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 sltrec ⊢ ( ( ( ( { 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 ) } ) ∧ ( { 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 ) } ) ) ∧ ( ( 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 ) } ) ) ∧ ( 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 ) } ) ) ) ) → ( ( 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 48 58 60 62 63 syl22anc ⊢ ( ( 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 ) ) ) )
65 64 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 ) ) ) )
66 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 ) )
67 eqeq1 ⊢ ( 𝑎 = 𝑝 → ( 𝑎 = ( 𝑥𝐿 +s 𝑧 ) ↔ 𝑝 = ( 𝑥𝐿 +s 𝑧 ) ) )
68 67 rexbidv ⊢ ( 𝑎 = 𝑝 → ( ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) 𝑎 = ( 𝑥𝐿 +s 𝑧 ) ↔ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) 𝑝 = ( 𝑥𝐿 +s 𝑧 ) ) )
69 68 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 ) )
70 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 ) )
71 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 ) )
72 71 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 ) )
73 70 72 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 ) )
74 ovex ⊢ ( 𝑥𝐿 +s 𝑧 ) ∈ V
75 breq2 ⊢ ( 𝑝 = ( 𝑥𝐿 +s 𝑧 ) → ( ( 𝑦 +s 𝑧 ) ≤s 𝑝 ↔ ( 𝑦 +s 𝑧 ) ≤s ( 𝑥𝐿 +s 𝑧 ) ) )
76 74 75 ceqsexv ⊢ ( ∃ 𝑝 ( 𝑝 = ( 𝑥𝐿 +s 𝑧 ) ∧ ( 𝑦 +s 𝑧 ) ≤s 𝑝 ) ↔ ( 𝑦 +s 𝑧 ) ≤s ( 𝑥𝐿 +s 𝑧 ) )
77 76 rexbii ⊢ ( ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑝 ( 𝑝 = ( 𝑥𝐿 +s 𝑧 ) ∧ ( 𝑦 +s 𝑧 ) ≤s 𝑝 ) ↔ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ( 𝑦 +s 𝑧 ) ≤s ( 𝑥𝐿 +s 𝑧 ) )
78 73 77 bitr3i ⊢ ( ∃ 𝑝 ( ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) 𝑝 = ( 𝑥𝐿 +s 𝑧 ) ∧ ( 𝑦 +s 𝑧 ) ≤s 𝑝 ) ↔ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ( 𝑦 +s 𝑧 ) ≤s ( 𝑥𝐿 +s 𝑧 ) )
79 69 78 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 ) )
80 eqeq1 ⊢ ( 𝑏 = 𝑝 → ( 𝑏 = ( 𝑥 +s 𝑧𝐿 ) ↔ 𝑝 = ( 𝑥 +s 𝑧𝐿 ) ) )
81 80 rexbidv ⊢ ( 𝑏 = 𝑝 → ( ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑏 = ( 𝑥 +s 𝑧𝐿 ) ↔ ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑝 = ( 𝑥 +s 𝑧𝐿 ) ) )
82 81 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 ) )
83 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 ) )
84 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 ) )
85 84 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 ) )
86 83 85 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 ) )
87 ovex ⊢ ( 𝑥 +s 𝑧𝐿 ) ∈ V
88 breq2 ⊢ ( 𝑝 = ( 𝑥 +s 𝑧𝐿 ) → ( ( 𝑦 +s 𝑧 ) ≤s 𝑝 ↔ ( 𝑦 +s 𝑧 ) ≤s ( 𝑥 +s 𝑧𝐿 ) ) )
89 87 88 ceqsexv ⊢ ( ∃ 𝑝 ( 𝑝 = ( 𝑥 +s 𝑧𝐿 ) ∧ ( 𝑦 +s 𝑧 ) ≤s 𝑝 ) ↔ ( 𝑦 +s 𝑧 ) ≤s ( 𝑥 +s 𝑧𝐿 ) )
90 89 rexbii ⊢ ( ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) ∃ 𝑝 ( 𝑝 = ( 𝑥 +s 𝑧𝐿 ) ∧ ( 𝑦 +s 𝑧 ) ≤s 𝑝 ) ↔ ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) ( 𝑦 +s 𝑧 ) ≤s ( 𝑥 +s 𝑧𝐿 ) )
91 86 90 bitr3i ⊢ ( ∃ 𝑝 ( ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑝 = ( 𝑥 +s 𝑧𝐿 ) ∧ ( 𝑦 +s 𝑧 ) ≤s 𝑝 ) ↔ ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) ( 𝑦 +s 𝑧 ) ≤s ( 𝑥 +s 𝑧𝐿 ) )
92 82 91 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 ) )
93 79 92 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 ) ) )
94 66 93 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 ) ) )
95 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 )
96 leftssno ⊢ ( L ‘ 𝑥 ) ⊆ No
97 96 sseli ⊢ ( 𝑥𝐿 ∈ ( L ‘ 𝑥 ) → 𝑥𝐿 ∈ No )
98 97 adantr ⊢ ( ( 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∧ ( 𝑦 +s 𝑧 ) ≤s ( 𝑥𝐿 +s 𝑧 ) ) → 𝑥𝐿 ∈ No )
99 98 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 )
100 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 )
101 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 ) )
102 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 )
103 sleadd1im ⊢ ( ( 𝑦 ∈ No ∧ 𝑥𝐿 ∈ No ∧ 𝑧 ∈ No ) → ( ( 𝑦 +s 𝑧 ) ≤s ( 𝑥𝐿 +s 𝑧 ) → 𝑦 ≤s 𝑥𝐿 ) )
104 95 99 102 103 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 ) )
105 101 104 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 )
106 leftlt ⊢ ( 𝑥𝐿 ∈ ( L ‘ 𝑥 ) → 𝑥𝐿 <s 𝑥 )
107 106 adantr ⊢ ( ( 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∧ ( 𝑦 +s 𝑧 ) ≤s ( 𝑥𝐿 +s 𝑧 ) ) → 𝑥𝐿 <s 𝑥 )
108 107 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 )
109 95 99 100 105 108 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 )
110 109 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 ) )
111 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 )
112 leftssno ⊢ ( L ‘ 𝑧 ) ⊆ No
113 112 sseli ⊢ ( 𝑧𝐿 ∈ ( L ‘ 𝑧 ) → 𝑧𝐿 ∈ No )
114 113 adantr ⊢ ( ( 𝑧𝐿 ∈ ( L ‘ 𝑧 ) ∧ ( 𝑦 +s 𝑧 ) ≤s ( 𝑥 +s 𝑧𝐿 ) ) → 𝑧𝐿 ∈ No )
115 114 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 )
116 111 115 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 )
117 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 )
118 111 117 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 )
119 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 )
120 119 115 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 )
121 leftlt ⊢ ( 𝑧𝐿 ∈ ( L ‘ 𝑧 ) → 𝑧𝐿 <s 𝑧 )
122 121 adantr ⊢ ( ( 𝑧𝐿 ∈ ( L ‘ 𝑧 ) ∧ ( 𝑦 +s 𝑧 ) ≤s ( 𝑥 +s 𝑧𝐿 ) ) → 𝑧𝐿 <s 𝑧 )
123 122 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 )
124 sltadd2im ⊢ ( ( 𝑧𝐿 ∈ No ∧ 𝑧 ∈ No ∧ 𝑦 ∈ No ) → ( 𝑧𝐿 <s 𝑧 → ( 𝑦 +s 𝑧𝐿 ) <s ( 𝑦 +s 𝑧 ) ) )
125 115 117 111 124 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 ) ) )
126 123 125 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 ) )
127 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 ) )
128 116 118 120 126 127 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 ) )
129 oveq2 ⊢ ( 𝑧𝑂 = 𝑧𝐿 → ( 𝑦 +s 𝑧𝑂 ) = ( 𝑦 +s 𝑧𝐿 ) )
130 oveq2 ⊢ ( 𝑧𝑂 = 𝑧𝐿 → ( 𝑥 +s 𝑧𝑂 ) = ( 𝑥 +s 𝑧𝐿 ) )
131 129 130 breq12d ⊢ ( 𝑧𝑂 = 𝑧𝐿 → ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) ↔ ( 𝑦 +s 𝑧𝐿 ) <s ( 𝑥 +s 𝑧𝐿 ) ) )
132 131 imbi1d ⊢ ( 𝑧𝑂 = 𝑧𝐿 → ( ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦 <s 𝑥 ) ↔ ( ( 𝑦 +s 𝑧𝐿 ) <s ( 𝑥 +s 𝑧𝐿 ) → 𝑦 <s 𝑥 ) ) )
133 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 ) )
134 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 ) )
135 elun1 ⊢ ( 𝑧𝐿 ∈ ( L ‘ 𝑧 ) → 𝑧𝐿 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) )
136 134 135 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 ) ) )
137 132 133 136 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 ) )
138 128 137 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 )
139 138 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 ) )
140 110 139 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 ) )
141 94 140 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 ) )
142 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 ) ) )
143 eqeq1 ⊢ ( 𝑐 = 𝑞 → ( 𝑐 = ( 𝑦𝑅 +s 𝑧 ) ↔ 𝑞 = ( 𝑦𝑅 +s 𝑧 ) ) )
144 143 rexbidv ⊢ ( 𝑐 = 𝑞 → ( ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) 𝑐 = ( 𝑦𝑅 +s 𝑧 ) ↔ ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) 𝑞 = ( 𝑦𝑅 +s 𝑧 ) ) )
145 144 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 ) ) )
146 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 ) ) )
147 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 ) ) )
148 147 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 ) ) )
149 146 148 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 ) ) )
150 ovex ⊢ ( 𝑦𝑅 +s 𝑧 ) ∈ V
151 breq1 ⊢ ( 𝑞 = ( 𝑦𝑅 +s 𝑧 ) → ( 𝑞 ≤s ( 𝑥 +s 𝑧 ) ↔ ( 𝑦𝑅 +s 𝑧 ) ≤s ( 𝑥 +s 𝑧 ) ) )
152 150 151 ceqsexv ⊢ ( ∃ 𝑞 ( 𝑞 = ( 𝑦𝑅 +s 𝑧 ) ∧ 𝑞 ≤s ( 𝑥 +s 𝑧 ) ) ↔ ( 𝑦𝑅 +s 𝑧 ) ≤s ( 𝑥 +s 𝑧 ) )
153 152 rexbii ⊢ ( ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ∃ 𝑞 ( 𝑞 = ( 𝑦𝑅 +s 𝑧 ) ∧ 𝑞 ≤s ( 𝑥 +s 𝑧 ) ) ↔ ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ( 𝑦𝑅 +s 𝑧 ) ≤s ( 𝑥 +s 𝑧 ) )
154 149 153 bitr3i ⊢ ( ∃ 𝑞 ( ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) 𝑞 = ( 𝑦𝑅 +s 𝑧 ) ∧ 𝑞 ≤s ( 𝑥 +s 𝑧 ) ) ↔ ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ( 𝑦𝑅 +s 𝑧 ) ≤s ( 𝑥 +s 𝑧 ) )
155 145 154 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 ) )
156 eqeq1 ⊢ ( 𝑑 = 𝑞 → ( 𝑑 = ( 𝑦 +s 𝑧𝑅 ) ↔ 𝑞 = ( 𝑦 +s 𝑧𝑅 ) ) )
157 156 rexbidv ⊢ ( 𝑑 = 𝑞 → ( ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑑 = ( 𝑦 +s 𝑧𝑅 ) ↔ ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑞 = ( 𝑦 +s 𝑧𝑅 ) ) )
158 157 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 ) ) )
159 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 ) ) )
160 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 ) ) )
161 160 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 ) ) )
162 159 161 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 ) ) )
163 ovex ⊢ ( 𝑦 +s 𝑧𝑅 ) ∈ V
164 breq1 ⊢ ( 𝑞 = ( 𝑦 +s 𝑧𝑅 ) → ( 𝑞 ≤s ( 𝑥 +s 𝑧 ) ↔ ( 𝑦 +s 𝑧𝑅 ) ≤s ( 𝑥 +s 𝑧 ) ) )
165 163 164 ceqsexv ⊢ ( ∃ 𝑞 ( 𝑞 = ( 𝑦 +s 𝑧𝑅 ) ∧ 𝑞 ≤s ( 𝑥 +s 𝑧 ) ) ↔ ( 𝑦 +s 𝑧𝑅 ) ≤s ( 𝑥 +s 𝑧 ) )
166 165 rexbii ⊢ ( ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) ∃ 𝑞 ( 𝑞 = ( 𝑦 +s 𝑧𝑅 ) ∧ 𝑞 ≤s ( 𝑥 +s 𝑧 ) ) ↔ ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) ( 𝑦 +s 𝑧𝑅 ) ≤s ( 𝑥 +s 𝑧 ) )
167 162 166 bitr3i ⊢ ( ∃ 𝑞 ( ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑞 = ( 𝑦 +s 𝑧𝑅 ) ∧ 𝑞 ≤s ( 𝑥 +s 𝑧 ) ) ↔ ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) ( 𝑦 +s 𝑧𝑅 ) ≤s ( 𝑥 +s 𝑧 ) )
168 158 167 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 ) )
169 155 168 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 ) ) )
170 142 169 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 ) ) )
171 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 )
172 rightssno ⊢ ( R ‘ 𝑦 ) ⊆ No
173 172 sseli ⊢ ( 𝑦𝑅 ∈ ( R ‘ 𝑦 ) → 𝑦𝑅 ∈ No )
174 173 adantr ⊢ ( ( 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ∧ ( 𝑦𝑅 +s 𝑧 ) ≤s ( 𝑥 +s 𝑧 ) ) → 𝑦𝑅 ∈ No )
175 174 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 )
176 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 )
177 rightgt ⊢ ( 𝑦𝑅 ∈ ( R ‘ 𝑦 ) → 𝑦 <s 𝑦𝑅 )
178 177 adantr ⊢ ( ( 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ∧ ( 𝑦𝑅 +s 𝑧 ) ≤s ( 𝑥 +s 𝑧 ) ) → 𝑦 <s 𝑦𝑅 )
179 178 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 )
180 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 ) )
181 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 )
182 sleadd1im ⊢ ( ( 𝑦𝑅 ∈ No ∧ 𝑥 ∈ No ∧ 𝑧 ∈ No ) → ( ( 𝑦𝑅 +s 𝑧 ) ≤s ( 𝑥 +s 𝑧 ) → 𝑦𝑅 ≤s 𝑥 ) )
183 175 176 181 182 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 ) )
184 180 183 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 )
185 171 175 176 179 184 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 )
186 185 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 ) )
187 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 )
188 rightssno ⊢ ( R ‘ 𝑧 ) ⊆ No
189 188 sseli ⊢ ( 𝑧𝑅 ∈ ( R ‘ 𝑧 ) → 𝑧𝑅 ∈ No )
190 189 adantr ⊢ ( ( 𝑧𝑅 ∈ ( R ‘ 𝑧 ) ∧ ( 𝑦 +s 𝑧𝑅 ) ≤s ( 𝑥 +s 𝑧 ) ) → 𝑧𝑅 ∈ No )
191 190 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 )
192 187 191 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 )
193 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 )
194 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 )
195 193 194 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 )
196 193 191 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 )
197 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 ) )
198 194 191 193 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 ) )
199 rightgt ⊢ ( 𝑧𝑅 ∈ ( R ‘ 𝑧 ) → 𝑧 <s 𝑧𝑅 )
200 199 adantr ⊢ ( ( 𝑧𝑅 ∈ ( R ‘ 𝑧 ) ∧ ( 𝑦 +s 𝑧𝑅 ) ≤s ( 𝑥 +s 𝑧 ) ) → 𝑧 <s 𝑧𝑅 )
201 200 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 )
202 sltadd2im ⊢ ( ( 𝑧 ∈ No ∧ 𝑧𝑅 ∈ No ∧ 𝑥 ∈ No ) → ( 𝑧 <s 𝑧𝑅 → ( 𝑥 +s 𝑧 ) <s ( 𝑥 +s 𝑧𝑅 ) ) )
203 198 201 202 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 ) )
204 192 195 196 197 203 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 ) )
205 oveq2 ⊢ ( 𝑧𝑂 = 𝑧𝑅 → ( 𝑦 +s 𝑧𝑂 ) = ( 𝑦 +s 𝑧𝑅 ) )
206 oveq2 ⊢ ( 𝑧𝑂 = 𝑧𝑅 → ( 𝑥 +s 𝑧𝑂 ) = ( 𝑥 +s 𝑧𝑅 ) )
207 205 206 breq12d ⊢ ( 𝑧𝑂 = 𝑧𝑅 → ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) ↔ ( 𝑦 +s 𝑧𝑅 ) <s ( 𝑥 +s 𝑧𝑅 ) ) )
208 207 imbi1d ⊢ ( 𝑧𝑂 = 𝑧𝑅 → ( ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦 <s 𝑥 ) ↔ ( ( 𝑦 +s 𝑧𝑅 ) <s ( 𝑥 +s 𝑧𝑅 ) → 𝑦 <s 𝑥 ) ) )
209 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 ) )
210 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 ) )
211 elun2 ⊢ ( 𝑧𝑅 ∈ ( R ‘ 𝑧 ) → 𝑧𝑅 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) )
212 210 211 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 ) ) )
213 208 209 212 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 ) )
214 204 213 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 )
215 214 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 ) )
216 186 215 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 ) )
217 170 216 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 ) )
218 141 217 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 ) )
219 65 218 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 ) )
220 219 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 ) ) )
221 4 8 12 16 20 22 25 29 33 37 220 no3inds ⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( ( 𝐵 +s 𝐶 ) <s ( 𝐴 +s 𝐶 ) → 𝐵 <s 𝐴 ) )
222 addscl ⊢ ( ( 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( 𝐵 +s 𝐶 ) ∈ No )
223 222 3adant1 ⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( 𝐵 +s 𝐶 ) ∈ No )
224 addscl ⊢ ( ( 𝐴 ∈ No ∧ 𝐶 ∈ No ) → ( 𝐴 +s 𝐶 ) ∈ No )
225 224 3adant2 ⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( 𝐴 +s 𝐶 ) ∈ No )
226 sltnle ⊢ ( ( ( 𝐵 +s 𝐶 ) ∈ No ∧ ( 𝐴 +s 𝐶 ) ∈ No ) → ( ( 𝐵 +s 𝐶 ) <s ( 𝐴 +s 𝐶 ) ↔ ¬ ( 𝐴 +s 𝐶 ) ≤s ( 𝐵 +s 𝐶 ) ) )
227 223 225 226 syl2anc ⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( ( 𝐵 +s 𝐶 ) <s ( 𝐴 +s 𝐶 ) ↔ ¬ ( 𝐴 +s 𝐶 ) ≤s ( 𝐵 +s 𝐶 ) ) )
228 sltnle ⊢ ( ( 𝐵 ∈ No ∧ 𝐴 ∈ No ) → ( 𝐵 <s 𝐴 ↔ ¬ 𝐴 ≤s 𝐵 ) )
229 228 ancoms ⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( 𝐵 <s 𝐴 ↔ ¬ 𝐴 ≤s 𝐵 ) )
230 229 3adant3 ⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( 𝐵 <s 𝐴 ↔ ¬ 𝐴 ≤s 𝐵 ) )
231 221 227 230 3imtr3d ⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( ¬ ( 𝐴 +s 𝐶 ) ≤s ( 𝐵 +s 𝐶 ) → ¬ 𝐴 ≤s 𝐵 ) )
232 231 con4d ⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( 𝐴 ≤s 𝐵 → ( 𝐴 +s 𝐶 ) ≤s ( 𝐵 +s 𝐶 ) ) )
233 sleadd1im ⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( ( 𝐴 +s 𝐶 ) ≤s ( 𝐵 +s 𝐶 ) → 𝐴 ≤s 𝐵 ) )
234 232 233 impbid ⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( 𝐴 ≤s 𝐵 ↔ ( 𝐴 +s 𝐶 ) ≤s ( 𝐵 +s 𝐶 ) ) )