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 leftval ⊢ ( L ‘ 𝑥 ) = { 𝑥𝐿 ∈ ( O ‘ ( bday ‘ 𝑥 ) ) ∣ 𝑥𝐿 <s 𝑥 }
107 106 reqabi ⊢ ( 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ↔ ( 𝑥𝐿 ∈ ( O ‘ ( bday ‘ 𝑥 ) ) ∧ 𝑥𝐿 <s 𝑥 ) )
108 107 simprbi ⊢ ( 𝑥𝐿 ∈ ( L ‘ 𝑥 ) → 𝑥𝐿 <s 𝑥 )
109 108 adantr ⊢ ( ( 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∧ ( 𝑦 +s 𝑧 ) ≤s ( 𝑥𝐿 +s 𝑧 ) ) → 𝑥𝐿 <s 𝑥 )
110 109 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 )
111 95 99 100 105 110 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 )
112 111 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 ) )
113 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 )
114 leftssno ⊢ ( L ‘ 𝑧 ) ⊆ No
115 114 sseli ⊢ ( 𝑧𝐿 ∈ ( L ‘ 𝑧 ) → 𝑧𝐿 ∈ No )
116 115 adantr ⊢ ( ( 𝑧𝐿 ∈ ( L ‘ 𝑧 ) ∧ ( 𝑦 +s 𝑧 ) ≤s ( 𝑥 +s 𝑧𝐿 ) ) → 𝑧𝐿 ∈ No )
117 116 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 )
118 113 117 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 𝑧𝐿 ) ∈ No )
119 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 )
120 113 119 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 )
121 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 )
122 121 117 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 ) ) ) → ( 𝑥 +s 𝑧𝐿 ) ∈ No )
123 leftval ⊢ ( L ‘ 𝑧 ) = { 𝑧𝐿 ∈ ( O ‘ ( bday ‘ 𝑧 ) ) ∣ 𝑧𝐿 <s 𝑧 }
124 123 reqabi ⊢ ( 𝑧𝐿 ∈ ( L ‘ 𝑧 ) ↔ ( 𝑧𝐿 ∈ ( O ‘ ( bday ‘ 𝑧 ) ) ∧ 𝑧𝐿 <s 𝑧 ) )
125 124 simprbi ⊢ ( 𝑧𝐿 ∈ ( L ‘ 𝑧 ) → 𝑧𝐿 <s 𝑧 )
126 125 adantr ⊢ ( ( 𝑧𝐿 ∈ ( L ‘ 𝑧 ) ∧ ( 𝑦 +s 𝑧 ) ≤s ( 𝑥 +s 𝑧𝐿 ) ) → 𝑧𝐿 <s 𝑧 )
127 126 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 )
128 sltadd2im ⊢ ( ( 𝑧𝐿 ∈ No ∧ 𝑧 ∈ No ∧ 𝑦 ∈ No ) → ( 𝑧𝐿 <s 𝑧 → ( 𝑦 +s 𝑧𝐿 ) <s ( 𝑦 +s 𝑧 ) ) )
129 117 119 113 128 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 ) ) )
130 127 129 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 ) )
131 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 ) )
132 118 120 122 130 131 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 ) )
133 oveq2 ⊢ ( 𝑧𝑂 = 𝑧𝐿 → ( 𝑦 +s 𝑧𝑂 ) = ( 𝑦 +s 𝑧𝐿 ) )
134 oveq2 ⊢ ( 𝑧𝑂 = 𝑧𝐿 → ( 𝑥 +s 𝑧𝑂 ) = ( 𝑥 +s 𝑧𝐿 ) )
135 133 134 breq12d ⊢ ( 𝑧𝑂 = 𝑧𝐿 → ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) ↔ ( 𝑦 +s 𝑧𝐿 ) <s ( 𝑥 +s 𝑧𝐿 ) ) )
136 135 imbi1d ⊢ ( 𝑧𝑂 = 𝑧𝐿 → ( ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦 <s 𝑥 ) ↔ ( ( 𝑦 +s 𝑧𝐿 ) <s ( 𝑥 +s 𝑧𝐿 ) → 𝑦 <s 𝑥 ) ) )
137 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 ) )
138 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 ) )
139 elun1 ⊢ ( 𝑧𝐿 ∈ ( L ‘ 𝑧 ) → 𝑧𝐿 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) )
140 138 139 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 ) ) )
141 136 137 140 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 ) )
142 132 141 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 )
143 142 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 ) )
144 112 143 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 ) )
145 94 144 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 ) )
146 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 ) ) )
147 eqeq1 ⊢ ( 𝑐 = 𝑞 → ( 𝑐 = ( 𝑦𝑅 +s 𝑧 ) ↔ 𝑞 = ( 𝑦𝑅 +s 𝑧 ) ) )
148 147 rexbidv ⊢ ( 𝑐 = 𝑞 → ( ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) 𝑐 = ( 𝑦𝑅 +s 𝑧 ) ↔ ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) 𝑞 = ( 𝑦𝑅 +s 𝑧 ) ) )
149 148 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 ) ) )
150 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 ) ) )
151 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 ) ) )
152 151 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 ) ) )
153 150 152 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 ) ) )
154 ovex ⊢ ( 𝑦𝑅 +s 𝑧 ) ∈ V
155 breq1 ⊢ ( 𝑞 = ( 𝑦𝑅 +s 𝑧 ) → ( 𝑞 ≤s ( 𝑥 +s 𝑧 ) ↔ ( 𝑦𝑅 +s 𝑧 ) ≤s ( 𝑥 +s 𝑧 ) ) )
156 154 155 ceqsexv ⊢ ( ∃ 𝑞 ( 𝑞 = ( 𝑦𝑅 +s 𝑧 ) ∧ 𝑞 ≤s ( 𝑥 +s 𝑧 ) ) ↔ ( 𝑦𝑅 +s 𝑧 ) ≤s ( 𝑥 +s 𝑧 ) )
157 156 rexbii ⊢ ( ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ∃ 𝑞 ( 𝑞 = ( 𝑦𝑅 +s 𝑧 ) ∧ 𝑞 ≤s ( 𝑥 +s 𝑧 ) ) ↔ ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ( 𝑦𝑅 +s 𝑧 ) ≤s ( 𝑥 +s 𝑧 ) )
158 153 157 bitr3i ⊢ ( ∃ 𝑞 ( ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) 𝑞 = ( 𝑦𝑅 +s 𝑧 ) ∧ 𝑞 ≤s ( 𝑥 +s 𝑧 ) ) ↔ ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ( 𝑦𝑅 +s 𝑧 ) ≤s ( 𝑥 +s 𝑧 ) )
159 149 158 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 ) )
160 eqeq1 ⊢ ( 𝑑 = 𝑞 → ( 𝑑 = ( 𝑦 +s 𝑧𝑅 ) ↔ 𝑞 = ( 𝑦 +s 𝑧𝑅 ) ) )
161 160 rexbidv ⊢ ( 𝑑 = 𝑞 → ( ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑑 = ( 𝑦 +s 𝑧𝑅 ) ↔ ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑞 = ( 𝑦 +s 𝑧𝑅 ) ) )
162 161 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 ) ) )
163 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 ) ) )
164 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 ) ) )
165 164 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 ) ) )
166 163 165 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 ) ) )
167 ovex ⊢ ( 𝑦 +s 𝑧𝑅 ) ∈ V
168 breq1 ⊢ ( 𝑞 = ( 𝑦 +s 𝑧𝑅 ) → ( 𝑞 ≤s ( 𝑥 +s 𝑧 ) ↔ ( 𝑦 +s 𝑧𝑅 ) ≤s ( 𝑥 +s 𝑧 ) ) )
169 167 168 ceqsexv ⊢ ( ∃ 𝑞 ( 𝑞 = ( 𝑦 +s 𝑧𝑅 ) ∧ 𝑞 ≤s ( 𝑥 +s 𝑧 ) ) ↔ ( 𝑦 +s 𝑧𝑅 ) ≤s ( 𝑥 +s 𝑧 ) )
170 169 rexbii ⊢ ( ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) ∃ 𝑞 ( 𝑞 = ( 𝑦 +s 𝑧𝑅 ) ∧ 𝑞 ≤s ( 𝑥 +s 𝑧 ) ) ↔ ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) ( 𝑦 +s 𝑧𝑅 ) ≤s ( 𝑥 +s 𝑧 ) )
171 166 170 bitr3i ⊢ ( ∃ 𝑞 ( ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑞 = ( 𝑦 +s 𝑧𝑅 ) ∧ 𝑞 ≤s ( 𝑥 +s 𝑧 ) ) ↔ ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) ( 𝑦 +s 𝑧𝑅 ) ≤s ( 𝑥 +s 𝑧 ) )
172 162 171 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 ) )
173 159 172 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 ) ) )
174 146 173 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 ) ) )
175 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 )
176 rightssno ⊢ ( R ‘ 𝑦 ) ⊆ No
177 176 sseli ⊢ ( 𝑦𝑅 ∈ ( R ‘ 𝑦 ) → 𝑦𝑅 ∈ No )
178 177 adantr ⊢ ( ( 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ∧ ( 𝑦𝑅 +s 𝑧 ) ≤s ( 𝑥 +s 𝑧 ) ) → 𝑦𝑅 ∈ No )
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 ) ) ) → 𝑦𝑅 ∈ No )
180 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 )
181 rightval ⊢ ( R ‘ 𝑦 ) = { 𝑦𝑅 ∈ ( O ‘ ( bday ‘ 𝑦 ) ) ∣ 𝑦 <s 𝑦𝑅 }
182 181 reqabi ⊢ ( 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ↔ ( 𝑦𝑅 ∈ ( O ‘ ( bday ‘ 𝑦 ) ) ∧ 𝑦 <s 𝑦𝑅 ) )
183 182 simprbi ⊢ ( 𝑦𝑅 ∈ ( R ‘ 𝑦 ) → 𝑦 <s 𝑦𝑅 )
184 183 adantr ⊢ ( ( 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ∧ ( 𝑦𝑅 +s 𝑧 ) ≤s ( 𝑥 +s 𝑧 ) ) → 𝑦 <s 𝑦𝑅 )
185 184 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 )
186 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 ) )
187 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 )
188 sleadd1im ⊢ ( ( 𝑦𝑅 ∈ No ∧ 𝑥 ∈ No ∧ 𝑧 ∈ No ) → ( ( 𝑦𝑅 +s 𝑧 ) ≤s ( 𝑥 +s 𝑧 ) → 𝑦𝑅 ≤s 𝑥 ) )
189 179 180 187 188 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 ) )
190 186 189 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 )
191 175 179 180 185 190 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 )
192 191 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 ) )
193 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 )
194 rightssno ⊢ ( R ‘ 𝑧 ) ⊆ No
195 194 sseli ⊢ ( 𝑧𝑅 ∈ ( R ‘ 𝑧 ) → 𝑧𝑅 ∈ No )
196 195 adantr ⊢ ( ( 𝑧𝑅 ∈ ( R ‘ 𝑧 ) ∧ ( 𝑦 +s 𝑧𝑅 ) ≤s ( 𝑥 +s 𝑧 ) ) → 𝑧𝑅 ∈ No )
197 196 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 )
198 193 197 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 )
199 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 )
200 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 )
201 199 200 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 )
202 199 197 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 )
203 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 ) )
204 200 197 199 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 ) )
205 rightval ⊢ ( R ‘ 𝑧 ) = { 𝑧𝑅 ∈ ( O ‘ ( bday ‘ 𝑧 ) ) ∣ 𝑧 <s 𝑧𝑅 }
206 205 reqabi ⊢ ( 𝑧𝑅 ∈ ( R ‘ 𝑧 ) ↔ ( 𝑧𝑅 ∈ ( O ‘ ( bday ‘ 𝑧 ) ) ∧ 𝑧 <s 𝑧𝑅 ) )
207 206 simprbi ⊢ ( 𝑧𝑅 ∈ ( R ‘ 𝑧 ) → 𝑧 <s 𝑧𝑅 )
208 207 adantr ⊢ ( ( 𝑧𝑅 ∈ ( R ‘ 𝑧 ) ∧ ( 𝑦 +s 𝑧𝑅 ) ≤s ( 𝑥 +s 𝑧 ) ) → 𝑧 <s 𝑧𝑅 )
209 208 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 )
210 sltadd2im ⊢ ( ( 𝑧 ∈ No ∧ 𝑧𝑅 ∈ No ∧ 𝑥 ∈ No ) → ( 𝑧 <s 𝑧𝑅 → ( 𝑥 +s 𝑧 ) <s ( 𝑥 +s 𝑧𝑅 ) ) )
211 204 209 210 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 ) )
212 198 201 202 203 211 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 ) )
213 oveq2 ⊢ ( 𝑧𝑂 = 𝑧𝑅 → ( 𝑦 +s 𝑧𝑂 ) = ( 𝑦 +s 𝑧𝑅 ) )
214 oveq2 ⊢ ( 𝑧𝑂 = 𝑧𝑅 → ( 𝑥 +s 𝑧𝑂 ) = ( 𝑥 +s 𝑧𝑅 ) )
215 213 214 breq12d ⊢ ( 𝑧𝑂 = 𝑧𝑅 → ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) ↔ ( 𝑦 +s 𝑧𝑅 ) <s ( 𝑥 +s 𝑧𝑅 ) ) )
216 215 imbi1d ⊢ ( 𝑧𝑂 = 𝑧𝑅 → ( ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦 <s 𝑥 ) ↔ ( ( 𝑦 +s 𝑧𝑅 ) <s ( 𝑥 +s 𝑧𝑅 ) → 𝑦 <s 𝑥 ) ) )
217 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 ) )
218 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 ) )
219 elun2 ⊢ ( 𝑧𝑅 ∈ ( R ‘ 𝑧 ) → 𝑧𝑅 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) )
220 218 219 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 ) ) )
221 216 217 220 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 ) )
222 212 221 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 )
223 222 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 ) )
224 192 223 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 ) )
225 174 224 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 ) )
226 145 225 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 ) )
227 65 226 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 ) )
228 227 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 ) ) )
229 4 8 12 16 20 22 25 29 33 37 228 no3inds ⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( ( 𝐵 +s 𝐶 ) <s ( 𝐴 +s 𝐶 ) → 𝐵 <s 𝐴 ) )
230 addscl ⊢ ( ( 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( 𝐵 +s 𝐶 ) ∈ No )
231 230 3adant1 ⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( 𝐵 +s 𝐶 ) ∈ No )
232 addscl ⊢ ( ( 𝐴 ∈ No ∧ 𝐶 ∈ No ) → ( 𝐴 +s 𝐶 ) ∈ No )
233 232 3adant2 ⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( 𝐴 +s 𝐶 ) ∈ No )
234 sltnle ⊢ ( ( ( 𝐵 +s 𝐶 ) ∈ No ∧ ( 𝐴 +s 𝐶 ) ∈ No ) → ( ( 𝐵 +s 𝐶 ) <s ( 𝐴 +s 𝐶 ) ↔ ¬ ( 𝐴 +s 𝐶 ) ≤s ( 𝐵 +s 𝐶 ) ) )
235 231 233 234 syl2anc ⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( ( 𝐵 +s 𝐶 ) <s ( 𝐴 +s 𝐶 ) ↔ ¬ ( 𝐴 +s 𝐶 ) ≤s ( 𝐵 +s 𝐶 ) ) )
236 sltnle ⊢ ( ( 𝐵 ∈ No ∧ 𝐴 ∈ No ) → ( 𝐵 <s 𝐴 ↔ ¬ 𝐴 ≤s 𝐵 ) )
237 236 ancoms ⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( 𝐵 <s 𝐴 ↔ ¬ 𝐴 ≤s 𝐵 ) )
238 237 3adant3 ⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( 𝐵 <s 𝐴 ↔ ¬ 𝐴 ≤s 𝐵 ) )
239 229 235 238 3imtr3d ⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( ¬ ( 𝐴 +s 𝐶 ) ≤s ( 𝐵 +s 𝐶 ) → ¬ 𝐴 ≤s 𝐵 ) )
240 239 con4d ⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( 𝐴 ≤s 𝐵 → ( 𝐴 +s 𝐶 ) ≤s ( 𝐵 +s 𝐶 ) ) )
241 sleadd1im ⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( ( 𝐴 +s 𝐶 ) ≤s ( 𝐵 +s 𝐶 ) → 𝐴 ≤s 𝐵 ) )
242 240 241 impbid ⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( 𝐴 ≤s 𝐵 ↔ ( 𝐴 +s 𝐶 ) ≤s ( 𝐵 +s 𝐶 ) ) )