Metamath Proof Explorer

Theorem leadds1

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 leadds1 ( ( 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 addcuts ⊢ ( ( 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 sltstr ⊢ ( ( ( { 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 addcuts ⊢ ( ( 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 sltstr ⊢ ( ( ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) 𝑎 = ( 𝑥𝐿 +s 𝑧 ) } ∪ { 𝑏 ∣ ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑏 = ( 𝑥 +s 𝑧𝐿 ) } ) <<s { ( 𝑥 +s 𝑧 ) } ∧ { ( 𝑥 +s 𝑧 ) } <<s ( { 𝑐 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) 𝑐 = ( 𝑥𝑅 +s 𝑧 ) } ∪ { 𝑑 ∣ ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑑 = ( 𝑥 +s 𝑧𝑅 ) } ) ∧ { ( 𝑥 +s 𝑧 ) } ≠ ∅ ) → ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) 𝑎 = ( 𝑥𝐿 +s 𝑧 ) } ∪ { 𝑏 ∣ ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑏 = ( 𝑥 +s 𝑧𝐿 ) } ) <<s ( { 𝑐 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) 𝑐 = ( 𝑥𝑅 +s 𝑧 ) } ∪ { 𝑑 ∣ ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑑 = ( 𝑥 +s 𝑧𝑅 ) } ) )
57 55 56 mp3an3 ⊢ ( ( ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) 𝑎 = ( 𝑥𝐿 +s 𝑧 ) } ∪ { 𝑏 ∣ ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑏 = ( 𝑥 +s 𝑧𝐿 ) } ) <<s { ( 𝑥 +s 𝑧 ) } ∧ { ( 𝑥 +s 𝑧 ) } <<s ( { 𝑐 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) 𝑐 = ( 𝑥𝑅 +s 𝑧 ) } ∪ { 𝑑 ∣ ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑑 = ( 𝑥 +s 𝑧𝑅 ) } ) ) → ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) 𝑎 = ( 𝑥𝐿 +s 𝑧 ) } ∪ { 𝑏 ∣ ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑏 = ( 𝑥 +s 𝑧𝐿 ) } ) <<s ( { 𝑐 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) 𝑐 = ( 𝑥𝑅 +s 𝑧 ) } ∪ { 𝑑 ∣ ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑑 = ( 𝑥 +s 𝑧𝑅 ) } ) )
58 52 53 57 syl2anc ⊢ ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) → ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) 𝑎 = ( 𝑥𝐿 +s 𝑧 ) } ∪ { 𝑏 ∣ ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑏 = ( 𝑥 +s 𝑧𝐿 ) } ) <<s ( { 𝑐 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) 𝑐 = ( 𝑥𝑅 +s 𝑧 ) } ∪ { 𝑑 ∣ ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑑 = ( 𝑥 +s 𝑧𝑅 ) } ) )
59 addsval2 ⊢ ( ( 𝑦 ∈ No ∧ 𝑧 ∈ No ) → ( 𝑦 +s 𝑧 ) = ( ( { 𝑎 ∣ ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) 𝑎 = ( 𝑦𝐿 +s 𝑧 ) } ∪ { 𝑏 ∣ ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑏 = ( 𝑦 +s 𝑧𝐿 ) } ) |s ( { 𝑐 ∣ ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) 𝑐 = ( 𝑦𝑅 +s 𝑧 ) } ∪ { 𝑑 ∣ ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑑 = ( 𝑦 +s 𝑧𝑅 ) } ) ) )
60 59 3adant1 ⊢ ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) → ( 𝑦 +s 𝑧 ) = ( ( { 𝑎 ∣ ∃ 𝑦𝐿 ∈ ( L ‘ 𝑦 ) 𝑎 = ( 𝑦𝐿 +s 𝑧 ) } ∪ { 𝑏 ∣ ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑏 = ( 𝑦 +s 𝑧𝐿 ) } ) |s ( { 𝑐 ∣ ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) 𝑐 = ( 𝑦𝑅 +s 𝑧 ) } ∪ { 𝑑 ∣ ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑑 = ( 𝑦 +s 𝑧𝑅 ) } ) ) )
61 addsval2 ⊢ ( ( 𝑥 ∈ No ∧ 𝑧 ∈ No ) → ( 𝑥 +s 𝑧 ) = ( ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) 𝑎 = ( 𝑥𝐿 +s 𝑧 ) } ∪ { 𝑏 ∣ ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑏 = ( 𝑥 +s 𝑧𝐿 ) } ) |s ( { 𝑐 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) 𝑐 = ( 𝑥𝑅 +s 𝑧 ) } ∪ { 𝑑 ∣ ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑑 = ( 𝑥 +s 𝑧𝑅 ) } ) ) )
62 61 3adant2 ⊢ ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) → ( 𝑥 +s 𝑧 ) = ( ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) 𝑎 = ( 𝑥𝐿 +s 𝑧 ) } ∪ { 𝑏 ∣ ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑏 = ( 𝑥 +s 𝑧𝐿 ) } ) |s ( { 𝑐 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝑥 ) 𝑐 = ( 𝑥𝑅 +s 𝑧 ) } ∪ { 𝑑 ∣ ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑑 = ( 𝑥 +s 𝑧𝑅 ) } ) ) )
63 48 58 60 62 ltsrecd ⊢ ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) → ( ( 𝑦 +s 𝑧 ) <s ( 𝑥 +s 𝑧 ) ↔ ( ∃ 𝑝 ∈ ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) 𝑎 = ( 𝑥𝐿 +s 𝑧 ) } ∪ { 𝑏 ∣ ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑏 = ( 𝑥 +s 𝑧𝐿 ) } ) ( 𝑦 +s 𝑧 ) ≤s 𝑝 ∨ ∃ 𝑞 ∈ ( { 𝑐 ∣ ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) 𝑐 = ( 𝑦𝑅 +s 𝑧 ) } ∪ { 𝑑 ∣ ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑑 = ( 𝑦 +s 𝑧𝑅 ) } ) 𝑞 ≤s ( 𝑥 +s 𝑧 ) ) ) )
64 63 adantr ⊢ ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦 <s 𝑥𝑂 ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( ( 𝑦 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦 <s 𝑥𝑂 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥 +s 𝑧 ) → 𝑦𝑂 <s 𝑥 ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦 <s 𝑥 ) ) ) → ( ( 𝑦 +s 𝑧 ) <s ( 𝑥 +s 𝑧 ) ↔ ( ∃ 𝑝 ∈ ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) 𝑎 = ( 𝑥𝐿 +s 𝑧 ) } ∪ { 𝑏 ∣ ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑏 = ( 𝑥 +s 𝑧𝐿 ) } ) ( 𝑦 +s 𝑧 ) ≤s 𝑝 ∨ ∃ 𝑞 ∈ ( { 𝑐 ∣ ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) 𝑐 = ( 𝑦𝑅 +s 𝑧 ) } ∪ { 𝑑 ∣ ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑑 = ( 𝑦 +s 𝑧𝑅 ) } ) 𝑞 ≤s ( 𝑥 +s 𝑧 ) ) ) )
65 rexun ⊢ ( ∃ 𝑝 ∈ ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) 𝑎 = ( 𝑥𝐿 +s 𝑧 ) } ∪ { 𝑏 ∣ ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑏 = ( 𝑥 +s 𝑧𝐿 ) } ) ( 𝑦 +s 𝑧 ) ≤s 𝑝 ↔ ( ∃ 𝑝 ∈ { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) 𝑎 = ( 𝑥𝐿 +s 𝑧 ) } ( 𝑦 +s 𝑧 ) ≤s 𝑝 ∨ ∃ 𝑝 ∈ { 𝑏 ∣ ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑏 = ( 𝑥 +s 𝑧𝐿 ) } ( 𝑦 +s 𝑧 ) ≤s 𝑝 ) )
66 eqeq1 ⊢ ( 𝑎 = 𝑝 → ( 𝑎 = ( 𝑥𝐿 +s 𝑧 ) ↔ 𝑝 = ( 𝑥𝐿 +s 𝑧 ) ) )
67 66 rexbidv ⊢ ( 𝑎 = 𝑝 → ( ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) 𝑎 = ( 𝑥𝐿 +s 𝑧 ) ↔ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) 𝑝 = ( 𝑥𝐿 +s 𝑧 ) ) )
68 67 rexab ⊢ ( ∃ 𝑝 ∈ { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) 𝑎 = ( 𝑥𝐿 +s 𝑧 ) } ( 𝑦 +s 𝑧 ) ≤s 𝑝 ↔ ∃ 𝑝 ( ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) 𝑝 = ( 𝑥𝐿 +s 𝑧 ) ∧ ( 𝑦 +s 𝑧 ) ≤s 𝑝 ) )
69 rexcom4 ⊢ ( ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑝 ( 𝑝 = ( 𝑥𝐿 +s 𝑧 ) ∧ ( 𝑦 +s 𝑧 ) ≤s 𝑝 ) ↔ ∃ 𝑝 ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ( 𝑝 = ( 𝑥𝐿 +s 𝑧 ) ∧ ( 𝑦 +s 𝑧 ) ≤s 𝑝 ) )
70 r19.41v ⊢ ( ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ( 𝑝 = ( 𝑥𝐿 +s 𝑧 ) ∧ ( 𝑦 +s 𝑧 ) ≤s 𝑝 ) ↔ ( ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) 𝑝 = ( 𝑥𝐿 +s 𝑧 ) ∧ ( 𝑦 +s 𝑧 ) ≤s 𝑝 ) )
71 70 exbii ⊢ ( ∃ 𝑝 ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ( 𝑝 = ( 𝑥𝐿 +s 𝑧 ) ∧ ( 𝑦 +s 𝑧 ) ≤s 𝑝 ) ↔ ∃ 𝑝 ( ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) 𝑝 = ( 𝑥𝐿 +s 𝑧 ) ∧ ( 𝑦 +s 𝑧 ) ≤s 𝑝 ) )
72 69 71 bitri ⊢ ( ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑝 ( 𝑝 = ( 𝑥𝐿 +s 𝑧 ) ∧ ( 𝑦 +s 𝑧 ) ≤s 𝑝 ) ↔ ∃ 𝑝 ( ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) 𝑝 = ( 𝑥𝐿 +s 𝑧 ) ∧ ( 𝑦 +s 𝑧 ) ≤s 𝑝 ) )
73 ovex ⊢ ( 𝑥𝐿 +s 𝑧 ) ∈ V
74 breq2 ⊢ ( 𝑝 = ( 𝑥𝐿 +s 𝑧 ) → ( ( 𝑦 +s 𝑧 ) ≤s 𝑝 ↔ ( 𝑦 +s 𝑧 ) ≤s ( 𝑥𝐿 +s 𝑧 ) ) )
75 73 74 ceqsexv ⊢ ( ∃ 𝑝 ( 𝑝 = ( 𝑥𝐿 +s 𝑧 ) ∧ ( 𝑦 +s 𝑧 ) ≤s 𝑝 ) ↔ ( 𝑦 +s 𝑧 ) ≤s ( 𝑥𝐿 +s 𝑧 ) )
76 75 rexbii ⊢ ( ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∃ 𝑝 ( 𝑝 = ( 𝑥𝐿 +s 𝑧 ) ∧ ( 𝑦 +s 𝑧 ) ≤s 𝑝 ) ↔ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ( 𝑦 +s 𝑧 ) ≤s ( 𝑥𝐿 +s 𝑧 ) )
77 72 76 bitr3i ⊢ ( ∃ 𝑝 ( ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) 𝑝 = ( 𝑥𝐿 +s 𝑧 ) ∧ ( 𝑦 +s 𝑧 ) ≤s 𝑝 ) ↔ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ( 𝑦 +s 𝑧 ) ≤s ( 𝑥𝐿 +s 𝑧 ) )
78 68 77 bitri ⊢ ( ∃ 𝑝 ∈ { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) 𝑎 = ( 𝑥𝐿 +s 𝑧 ) } ( 𝑦 +s 𝑧 ) ≤s 𝑝 ↔ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ( 𝑦 +s 𝑧 ) ≤s ( 𝑥𝐿 +s 𝑧 ) )
79 eqeq1 ⊢ ( 𝑏 = 𝑝 → ( 𝑏 = ( 𝑥 +s 𝑧𝐿 ) ↔ 𝑝 = ( 𝑥 +s 𝑧𝐿 ) ) )
80 79 rexbidv ⊢ ( 𝑏 = 𝑝 → ( ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑏 = ( 𝑥 +s 𝑧𝐿 ) ↔ ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑝 = ( 𝑥 +s 𝑧𝐿 ) ) )
81 80 rexab ⊢ ( ∃ 𝑝 ∈ { 𝑏 ∣ ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑏 = ( 𝑥 +s 𝑧𝐿 ) } ( 𝑦 +s 𝑧 ) ≤s 𝑝 ↔ ∃ 𝑝 ( ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑝 = ( 𝑥 +s 𝑧𝐿 ) ∧ ( 𝑦 +s 𝑧 ) ≤s 𝑝 ) )
82 rexcom4 ⊢ ( ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) ∃ 𝑝 ( 𝑝 = ( 𝑥 +s 𝑧𝐿 ) ∧ ( 𝑦 +s 𝑧 ) ≤s 𝑝 ) ↔ ∃ 𝑝 ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) ( 𝑝 = ( 𝑥 +s 𝑧𝐿 ) ∧ ( 𝑦 +s 𝑧 ) ≤s 𝑝 ) )
83 r19.41v ⊢ ( ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) ( 𝑝 = ( 𝑥 +s 𝑧𝐿 ) ∧ ( 𝑦 +s 𝑧 ) ≤s 𝑝 ) ↔ ( ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑝 = ( 𝑥 +s 𝑧𝐿 ) ∧ ( 𝑦 +s 𝑧 ) ≤s 𝑝 ) )
84 83 exbii ⊢ ( ∃ 𝑝 ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) ( 𝑝 = ( 𝑥 +s 𝑧𝐿 ) ∧ ( 𝑦 +s 𝑧 ) ≤s 𝑝 ) ↔ ∃ 𝑝 ( ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑝 = ( 𝑥 +s 𝑧𝐿 ) ∧ ( 𝑦 +s 𝑧 ) ≤s 𝑝 ) )
85 82 84 bitri ⊢ ( ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) ∃ 𝑝 ( 𝑝 = ( 𝑥 +s 𝑧𝐿 ) ∧ ( 𝑦 +s 𝑧 ) ≤s 𝑝 ) ↔ ∃ 𝑝 ( ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑝 = ( 𝑥 +s 𝑧𝐿 ) ∧ ( 𝑦 +s 𝑧 ) ≤s 𝑝 ) )
86 ovex ⊢ ( 𝑥 +s 𝑧𝐿 ) ∈ V
87 breq2 ⊢ ( 𝑝 = ( 𝑥 +s 𝑧𝐿 ) → ( ( 𝑦 +s 𝑧 ) ≤s 𝑝 ↔ ( 𝑦 +s 𝑧 ) ≤s ( 𝑥 +s 𝑧𝐿 ) ) )
88 86 87 ceqsexv ⊢ ( ∃ 𝑝 ( 𝑝 = ( 𝑥 +s 𝑧𝐿 ) ∧ ( 𝑦 +s 𝑧 ) ≤s 𝑝 ) ↔ ( 𝑦 +s 𝑧 ) ≤s ( 𝑥 +s 𝑧𝐿 ) )
89 88 rexbii ⊢ ( ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) ∃ 𝑝 ( 𝑝 = ( 𝑥 +s 𝑧𝐿 ) ∧ ( 𝑦 +s 𝑧 ) ≤s 𝑝 ) ↔ ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) ( 𝑦 +s 𝑧 ) ≤s ( 𝑥 +s 𝑧𝐿 ) )
90 85 89 bitr3i ⊢ ( ∃ 𝑝 ( ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑝 = ( 𝑥 +s 𝑧𝐿 ) ∧ ( 𝑦 +s 𝑧 ) ≤s 𝑝 ) ↔ ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) ( 𝑦 +s 𝑧 ) ≤s ( 𝑥 +s 𝑧𝐿 ) )
91 81 90 bitri ⊢ ( ∃ 𝑝 ∈ { 𝑏 ∣ ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑏 = ( 𝑥 +s 𝑧𝐿 ) } ( 𝑦 +s 𝑧 ) ≤s 𝑝 ↔ ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) ( 𝑦 +s 𝑧 ) ≤s ( 𝑥 +s 𝑧𝐿 ) )
92 78 91 orbi12i ⊢ ( ( ∃ 𝑝 ∈ { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) 𝑎 = ( 𝑥𝐿 +s 𝑧 ) } ( 𝑦 +s 𝑧 ) ≤s 𝑝 ∨ ∃ 𝑝 ∈ { 𝑏 ∣ ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑏 = ( 𝑥 +s 𝑧𝐿 ) } ( 𝑦 +s 𝑧 ) ≤s 𝑝 ) ↔ ( ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ( 𝑦 +s 𝑧 ) ≤s ( 𝑥𝐿 +s 𝑧 ) ∨ ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) ( 𝑦 +s 𝑧 ) ≤s ( 𝑥 +s 𝑧𝐿 ) ) )
93 65 92 bitri ⊢ ( ∃ 𝑝 ∈ ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) 𝑎 = ( 𝑥𝐿 +s 𝑧 ) } ∪ { 𝑏 ∣ ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) 𝑏 = ( 𝑥 +s 𝑧𝐿 ) } ) ( 𝑦 +s 𝑧 ) ≤s 𝑝 ↔ ( ∃ 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ( 𝑦 +s 𝑧 ) ≤s ( 𝑥𝐿 +s 𝑧 ) ∨ ∃ 𝑧𝐿 ∈ ( L ‘ 𝑧 ) ( 𝑦 +s 𝑧 ) ≤s ( 𝑥 +s 𝑧𝐿 ) ) )
94 simpll2 ⊢ ( ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦 <s 𝑥𝑂 ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( ( 𝑦 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦 <s 𝑥𝑂 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥 +s 𝑧 ) → 𝑦𝑂 <s 𝑥 ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦 <s 𝑥 ) ) ) ∧ ( 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∧ ( 𝑦 +s 𝑧 ) ≤s ( 𝑥𝐿 +s 𝑧 ) ) ) → 𝑦 ∈ No )
95 leftno ⊢ ( 𝑥𝐿 ∈ ( L ‘ 𝑥 ) → 𝑥𝐿 ∈ No )
96 95 adantr ⊢ ( ( 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∧ ( 𝑦 +s 𝑧 ) ≤s ( 𝑥𝐿 +s 𝑧 ) ) → 𝑥𝐿 ∈ No )
97 96 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 )
98 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 )
99 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 ) )
100 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 )
101 leadds1im ⊢ ( ( 𝑦 ∈ No ∧ 𝑥𝐿 ∈ No ∧ 𝑧 ∈ No ) → ( ( 𝑦 +s 𝑧 ) ≤s ( 𝑥𝐿 +s 𝑧 ) → 𝑦 ≤s 𝑥𝐿 ) )
102 94 97 100 101 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 ) )
103 99 102 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 )
104 leftlt ⊢ ( 𝑥𝐿 ∈ ( L ‘ 𝑥 ) → 𝑥𝐿 <s 𝑥 )
105 104 adantr ⊢ ( ( 𝑥𝐿 ∈ ( L ‘ 𝑥 ) ∧ ( 𝑦 +s 𝑧 ) ≤s ( 𝑥𝐿 +s 𝑧 ) ) → 𝑥𝐿 <s 𝑥 )
106 105 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 )
107 94 97 98 103 106 leltstrd ⊢ ( ( ( ( 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 )
108 107 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 ) )
109 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 )
110 leftno ⊢ ( 𝑧𝐿 ∈ ( L ‘ 𝑧 ) → 𝑧𝐿 ∈ No )
111 110 adantr ⊢ ( ( 𝑧𝐿 ∈ ( L ‘ 𝑧 ) ∧ ( 𝑦 +s 𝑧 ) ≤s ( 𝑥 +s 𝑧𝐿 ) ) → 𝑧𝐿 ∈ No )
112 111 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 )
113 109 112 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 )
114 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 )
115 109 114 addscld ⊢ ( ( ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) ∧ ( ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦𝑂 <s 𝑥𝑂 ) ∧ ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥𝑂 +s 𝑧𝑂 ) → 𝑦 <s 𝑥𝑂 ) ) ∧ ( ∀ 𝑥𝑂 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) ( ( 𝑦 +s 𝑧 ) <s ( 𝑥𝑂 +s 𝑧 ) → 𝑦 <s 𝑥𝑂 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦𝑂 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦𝑂 <s 𝑥 ) ∧ ∀ 𝑦𝑂 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) ( ( 𝑦𝑂 +s 𝑧 ) <s ( 𝑥 +s 𝑧 ) → 𝑦𝑂 <s 𝑥 ) ) ∧ ∀ 𝑧𝑂 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦 <s 𝑥 ) ) ) ∧ ( 𝑧𝐿 ∈ ( L ‘ 𝑧 ) ∧ ( 𝑦 +s 𝑧 ) ≤s ( 𝑥 +s 𝑧𝐿 ) ) ) → ( 𝑦 +s 𝑧 ) ∈ No )
116 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 )
117 116 112 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 )
118 leftlt ⊢ ( 𝑧𝐿 ∈ ( L ‘ 𝑧 ) → 𝑧𝐿 <s 𝑧 )
119 118 adantr ⊢ ( ( 𝑧𝐿 ∈ ( L ‘ 𝑧 ) ∧ ( 𝑦 +s 𝑧 ) ≤s ( 𝑥 +s 𝑧𝐿 ) ) → 𝑧𝐿 <s 𝑧 )
120 119 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 )
121 ltadds2im ⊢ ( ( 𝑧𝐿 ∈ No ∧ 𝑧 ∈ No ∧ 𝑦 ∈ No ) → ( 𝑧𝐿 <s 𝑧 → ( 𝑦 +s 𝑧𝐿 ) <s ( 𝑦 +s 𝑧 ) ) )
122 112 114 109 121 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 ) ) )
123 120 122 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 ) )
124 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 ) )
125 113 115 117 123 124 ltlestrd ⊢ ( ( ( ( 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 ) )
126 oveq2 ⊢ ( 𝑧𝑂 = 𝑧𝐿 → ( 𝑦 +s 𝑧𝑂 ) = ( 𝑦 +s 𝑧𝐿 ) )
127 oveq2 ⊢ ( 𝑧𝑂 = 𝑧𝐿 → ( 𝑥 +s 𝑧𝑂 ) = ( 𝑥 +s 𝑧𝐿 ) )
128 126 127 breq12d ⊢ ( 𝑧𝑂 = 𝑧𝐿 → ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) ↔ ( 𝑦 +s 𝑧𝐿 ) <s ( 𝑥 +s 𝑧𝐿 ) ) )
129 128 imbi1d ⊢ ( 𝑧𝑂 = 𝑧𝐿 → ( ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦 <s 𝑥 ) ↔ ( ( 𝑦 +s 𝑧𝐿 ) <s ( 𝑥 +s 𝑧𝐿 ) → 𝑦 <s 𝑥 ) ) )
130 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 ) )
131 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 ) )
132 elun1 ⊢ ( 𝑧𝐿 ∈ ( L ‘ 𝑧 ) → 𝑧𝐿 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) )
133 131 132 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 ) ) )
134 129 130 133 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 ) )
135 125 134 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 )
136 135 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 ) )
137 108 136 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 ) )
138 93 137 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 ) )
139 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 ) ) )
140 eqeq1 ⊢ ( 𝑐 = 𝑞 → ( 𝑐 = ( 𝑦𝑅 +s 𝑧 ) ↔ 𝑞 = ( 𝑦𝑅 +s 𝑧 ) ) )
141 140 rexbidv ⊢ ( 𝑐 = 𝑞 → ( ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) 𝑐 = ( 𝑦𝑅 +s 𝑧 ) ↔ ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) 𝑞 = ( 𝑦𝑅 +s 𝑧 ) ) )
142 141 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 ) ) )
143 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 ) ) )
144 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 ) ) )
145 144 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 ) ) )
146 143 145 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 ) ) )
147 ovex ⊢ ( 𝑦𝑅 +s 𝑧 ) ∈ V
148 breq1 ⊢ ( 𝑞 = ( 𝑦𝑅 +s 𝑧 ) → ( 𝑞 ≤s ( 𝑥 +s 𝑧 ) ↔ ( 𝑦𝑅 +s 𝑧 ) ≤s ( 𝑥 +s 𝑧 ) ) )
149 147 148 ceqsexv ⊢ ( ∃ 𝑞 ( 𝑞 = ( 𝑦𝑅 +s 𝑧 ) ∧ 𝑞 ≤s ( 𝑥 +s 𝑧 ) ) ↔ ( 𝑦𝑅 +s 𝑧 ) ≤s ( 𝑥 +s 𝑧 ) )
150 149 rexbii ⊢ ( ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ∃ 𝑞 ( 𝑞 = ( 𝑦𝑅 +s 𝑧 ) ∧ 𝑞 ≤s ( 𝑥 +s 𝑧 ) ) ↔ ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ( 𝑦𝑅 +s 𝑧 ) ≤s ( 𝑥 +s 𝑧 ) )
151 146 150 bitr3i ⊢ ( ∃ 𝑞 ( ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) 𝑞 = ( 𝑦𝑅 +s 𝑧 ) ∧ 𝑞 ≤s ( 𝑥 +s 𝑧 ) ) ↔ ∃ 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ( 𝑦𝑅 +s 𝑧 ) ≤s ( 𝑥 +s 𝑧 ) )
152 142 151 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 ) )
153 eqeq1 ⊢ ( 𝑑 = 𝑞 → ( 𝑑 = ( 𝑦 +s 𝑧𝑅 ) ↔ 𝑞 = ( 𝑦 +s 𝑧𝑅 ) ) )
154 153 rexbidv ⊢ ( 𝑑 = 𝑞 → ( ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑑 = ( 𝑦 +s 𝑧𝑅 ) ↔ ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑞 = ( 𝑦 +s 𝑧𝑅 ) ) )
155 154 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 ) ) )
156 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 ) ) )
157 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 ) ) )
158 157 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 ) ) )
159 156 158 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 ) ) )
160 ovex ⊢ ( 𝑦 +s 𝑧𝑅 ) ∈ V
161 breq1 ⊢ ( 𝑞 = ( 𝑦 +s 𝑧𝑅 ) → ( 𝑞 ≤s ( 𝑥 +s 𝑧 ) ↔ ( 𝑦 +s 𝑧𝑅 ) ≤s ( 𝑥 +s 𝑧 ) ) )
162 160 161 ceqsexv ⊢ ( ∃ 𝑞 ( 𝑞 = ( 𝑦 +s 𝑧𝑅 ) ∧ 𝑞 ≤s ( 𝑥 +s 𝑧 ) ) ↔ ( 𝑦 +s 𝑧𝑅 ) ≤s ( 𝑥 +s 𝑧 ) )
163 162 rexbii ⊢ ( ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) ∃ 𝑞 ( 𝑞 = ( 𝑦 +s 𝑧𝑅 ) ∧ 𝑞 ≤s ( 𝑥 +s 𝑧 ) ) ↔ ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) ( 𝑦 +s 𝑧𝑅 ) ≤s ( 𝑥 +s 𝑧 ) )
164 159 163 bitr3i ⊢ ( ∃ 𝑞 ( ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) 𝑞 = ( 𝑦 +s 𝑧𝑅 ) ∧ 𝑞 ≤s ( 𝑥 +s 𝑧 ) ) ↔ ∃ 𝑧𝑅 ∈ ( R ‘ 𝑧 ) ( 𝑦 +s 𝑧𝑅 ) ≤s ( 𝑥 +s 𝑧 ) )
165 155 164 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 ) )
166 152 165 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 ) ) )
167 139 166 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 ) ) )
168 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 )
169 rightno ⊢ ( 𝑦𝑅 ∈ ( R ‘ 𝑦 ) → 𝑦𝑅 ∈ No )
170 169 adantr ⊢ ( ( 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ∧ ( 𝑦𝑅 +s 𝑧 ) ≤s ( 𝑥 +s 𝑧 ) ) → 𝑦𝑅 ∈ No )
171 170 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 )
172 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 )
173 rightgt ⊢ ( 𝑦𝑅 ∈ ( R ‘ 𝑦 ) → 𝑦 <s 𝑦𝑅 )
174 173 adantr ⊢ ( ( 𝑦𝑅 ∈ ( R ‘ 𝑦 ) ∧ ( 𝑦𝑅 +s 𝑧 ) ≤s ( 𝑥 +s 𝑧 ) ) → 𝑦 <s 𝑦𝑅 )
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 ) ) ) → 𝑦 <s 𝑦𝑅 )
176 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 ) )
177 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 )
178 leadds1im ⊢ ( ( 𝑦𝑅 ∈ No ∧ 𝑥 ∈ No ∧ 𝑧 ∈ No ) → ( ( 𝑦𝑅 +s 𝑧 ) ≤s ( 𝑥 +s 𝑧 ) → 𝑦𝑅 ≤s 𝑥 ) )
179 171 172 177 178 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 ) )
180 176 179 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 )
181 168 171 172 175 180 ltlestrd ⊢ ( ( ( ( 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 )
182 181 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 ) )
183 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 )
184 rightno ⊢ ( 𝑧𝑅 ∈ ( R ‘ 𝑧 ) → 𝑧𝑅 ∈ No )
185 184 adantr ⊢ ( ( 𝑧𝑅 ∈ ( R ‘ 𝑧 ) ∧ ( 𝑦 +s 𝑧𝑅 ) ≤s ( 𝑥 +s 𝑧 ) ) → 𝑧𝑅 ∈ No )
186 185 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 )
187 183 186 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 )
188 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 )
189 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 )
190 188 189 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 )
191 188 186 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 )
192 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 ) )
193 189 186 188 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 ) )
194 rightgt ⊢ ( 𝑧𝑅 ∈ ( R ‘ 𝑧 ) → 𝑧 <s 𝑧𝑅 )
195 194 adantr ⊢ ( ( 𝑧𝑅 ∈ ( R ‘ 𝑧 ) ∧ ( 𝑦 +s 𝑧𝑅 ) ≤s ( 𝑥 +s 𝑧 ) ) → 𝑧 <s 𝑧𝑅 )
196 195 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 )
197 ltadds2im ⊢ ( ( 𝑧 ∈ No ∧ 𝑧𝑅 ∈ No ∧ 𝑥 ∈ No ) → ( 𝑧 <s 𝑧𝑅 → ( 𝑥 +s 𝑧 ) <s ( 𝑥 +s 𝑧𝑅 ) ) )
198 193 196 197 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 ) )
199 187 190 191 192 198 leltstrd ⊢ ( ( ( ( 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 ) )
200 oveq2 ⊢ ( 𝑧𝑂 = 𝑧𝑅 → ( 𝑦 +s 𝑧𝑂 ) = ( 𝑦 +s 𝑧𝑅 ) )
201 oveq2 ⊢ ( 𝑧𝑂 = 𝑧𝑅 → ( 𝑥 +s 𝑧𝑂 ) = ( 𝑥 +s 𝑧𝑅 ) )
202 200 201 breq12d ⊢ ( 𝑧𝑂 = 𝑧𝑅 → ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) ↔ ( 𝑦 +s 𝑧𝑅 ) <s ( 𝑥 +s 𝑧𝑅 ) ) )
203 202 imbi1d ⊢ ( 𝑧𝑂 = 𝑧𝑅 → ( ( ( 𝑦 +s 𝑧𝑂 ) <s ( 𝑥 +s 𝑧𝑂 ) → 𝑦 <s 𝑥 ) ↔ ( ( 𝑦 +s 𝑧𝑅 ) <s ( 𝑥 +s 𝑧𝑅 ) → 𝑦 <s 𝑥 ) ) )
204 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 ) )
205 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 ) )
206 elun2 ⊢ ( 𝑧𝑅 ∈ ( R ‘ 𝑧 ) → 𝑧𝑅 ∈ ( ( L ‘ 𝑧 ) ∪ ( R ‘ 𝑧 ) ) )
207 205 206 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 ) ) )
208 203 204 207 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 ) )
209 199 208 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 )
210 209 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 ) )
211 182 210 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 ) )
212 167 211 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 ) )
213 138 212 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 ) )
214 64 213 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 ) )
215 214 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 ) ) )
216 4 8 12 16 20 22 25 29 33 37 215 no3inds ⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( ( 𝐵 +s 𝐶 ) <s ( 𝐴 +s 𝐶 ) → 𝐵 <s 𝐴 ) )
217 addscl ⊢ ( ( 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( 𝐵 +s 𝐶 ) ∈ No )
218 217 3adant1 ⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( 𝐵 +s 𝐶 ) ∈ No )
219 addscl ⊢ ( ( 𝐴 ∈ No ∧ 𝐶 ∈ No ) → ( 𝐴 +s 𝐶 ) ∈ No )
220 219 3adant2 ⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( 𝐴 +s 𝐶 ) ∈ No )
221 ltnles ⊢ ( ( ( 𝐵 +s 𝐶 ) ∈ No ∧ ( 𝐴 +s 𝐶 ) ∈ No ) → ( ( 𝐵 +s 𝐶 ) <s ( 𝐴 +s 𝐶 ) ↔ ¬ ( 𝐴 +s 𝐶 ) ≤s ( 𝐵 +s 𝐶 ) ) )
222 218 220 221 syl2anc ⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( ( 𝐵 +s 𝐶 ) <s ( 𝐴 +s 𝐶 ) ↔ ¬ ( 𝐴 +s 𝐶 ) ≤s ( 𝐵 +s 𝐶 ) ) )
223 ltnles ⊢ ( ( 𝐵 ∈ No ∧ 𝐴 ∈ No ) → ( 𝐵 <s 𝐴 ↔ ¬ 𝐴 ≤s 𝐵 ) )
224 223 ancoms ⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( 𝐵 <s 𝐴 ↔ ¬ 𝐴 ≤s 𝐵 ) )
225 224 3adant3 ⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( 𝐵 <s 𝐴 ↔ ¬ 𝐴 ≤s 𝐵 ) )
226 216 222 225 3imtr3d ⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( ¬ ( 𝐴 +s 𝐶 ) ≤s ( 𝐵 +s 𝐶 ) → ¬ 𝐴 ≤s 𝐵 ) )
227 226 con4d ⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( 𝐴 ≤s 𝐵 → ( 𝐴 +s 𝐶 ) ≤s ( 𝐵 +s 𝐶 ) ) )
228 leadds1im ⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( ( 𝐴 +s 𝐶 ) ≤s ( 𝐵 +s 𝐶 ) → 𝐴 ≤s 𝐵 ) )
229 227 228 impbid ⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( 𝐴 ≤s 𝐵 ↔ ( 𝐴 +s 𝐶 ) ≤s ( 𝐵 +s 𝐶 ) ) )