | adantl 277 |
. . . . . . 7
β’ ((π β
ββ β§ π β Ο) β DECID
π =
β
) |
85 | | exmiddc 836 |
. . . . . . 7
β’
(DECID π = β
β (π = β
β¨ Β¬ π = β
)) |
86 | 84, 85 | syl 14 |
. . . . . 6
β’ ((π β
ββ β§ π β Ο) β (π = β
β¨ Β¬ π = β
)) |
87 | 38, 82, 86 | mpjaodan 798 |
. . . . 5
β’ ((π β
ββ β§ π β Ο) β if(suc π = β
, 1o,
(πββͺ suc π)) β if(π = β
, 1o, (πββͺ π))) |
88 | | eqeq1 2184 |
. . . . . . . 8
β’ (π = suc π β (π = β
β suc π = β
)) |
89 | | unieq 3818 |
. . . . . . . . 9
β’ (π = suc π β βͺ π = βͺ
suc π) |
90 | 89 | fveq2d 5519 |
. . . . . . . 8
β’ (π = suc π β (πββͺ π) = (πββͺ suc π)) |
91 | 88, 90 | ifbieq2d 3558 |
. . . . . . 7
β’ (π = suc π β if(π = β
, 1o, (πββͺ π))
= if(suc π = β
,
1o, (πββͺ suc π))) |
92 | 91, 12 | fvmptg 5592 |
. . . . . 6
β’ ((suc
π β Ο β§
if(suc π = β
,
1o, (πββͺ suc π)) β 2o) β
((π β Ο β¦
if(π = β
,
1o, (πββͺ π)))βsuc π) = if(suc π = β
, 1o, (πββͺ suc π))) |
93 | 24, 30, 92 | syl2anc 411 |
. . . . 5
β’ ((π β
ββ β§ π β Ο) β ((π β Ο β¦ if(π = β
, 1o, (πββͺ π)))βsuc π) = if(suc π = β
, 1o, (πββͺ suc π))) |
94 | 22, 72 | ffvelcdmd 5652 |
. . . . . . 7
β’ ((π β
ββ β§ π β Ο) β (πββͺ π) β
2o) |
95 | 21, 94, 84 | ifcldcd 3570 |
. . . . . 6
β’ ((π β
ββ β§ π β Ο) β if(π = β
, 1o, (πββͺ π))
β 2o) |
96 | | eqeq1 2184 |
. . . . . . . 8
β’ (π = π β (π = β
β π = β
)) |
97 | | unieq 3818 |
. . . . . . . . 9
β’ (π = π β βͺ π = βͺ
π) |
98 | 97 | fveq2d 5519 |
. . . . . . . 8
β’ (π = π β (πββͺ π) = (πββͺ π)) |
99 | 96, 98 | ifbieq2d 3558 |
. . . . . . 7
β’ (π = π β if(π = β
, 1o, (πββͺ π))
= if(π = β
,
1o, (πββͺ π))) |
100 | 99, 12 | fvmptg 5592 |
. . . . . 6
β’ ((π β Ο β§ if(π = β
, 1o,
(πββͺ π))
β 2o) β ((π β Ο β¦ if(π = β
, 1o, (πββͺ π)))βπ) = if(π = β
, 1o, (πββͺ π))) |
101 | 39, 95, 100 | syl2anc 411 |
. . . . 5
β’ ((π β
ββ β§ π β Ο) β ((π β Ο β¦ if(π = β
, 1o, (πββͺ π)))βπ) = if(π = β
, 1o, (πββͺ π))) |
102 | 87, 93, 101 | 3sstr4d 3200 |
. . . 4
β’ ((π β
ββ β§ π β Ο) β ((π β Ο β¦ if(π = β
, 1o, (πββͺ π)))βsuc π) β ((π β Ο β¦ if(π = β
, 1o, (πββͺ π)))βπ)) |
103 | 102 | ralrimiva 2550 |
. . 3
β’ (π β
ββ β βπ β Ο ((π β Ο β¦ if(π = β
, 1o, (πββͺ π)))βsuc π) β ((π β Ο β¦ if(π = β
, 1o, (πββͺ π)))βπ)) |
104 | | fveq1 5514 |
. . . . . 6
β’ (π = (π β Ο β¦ if(π = β
, 1o, (πββͺ π)))
β (πβsuc π) = ((π β Ο β¦ if(π = β
, 1o, (πββͺ π)))βsuc π)) |
105 | | fveq1 5514 |
. . . . . 6
β’ (π = (π β Ο β¦ if(π = β
, 1o, (πββͺ π)))
β (πβπ) = ((π β Ο β¦ if(π = β
, 1o, (πββͺ π)))βπ)) |
106 | 104, 105 | sseq12d 3186 |
. . . . 5
β’ (π = (π β Ο β¦ if(π = β
, 1o, (πββͺ π)))
β ((πβsuc π) β (πβπ) β ((π β Ο β¦ if(π = β
, 1o, (πββͺ π)))βsuc π) β ((π β Ο β¦ if(π = β
, 1o, (πββͺ π)))βπ))) |
107 | 106 | ralbidv 2477 |
. . . 4
β’ (π = (π β Ο β¦ if(π = β
, 1o, (πββͺ π)))
β (βπ β
Ο (πβsuc π) β (πβπ) β βπ β Ο ((π β Ο β¦ if(π = β
, 1o, (πββͺ π)))βsuc π) β ((π β Ο β¦ if(π = β
, 1o, (πββͺ π)))βπ))) |
108 | 107, 61 | elrab2 2896 |
. . 3
β’ ((π β Ο β¦ if(π = β
, 1o,
(πββͺ π)))
β ββ β ((π β Ο β¦ if(π = β
, 1o, (πββͺ π)))
β (2o βπ Ο) β§ βπ β Ο ((π β Ο β¦ if(π = β
, 1o,
(πββͺ π)))βsuc π) β ((π β Ο β¦ if(π = β
, 1o, (πββͺ π)))βπ))) |
109 | 18, 103, 108 | sylanbrc 417 |
. 2
β’ (π β
ββ β (π β Ο β¦ if(π = β
, 1o, (πββͺ π)))
β ββ) |
110 | 1, 109 | fmpti 5668 |
1
β’ π:βββΆββ |