Metamath Proof Explorer

Theorem sleadd1

Description: Addition to both sides of surreal less-than or equal. Theorem 5 of Conway p. 18. (Contributed by Scott Fenton, 21-Jan-2025)

Ref Expression
Assertion sleadd1 ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( 𝐴 ≤s 𝐵 ↔ ( 𝐴 +s 𝐶 ) ≤s ( 𝐵 +s 𝐶 ) ) )

Proof

Step Hyp Ref Expression
1 oveq1 ⊢ ( 𝑥 = 𝑥𝑂 → ( 𝑥 +s 𝑧 ) = ( 𝑥𝑂 +s 𝑧 ) )
2 1 breq2d ⊢ ( 𝑥 = 𝑥𝑂 → ( ( 𝑦 +s 𝑧 ) <s ( 𝑥 +s 𝑧 ) ↔ ( 𝑦 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) ) )
3 breq2 ⊢ ( 𝑥 = 𝑥𝑂 → ( 𝑦 <s 𝑥 ↔ 𝑦 <s 𝑥𝑂 ) )
4 2 3 imbi12d ⊢ ( 𝑥 = 𝑥𝑂 → ( ( ( 𝑦 +s 𝑧 ) <s ( 𝑥 +s 𝑧 ) → 𝑦 <s 𝑥 ) ↔ ( ( 𝑦 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦 <s 𝑥𝑂 ) ) )
5 oveq1 ⊢ ( 𝑦 = 𝑦𝑂 → ( 𝑦 +s 𝑧 ) = ( 𝑦𝑂 +s 𝑧 ) )
6 5 breq1d ⊢ ( 𝑦 = 𝑦𝑂 → ( ( 𝑦 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) ↔ ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) ) )
7 breq1 ⊢ ( 𝑦 = 𝑦𝑂 → ( 𝑦 <s 𝑥𝑂 ↔ 𝑦𝑂 <s 𝑥𝑂 ) )
8 6 7 imbi12d ⊢ ( 𝑦 = 𝑦𝑂 → ( ( ( 𝑦 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦 <s 𝑥𝑂 ) ↔ ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦𝑂 <s 𝑥𝑂 ) ) )
9 oveq2 ⊢ ( 𝑧 = 𝑧𝑂 → ( 𝑦𝑂 +s 𝑧 ) = ( 𝑦𝑂 +s 𝑧𝑂 ) )
10 oveq2 ⊢ ( 𝑧 = 𝑧𝑂 → ( 𝑥𝑂 +s 𝑧 ) = ( 𝑥𝑂 +s 𝑧𝑂 ) )
11 9 10 breq12d ⊢ ( 𝑧 = 𝑧𝑂 → ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) ↔ ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) ) )
12 11 imbi1d ⊢ ( 𝑧 = 𝑧𝑂 → ( ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦𝑂 <s 𝑥𝑂 ) ↔ ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥𝑂 ) ) )
13 oveq1 ⊢ ( 𝑥 = 𝑥𝑂 → ( 𝑥 +s 𝑧𝑂 ) = ( 𝑥𝑂 +s 𝑧𝑂 ) )
14 13 breq2d ⊢ ( 𝑥 = 𝑥𝑂 → ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) ↔ ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) ) )
15 breq2 ⊢ ( 𝑥 = 𝑥𝑂 → ( 𝑦𝑂 <s 𝑥 ↔ 𝑦𝑂 <s 𝑥𝑂 ) )
16 14 15 imbi12d ⊢ ( 𝑥 = 𝑥𝑂 → ( ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥 ) ↔ ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥𝑂 ) ) )
17 oveq1 ⊢ ( 𝑦 = 𝑦𝑂 → ( 𝑦 +s 𝑧𝑂 ) = ( 𝑦𝑂 +s 𝑧𝑂 ) )
18 17 breq1d ⊢ ( 𝑦 = 𝑦𝑂 → ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) ↔ ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) ) )
19 breq1 ⊢ ( 𝑦 = 𝑦𝑂 → ( 𝑦 <s 𝑥 ↔ 𝑦𝑂 <s 𝑥 ) )
20 18 19 imbi12d ⊢ ( 𝑦 = 𝑦𝑂 → ( ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦 <s 𝑥 ) ↔ ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥 ) ) )
21 17 breq1d ⊢ ( 𝑦 = 𝑦𝑂 → ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) ↔ ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) ) )
22 21 7 imbi12d ⊢ ( 𝑦 = 𝑦𝑂 → ( ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦 <s 𝑥𝑂 ) ↔ ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥𝑂 ) ) )
23 oveq2 ⊢ ( 𝑧 = 𝑧𝑂 → ( 𝑥 +s 𝑧 ) = ( 𝑥 +s 𝑧𝑂 ) )
24 9 23 breq12d ⊢ ( 𝑧 = 𝑧𝑂 → ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥 +s 𝑧 ) ↔ ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) ) )
25 24 imbi1d ⊢ ( 𝑧 = 𝑧𝑂 → ( ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥 +s 𝑧 ) → 𝑦𝑂 <s 𝑥 ) ↔ ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥 ) ) )
26 oveq1 ⊢ ( 𝑥 = 𝐴 → ( 𝑥 +s 𝑧 ) = ( 𝐴 +s 𝑧 ) )
27 26 breq2d ⊢ ( 𝑥 = 𝐴 → ( ( 𝑦 +s 𝑧 ) <s ( 𝑥 +s 𝑧 ) ↔ ( 𝑦 +s 𝑧 ) <s ( 𝐴 +s 𝑧 ) ) )
28 breq2 ⊢ ( 𝑥 = 𝐴 → ( 𝑦 <s 𝑥 ↔ 𝑦 <s 𝐴 ) )
29 27 28 imbi12d ⊢ ( 𝑥 = 𝐴 → ( ( ( 𝑦 +s 𝑧 ) <s ( 𝑥 +s 𝑧 ) → 𝑦 <s 𝑥 ) ↔ ( ( 𝑦 +s 𝑧 ) <s ( 𝐴 +s 𝑧 ) → 𝑦 <s 𝐴 ) ) )
30 oveq1 ⊢ ( 𝑦 = 𝐵 → ( 𝑦 +s 𝑧 ) = ( 𝐵 +s 𝑧 ) )
31 30 breq1d ⊢ ( 𝑦 = 𝐵 → ( ( 𝑦 +s 𝑧 ) <s ( 𝐴 +s 𝑧 ) ↔ ( 𝐵 +s 𝑧 ) <s ( 𝐴 +s 𝑧 ) ) )
32 breq1 ⊢ ( 𝑦 = 𝐵 → ( 𝑦 <s 𝐴 ↔ 𝐵 <s 𝐴 ) )
33 31 32 imbi12d ⊢ ( 𝑦 = 𝐵 → ( ( ( 𝑦 +s 𝑧 ) <s ( 𝐴 +s 𝑧 ) → 𝑦 <s 𝐴 ) ↔ ( ( 𝐵 +s 𝑧 ) <s ( 𝐴 +s 𝑧 ) → 𝐵 <s 𝐴 ) ) )
34 oveq2 ⊢ ( 𝑧 = 𝐶 → ( 𝐵 +s 𝑧 ) = ( 𝐵 +s 𝐶 ) )
35 oveq2 ⊢ ( 𝑧 = 𝐶 → ( 𝐴 +s 𝑧 ) = ( 𝐴 +s 𝐶 ) )
36 34 35 breq12d ⊢ ( 𝑧 = 𝐶 → ( ( 𝐵 +s 𝑧 ) <s ( 𝐴 +s 𝑧 ) ↔ ( 𝐵 +s 𝐶 ) <s ( 𝐴 +s 𝐶 ) ) )
37 36 imbi1d ⊢ ( 𝑧 = 𝐶 → ( ( ( 𝐵 +s 𝑧 ) <s ( 𝐴 +s 𝑧 ) → 𝐵 <s 𝐴 ) ↔ ( ( 𝐵 +s 𝐶 ) <s ( 𝐴 +s 𝐶 ) → 𝐵 <s 𝐴 ) ) )
38 simp2 ⊢ ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) → 𝑦 ∈ No )
39 simp3 ⊢ ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) → 𝑧 ∈ No )
40 38 39 addscut ⊢ ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) → ( ( 𝑦 +s 𝑧 ) ∈ No ∧ ( { 𝑎 ∣ ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) 𝑎 = ( 𝑦𝐿 +s 𝑧 ) } ∪ { 𝑏 ∣ ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑏 = ( 𝑦 +s 𝑧𝐿 ) } ) <<s { ( 𝑦 +s 𝑧 ) } ∧ { ( 𝑦 +s 𝑧 ) } <<s ( { 𝑐 ∣ ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) 𝑐 = ( 𝑦𝑅 +s 𝑧 ) } ∪ { 𝑑 ∣ ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑑 = ( 𝑦 +s 𝑧𝑅 ) } ) ) )
41 simp2 ⊢ ( ( ( 𝑦 +s 𝑧 ) ∈ No ∧ ( { 𝑎 ∣ ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) 𝑎 = ( 𝑦𝐿 +s 𝑧 ) } ∪ { 𝑏 ∣ ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑏 = ( 𝑦 +s 𝑧𝐿 ) } ) <<s { ( 𝑦 +s 𝑧 ) } ∧ { ( 𝑦 +s 𝑧 ) } <<s ( { 𝑐 ∣ ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) 𝑐 = ( 𝑦𝑅 +s 𝑧 ) } ∪ { 𝑑 ∣ ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑑 = ( 𝑦 +s 𝑧𝑅 ) } ) ) → ( { 𝑎 ∣ ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) 𝑎 = ( 𝑦𝐿 +s 𝑧 ) } ∪ { 𝑏 ∣ ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑏 = ( 𝑦 +s 𝑧𝐿 ) } ) <<s { ( 𝑦 +s 𝑧 ) } )
42 40 41 syl ⊢ ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) → ( { 𝑎 ∣ ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) 𝑎 = ( 𝑦𝐿 +s 𝑧 ) } ∪ { 𝑏 ∣ ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑏 = ( 𝑦 +s 𝑧𝐿 ) } ) <<s { ( 𝑦 +s 𝑧 ) } )
43 40 simp3d ⊢ ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) → { ( 𝑦 +s 𝑧 ) } <<s ( { 𝑐 ∣ ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) 𝑐 = ( 𝑦𝑅 +s 𝑧 ) } ∪ { 𝑑 ∣ ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑑 = ( 𝑦 +s 𝑧𝑅 ) } ) )
44 ovex ⊢ ( 𝑦 +s 𝑧 ) ∈ V
45 44 snnz ⊢ { ( 𝑦 +s 𝑧 ) } ≠ ∅
46 sslttr ⊢ ( ( ( { 𝑎 ∣ ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) 𝑎 = ( 𝑦𝐿 +s 𝑧 ) } ∪ { 𝑏 ∣ ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑏 = ( 𝑦 +s 𝑧𝐿 ) } ) <<s { ( 𝑦 +s 𝑧 ) } ∧ { ( 𝑦 +s 𝑧 ) } <<s ( { 𝑐 ∣ ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) 𝑐 = ( 𝑦𝑅 +s 𝑧 ) } ∪ { 𝑑 ∣ ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑑 = ( 𝑦 +s 𝑧𝑅 ) } ) ∧ { ( 𝑦 +s 𝑧 ) } ≠ ∅ ) → ( { 𝑎 ∣ ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) 𝑎 = ( 𝑦𝐿 +s 𝑧 ) } ∪ { 𝑏 ∣ ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑏 = ( 𝑦 +s 𝑧𝐿 ) } ) <<s ( { 𝑐 ∣ ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) 𝑐 = ( 𝑦𝑅 +s 𝑧 ) } ∪ { 𝑑 ∣ ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑑 = ( 𝑦 +s 𝑧𝑅 ) } ) )
47 45 46 mp3an3 ⊢ ( ( ( { 𝑎 ∣ ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) 𝑎 = ( 𝑦𝐿 +s 𝑧 ) } ∪ { 𝑏 ∣ ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑏 = ( 𝑦 +s 𝑧𝐿 ) } ) <<s { ( 𝑦 +s 𝑧 ) } ∧ { ( 𝑦 +s 𝑧 ) } <<s ( { 𝑐 ∣ ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) 𝑐 = ( 𝑦𝑅 +s 𝑧 ) } ∪ { 𝑑 ∣ ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑑 = ( 𝑦 +s 𝑧𝑅 ) } ) ) → ( { 𝑎 ∣ ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) 𝑎 = ( 𝑦𝐿 +s 𝑧 ) } ∪ { 𝑏 ∣ ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑏 = ( 𝑦 +s 𝑧𝐿 ) } ) <<s ( { 𝑐 ∣ ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) 𝑐 = ( 𝑦𝑅 +s 𝑧 ) } ∪ { 𝑑 ∣ ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑑 = ( 𝑦 +s 𝑧𝑅 ) } ) )
48 42 43 47 syl2anc ⊢ ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) → ( { 𝑎 ∣ ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) 𝑎 = ( 𝑦𝐿 +s 𝑧 ) } ∪ { 𝑏 ∣ ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑏 = ( 𝑦 +s 𝑧𝐿 ) } ) <<s ( { 𝑐 ∣ ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) 𝑐 = ( 𝑦𝑅 +s 𝑧 ) } ∪ { 𝑑 ∣ ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑑 = ( 𝑦 +s 𝑧𝑅 ) } ) )
49 simp1 ⊢ ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) → 𝑥 ∈ No )
50 49 39 addscut ⊢ ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) → ( ( 𝑥 +s 𝑧 ) ∈ No ∧ ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) 𝑎 = ( 𝑥𝐿 +s 𝑧 ) } ∪ { 𝑏 ∣ ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑏 = ( 𝑥 +s 𝑧𝐿 ) } ) <<s { ( 𝑥 +s 𝑧 ) } ∧ { ( 𝑥 +s 𝑧 ) } <<s ( { 𝑐 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) 𝑐 = ( 𝑥𝑅 +s 𝑧 ) } ∪ { 𝑑 ∣ ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑑 = ( 𝑥 +s 𝑧𝑅 ) } ) ) )
51 simp2 ⊢ ( ( ( 𝑥 +s 𝑧 ) ∈ No ∧ ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) 𝑎 = ( 𝑥𝐿 +s 𝑧 ) } ∪ { 𝑏 ∣ ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑏 = ( 𝑥 +s 𝑧𝐿 ) } ) <<s { ( 𝑥 +s 𝑧 ) } ∧ { ( 𝑥 +s 𝑧 ) } <<s ( { 𝑐 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) 𝑐 = ( 𝑥𝑅 +s 𝑧 ) } ∪ { 𝑑 ∣ ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑑 = ( 𝑥 +s 𝑧𝑅 ) } ) ) → ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) 𝑎 = ( 𝑥𝐿 +s 𝑧 ) } ∪ { 𝑏 ∣ ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑏 = ( 𝑥 +s 𝑧𝐿 ) } ) <<s { ( 𝑥 +s 𝑧 ) } )
52 50 51 syl ⊢ ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) → ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) 𝑎 = ( 𝑥𝐿 +s 𝑧 ) } ∪ { 𝑏 ∣ ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑏 = ( 𝑥 +s 𝑧𝐿 ) } ) <<s { ( 𝑥 +s 𝑧 ) } )
53 50 simp3d ⊢ ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) → { ( 𝑥 +s 𝑧 ) } <<s ( { 𝑐 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) 𝑐 = ( 𝑥𝑅 +s 𝑧 ) } ∪ { 𝑑 ∣ ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑑 = ( 𝑥 +s 𝑧𝑅 ) } ) )
54 ovex ⊢ ( 𝑥 +s 𝑧 ) ∈ V
55 54 snnz ⊢ { ( 𝑥 +s 𝑧 ) } ≠ ∅
56 sslttr ⊢ ( ( ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) 𝑎 = ( 𝑥𝐿 +s 𝑧 ) } ∪ { 𝑏 ∣ ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑏 = ( 𝑥 +s 𝑧𝐿 ) } ) <<s { ( 𝑥 +s 𝑧 ) } ∧ { ( 𝑥 +s 𝑧 ) } <<s ( { 𝑐 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) 𝑐 = ( 𝑥𝑅 +s 𝑧 ) } ∪ { 𝑑 ∣ ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑑 = ( 𝑥 +s 𝑧𝑅 ) } ) ∧ { ( 𝑥 +s 𝑧 ) } ≠ ∅ ) → ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) 𝑎 = ( 𝑥𝐿 +s 𝑧 ) } ∪ { 𝑏 ∣ ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑏 = ( 𝑥 +s 𝑧𝐿 ) } ) <<s ( { 𝑐 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) 𝑐 = ( 𝑥𝑅 +s 𝑧 ) } ∪ { 𝑑 ∣ ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑑 = ( 𝑥 +s 𝑧𝑅 ) } ) )
57 55 56 mp3an3 ⊢ ( ( ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) 𝑎 = ( 𝑥𝐿 +s 𝑧 ) } ∪ { 𝑏 ∣ ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑏 = ( 𝑥 +s 𝑧𝐿 ) } ) <<s { ( 𝑥 +s 𝑧 ) } ∧ { ( 𝑥 +s 𝑧 ) } <<s ( { 𝑐 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) 𝑐 = ( 𝑥𝑅 +s 𝑧 ) } ∪ { 𝑑 ∣ ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑑 = ( 𝑥 +s 𝑧𝑅 ) } ) ) → ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) 𝑎 = ( 𝑥𝐿 +s 𝑧 ) } ∪ { 𝑏 ∣ ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑏 = ( 𝑥 +s 𝑧𝐿 ) } ) <<s ( { 𝑐 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) 𝑐 = ( 𝑥𝑅 +s 𝑧 ) } ∪ { 𝑑 ∣ ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑑 = ( 𝑥 +s 𝑧𝑅 ) } ) )
58 52 53 57 syl2anc ⊢ ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) → ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) 𝑎 = ( 𝑥𝐿 +s 𝑧 ) } ∪ { 𝑏 ∣ ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑏 = ( 𝑥 +s 𝑧𝐿 ) } ) <<s ( { 𝑐 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) 𝑐 = ( 𝑥𝑅 +s 𝑧 ) } ∪ { 𝑑 ∣ ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑑 = ( 𝑥 +s 𝑧𝑅 ) } ) )
59 addsval2 ⊢ ( ( 𝑦 ∈ No ∧ 𝑧 ∈ No ) → ( 𝑦 +s 𝑧 ) = ( ( { 𝑎 ∣ ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) 𝑎 = ( 𝑦𝐿 +s 𝑧 ) } ∪ { 𝑏 ∣ ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑏 = ( 𝑦 +s 𝑧𝐿 ) } ) |s ( { 𝑐 ∣ ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) 𝑐 = ( 𝑦𝑅 +s 𝑧 ) } ∪ { 𝑑 ∣ ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑑 = ( 𝑦 +s 𝑧𝑅 ) } ) ) )
60 59 3adant1 ⊢ ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) → ( 𝑦 +s 𝑧 ) = ( ( { 𝑎 ∣ ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) 𝑎 = ( 𝑦𝐿 +s 𝑧 ) } ∪ { 𝑏 ∣ ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑏 = ( 𝑦 +s 𝑧𝐿 ) } ) |s ( { 𝑐 ∣ ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) 𝑐 = ( 𝑦𝑅 +s 𝑧 ) } ∪ { 𝑑 ∣ ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑑 = ( 𝑦 +s 𝑧𝑅 ) } ) ) )
61 addsval2 ⊢ ( ( 𝑥 ∈ No ∧ 𝑧 ∈ No ) → ( 𝑥 +s 𝑧 ) = ( ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) 𝑎 = ( 𝑥𝐿 +s 𝑧 ) } ∪ { 𝑏 ∣ ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑏 = ( 𝑥 +s 𝑧𝐿 ) } ) |s ( { 𝑐 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) 𝑐 = ( 𝑥𝑅 +s 𝑧 ) } ∪ { 𝑑 ∣ ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑑 = ( 𝑥 +s 𝑧𝑅 ) } ) ) )
62 61 3adant2 ⊢ ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) → ( 𝑥 +s 𝑧 ) = ( ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) 𝑎 = ( 𝑥𝐿 +s 𝑧 ) } ∪ { 𝑏 ∣ ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑏 = ( 𝑥 +s 𝑧𝐿 ) } ) |s ( { 𝑐 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) 𝑐 = ( 𝑥𝑅 +s 𝑧 ) } ∪ { 𝑑 ∣ ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑑 = ( 𝑥 +s 𝑧𝑅 ) } ) ) )
63 48 58 60 62 sltrecd ⊢ ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) → ( ( 𝑦 +s 𝑧 ) <s ( 𝑥 +s 𝑧 ) ↔ ( ∃ 𝑝 ∈ ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) 𝑎 = ( 𝑥𝐿 +s 𝑧 ) } ∪ { 𝑏 ∣ ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑏 = ( 𝑥 +s 𝑧𝐿 ) } ) ( 𝑦 +s 𝑧 ) ≤s 𝑝 ∨ ∃ 𝑞 ∈ ( { 𝑐 ∣ ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) 𝑐 = ( 𝑦𝑅 +s 𝑧 ) } ∪ { 𝑑 ∣ ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑑 = ( 𝑦 +s 𝑧𝑅 ) } ) 𝑞 ≤s ( 𝑥 +s 𝑧 ) ) ) )
64 63 adantr ⊢ ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦 <s 𝑥𝑂 ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( ( 𝑦 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦 <s 𝑥𝑂 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥 +s 𝑧 ) → 𝑦𝑂 <s 𝑥 ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦 <s 𝑥 ) ) ) → ( ( 𝑦 +s 𝑧 ) <s ( 𝑥 +s 𝑧 ) ↔ ( ∃ 𝑝 ∈ ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) 𝑎 = ( 𝑥𝐿 +s 𝑧 ) } ∪ { 𝑏 ∣ ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑏 = ( 𝑥 +s 𝑧𝐿 ) } ) ( 𝑦 +s 𝑧 ) ≤s 𝑝 ∨ ∃ 𝑞 ∈ ( { 𝑐 ∣ ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) 𝑐 = ( 𝑦𝑅 +s 𝑧 ) } ∪ { 𝑑 ∣ ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑑 = ( 𝑦 +s 𝑧𝑅 ) } ) 𝑞 ≤s ( 𝑥 +s 𝑧 ) ) ) )
65 rexun ⊢ ( ∃ 𝑝 ∈ ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) 𝑎 = ( 𝑥𝐿 +s 𝑧 ) } ∪ { 𝑏 ∣ ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑏 = ( 𝑥 +s 𝑧𝐿 ) } ) ( 𝑦 +s 𝑧 ) ≤s 𝑝 ↔ ( ∃ 𝑝 ∈ { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) 𝑎 = ( 𝑥𝐿 +s 𝑧 ) } ( 𝑦 +s 𝑧 ) ≤s 𝑝 ∨ ∃ 𝑝 ∈ { 𝑏 ∣ ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑏 = ( 𝑥 +s 𝑧𝐿 ) } ( 𝑦 +s 𝑧 ) ≤s 𝑝 ) )
66 eqeq1 ⊢ ( 𝑎 = 𝑝 → ( 𝑎 = ( 𝑥𝐿 +s 𝑧 ) ↔ 𝑝 = ( 𝑥𝐿 +s 𝑧 ) ) )
67 66 rexbidv ⊢ ( 𝑎 = 𝑝 → ( ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) 𝑎 = ( 𝑥𝐿 +s 𝑧 ) ↔ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) 𝑝 = ( 𝑥𝐿 +s 𝑧 ) ) )
68 67 rexab ⊢ ( ∃ 𝑝 ∈ { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) 𝑎 = ( 𝑥𝐿 +s 𝑧 ) } ( 𝑦 +s 𝑧 ) ≤s 𝑝 ↔ ∃ 𝑝 ( ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) 𝑝 = ( 𝑥𝐿 +s 𝑧 ) ∧ ( 𝑦 +s 𝑧 ) ≤s 𝑝 ) )
69 rexcom4 ⊢ ( ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑝 ( 𝑝 = ( 𝑥𝐿 +s 𝑧 ) ∧ ( 𝑦 +s 𝑧 ) ≤s 𝑝 ) ↔ ∃ 𝑝 ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ( 𝑝 = ( 𝑥𝐿 +s 𝑧 ) ∧ ( 𝑦 +s 𝑧 ) ≤s 𝑝 ) )
70 r19.41v ⊢ ( ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ( 𝑝 = ( 𝑥𝐿 +s 𝑧 ) ∧ ( 𝑦 +s 𝑧 ) ≤s 𝑝 ) ↔ ( ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) 𝑝 = ( 𝑥𝐿 +s 𝑧 ) ∧ ( 𝑦 +s 𝑧 ) ≤s 𝑝 ) )
71 70 exbii ⊢ ( ∃ 𝑝 ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ( 𝑝 = ( 𝑥𝐿 +s 𝑧 ) ∧ ( 𝑦 +s 𝑧 ) ≤s 𝑝 ) ↔ ∃ 𝑝 ( ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) 𝑝 = ( 𝑥𝐿 +s 𝑧 ) ∧ ( 𝑦 +s 𝑧 ) ≤s 𝑝 ) )
72 69 71 bitri ⊢ ( ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑝 ( 𝑝 = ( 𝑥𝐿 +s 𝑧 ) ∧ ( 𝑦 +s 𝑧 ) ≤s 𝑝 ) ↔ ∃ 𝑝 ( ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) 𝑝 = ( 𝑥𝐿 +s 𝑧 ) ∧ ( 𝑦 +s 𝑧 ) ≤s 𝑝 ) )
73 ovex ⊢ ( 𝑥𝐿 +s 𝑧 ) ∈ V
74 breq2 ⊢ ( 𝑝 = ( 𝑥𝐿 +s 𝑧 ) → ( ( 𝑦 +s 𝑧 ) ≤s 𝑝 ↔ ( 𝑦 +s 𝑧 ) ≤s ( 𝑥𝐿 +s 𝑧 ) ) )
75 73 74 ceqsexv ⊢ ( ∃ 𝑝 ( 𝑝 = ( 𝑥𝐿 +s 𝑧 ) ∧ ( 𝑦 +s 𝑧 ) ≤s 𝑝 ) ↔ ( 𝑦 +s 𝑧 ) ≤s ( 𝑥𝐿 +s 𝑧 ) )
76 75 rexbii ⊢ ( ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑝 ( 𝑝 = ( 𝑥𝐿 +s 𝑧 ) ∧ ( 𝑦 +s 𝑧 ) ≤s 𝑝 ) ↔ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ( 𝑦 +s 𝑧 ) ≤s ( 𝑥𝐿 +s 𝑧 ) )
77 72 76 bitr3i ⊢ ( ∃ 𝑝 ( ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) 𝑝 = ( 𝑥𝐿 +s 𝑧 ) ∧ ( 𝑦 +s 𝑧 ) ≤s 𝑝 ) ↔ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ( 𝑦 +s 𝑧 ) ≤s ( 𝑥𝐿 +s 𝑧 ) )
78 68 77 bitri ⊢ ( ∃ 𝑝 ∈ { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) 𝑎 = ( 𝑥𝐿 +s 𝑧 ) } ( 𝑦 +s 𝑧 ) ≤s 𝑝 ↔ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ( 𝑦 +s 𝑧 ) ≤s ( 𝑥𝐿 +s 𝑧 ) )
79 eqeq1 ⊢ ( 𝑏 = 𝑝 → ( 𝑏 = ( 𝑥 +s 𝑧𝐿 ) ↔ 𝑝 = ( 𝑥 +s 𝑧𝐿 ) ) )
80 79 rexbidv ⊢ ( 𝑏 = 𝑝 → ( ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑏 = ( 𝑥 +s 𝑧𝐿 ) ↔ ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑝 = ( 𝑥 +s 𝑧𝐿 ) ) )
81 80 rexab ⊢ ( ∃ 𝑝 ∈ { 𝑏 ∣ ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑏 = ( 𝑥 +s 𝑧𝐿 ) } ( 𝑦 +s 𝑧 ) ≤s 𝑝 ↔ ∃ 𝑝 ( ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑝 = ( 𝑥 +s 𝑧𝐿 ) ∧ ( 𝑦 +s 𝑧 ) ≤s 𝑝 ) )
82 rexcom4 ⊢ ( ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) ∃ 𝑝 ( 𝑝 = ( 𝑥 +s 𝑧𝐿 ) ∧ ( 𝑦 +s 𝑧 ) ≤s 𝑝 ) ↔ ∃ 𝑝 ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) ( 𝑝 = ( 𝑥 +s 𝑧𝐿 ) ∧ ( 𝑦 +s 𝑧 ) ≤s 𝑝 ) )
83 r19.41v ⊢ ( ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) ( 𝑝 = ( 𝑥 +s 𝑧𝐿 ) ∧ ( 𝑦 +s 𝑧 ) ≤s 𝑝 ) ↔ ( ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑝 = ( 𝑥 +s 𝑧𝐿 ) ∧ ( 𝑦 +s 𝑧 ) ≤s 𝑝 ) )
84 83 exbii ⊢ ( ∃ 𝑝 ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) ( 𝑝 = ( 𝑥 +s 𝑧𝐿 ) ∧ ( 𝑦 +s 𝑧 ) ≤s 𝑝 ) ↔ ∃ 𝑝 ( ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑝 = ( 𝑥 +s 𝑧𝐿 ) ∧ ( 𝑦 +s 𝑧 ) ≤s 𝑝 ) )
85 82 84 bitri ⊢ ( ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) ∃ 𝑝 ( 𝑝 = ( 𝑥 +s 𝑧𝐿 ) ∧ ( 𝑦 +s 𝑧 ) ≤s 𝑝 ) ↔ ∃ 𝑝 ( ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑝 = ( 𝑥 +s 𝑧𝐿 ) ∧ ( 𝑦 +s 𝑧 ) ≤s 𝑝 ) )
86 ovex ⊢ ( 𝑥 +s 𝑧𝐿 ) ∈ V
87 breq2 ⊢ ( 𝑝 = ( 𝑥 +s 𝑧𝐿 ) → ( ( 𝑦 +s 𝑧 ) ≤s 𝑝 ↔ ( 𝑦 +s 𝑧 ) ≤s ( 𝑥 +s 𝑧𝐿 ) ) )
88 86 87 ceqsexv ⊢ ( ∃ 𝑝 ( 𝑝 = ( 𝑥 +s 𝑧𝐿 ) ∧ ( 𝑦 +s 𝑧 ) ≤s 𝑝 ) ↔ ( 𝑦 +s 𝑧 ) ≤s ( 𝑥 +s 𝑧𝐿 ) )
89 88 rexbii ⊢ ( ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) ∃ 𝑝 ( 𝑝 = ( 𝑥 +s 𝑧𝐿 ) ∧ ( 𝑦 +s 𝑧 ) ≤s 𝑝 ) ↔ ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) ( 𝑦 +s 𝑧 ) ≤s ( 𝑥 +s 𝑧𝐿 ) )
90 85 89 bitr3i ⊢ ( ∃ 𝑝 ( ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑝 = ( 𝑥 +s 𝑧𝐿 ) ∧ ( 𝑦 +s 𝑧 ) ≤s 𝑝 ) ↔ ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) ( 𝑦 +s 𝑧 ) ≤s ( 𝑥 +s 𝑧𝐿 ) )
91 81 90 bitri ⊢ ( ∃ 𝑝 ∈ { 𝑏 ∣ ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑏 = ( 𝑥 +s 𝑧𝐿 ) } ( 𝑦 +s 𝑧 ) ≤s 𝑝 ↔ ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) ( 𝑦 +s 𝑧 ) ≤s ( 𝑥 +s 𝑧𝐿 ) )
92 78 91 orbi12i ⊢ ( ( ∃ 𝑝 ∈ { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) 𝑎 = ( 𝑥𝐿 +s 𝑧 ) } ( 𝑦 +s 𝑧 ) ≤s 𝑝 ∨ ∃ 𝑝 ∈ { 𝑏 ∣ ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑏 = ( 𝑥 +s 𝑧𝐿 ) } ( 𝑦 +s 𝑧 ) ≤s 𝑝 ) ↔ ( ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ( 𝑦 +s 𝑧 ) ≤s ( 𝑥𝐿 +s 𝑧 ) ∨ ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) ( 𝑦 +s 𝑧 ) ≤s ( 𝑥 +s 𝑧𝐿 ) ) )
93 65 92 bitri ⊢ ( ∃ 𝑝 ∈ ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) 𝑎 = ( 𝑥𝐿 +s 𝑧 ) } ∪ { 𝑏 ∣ ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑏 = ( 𝑥 +s 𝑧𝐿 ) } ) ( 𝑦 +s 𝑧 ) ≤s 𝑝 ↔ ( ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ( 𝑦 +s 𝑧 ) ≤s ( 𝑥𝐿 +s 𝑧 ) ∨ ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) ( 𝑦 +s 𝑧 ) ≤s ( 𝑥 +s 𝑧𝐿 ) ) )
94 simpll2 ⊢ ( ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦 <s 𝑥𝑂 ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( ( 𝑦 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦 <s 𝑥𝑂 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥 +s 𝑧 ) → 𝑦𝑂 <s 𝑥 ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦 <s 𝑥 ) ) ) ∧ ( 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∧ ( 𝑦 +s 𝑧 ) ≤s ( 𝑥𝐿 +s 𝑧 ) ) ) → 𝑦 ∈ No )
95 leftssno ⊢ ( L ‘ 𝑥 ) ⊆ No
96 95 sseli ⊢ ( 𝑥𝐿 ∈ ( L ‘ 𝑥 ) → 𝑥𝐿 ∈ No )
97 96 adantr ⊢ ( ( 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∧ ( 𝑦 +s 𝑧 ) ≤s ( 𝑥𝐿 +s 𝑧 ) ) → 𝑥𝐿 ∈ No )
98 97 adantl ⊢ ( ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦 <s 𝑥𝑂 ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( ( 𝑦 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦 <s 𝑥𝑂 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥 +s 𝑧 ) → 𝑦𝑂 <s 𝑥 ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦 <s 𝑥 ) ) ) ∧ ( 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∧ ( 𝑦 +s 𝑧 ) ≤s ( 𝑥𝐿 +s 𝑧 ) ) ) → 𝑥𝐿 ∈ No )
99 simpll1 ⊢ ( ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦 <s 𝑥𝑂 ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( ( 𝑦 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦 <s 𝑥𝑂 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥 +s 𝑧 ) → 𝑦𝑂 <s 𝑥 ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦 <s 𝑥 ) ) ) ∧ ( 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∧ ( 𝑦 +s 𝑧 ) ≤s ( 𝑥𝐿 +s 𝑧 ) ) ) → 𝑥 ∈ No )
100 simprr ⊢ ( ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦 <s 𝑥𝑂 ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( ( 𝑦 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦 <s 𝑥𝑂 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥 +s 𝑧 ) → 𝑦𝑂 <s 𝑥 ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦 <s 𝑥 ) ) ) ∧ ( 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∧ ( 𝑦 +s 𝑧 ) ≤s ( 𝑥𝐿 +s 𝑧 ) ) ) → ( 𝑦 +s 𝑧 ) ≤s ( 𝑥𝐿 +s 𝑧 ) )
101 simpll3 ⊢ ( ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦 <s 𝑥𝑂 ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( ( 𝑦 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦 <s 𝑥𝑂 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥 +s 𝑧 ) → 𝑦𝑂 <s 𝑥 ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦 <s 𝑥 ) ) ) ∧ ( 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∧ ( 𝑦 +s 𝑧 ) ≤s ( 𝑥𝐿 +s 𝑧 ) ) ) → 𝑧 ∈ No )
102 sleadd1im ⊢ ( ( 𝑦 ∈ No ∧ 𝑥𝐿 ∈ No ∧ 𝑧 ∈ No ) → ( ( 𝑦 +s 𝑧 ) ≤s ( 𝑥𝐿 +s 𝑧 ) → 𝑦 ≤s 𝑥𝐿 ) )
103 94 98 101 102 syl3anc ⊢ ( ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦 <s 𝑥𝑂 ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( ( 𝑦 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦 <s 𝑥𝑂 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥 +s 𝑧 ) → 𝑦𝑂 <s 𝑥 ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦 <s 𝑥 ) ) ) ∧ ( 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∧ ( 𝑦 +s 𝑧 ) ≤s ( 𝑥𝐿 +s 𝑧 ) ) ) → ( ( 𝑦 +s 𝑧 ) ≤s ( 𝑥𝐿 +s 𝑧 ) → 𝑦 ≤s 𝑥𝐿 ) )
104 100 103 mpd ⊢ ( ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦 <s 𝑥𝑂 ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( ( 𝑦 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦 <s 𝑥𝑂 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥 +s 𝑧 ) → 𝑦𝑂 <s 𝑥 ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦 <s 𝑥 ) ) ) ∧ ( 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∧ ( 𝑦 +s 𝑧 ) ≤s ( 𝑥𝐿 +s 𝑧 ) ) ) → 𝑦 ≤s 𝑥𝐿 )
105 leftlt ⊢ ( 𝑥𝐿 ∈ ( L ‘ 𝑥 ) → 𝑥𝐿 <s 𝑥 )
106 105 adantr ⊢ ( ( 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∧ ( 𝑦 +s 𝑧 ) ≤s ( 𝑥𝐿 +s 𝑧 ) ) → 𝑥𝐿 <s 𝑥 )
107 106 adantl ⊢ ( ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦 <s 𝑥𝑂 ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( ( 𝑦 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦 <s 𝑥𝑂 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥 +s 𝑧 ) → 𝑦𝑂 <s 𝑥 ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦 <s 𝑥 ) ) ) ∧ ( 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∧ ( 𝑦 +s 𝑧 ) ≤s ( 𝑥𝐿 +s 𝑧 ) ) ) → 𝑥𝐿 <s 𝑥 )
108 94 98 99 104 107 slelttrd ⊢ ( ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦 <s 𝑥𝑂 ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( ( 𝑦 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦 <s 𝑥𝑂 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥 +s 𝑧 ) → 𝑦𝑂 <s 𝑥 ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦 <s 𝑥 ) ) ) ∧ ( 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∧ ( 𝑦 +s 𝑧 ) ≤s ( 𝑥𝐿 +s 𝑧 ) ) ) → 𝑦 <s 𝑥 )
109 108 rexlimdvaa ⊢ ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦 <s 𝑥𝑂 ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( ( 𝑦 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦 <s 𝑥𝑂 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥 +s 𝑧 ) → 𝑦𝑂 <s 𝑥 ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦 <s 𝑥 ) ) ) → ( ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ( 𝑦 +s 𝑧 ) ≤s ( 𝑥𝐿 +s 𝑧 ) → 𝑦 <s 𝑥 ) )
110 simpll2 ⊢ ( ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦 <s 𝑥𝑂 ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( ( 𝑦 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦 <s 𝑥𝑂 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥 +s 𝑧 ) → 𝑦𝑂 <s 𝑥 ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦 <s 𝑥 ) ) ) ∧ ( 𝑧𝐿 ∈ ( L ‘ 𝑧 ) ∧ ( 𝑦 +s 𝑧 ) ≤s ( 𝑥 +s 𝑧𝐿 ) ) ) → 𝑦 ∈ No )
111 leftssno ⊢ ( L ‘ 𝑧 ) ⊆ No
112 111 sseli ⊢ ( 𝑧𝐿 ∈ ( L ‘ 𝑧 ) → 𝑧𝐿 ∈ No )
113 112 adantr ⊢ ( ( 𝑧𝐿 ∈ ( L ‘ 𝑧 ) ∧ ( 𝑦 +s 𝑧 ) ≤s ( 𝑥 +s 𝑧𝐿 ) ) → 𝑧𝐿 ∈ No )
114 113 adantl ⊢ ( ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦 <s 𝑥𝑂 ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( ( 𝑦 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦 <s 𝑥𝑂 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥 +s 𝑧 ) → 𝑦𝑂 <s 𝑥 ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦 <s 𝑥 ) ) ) ∧ ( 𝑧𝐿 ∈ ( L ‘ 𝑧 ) ∧ ( 𝑦 +s 𝑧 ) ≤s ( 𝑥 +s 𝑧𝐿 ) ) ) → 𝑧𝐿 ∈ No )
115 110 114 addscld ⊢ ( ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦 <s 𝑥𝑂 ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( ( 𝑦 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦 <s 𝑥𝑂 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥 +s 𝑧 ) → 𝑦𝑂 <s 𝑥 ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦 <s 𝑥 ) ) ) ∧ ( 𝑧𝐿 ∈ ( L ‘ 𝑧 ) ∧ ( 𝑦 +s 𝑧 ) ≤s ( 𝑥 +s 𝑧𝐿 ) ) ) → ( 𝑦 +s 𝑧𝐿 ) ∈ No )
116 simpll3 ⊢ ( ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦 <s 𝑥𝑂 ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( ( 𝑦 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦 <s 𝑥𝑂 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥 +s 𝑧 ) → 𝑦𝑂 <s 𝑥 ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦 <s 𝑥 ) ) ) ∧ ( 𝑧𝐿 ∈ ( L ‘ 𝑧 ) ∧ ( 𝑦 +s 𝑧 ) ≤s ( 𝑥 +s 𝑧𝐿 ) ) ) → 𝑧 ∈ No )
117 110 116 addscld ⊢ ( ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦 <s 𝑥𝑂 ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( ( 𝑦 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦 <s 𝑥𝑂 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥 +s 𝑧 ) → 𝑦𝑂 <s 𝑥 ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦 <s 𝑥 ) ) ) ∧ ( 𝑧𝐿 ∈ ( L ‘ 𝑧 ) ∧ ( 𝑦 +s 𝑧 ) ≤s ( 𝑥 +s 𝑧𝐿 ) ) ) → ( 𝑦 +s 𝑧 ) ∈ No )
118 simpll1 ⊢ ( ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦 <s 𝑥𝑂 ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( ( 𝑦 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦 <s 𝑥𝑂 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥 +s 𝑧 ) → 𝑦𝑂 <s 𝑥 ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦 <s 𝑥 ) ) ) ∧ ( 𝑧𝐿 ∈ ( L ‘ 𝑧 ) ∧ ( 𝑦 +s 𝑧 ) ≤s ( 𝑥 +s 𝑧𝐿 ) ) ) → 𝑥 ∈ No )
119 118 114 addscld ⊢ ( ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦 <s 𝑥𝑂 ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( ( 𝑦 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦 <s 𝑥𝑂 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥 +s 𝑧 ) → 𝑦𝑂 <s 𝑥 ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦 <s 𝑥 ) ) ) ∧ ( 𝑧𝐿 ∈ ( L ‘ 𝑧 ) ∧ ( 𝑦 +s 𝑧 ) ≤s ( 𝑥 +s 𝑧𝐿 ) ) ) → ( 𝑥 +s 𝑧𝐿 ) ∈ No )
120 leftlt ⊢ ( 𝑧𝐿 ∈ ( L ‘ 𝑧 ) → 𝑧𝐿 <s 𝑧 )
121 120 adantr ⊢ ( ( 𝑧𝐿 ∈ ( L ‘ 𝑧 ) ∧ ( 𝑦 +s 𝑧 ) ≤s ( 𝑥 +s 𝑧𝐿 ) ) → 𝑧𝐿 <s 𝑧 )
122 121 adantl ⊢ ( ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦 <s 𝑥𝑂 ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( ( 𝑦 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦 <s 𝑥𝑂 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥 +s 𝑧 ) → 𝑦𝑂 <s 𝑥 ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦 <s 𝑥 ) ) ) ∧ ( 𝑧𝐿 ∈ ( L ‘ 𝑧 ) ∧ ( 𝑦 +s 𝑧 ) ≤s ( 𝑥 +s 𝑧𝐿 ) ) ) → 𝑧𝐿 <s 𝑧 )
123 sltadd2im ⊢ ( ( 𝑧𝐿 ∈ No ∧ 𝑧 ∈ No ∧ 𝑦 ∈ No ) → ( 𝑧𝐿 <s 𝑧 → ( 𝑦 +s 𝑧𝐿 ) <s ( 𝑦 +s 𝑧 ) ) )
124 114 116 110 123 syl3anc ⊢ ( ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦 <s 𝑥𝑂 ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( ( 𝑦 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦 <s 𝑥𝑂 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥 +s 𝑧 ) → 𝑦𝑂 <s 𝑥 ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦 <s 𝑥 ) ) ) ∧ ( 𝑧𝐿 ∈ ( L ‘ 𝑧 ) ∧ ( 𝑦 +s 𝑧 ) ≤s ( 𝑥 +s 𝑧𝐿 ) ) ) → ( 𝑧𝐿 <s 𝑧 → ( 𝑦 +s 𝑧𝐿 ) <s ( 𝑦 +s 𝑧 ) ) )
125 122 124 mpd ⊢ ( ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦 <s 𝑥𝑂 ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( ( 𝑦 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦 <s 𝑥𝑂 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥 +s 𝑧 ) → 𝑦𝑂 <s 𝑥 ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦 <s 𝑥 ) ) ) ∧ ( 𝑧𝐿 ∈ ( L ‘ 𝑧 ) ∧ ( 𝑦 +s 𝑧 ) ≤s ( 𝑥 +s 𝑧𝐿 ) ) ) → ( 𝑦 +s 𝑧𝐿 ) <s ( 𝑦 +s 𝑧 ) )
126 simprr ⊢ ( ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦 <s 𝑥𝑂 ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( ( 𝑦 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦 <s 𝑥𝑂 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥 +s 𝑧 ) → 𝑦𝑂 <s 𝑥 ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦 <s 𝑥 ) ) ) ∧ ( 𝑧𝐿 ∈ ( L ‘ 𝑧 ) ∧ ( 𝑦 +s 𝑧 ) ≤s ( 𝑥 +s 𝑧𝐿 ) ) ) → ( 𝑦 +s 𝑧 ) ≤s ( 𝑥 +s 𝑧𝐿 ) )
127 115 117 119 125 126 sltletrd ⊢ ( ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦 <s 𝑥𝑂 ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( ( 𝑦 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦 <s 𝑥𝑂 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥 +s 𝑧 ) → 𝑦𝑂 <s 𝑥 ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦 <s 𝑥 ) ) ) ∧ ( 𝑧𝐿 ∈ ( L ‘ 𝑧 ) ∧ ( 𝑦 +s 𝑧 ) ≤s ( 𝑥 +s 𝑧𝐿 ) ) ) → ( 𝑦 +s 𝑧𝐿 ) <s ( 𝑥 +s 𝑧𝐿 ) )
128 oveq2 ⊢ ( 𝑧𝑂 = 𝑧𝐿 → ( 𝑦 +s 𝑧𝑂 ) = ( 𝑦 +s 𝑧𝐿 ) )
129 oveq2 ⊢ ( 𝑧𝑂 = 𝑧𝐿 → ( 𝑥 +s 𝑧𝑂 ) = ( 𝑥 +s 𝑧𝐿 ) )
130 128 129 breq12d ⊢ ( 𝑧𝑂 = 𝑧𝐿 → ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) ↔ ( 𝑦 +s 𝑧𝐿 ) <s ( 𝑥 +s 𝑧𝐿 ) ) )
131 130 imbi1d ⊢ ( 𝑧𝑂 = 𝑧𝐿 → ( ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦 <s 𝑥 ) ↔ ( ( 𝑦 +s 𝑧𝐿 ) <s ( 𝑥 +s 𝑧𝐿 ) → 𝑦 <s 𝑥 ) ) )
132 simplr3 ⊢ ( ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦 <s 𝑥𝑂 ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( ( 𝑦 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦 <s 𝑥𝑂 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥 +s 𝑧 ) → 𝑦𝑂 <s 𝑥 ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦 <s 𝑥 ) ) ) ∧ ( 𝑧𝐿 ∈ ( L ‘ 𝑧 ) ∧ ( 𝑦 +s 𝑧 ) ≤s ( 𝑥 +s 𝑧𝐿 ) ) ) → ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦 <s 𝑥 ) )
133 simprl ⊢ ( ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦 <s 𝑥𝑂 ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( ( 𝑦 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦 <s 𝑥𝑂 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥 +s 𝑧 ) → 𝑦𝑂 <s 𝑥 ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦 <s 𝑥 ) ) ) ∧ ( 𝑧𝐿 ∈ ( L ‘ 𝑧 ) ∧ ( 𝑦 +s 𝑧 ) ≤s ( 𝑥 +s 𝑧𝐿 ) ) ) → 𝑧𝐿 ∈ ( L ‘ 𝑧 ) )
134 elun1 ⊢ ( 𝑧𝐿 ∈ ( L ‘ 𝑧 ) → 𝑧𝐿 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) )
135 133 134 syl ⊢ ( ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦 <s 𝑥𝑂 ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( ( 𝑦 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦 <s 𝑥𝑂 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥 +s 𝑧 ) → 𝑦𝑂 <s 𝑥 ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦 <s 𝑥 ) ) ) ∧ ( 𝑧𝐿 ∈ ( L ‘ 𝑧 ) ∧ ( 𝑦 +s 𝑧 ) ≤s ( 𝑥 +s 𝑧𝐿 ) ) ) → 𝑧𝐿 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) )
136 131 132 135 rspcdva ⊢ ( ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦 <s 𝑥𝑂 ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( ( 𝑦 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦 <s 𝑥𝑂 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥 +s 𝑧 ) → 𝑦𝑂 <s 𝑥 ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦 <s 𝑥 ) ) ) ∧ ( 𝑧𝐿 ∈ ( L ‘ 𝑧 ) ∧ ( 𝑦 +s 𝑧 ) ≤s ( 𝑥 +s 𝑧𝐿 ) ) ) → ( ( 𝑦 +s 𝑧𝐿 ) <s ( 𝑥 +s 𝑧𝐿 ) → 𝑦 <s 𝑥 ) )
137 127 136 mpd ⊢ ( ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦 <s 𝑥𝑂 ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( ( 𝑦 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦 <s 𝑥𝑂 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥 +s 𝑧 ) → 𝑦𝑂 <s 𝑥 ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦 <s 𝑥 ) ) ) ∧ ( 𝑧𝐿 ∈ ( L ‘ 𝑧 ) ∧ ( 𝑦 +s 𝑧 ) ≤s ( 𝑥 +s 𝑧𝐿 ) ) ) → 𝑦 <s 𝑥 )
138 137 rexlimdvaa ⊢ ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦 <s 𝑥𝑂 ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( ( 𝑦 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦 <s 𝑥𝑂 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥 +s 𝑧 ) → 𝑦𝑂 <s 𝑥 ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦 <s 𝑥 ) ) ) → ( ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) ( 𝑦 +s 𝑧 ) ≤s ( 𝑥 +s 𝑧𝐿 ) → 𝑦 <s 𝑥 ) )
139 109 138 jaod ⊢ ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦 <s 𝑥𝑂 ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( ( 𝑦 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦 <s 𝑥𝑂 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥 +s 𝑧 ) → 𝑦𝑂 <s 𝑥 ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦 <s 𝑥 ) ) ) → ( ( ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ( 𝑦 +s 𝑧 ) ≤s ( 𝑥𝐿 +s 𝑧 ) ∨ ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) ( 𝑦 +s 𝑧 ) ≤s ( 𝑥 +s 𝑧𝐿 ) ) → 𝑦 <s 𝑥 ) )
140 93 139 biimtrid ⊢ ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦 <s 𝑥𝑂 ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( ( 𝑦 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦 <s 𝑥𝑂 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥 +s 𝑧 ) → 𝑦𝑂 <s 𝑥 ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦 <s 𝑥 ) ) ) → ( ∃ 𝑝 ∈ ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) 𝑎 = ( 𝑥𝐿 +s 𝑧 ) } ∪ { 𝑏 ∣ ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑏 = ( 𝑥 +s 𝑧𝐿 ) } ) ( 𝑦 +s 𝑧 ) ≤s 𝑝 → 𝑦 <s 𝑥 ) )
141 rexun ⊢ ( ∃ 𝑞 ∈ ( { 𝑐 ∣ ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) 𝑐 = ( 𝑦𝑅 +s 𝑧 ) } ∪ { 𝑑 ∣ ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑑 = ( 𝑦 +s 𝑧𝑅 ) } ) 𝑞 ≤s ( 𝑥 +s 𝑧 ) ↔ ( ∃ 𝑞 ∈ { 𝑐 ∣ ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) 𝑐 = ( 𝑦𝑅 +s 𝑧 ) } 𝑞 ≤s ( 𝑥 +s 𝑧 ) ∨ ∃ 𝑞 ∈ { 𝑑 ∣ ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑑 = ( 𝑦 +s 𝑧𝑅 ) } 𝑞 ≤s ( 𝑥 +s 𝑧 ) ) )
142 eqeq1 ⊢ ( 𝑐 = 𝑞 → ( 𝑐 = ( 𝑦𝑅 +s 𝑧 ) ↔ 𝑞 = ( 𝑦𝑅 +s 𝑧 ) ) )
143 142 rexbidv ⊢ ( 𝑐 = 𝑞 → ( ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) 𝑐 = ( 𝑦𝑅 +s 𝑧 ) ↔ ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) 𝑞 = ( 𝑦𝑅 +s 𝑧 ) ) )
144 143 rexab ⊢ ( ∃ 𝑞 ∈ { 𝑐 ∣ ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) 𝑐 = ( 𝑦𝑅 +s 𝑧 ) } 𝑞 ≤s ( 𝑥 +s 𝑧 ) ↔ ∃ 𝑞 ( ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) 𝑞 = ( 𝑦𝑅 +s 𝑧 ) ∧ 𝑞 ≤s ( 𝑥 +s 𝑧 ) ) )
145 rexcom4 ⊢ ( ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ∃ 𝑞 ( 𝑞 = ( 𝑦𝑅 +s 𝑧 ) ∧ 𝑞 ≤s ( 𝑥 +s 𝑧 ) ) ↔ ∃ 𝑞 ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ( 𝑞 = ( 𝑦𝑅 +s 𝑧 ) ∧ 𝑞 ≤s ( 𝑥 +s 𝑧 ) ) )
146 r19.41v ⊢ ( ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ( 𝑞 = ( 𝑦𝑅 +s 𝑧 ) ∧ 𝑞 ≤s ( 𝑥 +s 𝑧 ) ) ↔ ( ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) 𝑞 = ( 𝑦𝑅 +s 𝑧 ) ∧ 𝑞 ≤s ( 𝑥 +s 𝑧 ) ) )
147 146 exbii ⊢ ( ∃ 𝑞 ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ( 𝑞 = ( 𝑦𝑅 +s 𝑧 ) ∧ 𝑞 ≤s ( 𝑥 +s 𝑧 ) ) ↔ ∃ 𝑞 ( ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) 𝑞 = ( 𝑦𝑅 +s 𝑧 ) ∧ 𝑞 ≤s ( 𝑥 +s 𝑧 ) ) )
148 145 147 bitri ⊢ ( ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ∃ 𝑞 ( 𝑞 = ( 𝑦𝑅 +s 𝑧 ) ∧ 𝑞 ≤s ( 𝑥 +s 𝑧 ) ) ↔ ∃ 𝑞 ( ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) 𝑞 = ( 𝑦𝑅 +s 𝑧 ) ∧ 𝑞 ≤s ( 𝑥 +s 𝑧 ) ) )
149 ovex ⊢ ( 𝑦𝑅 +s 𝑧 ) ∈ V
150 breq1 ⊢ ( 𝑞 = ( 𝑦𝑅 +s 𝑧 ) → ( 𝑞 ≤s ( 𝑥 +s 𝑧 ) ↔ ( 𝑦𝑅 +s 𝑧 ) ≤s ( 𝑥 +s 𝑧 ) ) )
151 149 150 ceqsexv ⊢ ( ∃ 𝑞 ( 𝑞 = ( 𝑦𝑅 +s 𝑧 ) ∧ 𝑞 ≤s ( 𝑥 +s 𝑧 ) ) ↔ ( 𝑦𝑅 +s 𝑧 ) ≤s ( 𝑥 +s 𝑧 ) )
152 151 rexbii ⊢ ( ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ∃ 𝑞 ( 𝑞 = ( 𝑦𝑅 +s 𝑧 ) ∧ 𝑞 ≤s ( 𝑥 +s 𝑧 ) ) ↔ ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ( 𝑦𝑅 +s 𝑧 ) ≤s ( 𝑥 +s 𝑧 ) )
153 148 152 bitr3i ⊢ ( ∃ 𝑞 ( ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) 𝑞 = ( 𝑦𝑅 +s 𝑧 ) ∧ 𝑞 ≤s ( 𝑥 +s 𝑧 ) ) ↔ ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ( 𝑦𝑅 +s 𝑧 ) ≤s ( 𝑥 +s 𝑧 ) )
154 144 153 bitri ⊢ ( ∃ 𝑞 ∈ { 𝑐 ∣ ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) 𝑐 = ( 𝑦𝑅 +s 𝑧 ) } 𝑞 ≤s ( 𝑥 +s 𝑧 ) ↔ ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ( 𝑦𝑅 +s 𝑧 ) ≤s ( 𝑥 +s 𝑧 ) )
155 eqeq1 ⊢ ( 𝑑 = 𝑞 → ( 𝑑 = ( 𝑦 +s 𝑧𝑅 ) ↔ 𝑞 = ( 𝑦 +s 𝑧𝑅 ) ) )
156 155 rexbidv ⊢ ( 𝑑 = 𝑞 → ( ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑑 = ( 𝑦 +s 𝑧𝑅 ) ↔ ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑞 = ( 𝑦 +s 𝑧𝑅 ) ) )
157 156 rexab ⊢ ( ∃ 𝑞 ∈ { 𝑑 ∣ ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑑 = ( 𝑦 +s 𝑧𝑅 ) } 𝑞 ≤s ( 𝑥 +s 𝑧 ) ↔ ∃ 𝑞 ( ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑞 = ( 𝑦 +s 𝑧𝑅 ) ∧ 𝑞 ≤s ( 𝑥 +s 𝑧 ) ) )
158 rexcom4 ⊢ ( ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) ∃ 𝑞 ( 𝑞 = ( 𝑦 +s 𝑧𝑅 ) ∧ 𝑞 ≤s ( 𝑥 +s 𝑧 ) ) ↔ ∃ 𝑞 ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) ( 𝑞 = ( 𝑦 +s 𝑧𝑅 ) ∧ 𝑞 ≤s ( 𝑥 +s 𝑧 ) ) )
159 r19.41v ⊢ ( ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) ( 𝑞 = ( 𝑦 +s 𝑧𝑅 ) ∧ 𝑞 ≤s ( 𝑥 +s 𝑧 ) ) ↔ ( ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑞 = ( 𝑦 +s 𝑧𝑅 ) ∧ 𝑞 ≤s ( 𝑥 +s 𝑧 ) ) )
160 159 exbii ⊢ ( ∃ 𝑞 ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) ( 𝑞 = ( 𝑦 +s 𝑧𝑅 ) ∧ 𝑞 ≤s ( 𝑥 +s 𝑧 ) ) ↔ ∃ 𝑞 ( ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑞 = ( 𝑦 +s 𝑧𝑅 ) ∧ 𝑞 ≤s ( 𝑥 +s 𝑧 ) ) )
161 158 160 bitri ⊢ ( ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) ∃ 𝑞 ( 𝑞 = ( 𝑦 +s 𝑧𝑅 ) ∧ 𝑞 ≤s ( 𝑥 +s 𝑧 ) ) ↔ ∃ 𝑞 ( ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑞 = ( 𝑦 +s 𝑧𝑅 ) ∧ 𝑞 ≤s ( 𝑥 +s 𝑧 ) ) )
162 ovex ⊢ ( 𝑦 +s 𝑧𝑅 ) ∈ V
163 breq1 ⊢ ( 𝑞 = ( 𝑦 +s 𝑧𝑅 ) → ( 𝑞 ≤s ( 𝑥 +s 𝑧 ) ↔ ( 𝑦 +s 𝑧𝑅 ) ≤s ( 𝑥 +s 𝑧 ) ) )
164 162 163 ceqsexv ⊢ ( ∃ 𝑞 ( 𝑞 = ( 𝑦 +s 𝑧𝑅 ) ∧ 𝑞 ≤s ( 𝑥 +s 𝑧 ) ) ↔ ( 𝑦 +s 𝑧𝑅 ) ≤s ( 𝑥 +s 𝑧 ) )
165 164 rexbii ⊢ ( ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) ∃ 𝑞 ( 𝑞 = ( 𝑦 +s 𝑧𝑅 ) ∧ 𝑞 ≤s ( 𝑥 +s 𝑧 ) ) ↔ ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) ( 𝑦 +s 𝑧𝑅 ) ≤s ( 𝑥 +s 𝑧 ) )
166 161 165 bitr3i ⊢ ( ∃ 𝑞 ( ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑞 = ( 𝑦 +s 𝑧𝑅 ) ∧ 𝑞 ≤s ( 𝑥 +s 𝑧 ) ) ↔ ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) ( 𝑦 +s 𝑧𝑅 ) ≤s ( 𝑥 +s 𝑧 ) )
167 157 166 bitri ⊢ ( ∃ 𝑞 ∈ { 𝑑 ∣ ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑑 = ( 𝑦 +s 𝑧𝑅 ) } 𝑞 ≤s ( 𝑥 +s 𝑧 ) ↔ ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) ( 𝑦 +s 𝑧𝑅 ) ≤s ( 𝑥 +s 𝑧 ) )
168 154 167 orbi12i ⊢ ( ( ∃ 𝑞 ∈ { 𝑐 ∣ ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) 𝑐 = ( 𝑦𝑅 +s 𝑧 ) } 𝑞 ≤s ( 𝑥 +s 𝑧 ) ∨ ∃ 𝑞 ∈ { 𝑑 ∣ ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑑 = ( 𝑦 +s 𝑧𝑅 ) } 𝑞 ≤s ( 𝑥 +s 𝑧 ) ) ↔ ( ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ( 𝑦𝑅 +s 𝑧 ) ≤s ( 𝑥 +s 𝑧 ) ∨ ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) ( 𝑦 +s 𝑧𝑅 ) ≤s ( 𝑥 +s 𝑧 ) ) )
169 141 168 bitri ⊢ ( ∃ 𝑞 ∈ ( { 𝑐 ∣ ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) 𝑐 = ( 𝑦𝑅 +s 𝑧 ) } ∪ { 𝑑 ∣ ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑑 = ( 𝑦 +s 𝑧𝑅 ) } ) 𝑞 ≤s ( 𝑥 +s 𝑧 ) ↔ ( ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ( 𝑦𝑅 +s 𝑧 ) ≤s ( 𝑥 +s 𝑧 ) ∨ ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) ( 𝑦 +s 𝑧𝑅 ) ≤s ( 𝑥 +s 𝑧 ) ) )
170 simpll2 ⊢ ( ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦 <s 𝑥𝑂 ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( ( 𝑦 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦 <s 𝑥𝑂 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥 +s 𝑧 ) → 𝑦𝑂 <s 𝑥 ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦 <s 𝑥 ) ) ) ∧ ( 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ∧ ( 𝑦𝑅 +s 𝑧 ) ≤s ( 𝑥 +s 𝑧 ) ) ) → 𝑦 ∈ No )
171 rightssno ⊢ ( R ‘ 𝑦 ) ⊆ No
172 171 sseli ⊢ ( 𝑦𝑅 ∈ ( R ‘ 𝑦 ) → 𝑦𝑅 ∈ No )
173 172 adantr ⊢ ( ( 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ∧ ( 𝑦𝑅 +s 𝑧 ) ≤s ( 𝑥 +s 𝑧 ) ) → 𝑦𝑅 ∈ No )
174 173 adantl ⊢ ( ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦 <s 𝑥𝑂 ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( ( 𝑦 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦 <s 𝑥𝑂 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥 +s 𝑧 ) → 𝑦𝑂 <s 𝑥 ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦 <s 𝑥 ) ) ) ∧ ( 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ∧ ( 𝑦𝑅 +s 𝑧 ) ≤s ( 𝑥 +s 𝑧 ) ) ) → 𝑦𝑅 ∈ No )
175 simpll1 ⊢ ( ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦 <s 𝑥𝑂 ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( ( 𝑦 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦 <s 𝑥𝑂 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥 +s 𝑧 ) → 𝑦𝑂 <s 𝑥 ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦 <s 𝑥 ) ) ) ∧ ( 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ∧ ( 𝑦𝑅 +s 𝑧 ) ≤s ( 𝑥 +s 𝑧 ) ) ) → 𝑥 ∈ No )
176 rightgt ⊢ ( 𝑦𝑅 ∈ ( R ‘ 𝑦 ) → 𝑦 <s 𝑦𝑅 )
177 176 adantr ⊢ ( ( 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ∧ ( 𝑦𝑅 +s 𝑧 ) ≤s ( 𝑥 +s 𝑧 ) ) → 𝑦 <s 𝑦𝑅 )
178 177 adantl ⊢ ( ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦 <s 𝑥𝑂 ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( ( 𝑦 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦 <s 𝑥𝑂 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥 +s 𝑧 ) → 𝑦𝑂 <s 𝑥 ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦 <s 𝑥 ) ) ) ∧ ( 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ∧ ( 𝑦𝑅 +s 𝑧 ) ≤s ( 𝑥 +s 𝑧 ) ) ) → 𝑦 <s 𝑦𝑅 )
179 simprr ⊢ ( ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦 <s 𝑥𝑂 ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( ( 𝑦 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦 <s 𝑥𝑂 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥 +s 𝑧 ) → 𝑦𝑂 <s 𝑥 ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦 <s 𝑥 ) ) ) ∧ ( 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ∧ ( 𝑦𝑅 +s 𝑧 ) ≤s ( 𝑥 +s 𝑧 ) ) ) → ( 𝑦𝑅 +s 𝑧 ) ≤s ( 𝑥 +s 𝑧 ) )
180 simpll3 ⊢ ( ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦 <s 𝑥𝑂 ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( ( 𝑦 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦 <s 𝑥𝑂 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥 +s 𝑧 ) → 𝑦𝑂 <s 𝑥 ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦 <s 𝑥 ) ) ) ∧ ( 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ∧ ( 𝑦𝑅 +s 𝑧 ) ≤s ( 𝑥 +s 𝑧 ) ) ) → 𝑧 ∈ No )
181 sleadd1im ⊢ ( ( 𝑦𝑅 ∈ No ∧ 𝑥 ∈ No ∧ 𝑧 ∈ No ) → ( ( 𝑦𝑅 +s 𝑧 ) ≤s ( 𝑥 +s 𝑧 ) → 𝑦𝑅 ≤s 𝑥 ) )
182 174 175 180 181 syl3anc ⊢ ( ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦 <s 𝑥𝑂 ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( ( 𝑦 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦 <s 𝑥𝑂 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥 +s 𝑧 ) → 𝑦𝑂 <s 𝑥 ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦 <s 𝑥 ) ) ) ∧ ( 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ∧ ( 𝑦𝑅 +s 𝑧 ) ≤s ( 𝑥 +s 𝑧 ) ) ) → ( ( 𝑦𝑅 +s 𝑧 ) ≤s ( 𝑥 +s 𝑧 ) → 𝑦𝑅 ≤s 𝑥 ) )
183 179 182 mpd ⊢ ( ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦 <s 𝑥𝑂 ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( ( 𝑦 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦 <s 𝑥𝑂 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥 +s 𝑧 ) → 𝑦𝑂 <s 𝑥 ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦 <s 𝑥 ) ) ) ∧ ( 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ∧ ( 𝑦𝑅 +s 𝑧 ) ≤s ( 𝑥 +s 𝑧 ) ) ) → 𝑦𝑅 ≤s 𝑥 )
184 170 174 175 178 183 sltletrd ⊢ ( ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦 <s 𝑥𝑂 ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( ( 𝑦 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦 <s 𝑥𝑂 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥 +s 𝑧 ) → 𝑦𝑂 <s 𝑥 ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦 <s 𝑥 ) ) ) ∧ ( 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ∧ ( 𝑦𝑅 +s 𝑧 ) ≤s ( 𝑥 +s 𝑧 ) ) ) → 𝑦 <s 𝑥 )
185 184 rexlimdvaa ⊢ ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦 <s 𝑥𝑂 ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( ( 𝑦 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦 <s 𝑥𝑂 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥 +s 𝑧 ) → 𝑦𝑂 <s 𝑥 ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦 <s 𝑥 ) ) ) → ( ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ( 𝑦𝑅 +s 𝑧 ) ≤s ( 𝑥 +s 𝑧 ) → 𝑦 <s 𝑥 ) )
186 simpll2 ⊢ ( ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦 <s 𝑥𝑂 ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( ( 𝑦 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦 <s 𝑥𝑂 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥 +s 𝑧 ) → 𝑦𝑂 <s 𝑥 ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦 <s 𝑥 ) ) ) ∧ ( 𝑧𝑅 ∈ ( R ‘ 𝑧 ) ∧ ( 𝑦 +s 𝑧𝑅 ) ≤s ( 𝑥 +s 𝑧 ) ) ) → 𝑦 ∈ No )
187 rightssno ⊢ ( R ‘ 𝑧 ) ⊆ No
188 187 sseli ⊢ ( 𝑧𝑅 ∈ ( R ‘ 𝑧 ) → 𝑧𝑅 ∈ No )
189 188 adantr ⊢ ( ( 𝑧𝑅 ∈ ( R ‘ 𝑧 ) ∧ ( 𝑦 +s 𝑧𝑅 ) ≤s ( 𝑥 +s 𝑧 ) ) → 𝑧𝑅 ∈ No )
190 189 adantl ⊢ ( ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦 <s 𝑥𝑂 ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( ( 𝑦 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦 <s 𝑥𝑂 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥 +s 𝑧 ) → 𝑦𝑂 <s 𝑥 ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦 <s 𝑥 ) ) ) ∧ ( 𝑧𝑅 ∈ ( R ‘ 𝑧 ) ∧ ( 𝑦 +s 𝑧𝑅 ) ≤s ( 𝑥 +s 𝑧 ) ) ) → 𝑧𝑅 ∈ No )
191 186 190 addscld ⊢ ( ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦 <s 𝑥𝑂 ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( ( 𝑦 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦 <s 𝑥𝑂 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥 +s 𝑧 ) → 𝑦𝑂 <s 𝑥 ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦 <s 𝑥 ) ) ) ∧ ( 𝑧𝑅 ∈ ( R ‘ 𝑧 ) ∧ ( 𝑦 +s 𝑧𝑅 ) ≤s ( 𝑥 +s 𝑧 ) ) ) → ( 𝑦 +s 𝑧𝑅 ) ∈ No )
192 simpll1 ⊢ ( ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦 <s 𝑥𝑂 ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( ( 𝑦 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦 <s 𝑥𝑂 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥 +s 𝑧 ) → 𝑦𝑂 <s 𝑥 ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦 <s 𝑥 ) ) ) ∧ ( 𝑧𝑅 ∈ ( R ‘ 𝑧 ) ∧ ( 𝑦 +s 𝑧𝑅 ) ≤s ( 𝑥 +s 𝑧 ) ) ) → 𝑥 ∈ No )
193 simpll3 ⊢ ( ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦 <s 𝑥𝑂 ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( ( 𝑦 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦 <s 𝑥𝑂 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥 +s 𝑧 ) → 𝑦𝑂 <s 𝑥 ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦 <s 𝑥 ) ) ) ∧ ( 𝑧𝑅 ∈ ( R ‘ 𝑧 ) ∧ ( 𝑦 +s 𝑧𝑅 ) ≤s ( 𝑥 +s 𝑧 ) ) ) → 𝑧 ∈ No )
194 192 193 addscld ⊢ ( ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦 <s 𝑥𝑂 ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( ( 𝑦 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦 <s 𝑥𝑂 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥 +s 𝑧 ) → 𝑦𝑂 <s 𝑥 ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦 <s 𝑥 ) ) ) ∧ ( 𝑧𝑅 ∈ ( R ‘ 𝑧 ) ∧ ( 𝑦 +s 𝑧𝑅 ) ≤s ( 𝑥 +s 𝑧 ) ) ) → ( 𝑥 +s 𝑧 ) ∈ No )
195 192 190 addscld ⊢ ( ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦 <s 𝑥𝑂 ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( ( 𝑦 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦 <s 𝑥𝑂 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥 +s 𝑧 ) → 𝑦𝑂 <s 𝑥 ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦 <s 𝑥 ) ) ) ∧ ( 𝑧𝑅 ∈ ( R ‘ 𝑧 ) ∧ ( 𝑦 +s 𝑧𝑅 ) ≤s ( 𝑥 +s 𝑧 ) ) ) → ( 𝑥 +s 𝑧𝑅 ) ∈ No )
196 simprr ⊢ ( ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦 <s 𝑥𝑂 ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( ( 𝑦 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦 <s 𝑥𝑂 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥 +s 𝑧 ) → 𝑦𝑂 <s 𝑥 ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦 <s 𝑥 ) ) ) ∧ ( 𝑧𝑅 ∈ ( R ‘ 𝑧 ) ∧ ( 𝑦 +s 𝑧𝑅 ) ≤s ( 𝑥 +s 𝑧 ) ) ) → ( 𝑦 +s 𝑧𝑅 ) ≤s ( 𝑥 +s 𝑧 ) )
197 193 190 192 3jca ⊢ ( ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦 <s 𝑥𝑂 ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( ( 𝑦 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦 <s 𝑥𝑂 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥 +s 𝑧 ) → 𝑦𝑂 <s 𝑥 ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦 <s 𝑥 ) ) ) ∧ ( 𝑧𝑅 ∈ ( R ‘ 𝑧 ) ∧ ( 𝑦 +s 𝑧𝑅 ) ≤s ( 𝑥 +s 𝑧 ) ) ) → ( 𝑧 ∈ No ∧ 𝑧𝑅 ∈ No ∧ 𝑥 ∈ No ) )
198 rightgt ⊢ ( 𝑧𝑅 ∈ ( R ‘ 𝑧 ) → 𝑧 <s 𝑧𝑅 )
199 198 adantr ⊢ ( ( 𝑧𝑅 ∈ ( R ‘ 𝑧 ) ∧ ( 𝑦 +s 𝑧𝑅 ) ≤s ( 𝑥 +s 𝑧 ) ) → 𝑧 <s 𝑧𝑅 )
200 199 adantl ⊢ ( ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦 <s 𝑥𝑂 ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( ( 𝑦 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦 <s 𝑥𝑂 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥 +s 𝑧 ) → 𝑦𝑂 <s 𝑥 ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦 <s 𝑥 ) ) ) ∧ ( 𝑧𝑅 ∈ ( R ‘ 𝑧 ) ∧ ( 𝑦 +s 𝑧𝑅 ) ≤s ( 𝑥 +s 𝑧 ) ) ) → 𝑧 <s 𝑧𝑅 )
201 sltadd2im ⊢ ( ( 𝑧 ∈ No ∧ 𝑧𝑅 ∈ No ∧ 𝑥 ∈ No ) → ( 𝑧 <s 𝑧𝑅 → ( 𝑥 +s 𝑧 ) <s ( 𝑥 +s 𝑧𝑅 ) ) )
202 197 200 201 sylc ⊢ ( ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦 <s 𝑥𝑂 ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( ( 𝑦 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦 <s 𝑥𝑂 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥 +s 𝑧 ) → 𝑦𝑂 <s 𝑥 ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦 <s 𝑥 ) ) ) ∧ ( 𝑧𝑅 ∈ ( R ‘ 𝑧 ) ∧ ( 𝑦 +s 𝑧𝑅 ) ≤s ( 𝑥 +s 𝑧 ) ) ) → ( 𝑥 +s 𝑧 ) <s ( 𝑥 +s 𝑧𝑅 ) )
203 191 194 195 196 202 slelttrd ⊢ ( ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦 <s 𝑥𝑂 ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( ( 𝑦 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦 <s 𝑥𝑂 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥 +s 𝑧 ) → 𝑦𝑂 <s 𝑥 ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦 <s 𝑥 ) ) ) ∧ ( 𝑧𝑅 ∈ ( R ‘ 𝑧 ) ∧ ( 𝑦 +s 𝑧𝑅 ) ≤s ( 𝑥 +s 𝑧 ) ) ) → ( 𝑦 +s 𝑧𝑅 ) <s ( 𝑥 +s 𝑧𝑅 ) )
204 oveq2 ⊢ ( 𝑧𝑂 = 𝑧𝑅 → ( 𝑦 +s 𝑧𝑂 ) = ( 𝑦 +s 𝑧𝑅 ) )
205 oveq2 ⊢ ( 𝑧𝑂 = 𝑧𝑅 → ( 𝑥 +s 𝑧𝑂 ) = ( 𝑥 +s 𝑧𝑅 ) )
206 204 205 breq12d ⊢ ( 𝑧𝑂 = 𝑧𝑅 → ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) ↔ ( 𝑦 +s 𝑧𝑅 ) <s ( 𝑥 +s 𝑧𝑅 ) ) )
207 206 imbi1d ⊢ ( 𝑧𝑂 = 𝑧𝑅 → ( ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦 <s 𝑥 ) ↔ ( ( 𝑦 +s 𝑧𝑅 ) <s ( 𝑥 +s 𝑧𝑅 ) → 𝑦 <s 𝑥 ) ) )
208 simplr3 ⊢ ( ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦 <s 𝑥𝑂 ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( ( 𝑦 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦 <s 𝑥𝑂 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥 +s 𝑧 ) → 𝑦𝑂 <s 𝑥 ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦 <s 𝑥 ) ) ) ∧ ( 𝑧𝑅 ∈ ( R ‘ 𝑧 ) ∧ ( 𝑦 +s 𝑧𝑅 ) ≤s ( 𝑥 +s 𝑧 ) ) ) → ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦 <s 𝑥 ) )
209 simprl ⊢ ( ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦 <s 𝑥𝑂 ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( ( 𝑦 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦 <s 𝑥𝑂 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥 +s 𝑧 ) → 𝑦𝑂 <s 𝑥 ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦 <s 𝑥 ) ) ) ∧ ( 𝑧𝑅 ∈ ( R ‘ 𝑧 ) ∧ ( 𝑦 +s 𝑧𝑅 ) ≤s ( 𝑥 +s 𝑧 ) ) ) → 𝑧𝑅 ∈ ( R ‘ 𝑧 ) )
210 elun2 ⊢ ( 𝑧𝑅 ∈ ( R ‘ 𝑧 ) → 𝑧𝑅 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) )
211 209 210 syl ⊢ ( ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦 <s 𝑥𝑂 ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( ( 𝑦 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦 <s 𝑥𝑂 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥 +s 𝑧 ) → 𝑦𝑂 <s 𝑥 ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦 <s 𝑥 ) ) ) ∧ ( 𝑧𝑅 ∈ ( R ‘ 𝑧 ) ∧ ( 𝑦 +s 𝑧𝑅 ) ≤s ( 𝑥 +s 𝑧 ) ) ) → 𝑧𝑅 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) )
212 207 208 211 rspcdva ⊢ ( ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦 <s 𝑥𝑂 ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( ( 𝑦 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦 <s 𝑥𝑂 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥 +s 𝑧 ) → 𝑦𝑂 <s 𝑥 ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦 <s 𝑥 ) ) ) ∧ ( 𝑧𝑅 ∈ ( R ‘ 𝑧 ) ∧ ( 𝑦 +s 𝑧𝑅 ) ≤s ( 𝑥 +s 𝑧 ) ) ) → ( ( 𝑦 +s 𝑧𝑅 ) <s ( 𝑥 +s 𝑧𝑅 ) → 𝑦 <s 𝑥 ) )
213 203 212 mpd ⊢ ( ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦 <s 𝑥𝑂 ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( ( 𝑦 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦 <s 𝑥𝑂 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥 +s 𝑧 ) → 𝑦𝑂 <s 𝑥 ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦 <s 𝑥 ) ) ) ∧ ( 𝑧𝑅 ∈ ( R ‘ 𝑧 ) ∧ ( 𝑦 +s 𝑧𝑅 ) ≤s ( 𝑥 +s 𝑧 ) ) ) → 𝑦 <s 𝑥 )
214 213 rexlimdvaa ⊢ ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦 <s 𝑥𝑂 ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( ( 𝑦 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦 <s 𝑥𝑂 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥 +s 𝑧 ) → 𝑦𝑂 <s 𝑥 ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦 <s 𝑥 ) ) ) → ( ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) ( 𝑦 +s 𝑧𝑅 ) ≤s ( 𝑥 +s 𝑧 ) → 𝑦 <s 𝑥 ) )
215 185 214 jaod ⊢ ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦 <s 𝑥𝑂 ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( ( 𝑦 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦 <s 𝑥𝑂 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥 +s 𝑧 ) → 𝑦𝑂 <s 𝑥 ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦 <s 𝑥 ) ) ) → ( ( ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ( 𝑦𝑅 +s 𝑧 ) ≤s ( 𝑥 +s 𝑧 ) ∨ ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) ( 𝑦 +s 𝑧𝑅 ) ≤s ( 𝑥 +s 𝑧 ) ) → 𝑦 <s 𝑥 ) )
216 169 215 biimtrid ⊢ ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦 <s 𝑥𝑂 ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( ( 𝑦 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦 <s 𝑥𝑂 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥 +s 𝑧 ) → 𝑦𝑂 <s 𝑥 ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦 <s 𝑥 ) ) ) → ( ∃ 𝑞 ∈ ( { 𝑐 ∣ ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) 𝑐 = ( 𝑦𝑅 +s 𝑧 ) } ∪ { 𝑑 ∣ ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑑 = ( 𝑦 +s 𝑧𝑅 ) } ) 𝑞 ≤s ( 𝑥 +s 𝑧 ) → 𝑦 <s 𝑥 ) )
217 140 216 jaod ⊢ ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦 <s 𝑥𝑂 ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( ( 𝑦 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦 <s 𝑥𝑂 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥 +s 𝑧 ) → 𝑦𝑂 <s 𝑥 ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦 <s 𝑥 ) ) ) → ( ( ∃ 𝑝 ∈ ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) 𝑎 = ( 𝑥𝐿 +s 𝑧 ) } ∪ { 𝑏 ∣ ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑏 = ( 𝑥 +s 𝑧𝐿 ) } ) ( 𝑦 +s 𝑧 ) ≤s 𝑝 ∨ ∃ 𝑞 ∈ ( { 𝑐 ∣ ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) 𝑐 = ( 𝑦𝑅 +s 𝑧 ) } ∪ { 𝑑 ∣ ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑑 = ( 𝑦 +s 𝑧𝑅 ) } ) 𝑞 ≤s ( 𝑥 +s 𝑧 ) ) → 𝑦 <s 𝑥 ) )
218 64 217 sylbid ⊢ ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦 <s 𝑥𝑂 ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( ( 𝑦 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦 <s 𝑥𝑂 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥 +s 𝑧 ) → 𝑦𝑂 <s 𝑥 ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦 <s 𝑥 ) ) ) → ( ( 𝑦 +s 𝑧 ) <s ( 𝑥 +s 𝑧 ) → 𝑦 <s 𝑥 ) )
219 218 ex ⊢ ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) → ( ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦 <s 𝑥𝑂 ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( ( 𝑦 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦 <s 𝑥𝑂 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥 +s 𝑧 ) → 𝑦𝑂 <s 𝑥 ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦 <s 𝑥 ) ) → ( ( 𝑦 +s 𝑧 ) <s ( 𝑥 +s 𝑧 ) → 𝑦 <s 𝑥 ) ) )
220 4 8 12 16 20 22 25 29 33 37 219 no3inds ⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( ( 𝐵 +s 𝐶 ) <s ( 𝐴 +s 𝐶 ) → 𝐵 <s 𝐴 ) )
221 addscl ⊢ ( ( 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( 𝐵 +s 𝐶 ) ∈ No )
222 221 3adant1 ⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( 𝐵 +s 𝐶 ) ∈ No )
223 addscl ⊢ ( ( 𝐴 ∈ No ∧ 𝐶 ∈ No ) → ( 𝐴 +s 𝐶 ) ∈ No )
224 223 3adant2 ⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( 𝐴 +s 𝐶 ) ∈ No )
225 sltnle ⊢ ( ( ( 𝐵 +s 𝐶 ) ∈ No ∧ ( 𝐴 +s 𝐶 ) ∈ No ) → ( ( 𝐵 +s 𝐶 ) <s ( 𝐴 +s 𝐶 ) ↔ ¬ ( 𝐴 +s 𝐶 ) ≤s ( 𝐵 +s 𝐶 ) ) )
226 222 224 225 syl2anc ⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( ( 𝐵 +s 𝐶 ) <s ( 𝐴 +s 𝐶 ) ↔ ¬ ( 𝐴 +s 𝐶 ) ≤s ( 𝐵 +s 𝐶 ) ) )
227 sltnle ⊢ ( ( 𝐵 ∈ No ∧ 𝐴 ∈ No ) → ( 𝐵 <s 𝐴 ↔ ¬ 𝐴 ≤s 𝐵 ) )
228 227 ancoms ⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( 𝐵 <s 𝐴 ↔ ¬ 𝐴 ≤s 𝐵 ) )
229 228 3adant3 ⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( 𝐵 <s 𝐴 ↔ ¬ 𝐴 ≤s 𝐵 ) )
230 220 226 229 3imtr3d ⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( ¬ ( 𝐴 +s 𝐶 ) ≤s ( 𝐵 +s 𝐶 ) → ¬ 𝐴 ≤s 𝐵 ) )
231 230 con4d ⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( 𝐴 ≤s 𝐵 → ( 𝐴 +s 𝐶 ) ≤s ( 𝐵 +s 𝐶 ) ) )
232 sleadd1im ⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( ( 𝐴 +s 𝐶 ) ≤s ( 𝐵 +s 𝐶 ) → 𝐴 ≤s 𝐵 ) )
233 231 232 impbid ⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( 𝐴 ≤s 𝐵 ↔ ( 𝐴 +s 𝐶 ) ≤s ( 𝐵 +s 𝐶 ) ) )