| Step |
Hyp |
Ref |
Expression |
| 1 |
|
naddwordnex.a |
|- ( ph -> A = ( ( _om .o C ) +o M ) ) |
| 2 |
|
naddwordnex.b |
|- ( ph -> B = ( ( _om .o D ) +o N ) ) |
| 3 |
|
naddwordnex.c |
|- ( ph -> C e. D ) |
| 4 |
|
naddwordnex.d |
|- ( ph -> D e. On ) |
| 5 |
|
naddwordnex.m |
|- ( ph -> M e. _om ) |
| 6 |
|
naddwordnex.n |
|- ( ph -> N e. M ) |
| 7 |
|
naddwordnexlem4.s |
|- S = { y e. On | D C_ ( C +o y ) } |
| 8 |
7
|
ssrab3 |
|- S C_ On |
| 9 |
|
oveq2 |
|- ( y = D -> ( C +o y ) = ( C +o D ) ) |
| 10 |
9
|
sseq2d |
|- ( y = D -> ( D C_ ( C +o y ) <-> D C_ ( C +o D ) ) ) |
| 11 |
|
onelon |
|- ( ( D e. On /\ C e. D ) -> C e. On ) |
| 12 |
4 3 11
|
syl2anc |
|- ( ph -> C e. On ) |
| 13 |
|
oaword2 |
|- ( ( D e. On /\ C e. On ) -> D C_ ( C +o D ) ) |
| 14 |
4 12 13
|
syl2anc |
|- ( ph -> D C_ ( C +o D ) ) |
| 15 |
10 4 14
|
elrabd |
|- ( ph -> D e. { y e. On | D C_ ( C +o y ) } ) |
| 16 |
15 7
|
eleqtrrdi |
|- ( ph -> D e. S ) |
| 17 |
16
|
ne0d |
|- ( ph -> S =/= (/) ) |
| 18 |
|
oninton |
|- ( ( S C_ On /\ S =/= (/) ) -> |^| S e. On ) |
| 19 |
8 17 18
|
sylancr |
|- ( ph -> |^| S e. On ) |
| 20 |
|
oveq2 |
|- ( y = (/) -> ( C +o y ) = ( C +o (/) ) ) |
| 21 |
|
oa0 |
|- ( C e. On -> ( C +o (/) ) = C ) |
| 22 |
12 21
|
syl |
|- ( ph -> ( C +o (/) ) = C ) |
| 23 |
20 22
|
sylan9eqr |
|- ( ( ph /\ y = (/) ) -> ( C +o y ) = C ) |
| 24 |
3
|
adantr |
|- ( ( ph /\ y = (/) ) -> C e. D ) |
| 25 |
23 24
|
eqeltrd |
|- ( ( ph /\ y = (/) ) -> ( C +o y ) e. D ) |
| 26 |
25
|
ex |
|- ( ph -> ( y = (/) -> ( C +o y ) e. D ) ) |
| 27 |
26
|
adantr |
|- ( ( ph /\ y e. On ) -> ( y = (/) -> ( C +o y ) e. D ) ) |
| 28 |
27
|
con3d |
|- ( ( ph /\ y e. On ) -> ( -. ( C +o y ) e. D -> -. y = (/) ) ) |
| 29 |
|
oacl |
|- ( ( C e. On /\ y e. On ) -> ( C +o y ) e. On ) |
| 30 |
12 29
|
sylan |
|- ( ( ph /\ y e. On ) -> ( C +o y ) e. On ) |
| 31 |
|
ontri1 |
|- ( ( D e. On /\ ( C +o y ) e. On ) -> ( D C_ ( C +o y ) <-> -. ( C +o y ) e. D ) ) |
| 32 |
4 30 31
|
syl2an2r |
|- ( ( ph /\ y e. On ) -> ( D C_ ( C +o y ) <-> -. ( C +o y ) e. D ) ) |
| 33 |
|
on0eln0 |
|- ( y e. On -> ( (/) e. y <-> y =/= (/) ) ) |
| 34 |
|
df-ne |
|- ( y =/= (/) <-> -. y = (/) ) |
| 35 |
33 34
|
bitrdi |
|- ( y e. On -> ( (/) e. y <-> -. y = (/) ) ) |
| 36 |
35
|
adantl |
|- ( ( ph /\ y e. On ) -> ( (/) e. y <-> -. y = (/) ) ) |
| 37 |
28 32 36
|
3imtr4d |
|- ( ( ph /\ y e. On ) -> ( D C_ ( C +o y ) -> (/) e. y ) ) |
| 38 |
37
|
ex |
|- ( ph -> ( y e. On -> ( D C_ ( C +o y ) -> (/) e. y ) ) ) |
| 39 |
38
|
ralrimiv |
|- ( ph -> A. y e. On ( D C_ ( C +o y ) -> (/) e. y ) ) |
| 40 |
|
0ex |
|- (/) e. _V |
| 41 |
40
|
elintrab |
|- ( (/) e. |^| { y e. On | D C_ ( C +o y ) } <-> A. y e. On ( D C_ ( C +o y ) -> (/) e. y ) ) |
| 42 |
39 41
|
sylibr |
|- ( ph -> (/) e. |^| { y e. On | D C_ ( C +o y ) } ) |
| 43 |
7
|
inteqi |
|- |^| S = |^| { y e. On | D C_ ( C +o y ) } |
| 44 |
42 43
|
eleqtrrdi |
|- ( ph -> (/) e. |^| S ) |
| 45 |
|
ondif1 |
|- ( |^| S e. ( On \ 1o ) <-> ( |^| S e. On /\ (/) e. |^| S ) ) |
| 46 |
19 44 45
|
sylanbrc |
|- ( ph -> |^| S e. ( On \ 1o ) ) |
| 47 |
|
onzsl |
|- ( |^| S e. On <-> ( |^| S = (/) \/ E. z e. On |^| S = suc z \/ ( |^| S e. _V /\ Lim |^| S ) ) ) |
| 48 |
19 47
|
sylib |
|- ( ph -> ( |^| S = (/) \/ E. z e. On |^| S = suc z \/ ( |^| S e. _V /\ Lim |^| S ) ) ) |
| 49 |
|
oveq2 |
|- ( |^| S = (/) -> ( C +o |^| S ) = ( C +o (/) ) ) |
| 50 |
49 22
|
sylan9eqr |
|- ( ( ph /\ |^| S = (/) ) -> ( C +o |^| S ) = C ) |
| 51 |
|
onelpss |
|- ( ( C e. On /\ D e. On ) -> ( C e. D <-> ( C C_ D /\ C =/= D ) ) ) |
| 52 |
12 4 51
|
syl2anc |
|- ( ph -> ( C e. D <-> ( C C_ D /\ C =/= D ) ) ) |
| 53 |
3 52
|
mpbid |
|- ( ph -> ( C C_ D /\ C =/= D ) ) |
| 54 |
53
|
simpld |
|- ( ph -> C C_ D ) |
| 55 |
54
|
adantr |
|- ( ( ph /\ |^| S = (/) ) -> C C_ D ) |
| 56 |
50 55
|
eqsstrd |
|- ( ( ph /\ |^| S = (/) ) -> ( C +o |^| S ) C_ D ) |
| 57 |
56
|
ex |
|- ( ph -> ( |^| S = (/) -> ( C +o |^| S ) C_ D ) ) |
| 58 |
|
oveq2 |
|- ( |^| S = suc z -> ( C +o |^| S ) = ( C +o suc z ) ) |
| 59 |
|
oasuc |
|- ( ( C e. On /\ z e. On ) -> ( C +o suc z ) = suc ( C +o z ) ) |
| 60 |
12 59
|
sylan |
|- ( ( ph /\ z e. On ) -> ( C +o suc z ) = suc ( C +o z ) ) |
| 61 |
58 60
|
sylan9eqr |
|- ( ( ( ph /\ z e. On ) /\ |^| S = suc z ) -> ( C +o |^| S ) = suc ( C +o z ) ) |
| 62 |
|
vex |
|- z e. _V |
| 63 |
62
|
sucid |
|- z e. suc z |
| 64 |
|
eleq2 |
|- ( |^| S = suc z -> ( z e. |^| S <-> z e. suc z ) ) |
| 65 |
63 64
|
mpbiri |
|- ( |^| S = suc z -> z e. |^| S ) |
| 66 |
65
|
a1i |
|- ( ( ph /\ z e. On ) -> ( |^| S = suc z -> z e. |^| S ) ) |
| 67 |
43
|
eleq2i |
|- ( z e. |^| S <-> z e. |^| { y e. On | D C_ ( C +o y ) } ) |
| 68 |
|
oveq2 |
|- ( y = z -> ( C +o y ) = ( C +o z ) ) |
| 69 |
68
|
sseq2d |
|- ( y = z -> ( D C_ ( C +o y ) <-> D C_ ( C +o z ) ) ) |
| 70 |
69
|
onnminsb |
|- ( z e. On -> ( z e. |^| { y e. On | D C_ ( C +o y ) } -> -. D C_ ( C +o z ) ) ) |
| 71 |
70
|
adantl |
|- ( ( ph /\ z e. On ) -> ( z e. |^| { y e. On | D C_ ( C +o y ) } -> -. D C_ ( C +o z ) ) ) |
| 72 |
67 71
|
biimtrid |
|- ( ( ph /\ z e. On ) -> ( z e. |^| S -> -. D C_ ( C +o z ) ) ) |
| 73 |
|
oacl |
|- ( ( C e. On /\ z e. On ) -> ( C +o z ) e. On ) |
| 74 |
12 73
|
sylan |
|- ( ( ph /\ z e. On ) -> ( C +o z ) e. On ) |
| 75 |
|
ontri1 |
|- ( ( D e. On /\ ( C +o z ) e. On ) -> ( D C_ ( C +o z ) <-> -. ( C +o z ) e. D ) ) |
| 76 |
4 74 75
|
syl2an2r |
|- ( ( ph /\ z e. On ) -> ( D C_ ( C +o z ) <-> -. ( C +o z ) e. D ) ) |
| 77 |
76
|
con2bid |
|- ( ( ph /\ z e. On ) -> ( ( C +o z ) e. D <-> -. D C_ ( C +o z ) ) ) |
| 78 |
72 77
|
sylibrd |
|- ( ( ph /\ z e. On ) -> ( z e. |^| S -> ( C +o z ) e. D ) ) |
| 79 |
|
onsucss |
|- ( D e. On -> ( ( C +o z ) e. D -> suc ( C +o z ) C_ D ) ) |
| 80 |
4 79
|
syl |
|- ( ph -> ( ( C +o z ) e. D -> suc ( C +o z ) C_ D ) ) |
| 81 |
80
|
adantr |
|- ( ( ph /\ z e. On ) -> ( ( C +o z ) e. D -> suc ( C +o z ) C_ D ) ) |
| 82 |
66 78 81
|
3syld |
|- ( ( ph /\ z e. On ) -> ( |^| S = suc z -> suc ( C +o z ) C_ D ) ) |
| 83 |
82
|
imp |
|- ( ( ( ph /\ z e. On ) /\ |^| S = suc z ) -> suc ( C +o z ) C_ D ) |
| 84 |
61 83
|
eqsstrd |
|- ( ( ( ph /\ z e. On ) /\ |^| S = suc z ) -> ( C +o |^| S ) C_ D ) |
| 85 |
84
|
rexlimdva2 |
|- ( ph -> ( E. z e. On |^| S = suc z -> ( C +o |^| S ) C_ D ) ) |
| 86 |
|
oalim |
|- ( ( C e. On /\ ( |^| S e. _V /\ Lim |^| S ) ) -> ( C +o |^| S ) = U_ z e. |^| S ( C +o z ) ) |
| 87 |
12 86
|
sylan |
|- ( ( ph /\ ( |^| S e. _V /\ Lim |^| S ) ) -> ( C +o |^| S ) = U_ z e. |^| S ( C +o z ) ) |
| 88 |
|
onelon |
|- ( ( |^| S e. On /\ z e. |^| S ) -> z e. On ) |
| 89 |
19 88
|
sylan |
|- ( ( ph /\ z e. |^| S ) -> z e. On ) |
| 90 |
89
|
ex |
|- ( ph -> ( z e. |^| S -> z e. On ) ) |
| 91 |
90
|
ancrd |
|- ( ph -> ( z e. |^| S -> ( z e. On /\ z e. |^| S ) ) ) |
| 92 |
78
|
expimpd |
|- ( ph -> ( ( z e. On /\ z e. |^| S ) -> ( C +o z ) e. D ) ) |
| 93 |
|
onelss |
|- ( D e. On -> ( ( C +o z ) e. D -> ( C +o z ) C_ D ) ) |
| 94 |
4 93
|
syl |
|- ( ph -> ( ( C +o z ) e. D -> ( C +o z ) C_ D ) ) |
| 95 |
91 92 94
|
3syld |
|- ( ph -> ( z e. |^| S -> ( C +o z ) C_ D ) ) |
| 96 |
95
|
ralrimiv |
|- ( ph -> A. z e. |^| S ( C +o z ) C_ D ) |
| 97 |
|
iunss |
|- ( U_ z e. |^| S ( C +o z ) C_ D <-> A. z e. |^| S ( C +o z ) C_ D ) |
| 98 |
96 97
|
sylibr |
|- ( ph -> U_ z e. |^| S ( C +o z ) C_ D ) |
| 99 |
98
|
adantr |
|- ( ( ph /\ ( |^| S e. _V /\ Lim |^| S ) ) -> U_ z e. |^| S ( C +o z ) C_ D ) |
| 100 |
87 99
|
eqsstrd |
|- ( ( ph /\ ( |^| S e. _V /\ Lim |^| S ) ) -> ( C +o |^| S ) C_ D ) |
| 101 |
100
|
ex |
|- ( ph -> ( ( |^| S e. _V /\ Lim |^| S ) -> ( C +o |^| S ) C_ D ) ) |
| 102 |
57 85 101
|
3jaod |
|- ( ph -> ( ( |^| S = (/) \/ E. z e. On |^| S = suc z \/ ( |^| S e. _V /\ Lim |^| S ) ) -> ( C +o |^| S ) C_ D ) ) |
| 103 |
48 102
|
mpd |
|- ( ph -> ( C +o |^| S ) C_ D ) |
| 104 |
10
|
rspcev |
|- ( ( D e. On /\ D C_ ( C +o D ) ) -> E. y e. On D C_ ( C +o y ) ) |
| 105 |
4 14 104
|
syl2anc |
|- ( ph -> E. y e. On D C_ ( C +o y ) ) |
| 106 |
|
nfcv |
|- F/_ y D |
| 107 |
|
nfcv |
|- F/_ y C |
| 108 |
|
nfcv |
|- F/_ y +o |
| 109 |
|
nfrab1 |
|- F/_ y { y e. On | D C_ ( C +o y ) } |
| 110 |
109
|
nfint |
|- F/_ y |^| { y e. On | D C_ ( C +o y ) } |
| 111 |
107 108 110
|
nfov |
|- F/_ y ( C +o |^| { y e. On | D C_ ( C +o y ) } ) |
| 112 |
106 111
|
nfss |
|- F/ y D C_ ( C +o |^| { y e. On | D C_ ( C +o y ) } ) |
| 113 |
|
oveq2 |
|- ( y = |^| { y e. On | D C_ ( C +o y ) } -> ( C +o y ) = ( C +o |^| { y e. On | D C_ ( C +o y ) } ) ) |
| 114 |
113
|
sseq2d |
|- ( y = |^| { y e. On | D C_ ( C +o y ) } -> ( D C_ ( C +o y ) <-> D C_ ( C +o |^| { y e. On | D C_ ( C +o y ) } ) ) ) |
| 115 |
112 114
|
onminsb |
|- ( E. y e. On D C_ ( C +o y ) -> D C_ ( C +o |^| { y e. On | D C_ ( C +o y ) } ) ) |
| 116 |
105 115
|
syl |
|- ( ph -> D C_ ( C +o |^| { y e. On | D C_ ( C +o y ) } ) ) |
| 117 |
43
|
oveq2i |
|- ( C +o |^| S ) = ( C +o |^| { y e. On | D C_ ( C +o y ) } ) |
| 118 |
116 117
|
sseqtrrdi |
|- ( ph -> D C_ ( C +o |^| S ) ) |
| 119 |
103 118
|
eqssd |
|- ( ph -> ( C +o |^| S ) = D ) |
| 120 |
|
omelon |
|- _om e. On |
| 121 |
|
omcl |
|- ( ( _om e. On /\ D e. On ) -> ( _om .o D ) e. On ) |
| 122 |
120 4 121
|
sylancr |
|- ( ph -> ( _om .o D ) e. On ) |
| 123 |
120
|
a1i |
|- ( ph -> _om e. On ) |
| 124 |
6 5
|
jca |
|- ( ph -> ( N e. M /\ M e. _om ) ) |
| 125 |
|
ontr1 |
|- ( _om e. On -> ( ( N e. M /\ M e. _om ) -> N e. _om ) ) |
| 126 |
123 124 125
|
sylc |
|- ( ph -> N e. _om ) |
| 127 |
|
nnon |
|- ( N e. _om -> N e. On ) |
| 128 |
126 127
|
syl |
|- ( ph -> N e. On ) |
| 129 |
|
oaword1 |
|- ( ( ( _om .o D ) e. On /\ N e. On ) -> ( _om .o D ) C_ ( ( _om .o D ) +o N ) ) |
| 130 |
122 128 129
|
syl2anc |
|- ( ph -> ( _om .o D ) C_ ( ( _om .o D ) +o N ) ) |
| 131 |
1
|
oveq1d |
|- ( ph -> ( A +o ( _om .o |^| S ) ) = ( ( ( _om .o C ) +o M ) +o ( _om .o |^| S ) ) ) |
| 132 |
|
omcl |
|- ( ( _om e. On /\ C e. On ) -> ( _om .o C ) e. On ) |
| 133 |
120 12 132
|
sylancr |
|- ( ph -> ( _om .o C ) e. On ) |
| 134 |
|
nnon |
|- ( M e. _om -> M e. On ) |
| 135 |
5 134
|
syl |
|- ( ph -> M e. On ) |
| 136 |
|
omcl |
|- ( ( _om e. On /\ |^| S e. On ) -> ( _om .o |^| S ) e. On ) |
| 137 |
120 19 136
|
sylancr |
|- ( ph -> ( _om .o |^| S ) e. On ) |
| 138 |
|
oaass |
|- ( ( ( _om .o C ) e. On /\ M e. On /\ ( _om .o |^| S ) e. On ) -> ( ( ( _om .o C ) +o M ) +o ( _om .o |^| S ) ) = ( ( _om .o C ) +o ( M +o ( _om .o |^| S ) ) ) ) |
| 139 |
133 135 137 138
|
syl3anc |
|- ( ph -> ( ( ( _om .o C ) +o M ) +o ( _om .o |^| S ) ) = ( ( _om .o C ) +o ( M +o ( _om .o |^| S ) ) ) ) |
| 140 |
19 120
|
jctil |
|- ( ph -> ( _om e. On /\ |^| S e. On ) ) |
| 141 |
|
omword1 |
|- ( ( ( _om e. On /\ |^| S e. On ) /\ (/) e. |^| S ) -> _om C_ ( _om .o |^| S ) ) |
| 142 |
140 44 141
|
syl2anc |
|- ( ph -> _om C_ ( _om .o |^| S ) ) |
| 143 |
|
oaabs |
|- ( ( ( M e. _om /\ ( _om .o |^| S ) e. On ) /\ _om C_ ( _om .o |^| S ) ) -> ( M +o ( _om .o |^| S ) ) = ( _om .o |^| S ) ) |
| 144 |
5 137 142 143
|
syl21anc |
|- ( ph -> ( M +o ( _om .o |^| S ) ) = ( _om .o |^| S ) ) |
| 145 |
144
|
oveq2d |
|- ( ph -> ( ( _om .o C ) +o ( M +o ( _om .o |^| S ) ) ) = ( ( _om .o C ) +o ( _om .o |^| S ) ) ) |
| 146 |
|
odi |
|- ( ( _om e. On /\ C e. On /\ |^| S e. On ) -> ( _om .o ( C +o |^| S ) ) = ( ( _om .o C ) +o ( _om .o |^| S ) ) ) |
| 147 |
123 12 19 146
|
syl3anc |
|- ( ph -> ( _om .o ( C +o |^| S ) ) = ( ( _om .o C ) +o ( _om .o |^| S ) ) ) |
| 148 |
119
|
oveq2d |
|- ( ph -> ( _om .o ( C +o |^| S ) ) = ( _om .o D ) ) |
| 149 |
145 147 148
|
3eqtr2d |
|- ( ph -> ( ( _om .o C ) +o ( M +o ( _om .o |^| S ) ) ) = ( _om .o D ) ) |
| 150 |
131 139 149
|
3eqtrd |
|- ( ph -> ( A +o ( _om .o |^| S ) ) = ( _om .o D ) ) |
| 151 |
130 150 2
|
3sstr4d |
|- ( ph -> ( A +o ( _om .o |^| S ) ) C_ B ) |
| 152 |
|
naddcl |
|- ( ( ( _om .o C ) e. On /\ ( _om .o |^| S ) e. On ) -> ( ( _om .o C ) +no ( _om .o |^| S ) ) e. On ) |
| 153 |
133 137 152
|
syl2anc |
|- ( ph -> ( ( _om .o C ) +no ( _om .o |^| S ) ) e. On ) |
| 154 |
122 153 135
|
3jca |
|- ( ph -> ( ( _om .o D ) e. On /\ ( ( _om .o C ) +no ( _om .o |^| S ) ) e. On /\ M e. On ) ) |
| 155 |
148 147
|
eqtr3d |
|- ( ph -> ( _om .o D ) = ( ( _om .o C ) +o ( _om .o |^| S ) ) ) |
| 156 |
|
naddgeoa |
|- ( ( ( _om .o C ) e. On /\ ( _om .o |^| S ) e. On ) -> ( ( _om .o C ) +o ( _om .o |^| S ) ) C_ ( ( _om .o C ) +no ( _om .o |^| S ) ) ) |
| 157 |
133 137 156
|
syl2anc |
|- ( ph -> ( ( _om .o C ) +o ( _om .o |^| S ) ) C_ ( ( _om .o C ) +no ( _om .o |^| S ) ) ) |
| 158 |
155 157
|
eqsstrd |
|- ( ph -> ( _om .o D ) C_ ( ( _om .o C ) +no ( _om .o |^| S ) ) ) |
| 159 |
|
oawordri |
|- ( ( ( _om .o D ) e. On /\ ( ( _om .o C ) +no ( _om .o |^| S ) ) e. On /\ M e. On ) -> ( ( _om .o D ) C_ ( ( _om .o C ) +no ( _om .o |^| S ) ) -> ( ( _om .o D ) +o M ) C_ ( ( ( _om .o C ) +no ( _om .o |^| S ) ) +o M ) ) ) |
| 160 |
154 158 159
|
sylc |
|- ( ph -> ( ( _om .o D ) +o M ) C_ ( ( ( _om .o C ) +no ( _om .o |^| S ) ) +o M ) ) |
| 161 |
|
naddonnn |
|- ( ( ( _om .o C ) e. On /\ M e. _om ) -> ( ( _om .o C ) +o M ) = ( ( _om .o C ) +no M ) ) |
| 162 |
133 5 161
|
syl2anc |
|- ( ph -> ( ( _om .o C ) +o M ) = ( ( _om .o C ) +no M ) ) |
| 163 |
1 162
|
eqtrd |
|- ( ph -> A = ( ( _om .o C ) +no M ) ) |
| 164 |
163
|
oveq1d |
|- ( ph -> ( A +no ( _om .o |^| S ) ) = ( ( ( _om .o C ) +no M ) +no ( _om .o |^| S ) ) ) |
| 165 |
|
naddass |
|- ( ( ( _om .o C ) e. On /\ M e. On /\ ( _om .o |^| S ) e. On ) -> ( ( ( _om .o C ) +no M ) +no ( _om .o |^| S ) ) = ( ( _om .o C ) +no ( M +no ( _om .o |^| S ) ) ) ) |
| 166 |
133 135 137 165
|
syl3anc |
|- ( ph -> ( ( ( _om .o C ) +no M ) +no ( _om .o |^| S ) ) = ( ( _om .o C ) +no ( M +no ( _om .o |^| S ) ) ) ) |
| 167 |
|
naddcom |
|- ( ( M e. On /\ ( _om .o |^| S ) e. On ) -> ( M +no ( _om .o |^| S ) ) = ( ( _om .o |^| S ) +no M ) ) |
| 168 |
135 137 167
|
syl2anc |
|- ( ph -> ( M +no ( _om .o |^| S ) ) = ( ( _om .o |^| S ) +no M ) ) |
| 169 |
168
|
oveq2d |
|- ( ph -> ( ( _om .o C ) +no ( M +no ( _om .o |^| S ) ) ) = ( ( _om .o C ) +no ( ( _om .o |^| S ) +no M ) ) ) |
| 170 |
|
naddonnn |
|- ( ( ( ( _om .o C ) +no ( _om .o |^| S ) ) e. On /\ M e. _om ) -> ( ( ( _om .o C ) +no ( _om .o |^| S ) ) +o M ) = ( ( ( _om .o C ) +no ( _om .o |^| S ) ) +no M ) ) |
| 171 |
153 5 170
|
syl2anc |
|- ( ph -> ( ( ( _om .o C ) +no ( _om .o |^| S ) ) +o M ) = ( ( ( _om .o C ) +no ( _om .o |^| S ) ) +no M ) ) |
| 172 |
|
naddass |
|- ( ( ( _om .o C ) e. On /\ ( _om .o |^| S ) e. On /\ M e. On ) -> ( ( ( _om .o C ) +no ( _om .o |^| S ) ) +no M ) = ( ( _om .o C ) +no ( ( _om .o |^| S ) +no M ) ) ) |
| 173 |
133 137 135 172
|
syl3anc |
|- ( ph -> ( ( ( _om .o C ) +no ( _om .o |^| S ) ) +no M ) = ( ( _om .o C ) +no ( ( _om .o |^| S ) +no M ) ) ) |
| 174 |
171 173
|
eqtr2d |
|- ( ph -> ( ( _om .o C ) +no ( ( _om .o |^| S ) +no M ) ) = ( ( ( _om .o C ) +no ( _om .o |^| S ) ) +o M ) ) |
| 175 |
166 169 174
|
3eqtrd |
|- ( ph -> ( ( ( _om .o C ) +no M ) +no ( _om .o |^| S ) ) = ( ( ( _om .o C ) +no ( _om .o |^| S ) ) +o M ) ) |
| 176 |
164 175
|
eqtr2d |
|- ( ph -> ( ( ( _om .o C ) +no ( _om .o |^| S ) ) +o M ) = ( A +no ( _om .o |^| S ) ) ) |
| 177 |
160 176
|
sseqtrd |
|- ( ph -> ( ( _om .o D ) +o M ) C_ ( A +no ( _om .o |^| S ) ) ) |
| 178 |
135 122
|
jca |
|- ( ph -> ( M e. On /\ ( _om .o D ) e. On ) ) |
| 179 |
|
oaordi |
|- ( ( M e. On /\ ( _om .o D ) e. On ) -> ( N e. M -> ( ( _om .o D ) +o N ) e. ( ( _om .o D ) +o M ) ) ) |
| 180 |
178 6 179
|
sylc |
|- ( ph -> ( ( _om .o D ) +o N ) e. ( ( _om .o D ) +o M ) ) |
| 181 |
2 180
|
eqeltrd |
|- ( ph -> B e. ( ( _om .o D ) +o M ) ) |
| 182 |
177 181
|
sseldd |
|- ( ph -> B e. ( A +no ( _om .o |^| S ) ) ) |
| 183 |
119 151 182
|
3jca |
|- ( ph -> ( ( C +o |^| S ) = D /\ ( A +o ( _om .o |^| S ) ) C_ B /\ B e. ( A +no ( _om .o |^| S ) ) ) ) |
| 184 |
|
oveq2 |
|- ( x = |^| S -> ( C +o x ) = ( C +o |^| S ) ) |
| 185 |
184
|
eqeq1d |
|- ( x = |^| S -> ( ( C +o x ) = D <-> ( C +o |^| S ) = D ) ) |
| 186 |
|
oveq2 |
|- ( x = |^| S -> ( _om .o x ) = ( _om .o |^| S ) ) |
| 187 |
186
|
oveq2d |
|- ( x = |^| S -> ( A +o ( _om .o x ) ) = ( A +o ( _om .o |^| S ) ) ) |
| 188 |
187
|
sseq1d |
|- ( x = |^| S -> ( ( A +o ( _om .o x ) ) C_ B <-> ( A +o ( _om .o |^| S ) ) C_ B ) ) |
| 189 |
186
|
oveq2d |
|- ( x = |^| S -> ( A +no ( _om .o x ) ) = ( A +no ( _om .o |^| S ) ) ) |
| 190 |
189
|
eleq2d |
|- ( x = |^| S -> ( B e. ( A +no ( _om .o x ) ) <-> B e. ( A +no ( _om .o |^| S ) ) ) ) |
| 191 |
185 188 190
|
3anbi123d |
|- ( x = |^| S -> ( ( ( C +o x ) = D /\ ( A +o ( _om .o x ) ) C_ B /\ B e. ( A +no ( _om .o x ) ) ) <-> ( ( C +o |^| S ) = D /\ ( A +o ( _om .o |^| S ) ) C_ B /\ B e. ( A +no ( _om .o |^| S ) ) ) ) ) |
| 192 |
191
|
rspcev |
|- ( ( |^| S e. ( On \ 1o ) /\ ( ( C +o |^| S ) = D /\ ( A +o ( _om .o |^| S ) ) C_ B /\ B e. ( A +no ( _om .o |^| S ) ) ) ) -> E. x e. ( On \ 1o ) ( ( C +o x ) = D /\ ( A +o ( _om .o x ) ) C_ B /\ B e. ( A +no ( _om .o x ) ) ) ) |
| 193 |
46 183 192
|
syl2anc |
|- ( ph -> E. x e. ( On \ 1o ) ( ( C +o x ) = D /\ ( A +o ( _om .o x ) ) C_ B /\ B e. ( A +no ( _om .o x ) ) ) ) |