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 ⊢ ( 𝜑 → ∀ 𝑎 ∈ No ∀ 𝑏 ∈ No ∀ 𝑐 ∈ No ∀ 𝑑 ∈ No ∀ 𝑒 ∈ No ∀ 𝑓 ∈ No ( ( ( ( bday ‘ 𝑎 ) +no ( bday ‘ 𝑏 ) ) ∪ ( ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) ∈ ( ( ( bday ‘ 𝐴 ) +no ( bday ‘ 𝐵 ) ) ∪ ( ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝐸 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝐹 ) ) ) ∪ ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝐹 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝐸 ) ) ) ) ) → ( ( 𝑎 ·s 𝑏 ) ∈ No ∧ ( ( 𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑐 ·s 𝑓 ) -s ( 𝑐 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ) ) ) )
mulsproplem6.1 ⊢ ( 𝜑 → 𝐴 ∈ No )
mulsproplem6.2 ⊢ ( 𝜑 → 𝐵 ∈ No )
mulsproplem6.3 ⊢ ( 𝜑 → 𝑃 ∈ ( L ‘ 𝐴 ) )
mulsproplem6.4 ⊢ ( 𝜑 → 𝑄 ∈ ( L ‘ 𝐵 ) )
mulsproplem6.5 ⊢ ( 𝜑 → 𝑉 ∈ ( R ‘ 𝐴 ) )
mulsproplem6.6 ⊢ ( 𝜑 → 𝑊 ∈ ( L ‘ 𝐵 ) )
Assertion mulsproplem6 ( 𝜑 → ( ( ( 𝑃 ·s 𝐵 ) +s ( 𝐴 ·s 𝑄 ) ) -s ( 𝑃 ·s 𝑄 ) ) <s ( ( ( 𝑉 ·s 𝐵 ) +s ( 𝐴 ·s 𝑊 ) ) -s ( 𝑉 ·s 𝑊 ) ) )

Proof

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