Metamath Proof Explorer


Theorem ssltmul1

Description: One surreal set less-than relationship for cuts of A and B . (Contributed by Scott Fenton, 7-Mar-2025)

Ref Expression
Hypotheses ssltmul1.1 φ L s R
ssltmul1.2 φ M s S
ssltmul1.3 φ A = L | s R
ssltmul1.4 φ B = M | s S
Assertion ssltmul1 φ a | p L q M a = p s B + s A s q - s p s q b | r R s S b = r s B + s A s s - s r s s s A s B

Proof

Step Hyp Ref Expression
1 ssltmul1.1 φ L s R
2 ssltmul1.2 φ M s S
3 ssltmul1.3 φ A = L | s R
4 ssltmul1.4 φ B = M | s S
5 eqid p L , q M p s B + s A s q - s p s q = p L , q M p s B + s A s q - s p s q
6 5 rnmpo ran p L , q M p s B + s A s q - s p s q = a | p L q M a = p s B + s A s q - s p s q
7 ssltex1 L s R L V
8 1 7 syl φ L V
9 ssltex1 M s S M V
10 2 9 syl φ M V
11 5 mpoexg L V M V p L , q M p s B + s A s q - s p s q V
12 8 10 11 syl2anc φ p L , q M p s B + s A s q - s p s q V
13 rnexg p L , q M p s B + s A s q - s p s q V ran p L , q M p s B + s A s q - s p s q V
14 12 13 syl φ ran p L , q M p s B + s A s q - s p s q V
15 6 14 eqeltrrid φ a | p L q M a = p s B + s A s q - s p s q V
16 eqid r R , s S r s B + s A s s - s r s s = r R , s S r s B + s A s s - s r s s
17 16 rnmpo ran r R , s S r s B + s A s s - s r s s = b | r R s S b = r s B + s A s s - s r s s
18 ssltex2 L s R R V
19 1 18 syl φ R V
20 ssltex2 M s S S V
21 2 20 syl φ S V
22 16 mpoexg R V S V r R , s S r s B + s A s s - s r s s V
23 19 21 22 syl2anc φ r R , s S r s B + s A s s - s r s s V
24 rnexg r R , s S r s B + s A s s - s r s s V ran r R , s S r s B + s A s s - s r s s V
25 23 24 syl φ ran r R , s S r s B + s A s s - s r s s V
26 17 25 eqeltrrid φ b | r R s S b = r s B + s A s s - s r s s V
27 15 26 unexd φ a | p L q M a = p s B + s A s q - s p s q b | r R s S b = r s B + s A s s - s r s s V
28 snex A s B V
29 28 a1i φ A s B V
30 ssltss1 L s R L No
31 1 30 syl φ L No
32 31 adantr φ p L q M L No
33 simprl φ p L q M p L
34 32 33 sseldd φ p L q M p No
35 2 scutcld φ M | s S No
36 4 35 eqeltrd φ B No
37 36 adantr φ p L q M B No
38 34 37 mulscld φ p L q M p s B No
39 1 scutcld φ L | s R No
40 3 39 eqeltrd φ A No
41 40 adantr φ p L q M A No
42 ssltss1 M s S M No
43 2 42 syl φ M No
44 43 adantr φ p L q M M No
45 simprr φ p L q M q M
46 44 45 sseldd φ p L q M q No
47 41 46 mulscld φ p L q M A s q No
48 38 47 addscld φ p L q M p s B + s A s q No
49 34 46 mulscld φ p L q M p s q No
50 48 49 subscld φ p L q M p s B + s A s q - s p s q No
51 eleq1 a = p s B + s A s q - s p s q a No p s B + s A s q - s p s q No
52 50 51 syl5ibrcom φ p L q M a = p s B + s A s q - s p s q a No
53 52 rexlimdvva φ p L q M a = p s B + s A s q - s p s q a No
54 53 abssdv φ a | p L q M a = p s B + s A s q - s p s q No
55 ssltss2 L s R R No
56 1 55 syl φ R No
57 56 adantr φ r R s S R No
58 simprl φ r R s S r R
59 57 58 sseldd φ r R s S r No
60 36 adantr φ r R s S B No
61 59 60 mulscld φ r R s S r s B No
62 40 adantr φ r R s S A No
63 ssltss2 M s S S No
64 2 63 syl φ S No
65 64 adantr φ r R s S S No
66 simprr φ r R s S s S
67 65 66 sseldd φ r R s S s No
68 62 67 mulscld φ r R s S A s s No
69 61 68 addscld φ r R s S r s B + s A s s No
70 59 67 mulscld φ r R s S r s s No
71 69 70 subscld φ r R s S r s B + s A s s - s r s s No
72 eleq1 b = r s B + s A s s - s r s s b No r s B + s A s s - s r s s No
73 71 72 syl5ibrcom φ r R s S b = r s B + s A s s - s r s s b No
74 73 rexlimdvva φ r R s S b = r s B + s A s s - s r s s b No
75 74 abssdv φ b | r R s S b = r s B + s A s s - s r s s No
76 54 75 unssd φ a | p L q M a = p s B + s A s q - s p s q b | r R s S b = r s B + s A s s - s r s s No
77 40 36 mulscld φ A s B No
78 77 snssd φ A s B No
79 elun x a | p L q M a = p s B + s A s q - s p s q b | r R s S b = r s B + s A s s - s r s s x a | p L q M a = p s B + s A s q - s p s q x b | r R s S b = r s B + s A s s - s r s s
80 vex x V
81 eqeq1 a = x a = p s B + s A s q - s p s q x = p s B + s A s q - s p s q
82 81 2rexbidv a = x p L q M a = p s B + s A s q - s p s q p L q M x = p s B + s A s q - s p s q
83 80 82 elab x a | p L q M a = p s B + s A s q - s p s q p L q M x = p s B + s A s q - s p s q
84 eqeq1 b = x b = r s B + s A s s - s r s s x = r s B + s A s s - s r s s
85 84 2rexbidv b = x r R s S b = r s B + s A s s - s r s s r R s S x = r s B + s A s s - s r s s
86 80 85 elab x b | r R s S b = r s B + s A s s - s r s s r R s S x = r s B + s A s s - s r s s
87 83 86 orbi12i x a | p L q M a = p s B + s A s q - s p s q x b | r R s S b = r s B + s A s s - s r s s p L q M x = p s B + s A s q - s p s q r R s S x = r s B + s A s s - s r s s
88 79 87 bitri x a | p L q M a = p s B + s A s q - s p s q b | r R s S b = r s B + s A s s - s r s s p L q M x = p s B + s A s q - s p s q r R s S x = r s B + s A s s - s r s s
89 38 47 49 addsubsd φ p L q M p s B + s A s q - s p s q = p s B - s p s q + s A s q
90 scutcut L s R L | s R No L s L | s R L | s R s R
91 1 90 syl φ L | s R No L s L | s R L | s R s R
92 91 simp2d φ L s L | s R
93 92 adantr φ p L q M L s L | s R
94 ovex L | s R V
95 94 snid L | s R L | s R
96 3 95 eqeltrdi φ A L | s R
97 96 adantr φ p L q M A L | s R
98 93 33 97 ssltsepcd φ p L q M p < s A
99 scutcut M s S M | s S No M s M | s S M | s S s S
100 2 99 syl φ M | s S No M s M | s S M | s S s S
101 100 simp2d φ M s M | s S
102 101 adantr φ p L q M M s M | s S
103 ovex M | s S V
104 103 snid M | s S M | s S
105 4 104 eqeltrdi φ B M | s S
106 105 adantr φ p L q M B M | s S
107 102 45 106 ssltsepcd φ p L q M q < s B
108 34 41 46 37 98 107 sltmuld φ p L q M p s B - s p s q < s A s B - s A s q
109 38 49 subscld φ p L q M p s B - s p s q No
110 77 adantr φ p L q M A s B No
111 109 47 110 sltaddsubd φ p L q M p s B - s p s q + s A s q < s A s B p s B - s p s q < s A s B - s A s q
112 108 111 mpbird φ p L q M p s B - s p s q + s A s q < s A s B
113 89 112 eqbrtrd φ p L q M p s B + s A s q - s p s q < s A s B
114 breq1 x = p s B + s A s q - s p s q x < s A s B p s B + s A s q - s p s q < s A s B
115 113 114 syl5ibrcom φ p L q M x = p s B + s A s q - s p s q x < s A s B
116 115 rexlimdvva φ p L q M x = p s B + s A s q - s p s q x < s A s B
117 61 68 70 addsubsd φ r R s S r s B + s A s s - s r s s = r s B - s r s s + s A s s
118 1 adantr φ r R s S L s R
119 118 90 syl φ r R s S L | s R No L s L | s R L | s R s R
120 119 simp3d φ r R s S L | s R s R
121 3 adantr φ r R s S A = L | s R
122 121 95 eqeltrdi φ r R s S A L | s R
123 120 122 58 ssltsepcd φ r R s S A < s r
124 2 adantr φ r R s S M s S
125 124 99 syl φ r R s S M | s S No M s M | s S M | s S s S
126 125 simp3d φ r R s S M | s S s S
127 4 adantr φ r R s S B = M | s S
128 127 104 eqeltrdi φ r R s S B M | s S
129 126 128 66 ssltsepcd φ r R s S B < s s
130 62 59 60 67 123 129 sltmuld φ r R s S A s s - s A s B < s r s s - s r s B
131 61 70 subscld φ r R s S r s B - s r s s No
132 77 adantr φ r R s S A s B No
133 131 68 132 sltaddsubd φ r R s S r s B - s r s s + s A s s < s A s B r s B - s r s s < s A s B - s A s s
134 61 70 132 68 sltsubsub2bd φ r R s S r s B - s r s s < s A s B - s A s s A s s - s A s B < s r s s - s r s B
135 133 134 bitrd φ r R s S r s B - s r s s + s A s s < s A s B A s s - s A s B < s r s s - s r s B
136 130 135 mpbird φ r R s S r s B - s r s s + s A s s < s A s B
137 117 136 eqbrtrd φ r R s S r s B + s A s s - s r s s < s A s B
138 breq1 x = r s B + s A s s - s r s s x < s A s B r s B + s A s s - s r s s < s A s B
139 137 138 syl5ibrcom φ r R s S x = r s B + s A s s - s r s s x < s A s B
140 139 rexlimdvva φ r R s S x = r s B + s A s s - s r s s x < s A s B
141 116 140 jaod φ p L q M x = p s B + s A s q - s p s q r R s S x = r s B + s A s s - s r s s x < s A s B
142 88 141 biimtrid φ x a | p L q M a = p s B + s A s q - s p s q b | r R s S b = r s B + s A s s - s r s s x < s A s B
143 142 imp φ x a | p L q M a = p s B + s A s q - s p s q b | r R s S b = r s B + s A s s - s r s s x < s A s B
144 velsn y A s B y = A s B
145 breq2 y = A s B x < s y x < s A s B
146 144 145 sylbi y A s B x < s y x < s A s B
147 143 146 syl5ibrcom φ x a | p L q M a = p s B + s A s q - s p s q b | r R s S b = r s B + s A s s - s r s s y A s B x < s y
148 147 3impia φ x a | p L q M a = p s B + s A s q - s p s q b | r R s S b = r s B + s A s s - s r s s y A s B x < s y
149 27 29 76 78 148 ssltd φ a | p L q M a = p s B + s A s q - s p s q b | r R s S b = r s B + s A s s - s r s s s A s B