Metamath Proof Explorer

Theorem mulsproplem6

Description: Lemma for surreal multiplication. Show one of the inequalities involved in surreal multiplication's cuts. (Contributed by Scott Fenton, 5-Mar-2025)

Ref Expression
Hypotheses mulsproplem.1 
|- ( ph -> A. a e. No A. b e. No A. c e. No A. d e. No A. e e. No A. f e. No ( ( ( ( bday ` a ) +no ( bday ` b ) ) u. ( ( ( ( bday ` c ) +no ( bday ` e ) ) u. ( ( bday ` d ) +no ( bday ` f ) ) ) u. ( ( ( bday ` c ) +no ( bday ` f ) ) u. ( ( bday ` d ) +no ( bday ` e ) ) ) ) ) e. ( ( ( bday ` A ) +no ( bday ` B ) ) u. ( ( ( ( bday ` C ) +no ( bday ` E ) ) u. ( ( bday ` D ) +no ( bday ` F ) ) ) u. ( ( ( bday ` C ) +no ( bday ` F ) ) u. ( ( bday ` D ) +no ( bday ` E ) ) ) ) ) -> ( ( a x.s b ) e. No /\ ( ( c  ( ( c x.s f ) -s ( c x.s e ) )
mulsproplem6.1 
|- ( ph -> A e. No )
mulsproplem6.2 
|- ( ph -> B e. No )
mulsproplem6.3 
|- ( ph -> P e. ( _Left ` A ) )
mulsproplem6.4 
|- ( ph -> Q e. ( _Left ` B ) )
mulsproplem6.5 
|- ( ph -> V e. ( _Right ` A ) )
mulsproplem6.6 
|- ( ph -> W e. ( _Left ` B ) )
Assertion mulsproplem6 
|- ( ph -> ( ( ( P x.s B ) +s ( A x.s Q ) ) -s ( P x.s Q ) )

Proof

Step Hyp Ref Expression
1 mulsproplem.1 
 |-  ( ph -> A. a e. No A. b e. No A. c e. No A. d e. No A. e e. No A. f e. No ( ( ( ( bday ` a ) +no ( bday ` b ) ) u. ( ( ( ( bday ` c ) +no ( bday ` e ) ) u. ( ( bday ` d ) +no ( bday ` f ) ) ) u. ( ( ( bday ` c ) +no ( bday ` f ) ) u. ( ( bday ` d ) +no ( bday ` e ) ) ) ) ) e. ( ( ( bday ` A ) +no ( bday ` B ) ) u. ( ( ( ( bday ` C ) +no ( bday ` E ) ) u. ( ( bday ` D ) +no ( bday ` F ) ) ) u. ( ( ( bday ` C ) +no ( bday ` F ) ) u. ( ( bday ` D ) +no ( bday ` E ) ) ) ) ) -> ( ( a x.s b ) e. No /\ ( ( c  ( ( c x.s f ) -s ( c x.s e ) )
2 mulsproplem6.1 
 |-  ( ph -> A e. No )
3 mulsproplem6.2 
 |-  ( ph -> B e. No )
4 mulsproplem6.3 
 |-  ( ph -> P e. ( _Left ` A ) )
5 mulsproplem6.4 
 |-  ( ph -> Q e. ( _Left ` B ) )
6 mulsproplem6.5 
 |-  ( ph -> V e. ( _Right ` A ) )
7 mulsproplem6.6 
 |-  ( ph -> W e. ( _Left ` B ) )
8 5 leftnod 
 |-  ( ph -> Q e. No )
9 7 leftnod 
 |-  ( ph -> W e. No )
10 ltslin 
 |-  ( ( Q e. No /\ W e. No ) -> ( Q
11 8 9 10 syl2anc 
 |-  ( ph -> ( Q
12 4 leftoldd 
 |-  ( ph -> P e. ( _Old ` ( bday ` A ) ) )
13 1 12 3 mulsproplem2 
 |-  ( ph -> ( P x.s B ) e. No )
14 5 leftoldd 
 |-  ( ph -> Q e. ( _Old ` ( bday ` B ) ) )
15 1 2 14 mulsproplem3 
 |-  ( ph -> ( A x.s Q ) e. No )
16 13 15 addscld 
 |-  ( ph -> ( ( P x.s B ) +s ( A x.s Q ) ) e. No )
17 1 12 14 mulsproplem4 
 |-  ( ph -> ( P x.s Q ) e. No )
18 16 17 subscld 
 |-  ( ph -> ( ( ( P x.s B ) +s ( A x.s Q ) ) -s ( P x.s Q ) ) e. No )
19 18 adantr 
 |-  ( ( ph /\ Q  ( ( ( P x.s B ) +s ( A x.s Q ) ) -s ( P x.s Q ) ) e. No )
20 7 leftoldd 
 |-  ( ph -> W e. ( _Old ` ( bday ` B ) ) )
21 1 2 20 mulsproplem3 
 |-  ( ph -> ( A x.s W ) e. No )
22 13 21 addscld 
 |-  ( ph -> ( ( P x.s B ) +s ( A x.s W ) ) e. No )
23 1 12 20 mulsproplem4 
 |-  ( ph -> ( P x.s W ) e. No )
24 22 23 subscld 
 |-  ( ph -> ( ( ( P x.s B ) +s ( A x.s W ) ) -s ( P x.s W ) ) e. No )
25 24 adantr 
 |-  ( ( ph /\ Q  ( ( ( P x.s B ) +s ( A x.s W ) ) -s ( P x.s W ) ) e. No )
26 6 rightoldd 
 |-  ( ph -> V e. ( _Old ` ( bday ` A ) ) )
27 1 26 3 mulsproplem2 
 |-  ( ph -> ( V x.s B ) e. No )
28 27 21 addscld 
 |-  ( ph -> ( ( V x.s B ) +s ( A x.s W ) ) e. No )
29 1 26 20 mulsproplem4 
 |-  ( ph -> ( V x.s W ) e. No )
30 28 29 subscld 
 |-  ( ph -> ( ( ( V x.s B ) +s ( A x.s W ) ) -s ( V x.s W ) ) e. No )
31 30 adantr 
 |-  ( ( ph /\ Q  ( ( ( V x.s B ) +s ( A x.s W ) ) -s ( V x.s W ) ) e. No )
32 sltsleft 
 |-  ( A e. No -> ( _Left ` A ) <
33 2 32 syl 
 |-  ( ph -> ( _Left ` A ) <
34 snidg 
 |-  ( A e. No -> A e. { A } )
35 2 34 syl 
 |-  ( ph -> A e. { A } )
36 33 4 35 sltssepcd 
 |-  ( ph -> P
37 0no 
 |-  0s e. No
38 37 a1i 
 |-  ( ph -> 0s e. No )
39 4 leftnod 
 |-  ( ph -> P e. No )
40 bday0 
 |-  ( bday ` 0s ) = (/)
41 40 40 oveq12i 
 |-  ( ( bday ` 0s ) +no ( bday ` 0s ) ) = ( (/) +no (/) )
42 0elon 
 |-  (/) e. On
43 naddrid 
 |-  ( (/) e. On -> ( (/) +no (/) ) = (/) )
44 42 43 ax-mp 
 |-  ( (/) +no (/) ) = (/)
45 41 44 eqtri 
 |-  ( ( bday ` 0s ) +no ( bday ` 0s ) ) = (/)
46 45 uneq1i 
 |-  ( ( ( bday ` 0s ) +no ( bday ` 0s ) ) u. ( ( ( ( bday ` P ) +no ( bday ` Q ) ) u. ( ( bday ` A ) +no ( bday ` W ) ) ) u. ( ( ( bday ` P ) +no ( bday ` W ) ) u. ( ( bday ` A ) +no ( bday ` Q ) ) ) ) ) = ( (/) u. ( ( ( ( bday ` P ) +no ( bday ` Q ) ) u. ( ( bday ` A ) +no ( bday ` W ) ) ) u. ( ( ( bday ` P ) +no ( bday ` W ) ) u. ( ( bday ` A ) +no ( bday ` Q ) ) ) ) )
47 0un 
 |-  ( (/) u. ( ( ( ( bday ` P ) +no ( bday ` Q ) ) u. ( ( bday ` A ) +no ( bday ` W ) ) ) u. ( ( ( bday ` P ) +no ( bday ` W ) ) u. ( ( bday ` A ) +no ( bday ` Q ) ) ) ) ) = ( ( ( ( bday ` P ) +no ( bday ` Q ) ) u. ( ( bday ` A ) +no ( bday ` W ) ) ) u. ( ( ( bday ` P ) +no ( bday ` W ) ) u. ( ( bday ` A ) +no ( bday ` Q ) ) ) )
48 46 47 eqtri 
 |-  ( ( ( bday ` 0s ) +no ( bday ` 0s ) ) u. ( ( ( ( bday ` P ) +no ( bday ` Q ) ) u. ( ( bday ` A ) +no ( bday ` W ) ) ) u. ( ( ( bday ` P ) +no ( bday ` W ) ) u. ( ( bday ` A ) +no ( bday ` Q ) ) ) ) ) = ( ( ( ( bday ` P ) +no ( bday ` Q ) ) u. ( ( bday ` A ) +no ( bday ` W ) ) ) u. ( ( ( bday ` P ) +no ( bday ` W ) ) u. ( ( bday ` A ) +no ( bday ` Q ) ) ) )
49 oldbdayim 
 |-  ( P e. ( _Old ` ( bday ` A ) ) -> ( bday ` P ) e. ( bday ` A ) )
50 12 49 syl 
 |-  ( ph -> ( bday ` P ) e. ( bday ` A ) )
51 oldbdayim 
 |-  ( Q e. ( _Old ` ( bday ` B ) ) -> ( bday ` Q ) e. ( bday ` B ) )
52 14 51 syl 
 |-  ( ph -> ( bday ` Q ) e. ( bday ` B ) )
53 bdayon 
 |-  ( bday ` A ) e. On
54 bdayon 
 |-  ( bday ` B ) e. On
55 naddel12 
 |-  ( ( ( bday ` A ) e. On /\ ( bday ` B ) e. On ) -> ( ( ( bday ` P ) e. ( bday ` A ) /\ ( bday ` Q ) e. ( bday ` B ) ) -> ( ( bday ` P ) +no ( bday ` Q ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) )
56 53 54 55 mp2an 
 |-  ( ( ( bday ` P ) e. ( bday ` A ) /\ ( bday ` Q ) e. ( bday ` B ) ) -> ( ( bday ` P ) +no ( bday ` Q ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) )
57 50 52 56 syl2anc 
 |-  ( ph -> ( ( bday ` P ) +no ( bday ` Q ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) )
58 oldbdayim 
 |-  ( W e. ( _Old ` ( bday ` B ) ) -> ( bday ` W ) e. ( bday ` B ) )
59 20 58 syl 
 |-  ( ph -> ( bday ` W ) e. ( bday ` B ) )
60 bdayon 
 |-  ( bday ` W ) e. On
61 naddel2 
 |-  ( ( ( bday ` W ) e. On /\ ( bday ` B ) e. On /\ ( bday ` A ) e. On ) -> ( ( bday ` W ) e. ( bday ` B ) <-> ( ( bday ` A ) +no ( bday ` W ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) )
62 60 54 53 61 mp3an 
 |-  ( ( bday ` W ) e. ( bday ` B ) <-> ( ( bday ` A ) +no ( bday ` W ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) )
63 59 62 sylib 
 |-  ( ph -> ( ( bday ` A ) +no ( bday ` W ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) )
64 57 63 jca 
 |-  ( ph -> ( ( ( bday ` P ) +no ( bday ` Q ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) /\ ( ( bday ` A ) +no ( bday ` W ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) )
65 naddel12 
 |-  ( ( ( bday ` A ) e. On /\ ( bday ` B ) e. On ) -> ( ( ( bday ` P ) e. ( bday ` A ) /\ ( bday ` W ) e. ( bday ` B ) ) -> ( ( bday ` P ) +no ( bday ` W ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) )
66 53 54 65 mp2an 
 |-  ( ( ( bday ` P ) e. ( bday ` A ) /\ ( bday ` W ) e. ( bday ` B ) ) -> ( ( bday ` P ) +no ( bday ` W ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) )
67 50 59 66 syl2anc 
 |-  ( ph -> ( ( bday ` P ) +no ( bday ` W ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) )
68 bdayon 
 |-  ( bday ` Q ) e. On
69 naddel2 
 |-  ( ( ( bday ` Q ) e. On /\ ( bday ` B ) e. On /\ ( bday ` A ) e. On ) -> ( ( bday ` Q ) e. ( bday ` B ) <-> ( ( bday ` A ) +no ( bday ` Q ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) )
70 68 54 53 69 mp3an 
 |-  ( ( bday ` Q ) e. ( bday ` B ) <-> ( ( bday ` A ) +no ( bday ` Q ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) )
71 52 70 sylib 
 |-  ( ph -> ( ( bday ` A ) +no ( bday ` Q ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) )
72 67 71 jca 
 |-  ( ph -> ( ( ( bday ` P ) +no ( bday ` W ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) /\ ( ( bday ` A ) +no ( bday ` Q ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) )
73 bdayon 
 |-  ( bday ` P ) e. On
74 naddcl 
 |-  ( ( ( bday ` P ) e. On /\ ( bday ` Q ) e. On ) -> ( ( bday ` P ) +no ( bday ` Q ) ) e. On )
75 73 68 74 mp2an 
 |-  ( ( bday ` P ) +no ( bday ` Q ) ) e. On
76 naddcl 
 |-  ( ( ( bday ` A ) e. On /\ ( bday ` W ) e. On ) -> ( ( bday ` A ) +no ( bday ` W ) ) e. On )
77 53 60 76 mp2an 
 |-  ( ( bday ` A ) +no ( bday ` W ) ) e. On
78 75 77 onun2i 
 |-  ( ( ( bday ` P ) +no ( bday ` Q ) ) u. ( ( bday ` A ) +no ( bday ` W ) ) ) e. On
79 naddcl 
 |-  ( ( ( bday ` P ) e. On /\ ( bday ` W ) e. On ) -> ( ( bday ` P ) +no ( bday ` W ) ) e. On )
80 73 60 79 mp2an 
 |-  ( ( bday ` P ) +no ( bday ` W ) ) e. On
81 naddcl 
 |-  ( ( ( bday ` A ) e. On /\ ( bday ` Q ) e. On ) -> ( ( bday ` A ) +no ( bday ` Q ) ) e. On )
82 53 68 81 mp2an 
 |-  ( ( bday ` A ) +no ( bday ` Q ) ) e. On
83 80 82 onun2i 
 |-  ( ( ( bday ` P ) +no ( bday ` W ) ) u. ( ( bday ` A ) +no ( bday ` Q ) ) ) e. On
84 naddcl 
 |-  ( ( ( bday ` A ) e. On /\ ( bday ` B ) e. On ) -> ( ( bday ` A ) +no ( bday ` B ) ) e. On )
85 53 54 84 mp2an 
 |-  ( ( bday ` A ) +no ( bday ` B ) ) e. On
86 onunel 
 |-  ( ( ( ( ( bday ` P ) +no ( bday ` Q ) ) u. ( ( bday ` A ) +no ( bday ` W ) ) ) e. On /\ ( ( ( bday ` P ) +no ( bday ` W ) ) u. ( ( bday ` A ) +no ( bday ` Q ) ) ) e. On /\ ( ( bday ` A ) +no ( bday ` B ) ) e. On ) -> ( ( ( ( ( bday ` P ) +no ( bday ` Q ) ) u. ( ( bday ` A ) +no ( bday ` W ) ) ) u. ( ( ( bday ` P ) +no ( bday ` W ) ) u. ( ( bday ` A ) +no ( bday ` Q ) ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) <-> ( ( ( ( bday ` P ) +no ( bday ` Q ) ) u. ( ( bday ` A ) +no ( bday ` W ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) /\ ( ( ( bday ` P ) +no ( bday ` W ) ) u. ( ( bday ` A ) +no ( bday ` Q ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) ) )
87 78 83 85 86 mp3an 
 |-  ( ( ( ( ( bday ` P ) +no ( bday ` Q ) ) u. ( ( bday ` A ) +no ( bday ` W ) ) ) u. ( ( ( bday ` P ) +no ( bday ` W ) ) u. ( ( bday ` A ) +no ( bday ` Q ) ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) <-> ( ( ( ( bday ` P ) +no ( bday ` Q ) ) u. ( ( bday ` A ) +no ( bday ` W ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) /\ ( ( ( bday ` P ) +no ( bday ` W ) ) u. ( ( bday ` A ) +no ( bday ` Q ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) )
88 onunel 
 |-  ( ( ( ( bday ` P ) +no ( bday ` Q ) ) e. On /\ ( ( bday ` A ) +no ( bday ` W ) ) e. On /\ ( ( bday ` A ) +no ( bday ` B ) ) e. On ) -> ( ( ( ( bday ` P ) +no ( bday ` Q ) ) u. ( ( bday ` A ) +no ( bday ` W ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) <-> ( ( ( bday ` P ) +no ( bday ` Q ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) /\ ( ( bday ` A ) +no ( bday ` W ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) ) )
89 75 77 85 88 mp3an 
 |-  ( ( ( ( bday ` P ) +no ( bday ` Q ) ) u. ( ( bday ` A ) +no ( bday ` W ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) <-> ( ( ( bday ` P ) +no ( bday ` Q ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) /\ ( ( bday ` A ) +no ( bday ` W ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) )
90 onunel 
 |-  ( ( ( ( bday ` P ) +no ( bday ` W ) ) e. On /\ ( ( bday ` A ) +no ( bday ` Q ) ) e. On /\ ( ( bday ` A ) +no ( bday ` B ) ) e. On ) -> ( ( ( ( bday ` P ) +no ( bday ` W ) ) u. ( ( bday ` A ) +no ( bday ` Q ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) <-> ( ( ( bday ` P ) +no ( bday ` W ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) /\ ( ( bday ` A ) +no ( bday ` Q ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) ) )
91 80 82 85 90 mp3an 
 |-  ( ( ( ( bday ` P ) +no ( bday ` W ) ) u. ( ( bday ` A ) +no ( bday ` Q ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) <-> ( ( ( bday ` P ) +no ( bday ` W ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) /\ ( ( bday ` A ) +no ( bday ` Q ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) )
92 89 91 anbi12i 
 |-  ( ( ( ( ( bday ` P ) +no ( bday ` Q ) ) u. ( ( bday ` A ) +no ( bday ` W ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) /\ ( ( ( bday ` P ) +no ( bday ` W ) ) u. ( ( bday ` A ) +no ( bday ` Q ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) <-> ( ( ( ( bday ` P ) +no ( bday ` Q ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) /\ ( ( bday ` A ) +no ( bday ` W ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) /\ ( ( ( bday ` P ) +no ( bday ` W ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) /\ ( ( bday ` A ) +no ( bday ` Q ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) ) )
93 87 92 bitri 
 |-  ( ( ( ( ( bday ` P ) +no ( bday ` Q ) ) u. ( ( bday ` A ) +no ( bday ` W ) ) ) u. ( ( ( bday ` P ) +no ( bday ` W ) ) u. ( ( bday ` A ) +no ( bday ` Q ) ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) <-> ( ( ( ( bday ` P ) +no ( bday ` Q ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) /\ ( ( bday ` A ) +no ( bday ` W ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) /\ ( ( ( bday ` P ) +no ( bday ` W ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) /\ ( ( bday ` A ) +no ( bday ` Q ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) ) )
94 64 72 93 sylanbrc 
 |-  ( ph -> ( ( ( ( bday ` P ) +no ( bday ` Q ) ) u. ( ( bday ` A ) +no ( bday ` W ) ) ) u. ( ( ( bday ` P ) +no ( bday ` W ) ) u. ( ( bday ` A ) +no ( bday ` Q ) ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) )
95 elun1 
 |-  ( ( ( ( ( bday ` P ) +no ( bday ` Q ) ) u. ( ( bday ` A ) +no ( bday ` W ) ) ) u. ( ( ( bday ` P ) +no ( bday ` W ) ) u. ( ( bday ` A ) +no ( bday ` Q ) ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) -> ( ( ( ( bday ` P ) +no ( bday ` Q ) ) u. ( ( bday ` A ) +no ( bday ` W ) ) ) u. ( ( ( bday ` P ) +no ( bday ` W ) ) u. ( ( bday ` A ) +no ( bday ` Q ) ) ) ) e. ( ( ( bday ` A ) +no ( bday ` B ) ) u. ( ( ( ( bday ` C ) +no ( bday ` E ) ) u. ( ( bday ` D ) +no ( bday ` F ) ) ) u. ( ( ( bday ` C ) +no ( bday ` F ) ) u. ( ( bday ` D ) +no ( bday ` E ) ) ) ) ) )
96 94 95 syl 
 |-  ( ph -> ( ( ( ( bday ` P ) +no ( bday ` Q ) ) u. ( ( bday ` A ) +no ( bday ` W ) ) ) u. ( ( ( bday ` P ) +no ( bday ` W ) ) u. ( ( bday ` A ) +no ( bday ` Q ) ) ) ) e. ( ( ( bday ` A ) +no ( bday ` B ) ) u. ( ( ( ( bday ` C ) +no ( bday ` E ) ) u. ( ( bday ` D ) +no ( bday ` F ) ) ) u. ( ( ( bday ` C ) +no ( bday ` F ) ) u. ( ( bday ` D ) +no ( bday ` E ) ) ) ) ) )
97 48 96 eqeltrid 
 |-  ( ph -> ( ( ( bday ` 0s ) +no ( bday ` 0s ) ) u. ( ( ( ( bday ` P ) +no ( bday ` Q ) ) u. ( ( bday ` A ) +no ( bday ` W ) ) ) u. ( ( ( bday ` P ) +no ( bday ` W ) ) u. ( ( bday ` A ) +no ( bday ` Q ) ) ) ) ) e. ( ( ( bday ` A ) +no ( bday ` B ) ) u. ( ( ( ( bday ` C ) +no ( bday ` E ) ) u. ( ( bday ` D ) +no ( bday ` F ) ) ) u. ( ( ( bday ` C ) +no ( bday ` F ) ) u. ( ( bday ` D ) +no ( bday ` E ) ) ) ) ) )
98 1 38 38 39 2 8 9 97 mulsproplem1 
 |-  ( ph -> ( ( 0s x.s 0s ) e. No /\ ( ( P  ( ( P x.s W ) -s ( P x.s Q ) )
99 98 simprd 
 |-  ( ph -> ( ( P  ( ( P x.s W ) -s ( P x.s Q ) )
100 36 99 mpand 
 |-  ( ph -> ( Q  ( ( P x.s W ) -s ( P x.s Q ) )
101 100 imp 
 |-  ( ( ph /\ Q  ( ( P x.s W ) -s ( P x.s Q ) )
102 23 21 17 15 ltsubsubs3bd 
 |-  ( ph -> ( ( ( P x.s W ) -s ( P x.s Q ) )  ( ( A x.s Q ) -s ( P x.s Q ) )
103 15 17 subscld 
 |-  ( ph -> ( ( A x.s Q ) -s ( P x.s Q ) ) e. No )
104 21 23 subscld 
 |-  ( ph -> ( ( A x.s W ) -s ( P x.s W ) ) e. No )
105 103 104 13 ltadds2d 
 |-  ( ph -> ( ( ( A x.s Q ) -s ( P x.s Q ) )  ( ( P x.s B ) +s ( ( A x.s Q ) -s ( P x.s Q ) ) )
106 102 105 bitrd 
 |-  ( ph -> ( ( ( P x.s W ) -s ( P x.s Q ) )  ( ( P x.s B ) +s ( ( A x.s Q ) -s ( P x.s Q ) ) )
107 106 adantr 
 |-  ( ( ph /\ Q  ( ( ( P x.s W ) -s ( P x.s Q ) )  ( ( P x.s B ) +s ( ( A x.s Q ) -s ( P x.s Q ) ) )
108 101 107 mpbid 
 |-  ( ( ph /\ Q  ( ( P x.s B ) +s ( ( A x.s Q ) -s ( P x.s Q ) ) )
109 13 15 17 addsubsassd 
 |-  ( ph -> ( ( ( P x.s B ) +s ( A x.s Q ) ) -s ( P x.s Q ) ) = ( ( P x.s B ) +s ( ( A x.s Q ) -s ( P x.s Q ) ) ) )
110 109 adantr 
 |-  ( ( ph /\ Q  ( ( ( P x.s B ) +s ( A x.s Q ) ) -s ( P x.s Q ) ) = ( ( P x.s B ) +s ( ( A x.s Q ) -s ( P x.s Q ) ) ) )
111 13 21 23 addsubsassd 
 |-  ( ph -> ( ( ( P x.s B ) +s ( A x.s W ) ) -s ( P x.s W ) ) = ( ( P x.s B ) +s ( ( A x.s W ) -s ( P x.s W ) ) ) )
112 111 adantr 
 |-  ( ( ph /\ Q  ( ( ( P x.s B ) +s ( A x.s W ) ) -s ( P x.s W ) ) = ( ( P x.s B ) +s ( ( A x.s W ) -s ( P x.s W ) ) ) )
113 108 110 112 3brtr4d 
 |-  ( ( ph /\ Q  ( ( ( P x.s B ) +s ( A x.s Q ) ) -s ( P x.s Q ) )
114 lltr 
 |-  ( _Left ` A ) <
115 114 a1i 
 |-  ( ph -> ( _Left ` A ) <
116 115 4 6 sltssepcd 
 |-  ( ph -> P
117 sltsleft 
 |-  ( B e. No -> ( _Left ` B ) <
118 3 117 syl 
 |-  ( ph -> ( _Left ` B ) <
119 snidg 
 |-  ( B e. No -> B e. { B } )
120 3 119 syl 
 |-  ( ph -> B e. { B } )
121 118 7 120 sltssepcd 
 |-  ( ph -> W
122 6 rightnod 
 |-  ( ph -> V e. No )
123 45 uneq1i 
 |-  ( ( ( bday ` 0s ) +no ( bday ` 0s ) ) u. ( ( ( ( bday ` P ) +no ( bday ` W ) ) u. ( ( bday ` V ) +no ( bday ` B ) ) ) u. ( ( ( bday ` P ) +no ( bday ` B ) ) u. ( ( bday ` V ) +no ( bday ` W ) ) ) ) ) = ( (/) u. ( ( ( ( bday ` P ) +no ( bday ` W ) ) u. ( ( bday ` V ) +no ( bday ` B ) ) ) u. ( ( ( bday ` P ) +no ( bday ` B ) ) u. ( ( bday ` V ) +no ( bday ` W ) ) ) ) )
124 0un 
 |-  ( (/) u. ( ( ( ( bday ` P ) +no ( bday ` W ) ) u. ( ( bday ` V ) +no ( bday ` B ) ) ) u. ( ( ( bday ` P ) +no ( bday ` B ) ) u. ( ( bday ` V ) +no ( bday ` W ) ) ) ) ) = ( ( ( ( bday ` P ) +no ( bday ` W ) ) u. ( ( bday ` V ) +no ( bday ` B ) ) ) u. ( ( ( bday ` P ) +no ( bday ` B ) ) u. ( ( bday ` V ) +no ( bday ` W ) ) ) )
125 123 124 eqtri 
 |-  ( ( ( bday ` 0s ) +no ( bday ` 0s ) ) u. ( ( ( ( bday ` P ) +no ( bday ` W ) ) u. ( ( bday ` V ) +no ( bday ` B ) ) ) u. ( ( ( bday ` P ) +no ( bday ` B ) ) u. ( ( bday ` V ) +no ( bday ` W ) ) ) ) ) = ( ( ( ( bday ` P ) +no ( bday ` W ) ) u. ( ( bday ` V ) +no ( bday ` B ) ) ) u. ( ( ( bday ` P ) +no ( bday ` B ) ) u. ( ( bday ` V ) +no ( bday ` W ) ) ) )
126 oldbdayim 
 |-  ( V e. ( _Old ` ( bday ` A ) ) -> ( bday ` V ) e. ( bday ` A ) )
127 26 126 syl 
 |-  ( ph -> ( bday ` V ) e. ( bday ` A ) )
128 bdayon 
 |-  ( bday ` V ) e. On
129 naddel1 
 |-  ( ( ( bday ` V ) e. On /\ ( bday ` A ) e. On /\ ( bday ` B ) e. On ) -> ( ( bday ` V ) e. ( bday ` A ) <-> ( ( bday ` V ) +no ( bday ` B ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) )
130 128 53 54 129 mp3an 
 |-  ( ( bday ` V ) e. ( bday ` A ) <-> ( ( bday ` V ) +no ( bday ` B ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) )
131 127 130 sylib 
 |-  ( ph -> ( ( bday ` V ) +no ( bday ` B ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) )
132 67 131 jca 
 |-  ( ph -> ( ( ( bday ` P ) +no ( bday ` W ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) /\ ( ( bday ` V ) +no ( bday ` B ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) )
133 naddel1 
 |-  ( ( ( bday ` P ) e. On /\ ( bday ` A ) e. On /\ ( bday ` B ) e. On ) -> ( ( bday ` P ) e. ( bday ` A ) <-> ( ( bday ` P ) +no ( bday ` B ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) )
134 73 53 54 133 mp3an 
 |-  ( ( bday ` P ) e. ( bday ` A ) <-> ( ( bday ` P ) +no ( bday ` B ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) )
135 50 134 sylib 
 |-  ( ph -> ( ( bday ` P ) +no ( bday ` B ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) )
136 naddel12 
 |-  ( ( ( bday ` A ) e. On /\ ( bday ` B ) e. On ) -> ( ( ( bday ` V ) e. ( bday ` A ) /\ ( bday ` W ) e. ( bday ` B ) ) -> ( ( bday ` V ) +no ( bday ` W ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) )
137 53 54 136 mp2an 
 |-  ( ( ( bday ` V ) e. ( bday ` A ) /\ ( bday ` W ) e. ( bday ` B ) ) -> ( ( bday ` V ) +no ( bday ` W ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) )
138 127 59 137 syl2anc 
 |-  ( ph -> ( ( bday ` V ) +no ( bday ` W ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) )
139 135 138 jca 
 |-  ( ph -> ( ( ( bday ` P ) +no ( bday ` B ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) /\ ( ( bday ` V ) +no ( bday ` W ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) )
140 naddcl 
 |-  ( ( ( bday ` V ) e. On /\ ( bday ` B ) e. On ) -> ( ( bday ` V ) +no ( bday ` B ) ) e. On )
141 128 54 140 mp2an 
 |-  ( ( bday ` V ) +no ( bday ` B ) ) e. On
142 80 141 onun2i 
 |-  ( ( ( bday ` P ) +no ( bday ` W ) ) u. ( ( bday ` V ) +no ( bday ` B ) ) ) e. On
143 naddcl 
 |-  ( ( ( bday ` P ) e. On /\ ( bday ` B ) e. On ) -> ( ( bday ` P ) +no ( bday ` B ) ) e. On )
144 73 54 143 mp2an 
 |-  ( ( bday ` P ) +no ( bday ` B ) ) e. On
145 naddcl 
 |-  ( ( ( bday ` V ) e. On /\ ( bday ` W ) e. On ) -> ( ( bday ` V ) +no ( bday ` W ) ) e. On )
146 128 60 145 mp2an 
 |-  ( ( bday ` V ) +no ( bday ` W ) ) e. On
147 144 146 onun2i 
 |-  ( ( ( bday ` P ) +no ( bday ` B ) ) u. ( ( bday ` V ) +no ( bday ` W ) ) ) e. On
148 onunel 
 |-  ( ( ( ( ( bday ` P ) +no ( bday ` W ) ) u. ( ( bday ` V ) +no ( bday ` B ) ) ) e. On /\ ( ( ( bday ` P ) +no ( bday ` B ) ) u. ( ( bday ` V ) +no ( bday ` W ) ) ) e. On /\ ( ( bday ` A ) +no ( bday ` B ) ) e. On ) -> ( ( ( ( ( bday ` P ) +no ( bday ` W ) ) u. ( ( bday ` V ) +no ( bday ` B ) ) ) u. ( ( ( bday ` P ) +no ( bday ` B ) ) u. ( ( bday ` V ) +no ( bday ` W ) ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) <-> ( ( ( ( bday ` P ) +no ( bday ` W ) ) u. ( ( bday ` V ) +no ( bday ` B ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) /\ ( ( ( bday ` P ) +no ( bday ` B ) ) u. ( ( bday ` V ) +no ( bday ` W ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) ) )
149 142 147 85 148 mp3an 
 |-  ( ( ( ( ( bday ` P ) +no ( bday ` W ) ) u. ( ( bday ` V ) +no ( bday ` B ) ) ) u. ( ( ( bday ` P ) +no ( bday ` B ) ) u. ( ( bday ` V ) +no ( bday ` W ) ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) <-> ( ( ( ( bday ` P ) +no ( bday ` W ) ) u. ( ( bday ` V ) +no ( bday ` B ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) /\ ( ( ( bday ` P ) +no ( bday ` B ) ) u. ( ( bday ` V ) +no ( bday ` W ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) )
150 onunel 
 |-  ( ( ( ( bday ` P ) +no ( bday ` W ) ) e. On /\ ( ( bday ` V ) +no ( bday ` B ) ) e. On /\ ( ( bday ` A ) +no ( bday ` B ) ) e. On ) -> ( ( ( ( bday ` P ) +no ( bday ` W ) ) u. ( ( bday ` V ) +no ( bday ` B ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) <-> ( ( ( bday ` P ) +no ( bday ` W ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) /\ ( ( bday ` V ) +no ( bday ` B ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) ) )
151 80 141 85 150 mp3an 
 |-  ( ( ( ( bday ` P ) +no ( bday ` W ) ) u. ( ( bday ` V ) +no ( bday ` B ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) <-> ( ( ( bday ` P ) +no ( bday ` W ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) /\ ( ( bday ` V ) +no ( bday ` B ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) )
152 onunel 
 |-  ( ( ( ( bday ` P ) +no ( bday ` B ) ) e. On /\ ( ( bday ` V ) +no ( bday ` W ) ) e. On /\ ( ( bday ` A ) +no ( bday ` B ) ) e. On ) -> ( ( ( ( bday ` P ) +no ( bday ` B ) ) u. ( ( bday ` V ) +no ( bday ` W ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) <-> ( ( ( bday ` P ) +no ( bday ` B ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) /\ ( ( bday ` V ) +no ( bday ` W ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) ) )
153 144 146 85 152 mp3an 
 |-  ( ( ( ( bday ` P ) +no ( bday ` B ) ) u. ( ( bday ` V ) +no ( bday ` W ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) <-> ( ( ( bday ` P ) +no ( bday ` B ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) /\ ( ( bday ` V ) +no ( bday ` W ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) )
154 151 153 anbi12i 
 |-  ( ( ( ( ( bday ` P ) +no ( bday ` W ) ) u. ( ( bday ` V ) +no ( bday ` B ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) /\ ( ( ( bday ` P ) +no ( bday ` B ) ) u. ( ( bday ` V ) +no ( bday ` W ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) <-> ( ( ( ( bday ` P ) +no ( bday ` W ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) /\ ( ( bday ` V ) +no ( bday ` B ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) /\ ( ( ( bday ` P ) +no ( bday ` B ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) /\ ( ( bday ` V ) +no ( bday ` W ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) ) )
155 149 154 bitri 
 |-  ( ( ( ( ( bday ` P ) +no ( bday ` W ) ) u. ( ( bday ` V ) +no ( bday ` B ) ) ) u. ( ( ( bday ` P ) +no ( bday ` B ) ) u. ( ( bday ` V ) +no ( bday ` W ) ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) <-> ( ( ( ( bday ` P ) +no ( bday ` W ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) /\ ( ( bday ` V ) +no ( bday ` B ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) /\ ( ( ( bday ` P ) +no ( bday ` B ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) /\ ( ( bday ` V ) +no ( bday ` W ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) ) )
156 132 139 155 sylanbrc 
 |-  ( ph -> ( ( ( ( bday ` P ) +no ( bday ` W ) ) u. ( ( bday ` V ) +no ( bday ` B ) ) ) u. ( ( ( bday ` P ) +no ( bday ` B ) ) u. ( ( bday ` V ) +no ( bday ` W ) ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) )
157 elun1 
 |-  ( ( ( ( ( bday ` P ) +no ( bday ` W ) ) u. ( ( bday ` V ) +no ( bday ` B ) ) ) u. ( ( ( bday ` P ) +no ( bday ` B ) ) u. ( ( bday ` V ) +no ( bday ` W ) ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) -> ( ( ( ( bday ` P ) +no ( bday ` W ) ) u. ( ( bday ` V ) +no ( bday ` B ) ) ) u. ( ( ( bday ` P ) +no ( bday ` B ) ) u. ( ( bday ` V ) +no ( bday ` W ) ) ) ) e. ( ( ( bday ` A ) +no ( bday ` B ) ) u. ( ( ( ( bday ` C ) +no ( bday ` E ) ) u. ( ( bday ` D ) +no ( bday ` F ) ) ) u. ( ( ( bday ` C ) +no ( bday ` F ) ) u. ( ( bday ` D ) +no ( bday ` E ) ) ) ) ) )
158 156 157 syl 
 |-  ( ph -> ( ( ( ( bday ` P ) +no ( bday ` W ) ) u. ( ( bday ` V ) +no ( bday ` B ) ) ) u. ( ( ( bday ` P ) +no ( bday ` B ) ) u. ( ( bday ` V ) +no ( bday ` W ) ) ) ) e. ( ( ( bday ` A ) +no ( bday ` B ) ) u. ( ( ( ( bday ` C ) +no ( bday ` E ) ) u. ( ( bday ` D ) +no ( bday ` F ) ) ) u. ( ( ( bday ` C ) +no ( bday ` F ) ) u. ( ( bday ` D ) +no ( bday ` E ) ) ) ) ) )
159 125 158 eqeltrid 
 |-  ( ph -> ( ( ( bday ` 0s ) +no ( bday ` 0s ) ) u. ( ( ( ( bday ` P ) +no ( bday ` W ) ) u. ( ( bday ` V ) +no ( bday ` B ) ) ) u. ( ( ( bday ` P ) +no ( bday ` B ) ) u. ( ( bday ` V ) +no ( bday ` W ) ) ) ) ) e. ( ( ( bday ` A ) +no ( bday ` B ) ) u. ( ( ( ( bday ` C ) +no ( bday ` E ) ) u. ( ( bday ` D ) +no ( bday ` F ) ) ) u. ( ( ( bday ` C ) +no ( bday ` F ) ) u. ( ( bday ` D ) +no ( bday ` E ) ) ) ) ) )
160 1 38 38 39 122 9 3 159 mulsproplem1 
 |-  ( ph -> ( ( 0s x.s 0s ) e. No /\ ( ( P  ( ( P x.s B ) -s ( P x.s W ) )
161 160 simprd 
 |-  ( ph -> ( ( P  ( ( P x.s B ) -s ( P x.s W ) )
162 116 121 161 mp2and 
 |-  ( ph -> ( ( P x.s B ) -s ( P x.s W ) )
163 13 23 subscld 
 |-  ( ph -> ( ( P x.s B ) -s ( P x.s W ) ) e. No )
164 27 29 subscld 
 |-  ( ph -> ( ( V x.s B ) -s ( V x.s W ) ) e. No )
165 163 164 21 ltadds1d 
 |-  ( ph -> ( ( ( P x.s B ) -s ( P x.s W ) )  ( ( ( P x.s B ) -s ( P x.s W ) ) +s ( A x.s W ) )
166 162 165 mpbid 
 |-  ( ph -> ( ( ( P x.s B ) -s ( P x.s W ) ) +s ( A x.s W ) )
167 13 21 23 addsubsd 
 |-  ( ph -> ( ( ( P x.s B ) +s ( A x.s W ) ) -s ( P x.s W ) ) = ( ( ( P x.s B ) -s ( P x.s W ) ) +s ( A x.s W ) ) )
168 27 21 29 addsubsd 
 |-  ( ph -> ( ( ( V x.s B ) +s ( A x.s W ) ) -s ( V x.s W ) ) = ( ( ( V x.s B ) -s ( V x.s W ) ) +s ( A x.s W ) ) )
169 166 167 168 3brtr4d 
 |-  ( ph -> ( ( ( P x.s B ) +s ( A x.s W ) ) -s ( P x.s W ) )
170 169 adantr 
 |-  ( ( ph /\ Q  ( ( ( P x.s B ) +s ( A x.s W ) ) -s ( P x.s W ) )
171 19 25 31 113 170 ltstrd 
 |-  ( ( ph /\ Q  ( ( ( P x.s B ) +s ( A x.s Q ) ) -s ( P x.s Q ) )
172 171 ex 
 |-  ( ph -> ( Q  ( ( ( P x.s B ) +s ( A x.s Q ) ) -s ( P x.s Q ) )
173 oveq2 
 |-  ( Q = W -> ( A x.s Q ) = ( A x.s W ) )
174 173 oveq2d 
 |-  ( Q = W -> ( ( P x.s B ) +s ( A x.s Q ) ) = ( ( P x.s B ) +s ( A x.s W ) ) )
175 oveq2 
 |-  ( Q = W -> ( P x.s Q ) = ( P x.s W ) )
176 174 175 oveq12d 
 |-  ( Q = W -> ( ( ( P x.s B ) +s ( A x.s Q ) ) -s ( P x.s Q ) ) = ( ( ( P x.s B ) +s ( A x.s W ) ) -s ( P x.s W ) ) )
177 176 breq1d 
 |-  ( Q = W -> ( ( ( ( P x.s B ) +s ( A x.s Q ) ) -s ( P x.s Q ) )  ( ( ( P x.s B ) +s ( A x.s W ) ) -s ( P x.s W ) )
178 169 177 syl5ibrcom 
 |-  ( ph -> ( Q = W -> ( ( ( P x.s B ) +s ( A x.s Q ) ) -s ( P x.s Q ) )
179 18 adantr 
 |-  ( ( ph /\ W  ( ( ( P x.s B ) +s ( A x.s Q ) ) -s ( P x.s Q ) ) e. No )
180 27 15 addscld 
 |-  ( ph -> ( ( V x.s B ) +s ( A x.s Q ) ) e. No )
181 1 26 14 mulsproplem4 
 |-  ( ph -> ( V x.s Q ) e. No )
182 180 181 subscld 
 |-  ( ph -> ( ( ( V x.s B ) +s ( A x.s Q ) ) -s ( V x.s Q ) ) e. No )
183 182 adantr 
 |-  ( ( ph /\ W  ( ( ( V x.s B ) +s ( A x.s Q ) ) -s ( V x.s Q ) ) e. No )
184 30 adantr 
 |-  ( ( ph /\ W  ( ( ( V x.s B ) +s ( A x.s W ) ) -s ( V x.s W ) ) e. No )
185 118 5 120 sltssepcd 
 |-  ( ph -> Q
186 45 uneq1i 
 |-  ( ( ( bday ` 0s ) +no ( bday ` 0s ) ) u. ( ( ( ( bday ` P ) +no ( bday ` Q ) ) u. ( ( bday ` V ) +no ( bday ` B ) ) ) u. ( ( ( bday ` P ) +no ( bday ` B ) ) u. ( ( bday ` V ) +no ( bday ` Q ) ) ) ) ) = ( (/) u. ( ( ( ( bday ` P ) +no ( bday ` Q ) ) u. ( ( bday ` V ) +no ( bday ` B ) ) ) u. ( ( ( bday ` P ) +no ( bday ` B ) ) u. ( ( bday ` V ) +no ( bday ` Q ) ) ) ) )
187 0un 
 |-  ( (/) u. ( ( ( ( bday ` P ) +no ( bday ` Q ) ) u. ( ( bday ` V ) +no ( bday ` B ) ) ) u. ( ( ( bday ` P ) +no ( bday ` B ) ) u. ( ( bday ` V ) +no ( bday ` Q ) ) ) ) ) = ( ( ( ( bday ` P ) +no ( bday ` Q ) ) u. ( ( bday ` V ) +no ( bday ` B ) ) ) u. ( ( ( bday ` P ) +no ( bday ` B ) ) u. ( ( bday ` V ) +no ( bday ` Q ) ) ) )
188 186 187 eqtri 
 |-  ( ( ( bday ` 0s ) +no ( bday ` 0s ) ) u. ( ( ( ( bday ` P ) +no ( bday ` Q ) ) u. ( ( bday ` V ) +no ( bday ` B ) ) ) u. ( ( ( bday ` P ) +no ( bday ` B ) ) u. ( ( bday ` V ) +no ( bday ` Q ) ) ) ) ) = ( ( ( ( bday ` P ) +no ( bday ` Q ) ) u. ( ( bday ` V ) +no ( bday ` B ) ) ) u. ( ( ( bday ` P ) +no ( bday ` B ) ) u. ( ( bday ` V ) +no ( bday ` Q ) ) ) )
189 57 131 jca 
 |-  ( ph -> ( ( ( bday ` P ) +no ( bday ` Q ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) /\ ( ( bday ` V ) +no ( bday ` B ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) )
190 naddel12 
 |-  ( ( ( bday ` A ) e. On /\ ( bday ` B ) e. On ) -> ( ( ( bday ` V ) e. ( bday ` A ) /\ ( bday ` Q ) e. ( bday ` B ) ) -> ( ( bday ` V ) +no ( bday ` Q ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) )
191 53 54 190 mp2an 
 |-  ( ( ( bday ` V ) e. ( bday ` A ) /\ ( bday ` Q ) e. ( bday ` B ) ) -> ( ( bday ` V ) +no ( bday ` Q ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) )
192 127 52 191 syl2anc 
 |-  ( ph -> ( ( bday ` V ) +no ( bday ` Q ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) )
193 135 192 jca 
 |-  ( ph -> ( ( ( bday ` P ) +no ( bday ` B ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) /\ ( ( bday ` V ) +no ( bday ` Q ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) )
194 75 141 onun2i 
 |-  ( ( ( bday ` P ) +no ( bday ` Q ) ) u. ( ( bday ` V ) +no ( bday ` B ) ) ) e. On
195 naddcl 
 |-  ( ( ( bday ` V ) e. On /\ ( bday ` Q ) e. On ) -> ( ( bday ` V ) +no ( bday ` Q ) ) e. On )
196 128 68 195 mp2an 
 |-  ( ( bday ` V ) +no ( bday ` Q ) ) e. On
197 144 196 onun2i 
 |-  ( ( ( bday ` P ) +no ( bday ` B ) ) u. ( ( bday ` V ) +no ( bday ` Q ) ) ) e. On
198 onunel 
 |-  ( ( ( ( ( bday ` P ) +no ( bday ` Q ) ) u. ( ( bday ` V ) +no ( bday ` B ) ) ) e. On /\ ( ( ( bday ` P ) +no ( bday ` B ) ) u. ( ( bday ` V ) +no ( bday ` Q ) ) ) e. On /\ ( ( bday ` A ) +no ( bday ` B ) ) e. On ) -> ( ( ( ( ( bday ` P ) +no ( bday ` Q ) ) u. ( ( bday ` V ) +no ( bday ` B ) ) ) u. ( ( ( bday ` P ) +no ( bday ` B ) ) u. ( ( bday ` V ) +no ( bday ` Q ) ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) <-> ( ( ( ( bday ` P ) +no ( bday ` Q ) ) u. ( ( bday ` V ) +no ( bday ` B ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) /\ ( ( ( bday ` P ) +no ( bday ` B ) ) u. ( ( bday ` V ) +no ( bday ` Q ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) ) )
199 194 197 85 198 mp3an 
 |-  ( ( ( ( ( bday ` P ) +no ( bday ` Q ) ) u. ( ( bday ` V ) +no ( bday ` B ) ) ) u. ( ( ( bday ` P ) +no ( bday ` B ) ) u. ( ( bday ` V ) +no ( bday ` Q ) ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) <-> ( ( ( ( bday ` P ) +no ( bday ` Q ) ) u. ( ( bday ` V ) +no ( bday ` B ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) /\ ( ( ( bday ` P ) +no ( bday ` B ) ) u. ( ( bday ` V ) +no ( bday ` Q ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) )
200 onunel 
 |-  ( ( ( ( bday ` P ) +no ( bday ` Q ) ) e. On /\ ( ( bday ` V ) +no ( bday ` B ) ) e. On /\ ( ( bday ` A ) +no ( bday ` B ) ) e. On ) -> ( ( ( ( bday ` P ) +no ( bday ` Q ) ) u. ( ( bday ` V ) +no ( bday ` B ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) <-> ( ( ( bday ` P ) +no ( bday ` Q ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) /\ ( ( bday ` V ) +no ( bday ` B ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) ) )
201 75 141 85 200 mp3an 
 |-  ( ( ( ( bday ` P ) +no ( bday ` Q ) ) u. ( ( bday ` V ) +no ( bday ` B ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) <-> ( ( ( bday ` P ) +no ( bday ` Q ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) /\ ( ( bday ` V ) +no ( bday ` B ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) )
202 onunel 
 |-  ( ( ( ( bday ` P ) +no ( bday ` B ) ) e. On /\ ( ( bday ` V ) +no ( bday ` Q ) ) e. On /\ ( ( bday ` A ) +no ( bday ` B ) ) e. On ) -> ( ( ( ( bday ` P ) +no ( bday ` B ) ) u. ( ( bday ` V ) +no ( bday ` Q ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) <-> ( ( ( bday ` P ) +no ( bday ` B ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) /\ ( ( bday ` V ) +no ( bday ` Q ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) ) )
203 144 196 85 202 mp3an 
 |-  ( ( ( ( bday ` P ) +no ( bday ` B ) ) u. ( ( bday ` V ) +no ( bday ` Q ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) <-> ( ( ( bday ` P ) +no ( bday ` B ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) /\ ( ( bday ` V ) +no ( bday ` Q ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) )
204 201 203 anbi12i 
 |-  ( ( ( ( ( bday ` P ) +no ( bday ` Q ) ) u. ( ( bday ` V ) +no ( bday ` B ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) /\ ( ( ( bday ` P ) +no ( bday ` B ) ) u. ( ( bday ` V ) +no ( bday ` Q ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) <-> ( ( ( ( bday ` P ) +no ( bday ` Q ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) /\ ( ( bday ` V ) +no ( bday ` B ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) /\ ( ( ( bday ` P ) +no ( bday ` B ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) /\ ( ( bday ` V ) +no ( bday ` Q ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) ) )
205 199 204 bitri 
 |-  ( ( ( ( ( bday ` P ) +no ( bday ` Q ) ) u. ( ( bday ` V ) +no ( bday ` B ) ) ) u. ( ( ( bday ` P ) +no ( bday ` B ) ) u. ( ( bday ` V ) +no ( bday ` Q ) ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) <-> ( ( ( ( bday ` P ) +no ( bday ` Q ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) /\ ( ( bday ` V ) +no ( bday ` B ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) /\ ( ( ( bday ` P ) +no ( bday ` B ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) /\ ( ( bday ` V ) +no ( bday ` Q ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) ) )
206 189 193 205 sylanbrc 
 |-  ( ph -> ( ( ( ( bday ` P ) +no ( bday ` Q ) ) u. ( ( bday ` V ) +no ( bday ` B ) ) ) u. ( ( ( bday ` P ) +no ( bday ` B ) ) u. ( ( bday ` V ) +no ( bday ` Q ) ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) )
207 elun1 
 |-  ( ( ( ( ( bday ` P ) +no ( bday ` Q ) ) u. ( ( bday ` V ) +no ( bday ` B ) ) ) u. ( ( ( bday ` P ) +no ( bday ` B ) ) u. ( ( bday ` V ) +no ( bday ` Q ) ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) -> ( ( ( ( bday ` P ) +no ( bday ` Q ) ) u. ( ( bday ` V ) +no ( bday ` B ) ) ) u. ( ( ( bday ` P ) +no ( bday ` B ) ) u. ( ( bday ` V ) +no ( bday ` Q ) ) ) ) e. ( ( ( bday ` A ) +no ( bday ` B ) ) u. ( ( ( ( bday ` C ) +no ( bday ` E ) ) u. ( ( bday ` D ) +no ( bday ` F ) ) ) u. ( ( ( bday ` C ) +no ( bday ` F ) ) u. ( ( bday ` D ) +no ( bday ` E ) ) ) ) ) )
208 206 207 syl 
 |-  ( ph -> ( ( ( ( bday ` P ) +no ( bday ` Q ) ) u. ( ( bday ` V ) +no ( bday ` B ) ) ) u. ( ( ( bday ` P ) +no ( bday ` B ) ) u. ( ( bday ` V ) +no ( bday ` Q ) ) ) ) e. ( ( ( bday ` A ) +no ( bday ` B ) ) u. ( ( ( ( bday ` C ) +no ( bday ` E ) ) u. ( ( bday ` D ) +no ( bday ` F ) ) ) u. ( ( ( bday ` C ) +no ( bday ` F ) ) u. ( ( bday ` D ) +no ( bday ` E ) ) ) ) ) )
209 188 208 eqeltrid 
 |-  ( ph -> ( ( ( bday ` 0s ) +no ( bday ` 0s ) ) u. ( ( ( ( bday ` P ) +no ( bday ` Q ) ) u. ( ( bday ` V ) +no ( bday ` B ) ) ) u. ( ( ( bday ` P ) +no ( bday ` B ) ) u. ( ( bday ` V ) +no ( bday ` Q ) ) ) ) ) e. ( ( ( bday ` A ) +no ( bday ` B ) ) u. ( ( ( ( bday ` C ) +no ( bday ` E ) ) u. ( ( bday ` D ) +no ( bday ` F ) ) ) u. ( ( ( bday ` C ) +no ( bday ` F ) ) u. ( ( bday ` D ) +no ( bday ` E ) ) ) ) ) )
210 1 38 38 39 122 8 3 209 mulsproplem1 
 |-  ( ph -> ( ( 0s x.s 0s ) e. No /\ ( ( P  ( ( P x.s B ) -s ( P x.s Q ) )
211 210 simprd 
 |-  ( ph -> ( ( P  ( ( P x.s B ) -s ( P x.s Q ) )
212 116 185 211 mp2and 
 |-  ( ph -> ( ( P x.s B ) -s ( P x.s Q ) )
213 13 17 subscld 
 |-  ( ph -> ( ( P x.s B ) -s ( P x.s Q ) ) e. No )
214 27 181 subscld 
 |-  ( ph -> ( ( V x.s B ) -s ( V x.s Q ) ) e. No )
215 213 214 15 ltadds1d 
 |-  ( ph -> ( ( ( P x.s B ) -s ( P x.s Q ) )  ( ( ( P x.s B ) -s ( P x.s Q ) ) +s ( A x.s Q ) )
216 212 215 mpbid 
 |-  ( ph -> ( ( ( P x.s B ) -s ( P x.s Q ) ) +s ( A x.s Q ) )
217 13 15 17 addsubsd 
 |-  ( ph -> ( ( ( P x.s B ) +s ( A x.s Q ) ) -s ( P x.s Q ) ) = ( ( ( P x.s B ) -s ( P x.s Q ) ) +s ( A x.s Q ) ) )
218 27 15 181 addsubsd 
 |-  ( ph -> ( ( ( V x.s B ) +s ( A x.s Q ) ) -s ( V x.s Q ) ) = ( ( ( V x.s B ) -s ( V x.s Q ) ) +s ( A x.s Q ) ) )
219 216 217 218 3brtr4d 
 |-  ( ph -> ( ( ( P x.s B ) +s ( A x.s Q ) ) -s ( P x.s Q ) )
220 219 adantr 
 |-  ( ( ph /\ W  ( ( ( P x.s B ) +s ( A x.s Q ) ) -s ( P x.s Q ) )
221 sltsright 
 |-  ( A e. No -> { A } <
222 2 221 syl 
 |-  ( ph -> { A } <
223 222 35 6 sltssepcd 
 |-  ( ph -> A
224 45 uneq1i 
 |-  ( ( ( bday ` 0s ) +no ( bday ` 0s ) ) u. ( ( ( ( bday ` A ) +no ( bday ` W ) ) u. ( ( bday ` V ) +no ( bday ` Q ) ) ) u. ( ( ( bday ` A ) +no ( bday ` Q ) ) u. ( ( bday ` V ) +no ( bday ` W ) ) ) ) ) = ( (/) u. ( ( ( ( bday ` A ) +no ( bday ` W ) ) u. ( ( bday ` V ) +no ( bday ` Q ) ) ) u. ( ( ( bday ` A ) +no ( bday ` Q ) ) u. ( ( bday ` V ) +no ( bday ` W ) ) ) ) )
225 0un 
 |-  ( (/) u. ( ( ( ( bday ` A ) +no ( bday ` W ) ) u. ( ( bday ` V ) +no ( bday ` Q ) ) ) u. ( ( ( bday ` A ) +no ( bday ` Q ) ) u. ( ( bday ` V ) +no ( bday ` W ) ) ) ) ) = ( ( ( ( bday ` A ) +no ( bday ` W ) ) u. ( ( bday ` V ) +no ( bday ` Q ) ) ) u. ( ( ( bday ` A ) +no ( bday ` Q ) ) u. ( ( bday ` V ) +no ( bday ` W ) ) ) )
226 224 225 eqtri 
 |-  ( ( ( bday ` 0s ) +no ( bday ` 0s ) ) u. ( ( ( ( bday ` A ) +no ( bday ` W ) ) u. ( ( bday ` V ) +no ( bday ` Q ) ) ) u. ( ( ( bday ` A ) +no ( bday ` Q ) ) u. ( ( bday ` V ) +no ( bday ` W ) ) ) ) ) = ( ( ( ( bday ` A ) +no ( bday ` W ) ) u. ( ( bday ` V ) +no ( bday ` Q ) ) ) u. ( ( ( bday ` A ) +no ( bday ` Q ) ) u. ( ( bday ` V ) +no ( bday ` W ) ) ) )
227 63 192 jca 
 |-  ( ph -> ( ( ( bday ` A ) +no ( bday ` W ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) /\ ( ( bday ` V ) +no ( bday ` Q ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) )
228 71 138 jca 
 |-  ( ph -> ( ( ( bday ` A ) +no ( bday ` Q ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) /\ ( ( bday ` V ) +no ( bday ` W ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) )
229 77 196 onun2i 
 |-  ( ( ( bday ` A ) +no ( bday ` W ) ) u. ( ( bday ` V ) +no ( bday ` Q ) ) ) e. On
230 82 146 onun2i 
 |-  ( ( ( bday ` A ) +no ( bday ` Q ) ) u. ( ( bday ` V ) +no ( bday ` W ) ) ) e. On
231 onunel 
 |-  ( ( ( ( ( bday ` A ) +no ( bday ` W ) ) u. ( ( bday ` V ) +no ( bday ` Q ) ) ) e. On /\ ( ( ( bday ` A ) +no ( bday ` Q ) ) u. ( ( bday ` V ) +no ( bday ` W ) ) ) e. On /\ ( ( bday ` A ) +no ( bday ` B ) ) e. On ) -> ( ( ( ( ( bday ` A ) +no ( bday ` W ) ) u. ( ( bday ` V ) +no ( bday ` Q ) ) ) u. ( ( ( bday ` A ) +no ( bday ` Q ) ) u. ( ( bday ` V ) +no ( bday ` W ) ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) <-> ( ( ( ( bday ` A ) +no ( bday ` W ) ) u. ( ( bday ` V ) +no ( bday ` Q ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) /\ ( ( ( bday ` A ) +no ( bday ` Q ) ) u. ( ( bday ` V ) +no ( bday ` W ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) ) )
232 229 230 85 231 mp3an 
 |-  ( ( ( ( ( bday ` A ) +no ( bday ` W ) ) u. ( ( bday ` V ) +no ( bday ` Q ) ) ) u. ( ( ( bday ` A ) +no ( bday ` Q ) ) u. ( ( bday ` V ) +no ( bday ` W ) ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) <-> ( ( ( ( bday ` A ) +no ( bday ` W ) ) u. ( ( bday ` V ) +no ( bday ` Q ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) /\ ( ( ( bday ` A ) +no ( bday ` Q ) ) u. ( ( bday ` V ) +no ( bday ` W ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) )
233 onunel 
 |-  ( ( ( ( bday ` A ) +no ( bday ` W ) ) e. On /\ ( ( bday ` V ) +no ( bday ` Q ) ) e. On /\ ( ( bday ` A ) +no ( bday ` B ) ) e. On ) -> ( ( ( ( bday ` A ) +no ( bday ` W ) ) u. ( ( bday ` V ) +no ( bday ` Q ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) <-> ( ( ( bday ` A ) +no ( bday ` W ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) /\ ( ( bday ` V ) +no ( bday ` Q ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) ) )
234 77 196 85 233 mp3an 
 |-  ( ( ( ( bday ` A ) +no ( bday ` W ) ) u. ( ( bday ` V ) +no ( bday ` Q ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) <-> ( ( ( bday ` A ) +no ( bday ` W ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) /\ ( ( bday ` V ) +no ( bday ` Q ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) )
235 onunel 
 |-  ( ( ( ( bday ` A ) +no ( bday ` Q ) ) e. On /\ ( ( bday ` V ) +no ( bday ` W ) ) e. On /\ ( ( bday ` A ) +no ( bday ` B ) ) e. On ) -> ( ( ( ( bday ` A ) +no ( bday ` Q ) ) u. ( ( bday ` V ) +no ( bday ` W ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) <-> ( ( ( bday ` A ) +no ( bday ` Q ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) /\ ( ( bday ` V ) +no ( bday ` W ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) ) )
236 82 146 85 235 mp3an 
 |-  ( ( ( ( bday ` A ) +no ( bday ` Q ) ) u. ( ( bday ` V ) +no ( bday ` W ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) <-> ( ( ( bday ` A ) +no ( bday ` Q ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) /\ ( ( bday ` V ) +no ( bday ` W ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) )
237 234 236 anbi12i 
 |-  ( ( ( ( ( bday ` A ) +no ( bday ` W ) ) u. ( ( bday ` V ) +no ( bday ` Q ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) /\ ( ( ( bday ` A ) +no ( bday ` Q ) ) u. ( ( bday ` V ) +no ( bday ` W ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) <-> ( ( ( ( bday ` A ) +no ( bday ` W ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) /\ ( ( bday ` V ) +no ( bday ` Q ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) /\ ( ( ( bday ` A ) +no ( bday ` Q ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) /\ ( ( bday ` V ) +no ( bday ` W ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) ) )
238 232 237 bitri 
 |-  ( ( ( ( ( bday ` A ) +no ( bday ` W ) ) u. ( ( bday ` V ) +no ( bday ` Q ) ) ) u. ( ( ( bday ` A ) +no ( bday ` Q ) ) u. ( ( bday ` V ) +no ( bday ` W ) ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) <-> ( ( ( ( bday ` A ) +no ( bday ` W ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) /\ ( ( bday ` V ) +no ( bday ` Q ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) /\ ( ( ( bday ` A ) +no ( bday ` Q ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) /\ ( ( bday ` V ) +no ( bday ` W ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) ) )
239 227 228 238 sylanbrc 
 |-  ( ph -> ( ( ( ( bday ` A ) +no ( bday ` W ) ) u. ( ( bday ` V ) +no ( bday ` Q ) ) ) u. ( ( ( bday ` A ) +no ( bday ` Q ) ) u. ( ( bday ` V ) +no ( bday ` W ) ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) )
240 elun1 
 |-  ( ( ( ( ( bday ` A ) +no ( bday ` W ) ) u. ( ( bday ` V ) +no ( bday ` Q ) ) ) u. ( ( ( bday ` A ) +no ( bday ` Q ) ) u. ( ( bday ` V ) +no ( bday ` W ) ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) -> ( ( ( ( bday ` A ) +no ( bday ` W ) ) u. ( ( bday ` V ) +no ( bday ` Q ) ) ) u. ( ( ( bday ` A ) +no ( bday ` Q ) ) u. ( ( bday ` V ) +no ( bday ` W ) ) ) ) e. ( ( ( bday ` A ) +no ( bday ` B ) ) u. ( ( ( ( bday ` C ) +no ( bday ` E ) ) u. ( ( bday ` D ) +no ( bday ` F ) ) ) u. ( ( ( bday ` C ) +no ( bday ` F ) ) u. ( ( bday ` D ) +no ( bday ` E ) ) ) ) ) )
241 239 240 syl 
 |-  ( ph -> ( ( ( ( bday ` A ) +no ( bday ` W ) ) u. ( ( bday ` V ) +no ( bday ` Q ) ) ) u. ( ( ( bday ` A ) +no ( bday ` Q ) ) u. ( ( bday ` V ) +no ( bday ` W ) ) ) ) e. ( ( ( bday ` A ) +no ( bday ` B ) ) u. ( ( ( ( bday ` C ) +no ( bday ` E ) ) u. ( ( bday ` D ) +no ( bday ` F ) ) ) u. ( ( ( bday ` C ) +no ( bday ` F ) ) u. ( ( bday ` D ) +no ( bday ` E ) ) ) ) ) )
242 226 241 eqeltrid 
 |-  ( ph -> ( ( ( bday ` 0s ) +no ( bday ` 0s ) ) u. ( ( ( ( bday ` A ) +no ( bday ` W ) ) u. ( ( bday ` V ) +no ( bday ` Q ) ) ) u. ( ( ( bday ` A ) +no ( bday ` Q ) ) u. ( ( bday ` V ) +no ( bday ` W ) ) ) ) ) e. ( ( ( bday ` A ) +no ( bday ` B ) ) u. ( ( ( ( bday ` C ) +no ( bday ` E ) ) u. ( ( bday ` D ) +no ( bday ` F ) ) ) u. ( ( ( bday ` C ) +no ( bday ` F ) ) u. ( ( bday ` D ) +no ( bday ` E ) ) ) ) ) )
243 1 38 38 2 122 9 8 242 mulsproplem1 
 |-  ( ph -> ( ( 0s x.s 0s ) e. No /\ ( ( A  ( ( A x.s Q ) -s ( A x.s W ) )
244 243 simprd 
 |-  ( ph -> ( ( A  ( ( A x.s Q ) -s ( A x.s W ) )
245 223 244 mpand 
 |-  ( ph -> ( W  ( ( A x.s Q ) -s ( A x.s W ) )
246 245 imp 
 |-  ( ( ph /\ W  ( ( A x.s Q ) -s ( A x.s W ) )
247 15 181 21 29 ltsubsubsbd 
 |-  ( ph -> ( ( ( A x.s Q ) -s ( A x.s W ) )  ( ( A x.s Q ) -s ( V x.s Q ) )
248 15 181 subscld 
 |-  ( ph -> ( ( A x.s Q ) -s ( V x.s Q ) ) e. No )
249 21 29 subscld 
 |-  ( ph -> ( ( A x.s W ) -s ( V x.s W ) ) e. No )
250 248 249 27 ltadds2d 
 |-  ( ph -> ( ( ( A x.s Q ) -s ( V x.s Q ) )  ( ( V x.s B ) +s ( ( A x.s Q ) -s ( V x.s Q ) ) )
251 247 250 bitrd 
 |-  ( ph -> ( ( ( A x.s Q ) -s ( A x.s W ) )  ( ( V x.s B ) +s ( ( A x.s Q ) -s ( V x.s Q ) ) )
252 251 adantr 
 |-  ( ( ph /\ W  ( ( ( A x.s Q ) -s ( A x.s W ) )  ( ( V x.s B ) +s ( ( A x.s Q ) -s ( V x.s Q ) ) )
253 246 252 mpbid 
 |-  ( ( ph /\ W  ( ( V x.s B ) +s ( ( A x.s Q ) -s ( V x.s Q ) ) )
254 27 15 181 addsubsassd 
 |-  ( ph -> ( ( ( V x.s B ) +s ( A x.s Q ) ) -s ( V x.s Q ) ) = ( ( V x.s B ) +s ( ( A x.s Q ) -s ( V x.s Q ) ) ) )
255 254 adantr 
 |-  ( ( ph /\ W  ( ( ( V x.s B ) +s ( A x.s Q ) ) -s ( V x.s Q ) ) = ( ( V x.s B ) +s ( ( A x.s Q ) -s ( V x.s Q ) ) ) )
256 27 21 29 addsubsassd 
 |-  ( ph -> ( ( ( V x.s B ) +s ( A x.s W ) ) -s ( V x.s W ) ) = ( ( V x.s B ) +s ( ( A x.s W ) -s ( V x.s W ) ) ) )
257 256 adantr 
 |-  ( ( ph /\ W  ( ( ( V x.s B ) +s ( A x.s W ) ) -s ( V x.s W ) ) = ( ( V x.s B ) +s ( ( A x.s W ) -s ( V x.s W ) ) ) )
258 253 255 257 3brtr4d 
 |-  ( ( ph /\ W  ( ( ( V x.s B ) +s ( A x.s Q ) ) -s ( V x.s Q ) )
259 179 183 184 220 258 ltstrd 
 |-  ( ( ph /\ W  ( ( ( P x.s B ) +s ( A x.s Q ) ) -s ( P x.s Q ) )
260 259 ex 
 |-  ( ph -> ( W  ( ( ( P x.s B ) +s ( A x.s Q ) ) -s ( P x.s Q ) )
261 172 178 260 3jaod 
 |-  ( ph -> ( ( Q  ( ( ( P x.s B ) +s ( A x.s Q ) ) -s ( P x.s Q ) )
262 11 261 mpd 
 |-  ( ph -> ( ( ( P x.s B ) +s ( A x.s Q ) ) -s ( P x.s Q ) )