Step |
Hyp |
Ref |
Expression |
1 |
|
pmltpc.1 |
|- ( ph -> F e. ( RR ^pm RR ) ) |
2 |
|
pmltpc.2 |
|- ( ph -> A C_ dom F ) |
3 |
|
pmltpc.3 |
|- ( ph -> U e. A ) |
4 |
|
pmltpc.4 |
|- ( ph -> V e. A ) |
5 |
|
pmltpc.5 |
|- ( ph -> W e. A ) |
6 |
|
pmltpc.6 |
|- ( ph -> X e. A ) |
7 |
|
pmltpc.7 |
|- ( ph -> U <_ V ) |
8 |
|
pmltpc.8 |
|- ( ph -> W <_ X ) |
9 |
|
pmltpc.9 |
|- ( ph -> -. ( F ` U ) <_ ( F ` V ) ) |
10 |
|
pmltpc.10 |
|- ( ph -> -. ( F ` X ) <_ ( F ` W ) ) |
11 |
5
|
ad2antrr |
|- ( ( ( ph /\ ( F ` W ) < ( F ` U ) ) /\ W < U ) -> W e. A ) |
12 |
3
|
ad2antrr |
|- ( ( ( ph /\ ( F ` W ) < ( F ` U ) ) /\ W < U ) -> U e. A ) |
13 |
4
|
ad2antrr |
|- ( ( ( ph /\ ( F ` W ) < ( F ` U ) ) /\ W < U ) -> V e. A ) |
14 |
|
simpr |
|- ( ( ( ph /\ ( F ` W ) < ( F ` U ) ) /\ W < U ) -> W < U ) |
15 |
|
reex |
|- RR e. _V |
16 |
15 15
|
elpm2 |
|- ( F e. ( RR ^pm RR ) <-> ( F : dom F --> RR /\ dom F C_ RR ) ) |
17 |
1 16
|
sylib |
|- ( ph -> ( F : dom F --> RR /\ dom F C_ RR ) ) |
18 |
17
|
simprd |
|- ( ph -> dom F C_ RR ) |
19 |
2 3
|
sseldd |
|- ( ph -> U e. dom F ) |
20 |
18 19
|
sseldd |
|- ( ph -> U e. RR ) |
21 |
2 4
|
sseldd |
|- ( ph -> V e. dom F ) |
22 |
18 21
|
sseldd |
|- ( ph -> V e. RR ) |
23 |
17
|
simpld |
|- ( ph -> F : dom F --> RR ) |
24 |
23 21
|
ffvelrnd |
|- ( ph -> ( F ` V ) e. RR ) |
25 |
23 19
|
ffvelrnd |
|- ( ph -> ( F ` U ) e. RR ) |
26 |
24 25
|
ltnled |
|- ( ph -> ( ( F ` V ) < ( F ` U ) <-> -. ( F ` U ) <_ ( F ` V ) ) ) |
27 |
9 26
|
mpbird |
|- ( ph -> ( F ` V ) < ( F ` U ) ) |
28 |
24 27
|
gtned |
|- ( ph -> ( F ` U ) =/= ( F ` V ) ) |
29 |
|
fveq2 |
|- ( V = U -> ( F ` V ) = ( F ` U ) ) |
30 |
29
|
eqcomd |
|- ( V = U -> ( F ` U ) = ( F ` V ) ) |
31 |
30
|
necon3i |
|- ( ( F ` U ) =/= ( F ` V ) -> V =/= U ) |
32 |
28 31
|
syl |
|- ( ph -> V =/= U ) |
33 |
20 22 7 32
|
leneltd |
|- ( ph -> U < V ) |
34 |
33
|
ad2antrr |
|- ( ( ( ph /\ ( F ` W ) < ( F ` U ) ) /\ W < U ) -> U < V ) |
35 |
|
simplr |
|- ( ( ( ph /\ ( F ` W ) < ( F ` U ) ) /\ W < U ) -> ( F ` W ) < ( F ` U ) ) |
36 |
27
|
ad2antrr |
|- ( ( ( ph /\ ( F ` W ) < ( F ` U ) ) /\ W < U ) -> ( F ` V ) < ( F ` U ) ) |
37 |
35 36
|
jca |
|- ( ( ( ph /\ ( F ` W ) < ( F ` U ) ) /\ W < U ) -> ( ( F ` W ) < ( F ` U ) /\ ( F ` V ) < ( F ` U ) ) ) |
38 |
37
|
orcd |
|- ( ( ( ph /\ ( F ` W ) < ( F ` U ) ) /\ W < U ) -> ( ( ( F ` W ) < ( F ` U ) /\ ( F ` V ) < ( F ` U ) ) \/ ( ( F ` U ) < ( F ` W ) /\ ( F ` U ) < ( F ` V ) ) ) ) |
39 |
11 12 13 14 34 38
|
pmltpclem1 |
|- ( ( ( ph /\ ( F ` W ) < ( F ` U ) ) /\ W < U ) -> E. a e. A E. b e. A E. c e. A ( a < b /\ b < c /\ ( ( ( F ` a ) < ( F ` b ) /\ ( F ` c ) < ( F ` b ) ) \/ ( ( F ` b ) < ( F ` a ) /\ ( F ` b ) < ( F ` c ) ) ) ) ) |
40 |
3
|
ad2antrr |
|- ( ( ( ph /\ ( F ` W ) < ( F ` U ) ) /\ U <_ W ) -> U e. A ) |
41 |
5
|
ad2antrr |
|- ( ( ( ph /\ ( F ` W ) < ( F ` U ) ) /\ U <_ W ) -> W e. A ) |
42 |
6
|
ad2antrr |
|- ( ( ( ph /\ ( F ` W ) < ( F ` U ) ) /\ U <_ W ) -> X e. A ) |
43 |
20
|
ad2antrr |
|- ( ( ( ph /\ ( F ` W ) < ( F ` U ) ) /\ U <_ W ) -> U e. RR ) |
44 |
2 5
|
sseldd |
|- ( ph -> W e. dom F ) |
45 |
18 44
|
sseldd |
|- ( ph -> W e. RR ) |
46 |
45
|
ad2antrr |
|- ( ( ( ph /\ ( F ` W ) < ( F ` U ) ) /\ U <_ W ) -> W e. RR ) |
47 |
|
simpr |
|- ( ( ( ph /\ ( F ` W ) < ( F ` U ) ) /\ U <_ W ) -> U <_ W ) |
48 |
23 44
|
ffvelrnd |
|- ( ph -> ( F ` W ) e. RR ) |
49 |
48
|
ad2antrr |
|- ( ( ( ph /\ ( F ` W ) < ( F ` U ) ) /\ U <_ W ) -> ( F ` W ) e. RR ) |
50 |
|
simplr |
|- ( ( ( ph /\ ( F ` W ) < ( F ` U ) ) /\ U <_ W ) -> ( F ` W ) < ( F ` U ) ) |
51 |
49 50
|
gtned |
|- ( ( ( ph /\ ( F ` W ) < ( F ` U ) ) /\ U <_ W ) -> ( F ` U ) =/= ( F ` W ) ) |
52 |
|
fveq2 |
|- ( W = U -> ( F ` W ) = ( F ` U ) ) |
53 |
52
|
eqcomd |
|- ( W = U -> ( F ` U ) = ( F ` W ) ) |
54 |
53
|
necon3i |
|- ( ( F ` U ) =/= ( F ` W ) -> W =/= U ) |
55 |
51 54
|
syl |
|- ( ( ( ph /\ ( F ` W ) < ( F ` U ) ) /\ U <_ W ) -> W =/= U ) |
56 |
43 46 47 55
|
leneltd |
|- ( ( ( ph /\ ( F ` W ) < ( F ` U ) ) /\ U <_ W ) -> U < W ) |
57 |
2 6
|
sseldd |
|- ( ph -> X e. dom F ) |
58 |
18 57
|
sseldd |
|- ( ph -> X e. RR ) |
59 |
23 57
|
ffvelrnd |
|- ( ph -> ( F ` X ) e. RR ) |
60 |
48 59
|
ltnled |
|- ( ph -> ( ( F ` W ) < ( F ` X ) <-> -. ( F ` X ) <_ ( F ` W ) ) ) |
61 |
10 60
|
mpbird |
|- ( ph -> ( F ` W ) < ( F ` X ) ) |
62 |
48 61
|
gtned |
|- ( ph -> ( F ` X ) =/= ( F ` W ) ) |
63 |
|
fveq2 |
|- ( X = W -> ( F ` X ) = ( F ` W ) ) |
64 |
63
|
necon3i |
|- ( ( F ` X ) =/= ( F ` W ) -> X =/= W ) |
65 |
62 64
|
syl |
|- ( ph -> X =/= W ) |
66 |
45 58 8 65
|
leneltd |
|- ( ph -> W < X ) |
67 |
66
|
ad2antrr |
|- ( ( ( ph /\ ( F ` W ) < ( F ` U ) ) /\ U <_ W ) -> W < X ) |
68 |
61
|
ad2antrr |
|- ( ( ( ph /\ ( F ` W ) < ( F ` U ) ) /\ U <_ W ) -> ( F ` W ) < ( F ` X ) ) |
69 |
50 68
|
jca |
|- ( ( ( ph /\ ( F ` W ) < ( F ` U ) ) /\ U <_ W ) -> ( ( F ` W ) < ( F ` U ) /\ ( F ` W ) < ( F ` X ) ) ) |
70 |
69
|
olcd |
|- ( ( ( ph /\ ( F ` W ) < ( F ` U ) ) /\ U <_ W ) -> ( ( ( F ` U ) < ( F ` W ) /\ ( F ` X ) < ( F ` W ) ) \/ ( ( F ` W ) < ( F ` U ) /\ ( F ` W ) < ( F ` X ) ) ) ) |
71 |
40 41 42 56 67 70
|
pmltpclem1 |
|- ( ( ( ph /\ ( F ` W ) < ( F ` U ) ) /\ U <_ W ) -> E. a e. A E. b e. A E. c e. A ( a < b /\ b < c /\ ( ( ( F ` a ) < ( F ` b ) /\ ( F ` c ) < ( F ` b ) ) \/ ( ( F ` b ) < ( F ` a ) /\ ( F ` b ) < ( F ` c ) ) ) ) ) |
72 |
45
|
adantr |
|- ( ( ph /\ ( F ` W ) < ( F ` U ) ) -> W e. RR ) |
73 |
20
|
adantr |
|- ( ( ph /\ ( F ` W ) < ( F ` U ) ) -> U e. RR ) |
74 |
39 71 72 73
|
ltlecasei |
|- ( ( ph /\ ( F ` W ) < ( F ` U ) ) -> E. a e. A E. b e. A E. c e. A ( a < b /\ b < c /\ ( ( ( F ` a ) < ( F ` b ) /\ ( F ` c ) < ( F ` b ) ) \/ ( ( F ` b ) < ( F ` a ) /\ ( F ` b ) < ( F ` c ) ) ) ) ) |
75 |
3
|
ad2antrr |
|- ( ( ( ph /\ ( F ` U ) <_ ( F ` W ) ) /\ V < X ) -> U e. A ) |
76 |
4
|
ad2antrr |
|- ( ( ( ph /\ ( F ` U ) <_ ( F ` W ) ) /\ V < X ) -> V e. A ) |
77 |
6
|
ad2antrr |
|- ( ( ( ph /\ ( F ` U ) <_ ( F ` W ) ) /\ V < X ) -> X e. A ) |
78 |
33
|
ad2antrr |
|- ( ( ( ph /\ ( F ` U ) <_ ( F ` W ) ) /\ V < X ) -> U < V ) |
79 |
|
simpr |
|- ( ( ( ph /\ ( F ` U ) <_ ( F ` W ) ) /\ V < X ) -> V < X ) |
80 |
27
|
ad2antrr |
|- ( ( ( ph /\ ( F ` U ) <_ ( F ` W ) ) /\ V < X ) -> ( F ` V ) < ( F ` U ) ) |
81 |
24
|
adantr |
|- ( ( ph /\ ( F ` U ) <_ ( F ` W ) ) -> ( F ` V ) e. RR ) |
82 |
25
|
adantr |
|- ( ( ph /\ ( F ` U ) <_ ( F ` W ) ) -> ( F ` U ) e. RR ) |
83 |
59
|
adantr |
|- ( ( ph /\ ( F ` U ) <_ ( F ` W ) ) -> ( F ` X ) e. RR ) |
84 |
27
|
adantr |
|- ( ( ph /\ ( F ` U ) <_ ( F ` W ) ) -> ( F ` V ) < ( F ` U ) ) |
85 |
48
|
adantr |
|- ( ( ph /\ ( F ` U ) <_ ( F ` W ) ) -> ( F ` W ) e. RR ) |
86 |
|
simpr |
|- ( ( ph /\ ( F ` U ) <_ ( F ` W ) ) -> ( F ` U ) <_ ( F ` W ) ) |
87 |
61
|
adantr |
|- ( ( ph /\ ( F ` U ) <_ ( F ` W ) ) -> ( F ` W ) < ( F ` X ) ) |
88 |
82 85 83 86 87
|
lelttrd |
|- ( ( ph /\ ( F ` U ) <_ ( F ` W ) ) -> ( F ` U ) < ( F ` X ) ) |
89 |
81 82 83 84 88
|
lttrd |
|- ( ( ph /\ ( F ` U ) <_ ( F ` W ) ) -> ( F ` V ) < ( F ` X ) ) |
90 |
89
|
adantr |
|- ( ( ( ph /\ ( F ` U ) <_ ( F ` W ) ) /\ V < X ) -> ( F ` V ) < ( F ` X ) ) |
91 |
80 90
|
jca |
|- ( ( ( ph /\ ( F ` U ) <_ ( F ` W ) ) /\ V < X ) -> ( ( F ` V ) < ( F ` U ) /\ ( F ` V ) < ( F ` X ) ) ) |
92 |
91
|
olcd |
|- ( ( ( ph /\ ( F ` U ) <_ ( F ` W ) ) /\ V < X ) -> ( ( ( F ` U ) < ( F ` V ) /\ ( F ` X ) < ( F ` V ) ) \/ ( ( F ` V ) < ( F ` U ) /\ ( F ` V ) < ( F ` X ) ) ) ) |
93 |
75 76 77 78 79 92
|
pmltpclem1 |
|- ( ( ( ph /\ ( F ` U ) <_ ( F ` W ) ) /\ V < X ) -> E. a e. A E. b e. A E. c e. A ( a < b /\ b < c /\ ( ( ( F ` a ) < ( F ` b ) /\ ( F ` c ) < ( F ` b ) ) \/ ( ( F ` b ) < ( F ` a ) /\ ( F ` b ) < ( F ` c ) ) ) ) ) |
94 |
5
|
ad2antrr |
|- ( ( ( ph /\ ( F ` U ) <_ ( F ` W ) ) /\ X <_ V ) -> W e. A ) |
95 |
6
|
ad2antrr |
|- ( ( ( ph /\ ( F ` U ) <_ ( F ` W ) ) /\ X <_ V ) -> X e. A ) |
96 |
4
|
ad2antrr |
|- ( ( ( ph /\ ( F ` U ) <_ ( F ` W ) ) /\ X <_ V ) -> V e. A ) |
97 |
66
|
ad2antrr |
|- ( ( ( ph /\ ( F ` U ) <_ ( F ` W ) ) /\ X <_ V ) -> W < X ) |
98 |
58
|
ad2antrr |
|- ( ( ( ph /\ ( F ` U ) <_ ( F ` W ) ) /\ X <_ V ) -> X e. RR ) |
99 |
22
|
ad2antrr |
|- ( ( ( ph /\ ( F ` U ) <_ ( F ` W ) ) /\ X <_ V ) -> V e. RR ) |
100 |
|
simpr |
|- ( ( ( ph /\ ( F ` U ) <_ ( F ` W ) ) /\ X <_ V ) -> X <_ V ) |
101 |
24
|
ad2antrr |
|- ( ( ( ph /\ ( F ` U ) <_ ( F ` W ) ) /\ X <_ V ) -> ( F ` V ) e. RR ) |
102 |
89
|
adantr |
|- ( ( ( ph /\ ( F ` U ) <_ ( F ` W ) ) /\ X <_ V ) -> ( F ` V ) < ( F ` X ) ) |
103 |
101 102
|
gtned |
|- ( ( ( ph /\ ( F ` U ) <_ ( F ` W ) ) /\ X <_ V ) -> ( F ` X ) =/= ( F ` V ) ) |
104 |
|
fveq2 |
|- ( V = X -> ( F ` V ) = ( F ` X ) ) |
105 |
104
|
eqcomd |
|- ( V = X -> ( F ` X ) = ( F ` V ) ) |
106 |
105
|
necon3i |
|- ( ( F ` X ) =/= ( F ` V ) -> V =/= X ) |
107 |
103 106
|
syl |
|- ( ( ( ph /\ ( F ` U ) <_ ( F ` W ) ) /\ X <_ V ) -> V =/= X ) |
108 |
98 99 100 107
|
leneltd |
|- ( ( ( ph /\ ( F ` U ) <_ ( F ` W ) ) /\ X <_ V ) -> X < V ) |
109 |
61
|
ad2antrr |
|- ( ( ( ph /\ ( F ` U ) <_ ( F ` W ) ) /\ X <_ V ) -> ( F ` W ) < ( F ` X ) ) |
110 |
109 102
|
jca |
|- ( ( ( ph /\ ( F ` U ) <_ ( F ` W ) ) /\ X <_ V ) -> ( ( F ` W ) < ( F ` X ) /\ ( F ` V ) < ( F ` X ) ) ) |
111 |
110
|
orcd |
|- ( ( ( ph /\ ( F ` U ) <_ ( F ` W ) ) /\ X <_ V ) -> ( ( ( F ` W ) < ( F ` X ) /\ ( F ` V ) < ( F ` X ) ) \/ ( ( F ` X ) < ( F ` W ) /\ ( F ` X ) < ( F ` V ) ) ) ) |
112 |
94 95 96 97 108 111
|
pmltpclem1 |
|- ( ( ( ph /\ ( F ` U ) <_ ( F ` W ) ) /\ X <_ V ) -> E. a e. A E. b e. A E. c e. A ( a < b /\ b < c /\ ( ( ( F ` a ) < ( F ` b ) /\ ( F ` c ) < ( F ` b ) ) \/ ( ( F ` b ) < ( F ` a ) /\ ( F ` b ) < ( F ` c ) ) ) ) ) |
113 |
22
|
adantr |
|- ( ( ph /\ ( F ` U ) <_ ( F ` W ) ) -> V e. RR ) |
114 |
58
|
adantr |
|- ( ( ph /\ ( F ` U ) <_ ( F ` W ) ) -> X e. RR ) |
115 |
93 112 113 114
|
ltlecasei |
|- ( ( ph /\ ( F ` U ) <_ ( F ` W ) ) -> E. a e. A E. b e. A E. c e. A ( a < b /\ b < c /\ ( ( ( F ` a ) < ( F ` b ) /\ ( F ` c ) < ( F ` b ) ) \/ ( ( F ` b ) < ( F ` a ) /\ ( F ` b ) < ( F ` c ) ) ) ) ) |
116 |
74 115 48 25
|
ltlecasei |
|- ( ph -> E. a e. A E. b e. A E. c e. A ( a < b /\ b < c /\ ( ( ( F ` a ) < ( F ` b ) /\ ( F ` c ) < ( F ` b ) ) \/ ( ( F ` b ) < ( F ` a ) /\ ( F ` b ) < ( F ` c ) ) ) ) ) |