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Theorem leadds1 28059
Description: Addition to both sides of surreal less-than or equal. Theorem 5 of [Conway] p. 18. (Contributed by Scott Fenton, 21-Jan-2025.)
Assertion
Ref Expression
leadds1 ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → (𝐴 ≤s 𝐵 ↔ (𝐴 +s 𝐶) ≤s (𝐵 +s 𝐶)))
Proof of Theorem leadds1
Dummy variables 𝑥 𝑦 𝑧 𝑎 𝑏 𝑐 𝑑 𝑝 𝑞 𝑥𝐿 𝑦𝐿 𝑧𝐿 𝑥𝑅 𝑦𝑅 𝑧𝑅 𝑥𝑂 𝑦𝑂 𝑧𝑂 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq1 7399 . . . . . . 7 (𝑥 = 𝑥𝑂 → (𝑥 +s 𝑧) = (𝑥𝑂 +s 𝑧))
21breq2d 5111 . . . . . 6 (𝑥 = 𝑥𝑂 → ((𝑦 +s 𝑧) <s (𝑥 +s 𝑧) ↔ (𝑦 +s 𝑧) <s (𝑥𝑂 +s 𝑧)))
3 breq2 5103 . . . . . 6 (𝑥 = 𝑥𝑂 → (𝑦 <s 𝑥 ↔ 𝑦 <s 𝑥𝑂))
42, 3imbi12d 346 . . . . 5 (𝑥 = 𝑥𝑂 → (((𝑦 +s 𝑧) <s (𝑥 +s 𝑧) → 𝑦 <s 𝑥) ↔ ((𝑦 +s 𝑧) <s (𝑥𝑂 +s 𝑧) → 𝑦 <s 𝑥𝑂)))
5 oveq1 7399 . . . . . . 7 (𝑦 = 𝑦𝑂 → (𝑦 +s 𝑧) = (𝑦𝑂 +s 𝑧))
65breq1d 5109 . . . . . 6 (𝑦 = 𝑦𝑂 → ((𝑦 +s 𝑧) <s (𝑥𝑂 +s 𝑧) ↔ (𝑦𝑂 +s 𝑧) <s (𝑥𝑂 +s 𝑧)))
7 breq1 5102 . . . . . 6 (𝑦 = 𝑦𝑂 → (𝑦 <s 𝑥𝑂 ↔ 𝑦𝑂 <s 𝑥𝑂))
86, 7imbi12d 346 . . . . 5 (𝑦 = 𝑦𝑂 → (((𝑦 +s 𝑧) <s (𝑥𝑂 +s 𝑧) → 𝑦 <s 𝑥𝑂) ↔ ((𝑦𝑂 +s 𝑧) <s (𝑥𝑂 +s 𝑧) → 𝑦𝑂 <s 𝑥𝑂)))
9 oveq2 7400 . . . . . . 7 (𝑧 = 𝑧𝑂 → (𝑦𝑂 +s 𝑧) = (𝑦𝑂 +s 𝑧𝑂))
10 oveq2 7400 . . . . . . 7 (𝑧 = 𝑧𝑂 → (𝑥𝑂 +s 𝑧) = (𝑥𝑂 +s 𝑧𝑂))
119, 10breq12d 5112 . . . . . 6 (𝑧 = 𝑧𝑂 → ((𝑦𝑂 +s 𝑧) <s (𝑥𝑂 +s 𝑧) ↔ (𝑦𝑂 +s 𝑧𝑂) <s (𝑥𝑂 +s 𝑧𝑂)))
1211imbi1d 343 . . . . 5 (𝑧 = 𝑧𝑂 → (((𝑦𝑂 +s 𝑧) <s (𝑥𝑂 +s 𝑧) → 𝑦𝑂 <s 𝑥𝑂) ↔ ((𝑦𝑂 +s 𝑧𝑂) <s (𝑥𝑂 +s 𝑧𝑂) → 𝑦𝑂 <s 𝑥𝑂)))
13 oveq1 7399 . . . . . . 7 (𝑥 = 𝑥𝑂 → (𝑥 +s 𝑧𝑂) = (𝑥𝑂 +s 𝑧𝑂))
1413breq2d 5111 . . . . . 6 (𝑥 = 𝑥𝑂 → ((𝑦𝑂 +s 𝑧𝑂) <s (𝑥 +s 𝑧𝑂) ↔ (𝑦𝑂 +s 𝑧𝑂) <s (𝑥𝑂 +s 𝑧𝑂)))
15 breq2 5103 . . . . . 6 (𝑥 = 𝑥𝑂 → (𝑦𝑂 <s 𝑥 ↔ 𝑦𝑂 <s 𝑥𝑂))
1614, 15imbi12d 346 . . . . 5 (𝑥 = 𝑥𝑂 → (((𝑦𝑂 +s 𝑧𝑂) <s (𝑥 +s 𝑧𝑂) → 𝑦𝑂 <s 𝑥) ↔ ((𝑦𝑂 +s 𝑧𝑂) <s (𝑥𝑂 +s 𝑧𝑂) → 𝑦𝑂 <s 𝑥𝑂)))
17 oveq1 7399 . . . . . . 7 (𝑦 = 𝑦𝑂 → (𝑦 +s 𝑧𝑂) = (𝑦𝑂 +s 𝑧𝑂))
1817breq1d 5109 . . . . . 6 (𝑦 = 𝑦𝑂 → ((𝑦 +s 𝑧𝑂) <s (𝑥 +s 𝑧𝑂) ↔ (𝑦𝑂 +s 𝑧𝑂) <s (𝑥 +s 𝑧𝑂)))
19 breq1 5102 . . . . . 6 (𝑦 = 𝑦𝑂 → (𝑦 <s 𝑥 ↔ 𝑦𝑂 <s 𝑥))
2018, 19imbi12d 346 . . . . 5 (𝑦 = 𝑦𝑂 → (((𝑦 +s 𝑧𝑂) <s (𝑥 +s 𝑧𝑂) → 𝑦 <s 𝑥) ↔ ((𝑦𝑂 +s 𝑧𝑂) <s (𝑥 +s 𝑧𝑂) → 𝑦𝑂 <s 𝑥)))
2117breq1d 5109 . . . . . 6 (𝑦 = 𝑦𝑂 → ((𝑦 +s 𝑧𝑂) <s (𝑥𝑂 +s 𝑧𝑂) ↔ (𝑦𝑂 +s 𝑧𝑂) <s (𝑥𝑂 +s 𝑧𝑂)))
2221, 7imbi12d 346 . . . . 5 (𝑦 = 𝑦𝑂 → (((𝑦 +s 𝑧𝑂) <s (𝑥𝑂 +s 𝑧𝑂) → 𝑦 <s 𝑥𝑂) ↔ ((𝑦𝑂 +s 𝑧𝑂) <s (𝑥𝑂 +s 𝑧𝑂) → 𝑦𝑂 <s 𝑥𝑂)))
23 oveq2 7400 . . . . . . 7 (𝑧 = 𝑧𝑂 → (𝑥 +s 𝑧) = (𝑥 +s 𝑧𝑂))
249, 23breq12d 5112 . . . . . 6 (𝑧 = 𝑧𝑂 → ((𝑦𝑂 +s 𝑧) <s (𝑥 +s 𝑧) ↔ (𝑦𝑂 +s 𝑧𝑂) <s (𝑥 +s 𝑧𝑂)))
2524imbi1d 343 . . . . 5 (𝑧 = 𝑧𝑂 → (((𝑦𝑂 +s 𝑧) <s (𝑥 +s 𝑧) → 𝑦𝑂 <s 𝑥) ↔ ((𝑦𝑂 +s 𝑧𝑂) <s (𝑥 +s 𝑧𝑂) → 𝑦𝑂 <s 𝑥)))
26 oveq1 7399 . . . . . . 7 (𝑥 = 𝐴 → (𝑥 +s 𝑧) = (𝐴 +s 𝑧))
2726breq2d 5111 . . . . . 6 (𝑥 = 𝐴 → ((𝑦 +s 𝑧) <s (𝑥 +s 𝑧) ↔ (𝑦 +s 𝑧) <s (𝐴 +s 𝑧)))
28 breq2 5103 . . . . . 6 (𝑥 = 𝐴 → (𝑦 <s 𝑥 ↔ 𝑦 <s 𝐴))
2927, 28imbi12d 346 . . . . 5 (𝑥 = 𝐴 → (((𝑦 +s 𝑧) <s (𝑥 +s 𝑧) → 𝑦 <s 𝑥) ↔ ((𝑦 +s 𝑧) <s (𝐴 +s 𝑧) → 𝑦 <s 𝐴)))
30 oveq1 7399 . . . . . . 7 (𝑦 = 𝐵 → (𝑦 +s 𝑧) = (𝐵 +s 𝑧))
3130breq1d 5109 . . . . . 6 (𝑦 = 𝐵 → ((𝑦 +s 𝑧) <s (𝐴 +s 𝑧) ↔ (𝐵 +s 𝑧) <s (𝐴 +s 𝑧)))
32 breq1 5102 . . . . . 6 (𝑦 = 𝐵 → (𝑦 <s 𝐴 ↔ 𝐵 <s 𝐴))
3331, 32imbi12d 346 . . . . 5 (𝑦 = 𝐵 → (((𝑦 +s 𝑧) <s (𝐴 +s 𝑧) → 𝑦 <s 𝐴) ↔ ((𝐵 +s 𝑧) <s (𝐴 +s 𝑧) → 𝐵 <s 𝐴)))
34 oveq2 7400 . . . . . . 7 (𝑧 = 𝐶 → (𝐵 +s 𝑧) = (𝐵 +s 𝐶))
35 oveq2 7400 . . . . . . 7 (𝑧 = 𝐶 → (𝐴 +s 𝑧) = (𝐴 +s 𝐶))
3634, 35breq12d 5112 . . . . . 6 (𝑧 = 𝐶 → ((𝐵 +s 𝑧) <s (𝐴 +s 𝑧) ↔ (𝐵 +s 𝐶) <s (𝐴 +s 𝐶)))
3736imbi1d 343 . . . . 5 (𝑧 = 𝐶 → (((𝐵 +s 𝑧) <s (𝐴 +s 𝑧) → 𝐵 <s 𝐴) ↔ ((𝐵 +s 𝐶) <s (𝐴 +s 𝐶) → 𝐵 <s 𝐴)))
38 simp2 1149 . . . . . . . . . . . 12 ((𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) → 𝑦 ∈ No )
39 simp3 1150 . . . . . . . . . . . 12 ((𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) → 𝑧 ∈ No )
4038, 39addcuts 28048 . . . . . . . . . . 11 ((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 1149 . . . . . . . . . . 11 (((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)})
4240, 41syl 17 . . . . . . . . . 10 ((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)})
4340simp3d 1156 . . . . . . . . . 10 ((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 7425 . . . . . . . . . . . 12 (𝑦 +s 𝑧) ∈ V
4544snnz 4734 . . . . . . . . . . 11 {(𝑦 +s 𝑧)} ≠ ∅
46 sltstr 27857 . . . . . . . . . . 11 ((({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)}))
4745, 46mp3an3 1470 . . . . . . . . . 10 ((({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)}))
4842, 43, 47syl2anc 593 . . . . . . . . 9 ((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 1148 . . . . . . . . . . . 12 ((𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) → 𝑥 ∈ No )
5049, 39addcuts 28048 . . . . . . . . . . 11 ((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 1149 . . . . . . . . . . 11 (((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)})
5250, 51syl 17 . . . . . . . . . 10 ((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)})
5350simp3d 1156 . . . . . . . . . 10 ((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 7425 . . . . . . . . . . . 12 (𝑥 +s 𝑧) ∈ V
5554snnz 4734 . . . . . . . . . . 11 {(𝑥 +s 𝑧)} ≠ ∅
56 sltstr 27857 . . . . . . . . . . 11 ((({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)}))
5755, 56mp3an3 1470 . . . . . . . . . 10 ((({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)}))
5852, 53, 57syl2anc 593 . . . . . . . . 9 ((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 28033 . . . . . . . . . 10 ((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)})))
60593adant1 1142 . . . . . . . . 9 ((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 28033 . . . . . . . . . 10 ((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)})))
62613adant2 1143 . . . . . . . . 9 ((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)})))
6348, 58, 60, 62ltsrecd 27872 . . . . . . . 8 ((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))))
6463adantr 484 . . . . . . 7 (((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 4148 . . . . . . . . . 10 (∃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 2765 . . . . . . . . . . . . . 14 (𝑎 = 𝑝 → (𝑎 = (𝑥𝐿 +s 𝑧) ↔ 𝑝 = (𝑥𝐿 +s 𝑧)))
6766rexbidv 3185 . . . . . . . . . . . . 13 (𝑎 = 𝑝 → (∃𝑥𝐿 ∈ ( L ‘𝑥)𝑎 = (𝑥𝐿 +s 𝑧) ↔ ∃𝑥𝐿 ∈ ( L ‘𝑥)𝑝 = (𝑥𝐿 +s 𝑧)))
6867rexab 3657 . . . . . . . . . . . 12 (∃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 3288 . . . . . . . . . . . . . 14 (∃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 3191 . . . . . . . . . . . . . . 15 (∃𝑥𝐿 ∈ ( L ‘𝑥)(𝑝 = (𝑥𝐿 +s 𝑧) ∧ (𝑦 +s 𝑧) ≤s 𝑝) ↔ (∃𝑥𝐿 ∈ ( L ‘𝑥)𝑝 = (𝑥𝐿 +s 𝑧) ∧ (𝑦 +s 𝑧) ≤s 𝑝))
7170exbii 1867 . . . . . . . . . . . . . 14 (∃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))
7269, 71bitri 277 . . . . . . . . . . . . 13 (∃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 7425 . . . . . . . . . . . . . . 15 (𝑥𝐿 +s 𝑧) ∈ V
74 breq2 5103 . . . . . . . . . . . . . . 15 (𝑝 = (𝑥𝐿 +s 𝑧) → ((𝑦 +s 𝑧) ≤s 𝑝 ↔ (𝑦 +s 𝑧) ≤s (𝑥𝐿 +s 𝑧)))
7573, 74ceqsexv 3501 . . . . . . . . . . . . . 14 (∃𝑝(𝑝 = (𝑥𝐿 +s 𝑧) ∧ (𝑦 +s 𝑧) ≤s 𝑝) ↔ (𝑦 +s 𝑧) ≤s (𝑥𝐿 +s 𝑧))
7675rexbii 3108 . . . . . . . . . . . . 13 (∃𝑥𝐿 ∈ ( L ‘𝑥)∃𝑝(𝑝 = (𝑥𝐿 +s 𝑧) ∧ (𝑦 +s 𝑧) ≤s 𝑝) ↔ ∃𝑥𝐿 ∈ ( L ‘𝑥)(𝑦 +s 𝑧) ≤s (𝑥𝐿 +s 𝑧))
7772, 76bitr3i 279 . . . . . . . . . . . 12 (∃𝑝(∃𝑥𝐿 ∈ ( L ‘𝑥)𝑝 = (𝑥𝐿 +s 𝑧) ∧ (𝑦 +s 𝑧) ≤s 𝑝) ↔ ∃𝑥𝐿 ∈ ( L ‘𝑥)(𝑦 +s 𝑧) ≤s (𝑥𝐿 +s 𝑧))
7868, 77bitri 277 . . . . . . . . . . 11 (∃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 2765 . . . . . . . . . . . . . 14 (𝑏 = 𝑝 → (𝑏 = (𝑥 +s 𝑧𝐿) ↔ 𝑝 = (𝑥 +s 𝑧𝐿)))
8079rexbidv 3185 . . . . . . . . . . . . 13 (𝑏 = 𝑝 → (∃𝑧𝐿 ∈ ( L ‘𝑧)𝑏 = (𝑥 +s 𝑧𝐿) ↔ ∃𝑧𝐿 ∈ ( L ‘𝑧)𝑝 = (𝑥 +s 𝑧𝐿)))
8180rexab 3657 . . . . . . . . . . . 12 (∃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 3288 . . . . . . . . . . . . . 14 (∃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 3191 . . . . . . . . . . . . . . 15 (∃𝑧𝐿 ∈ ( L ‘𝑧)(𝑝 = (𝑥 +s 𝑧𝐿) ∧ (𝑦 +s 𝑧) ≤s 𝑝) ↔ (∃𝑧𝐿 ∈ ( L ‘𝑧)𝑝 = (𝑥 +s 𝑧𝐿) ∧ (𝑦 +s 𝑧) ≤s 𝑝))
8483exbii 1867 . . . . . . . . . . . . . 14 (∃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))
8582, 84bitri 277 . . . . . . . . . . . . 13 (∃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 7425 . . . . . . . . . . . . . . 15 (𝑥 +s 𝑧𝐿) ∈ V
87 breq2 5103 . . . . . . . . . . . . . . 15 (𝑝 = (𝑥 +s 𝑧𝐿) → ((𝑦 +s 𝑧) ≤s 𝑝 ↔ (𝑦 +s 𝑧) ≤s (𝑥 +s 𝑧𝐿)))
8886, 87ceqsexv 3501 . . . . . . . . . . . . . 14 (∃𝑝(𝑝 = (𝑥 +s 𝑧𝐿) ∧ (𝑦 +s 𝑧) ≤s 𝑝) ↔ (𝑦 +s 𝑧) ≤s (𝑥 +s 𝑧𝐿))
8988rexbii 3108 . . . . . . . . . . . . 13 (∃𝑧𝐿 ∈ ( L ‘𝑧)∃𝑝(𝑝 = (𝑥 +s 𝑧𝐿) ∧ (𝑦 +s 𝑧) ≤s 𝑝) ↔ ∃𝑧𝐿 ∈ ( L ‘𝑧)(𝑦 +s 𝑧) ≤s (𝑥 +s 𝑧𝐿))
9085, 89bitr3i 279 . . . . . . . . . . . 12 (∃𝑝(∃𝑧𝐿 ∈ ( L ‘𝑧)𝑝 = (𝑥 +s 𝑧𝐿) ∧ (𝑦 +s 𝑧) ≤s 𝑝) ↔ ∃𝑧𝐿 ∈ ( L ‘𝑧)(𝑦 +s 𝑧) ≤s (𝑥 +s 𝑧𝐿))
9181, 90bitri 277 . . . . . . . . . . 11 (∃𝑝 ∈ {𝑏 ∣ ∃𝑧𝐿 ∈ ( L ‘𝑧)𝑏 = (𝑥 +s 𝑧𝐿)} (𝑦 +s 𝑧) ≤s 𝑝 ↔ ∃𝑧𝐿 ∈ ( L ‘𝑧)(𝑦 +s 𝑧) ≤s (𝑥 +s 𝑧𝐿))
9278, 91orbi12i 925 . . . . . . . . . 10 ((∃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)))
9365, 92bitri 277 . . . . . . . . 9 (∃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 1226 . . . . . . . . . . . 12 ((((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 27947 . . . . . . . . . . . . . 14 (𝑥𝐿 ∈ ( L ‘𝑥) → 𝑥𝐿 ∈ No )
9695adantr 484 . . . . . . . . . . . . 13 ((𝑥𝐿 ∈ ( L ‘𝑥) ∧ (𝑦 +s 𝑧) ≤s (𝑥𝐿 +s 𝑧)) → 𝑥𝐿 ∈ No )
9796adantl 485 . . . . . . . . . . . 12 ((((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 1225 . . . . . . . . . . . 12 ((((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 782 . . . . . . . . . . . . 13 ((((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 1227 . . . . . . . . . . . . . 14 ((((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 28057 . . . . . . . . . . . . . 14 ((𝑦 ∈ No ∧ 𝑥𝐿 ∈ No ∧ 𝑧 ∈ No ) → ((𝑦 +s 𝑧) ≤s (𝑥𝐿 +s 𝑧) → 𝑦 ≤s 𝑥𝐿))
10294, 97, 100, 101syl3anc 1389 . . . . . . . . . . . . 13 ((((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))
10399, 102mpd 15 . . . . . . . . . . . 12 ((((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 27923 . . . . . . . . . . . . . 14 (𝑥𝐿 ∈ ( L ‘𝑥) → 𝑥𝐿 <s 𝑥)
105104adantr 484 . . . . . . . . . . . . 13 ((𝑥𝐿 ∈ ( L ‘𝑥) ∧ (𝑦 +s 𝑧) ≤s (𝑥𝐿 +s 𝑧)) → 𝑥𝐿 <s 𝑥)
106105adantl 485 . . . . . . . . . . . 12 ((((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)
10794, 97, 98, 103, 106leltstrd 27806 . . . . . . . . . . 11 ((((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)
108107rexlimdvaa 3163 . . . . . . . . . 10 (((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 1226 . . . . . . . . . . . . . 14 ((((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 27947 . . . . . . . . . . . . . . . 16 (𝑧𝐿 ∈ ( L ‘𝑧) → 𝑧𝐿 ∈ No )
111110adantr 484 . . . . . . . . . . . . . . 15 ((𝑧𝐿 ∈ ( L ‘𝑧) ∧ (𝑦 +s 𝑧) ≤s (𝑥 +s 𝑧𝐿)) → 𝑧𝐿 ∈ No )
112111adantl 485 . . . . . . . . . . . . . 14 ((((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 )
113109, 112addscld 28050 . . . . . . . . . . . . 13 ((((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 1227 . . . . . . . . . . . . . 14 ((((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 )
115109, 114addscld 28050 . . . . . . . . . . . . 13 ((((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 1225 . . . . . . . . . . . . . 14 ((((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 )
117116, 112addscld 28050 . . . . . . . . . . . . 13 ((((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 27923 . . . . . . . . . . . . . . . 16 (𝑧𝐿 ∈ ( L ‘𝑧) → 𝑧𝐿 <s 𝑧)
119118adantr 484 . . . . . . . . . . . . . . 15 ((𝑧𝐿 ∈ ( L ‘𝑧) ∧ (𝑦 +s 𝑧) ≤s (𝑥 +s 𝑧𝐿)) → 𝑧𝐿 <s 𝑧)
120119adantl 485 . . . . . . . . . . . . . 14 ((((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 28056 . . . . . . . . . . . . . . 15 ((𝑧𝐿 ∈ No ∧ 𝑧 ∈ No ∧ 𝑦 ∈ No ) → (𝑧𝐿 <s 𝑧 → (𝑦 +s 𝑧𝐿) <s (𝑦 +s 𝑧)))
122112, 114, 109, 121syl3anc 1389 . . . . . . . . . . . . . 14 ((((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)))
123120, 122mpd 15 . . . . . . . . . . . . 13 ((((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 782 . . . . . . . . . . . . 13 ((((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))
125113, 115, 117, 123, 124ltlestrd 27805 . . . . . . . . . . . 12 ((((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 7400 . . . . . . . . . . . . . . 15 (𝑧𝑂 = 𝑧𝐿 → (𝑦 +s 𝑧𝑂) = (𝑦 +s 𝑧𝐿))
127 oveq2 7400 . . . . . . . . . . . . . . 15 (𝑧𝑂 = 𝑧𝐿 → (𝑥 +s 𝑧𝑂) = (𝑥 +s 𝑧𝐿))
128126, 127breq12d 5112 . . . . . . . . . . . . . 14 (𝑧𝑂 = 𝑧𝐿 → ((𝑦 +s 𝑧𝑂) <s (𝑥 +s 𝑧𝑂) ↔ (𝑦 +s 𝑧𝐿) <s (𝑥 +s 𝑧𝐿)))
129128imbi1d 343 . . . . . . . . . . . . 13 (𝑧𝑂 = 𝑧𝐿 → (((𝑦 +s 𝑧𝑂) <s (𝑥 +s 𝑧𝑂) → 𝑦 <s 𝑥) ↔ ((𝑦 +s 𝑧𝐿) <s (𝑥 +s 𝑧𝐿) → 𝑦 <s 𝑥)))
130 simplr3 1230 . . . . . . . . . . . . 13 ((((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 780 . . . . . . . . . . . . . 14 ((((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 4134 . . . . . . . . . . . . . 14 (𝑧𝐿 ∈ ( L ‘𝑧) → 𝑧𝐿 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧)))
133131, 132syl 17 . . . . . . . . . . . . 13 ((((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)))
134129, 130, 133rspcdva 3582 . . . . . . . . . . . 12 ((((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))
135125, 134mpd 15 . . . . . . . . . . 11 ((((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)
136135rexlimdvaa 3163 . . . . . . . . . 10 (((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))
137108, 136jaod 870 . . . . . . . . 9 (((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))
13893, 137biimtrid 244 . . . . . . . 8 (((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 4148 . . . . . . . . . 10 (∃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 2765 . . . . . . . . . . . . . 14 (𝑐 = 𝑞 → (𝑐 = (𝑦𝑅 +s 𝑧) ↔ 𝑞 = (𝑦𝑅 +s 𝑧)))
141140rexbidv 3185 . . . . . . . . . . . . 13 (𝑐 = 𝑞 → (∃𝑦𝑅 ∈ ( R ‘𝑦)𝑐 = (𝑦𝑅 +s 𝑧) ↔ ∃𝑦𝑅 ∈ ( R ‘𝑦)𝑞 = (𝑦𝑅 +s 𝑧)))
142141rexab 3657 . . . . . . . . . . . 12 (∃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 3288 . . . . . . . . . . . . . 14 (∃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 3191 . . . . . . . . . . . . . . 15 (∃𝑦𝑅 ∈ ( R ‘𝑦)(𝑞 = (𝑦𝑅 +s 𝑧) ∧ 𝑞 ≤s (𝑥 +s 𝑧)) ↔ (∃𝑦𝑅 ∈ ( R ‘𝑦)𝑞 = (𝑦𝑅 +s 𝑧) ∧ 𝑞 ≤s (𝑥 +s 𝑧)))
145144exbii 1867 . . . . . . . . . . . . . 14 (∃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)))
146143, 145bitri 277 . . . . . . . . . . . . 13 (∃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 7425 . . . . . . . . . . . . . . 15 (𝑦𝑅 +s 𝑧) ∈ V
148 breq1 5102 . . . . . . . . . . . . . . 15 (𝑞 = (𝑦𝑅 +s 𝑧) → (𝑞 ≤s (𝑥 +s 𝑧) ↔ (𝑦𝑅 +s 𝑧) ≤s (𝑥 +s 𝑧)))
149147, 148ceqsexv 3501 . . . . . . . . . . . . . 14 (∃𝑞(𝑞 = (𝑦𝑅 +s 𝑧) ∧ 𝑞 ≤s (𝑥 +s 𝑧)) ↔ (𝑦𝑅 +s 𝑧) ≤s (𝑥 +s 𝑧))
150149rexbii 3108 . . . . . . . . . . . . 13 (∃𝑦𝑅 ∈ ( R ‘𝑦)∃𝑞(𝑞 = (𝑦𝑅 +s 𝑧) ∧ 𝑞 ≤s (𝑥 +s 𝑧)) ↔ ∃𝑦𝑅 ∈ ( R ‘𝑦)(𝑦𝑅 +s 𝑧) ≤s (𝑥 +s 𝑧))
151146, 150bitr3i 279 . . . . . . . . . . . 12 (∃𝑞(∃𝑦𝑅 ∈ ( R ‘𝑦)𝑞 = (𝑦𝑅 +s 𝑧) ∧ 𝑞 ≤s (𝑥 +s 𝑧)) ↔ ∃𝑦𝑅 ∈ ( R ‘𝑦)(𝑦𝑅 +s 𝑧) ≤s (𝑥 +s 𝑧))
152142, 151bitri 277 . . . . . . . . . . 11 (∃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 2765 . . . . . . . . . . . . . 14 (𝑑 = 𝑞 → (𝑑 = (𝑦 +s 𝑧𝑅) ↔ 𝑞 = (𝑦 +s 𝑧𝑅)))
154153rexbidv 3185 . . . . . . . . . . . . 13 (𝑑 = 𝑞 → (∃𝑧𝑅 ∈ ( R ‘𝑧)𝑑 = (𝑦 +s 𝑧𝑅) ↔ ∃𝑧𝑅 ∈ ( R ‘𝑧)𝑞 = (𝑦 +s 𝑧𝑅)))
155154rexab 3657 . . . . . . . . . . . 12 (∃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 3288 . . . . . . . . . . . . . 14 (∃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 3191 . . . . . . . . . . . . . . 15 (∃𝑧𝑅 ∈ ( R ‘𝑧)(𝑞 = (𝑦 +s 𝑧𝑅) ∧ 𝑞 ≤s (𝑥 +s 𝑧)) ↔ (∃𝑧𝑅 ∈ ( R ‘𝑧)𝑞 = (𝑦 +s 𝑧𝑅) ∧ 𝑞 ≤s (𝑥 +s 𝑧)))
158157exbii 1867 . . . . . . . . . . . . . 14 (∃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)))
159156, 158bitri 277 . . . . . . . . . . . . 13 (∃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 7425 . . . . . . . . . . . . . . 15 (𝑦 +s 𝑧𝑅) ∈ V
161 breq1 5102 . . . . . . . . . . . . . . 15 (𝑞 = (𝑦 +s 𝑧𝑅) → (𝑞 ≤s (𝑥 +s 𝑧) ↔ (𝑦 +s 𝑧𝑅) ≤s (𝑥 +s 𝑧)))
162160, 161ceqsexv 3501 . . . . . . . . . . . . . 14 (∃𝑞(𝑞 = (𝑦 +s 𝑧𝑅) ∧ 𝑞 ≤s (𝑥 +s 𝑧)) ↔ (𝑦 +s 𝑧𝑅) ≤s (𝑥 +s 𝑧))
163162rexbii 3108 . . . . . . . . . . . . 13 (∃𝑧𝑅 ∈ ( R ‘𝑧)∃𝑞(𝑞 = (𝑦 +s 𝑧𝑅) ∧ 𝑞 ≤s (𝑥 +s 𝑧)) ↔ ∃𝑧𝑅 ∈ ( R ‘𝑧)(𝑦 +s 𝑧𝑅) ≤s (𝑥 +s 𝑧))
164159, 163bitr3i 279 . . . . . . . . . . . 12 (∃𝑞(∃𝑧𝑅 ∈ ( R ‘𝑧)𝑞 = (𝑦 +s 𝑧𝑅) ∧ 𝑞 ≤s (𝑥 +s 𝑧)) ↔ ∃𝑧𝑅 ∈ ( R ‘𝑧)(𝑦 +s 𝑧𝑅) ≤s (𝑥 +s 𝑧))
165155, 164bitri 277 . . . . . . . . . . 11 (∃𝑞 ∈ {𝑑 ∣ ∃𝑧𝑅 ∈ ( R ‘𝑧)𝑑 = (𝑦 +s 𝑧𝑅)}𝑞 ≤s (𝑥 +s 𝑧) ↔ ∃𝑧𝑅 ∈ ( R ‘𝑧)(𝑦 +s 𝑧𝑅) ≤s (𝑥 +s 𝑧))
166152, 165orbi12i 925 . . . . . . . . . 10 ((∃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)))
167139, 166bitri 277 . . . . . . . . 9 (∃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 1226 . . . . . . . . . . . 12 ((((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 27948 . . . . . . . . . . . . . 14 (𝑦𝑅 ∈ ( R ‘𝑦) → 𝑦𝑅 ∈ No )
170169adantr 484 . . . . . . . . . . . . 13 ((𝑦𝑅 ∈ ( R ‘𝑦) ∧ (𝑦𝑅 +s 𝑧) ≤s (𝑥 +s 𝑧)) → 𝑦𝑅 ∈ No )
171170adantl 485 . . . . . . . . . . . 12 ((((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 1225 . . . . . . . . . . . 12 ((((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 27924 . . . . . . . . . . . . . 14 (𝑦𝑅 ∈ ( R ‘𝑦) → 𝑦 <s 𝑦𝑅)
174173adantr 484 . . . . . . . . . . . . 13 ((𝑦𝑅 ∈ ( R ‘𝑦) ∧ (𝑦𝑅 +s 𝑧) ≤s (𝑥 +s 𝑧)) → 𝑦 <s 𝑦𝑅)
175174adantl 485 . . . . . . . . . . . 12 ((((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 782 . . . . . . . . . . . . 13 ((((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 1227 . . . . . . . . . . . . . 14 ((((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 28057 . . . . . . . . . . . . . 14 ((𝑦𝑅 ∈ No ∧ 𝑥 ∈ No ∧ 𝑧 ∈ No ) → ((𝑦𝑅 +s 𝑧) ≤s (𝑥 +s 𝑧) → 𝑦𝑅 ≤s 𝑥))
179171, 172, 177, 178syl3anc 1389 . . . . . . . . . . . . 13 ((((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))
180176, 179mpd 15 . . . . . . . . . . . 12 ((((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)
181168, 171, 172, 175, 180ltlestrd 27805 . . . . . . . . . . 11 ((((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)
182181rexlimdvaa 3163 . . . . . . . . . 10 (((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 1226 . . . . . . . . . . . . . 14 ((((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 27948 . . . . . . . . . . . . . . . 16 (𝑧𝑅 ∈ ( R ‘𝑧) → 𝑧𝑅 ∈ No )
185184adantr 484 . . . . . . . . . . . . . . 15 ((𝑧𝑅 ∈ ( R ‘𝑧) ∧ (𝑦 +s 𝑧𝑅) ≤s (𝑥 +s 𝑧)) → 𝑧𝑅 ∈ No )
186185adantl 485 . . . . . . . . . . . . . 14 ((((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 )
187183, 186addscld 28050 . . . . . . . . . . . . 13 ((((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 1225 . . . . . . . . . . . . . 14 ((((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 1227 . . . . . . . . . . . . . 14 ((((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 )
190188, 189addscld 28050 . . . . . . . . . . . . 13 ((((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 )
191188, 186addscld 28050 . . . . . . . . . . . . 13 ((((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 782 . . . . . . . . . . . . 13 ((((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))
193189, 186, 1883jca 1140 . . . . . . . . . . . . . 14 ((((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 27924 . . . . . . . . . . . . . . . 16 (𝑧𝑅 ∈ ( R ‘𝑧) → 𝑧 <s 𝑧𝑅)
195194adantr 484 . . . . . . . . . . . . . . 15 ((𝑧𝑅 ∈ ( R ‘𝑧) ∧ (𝑦 +s 𝑧𝑅) ≤s (𝑥 +s 𝑧)) → 𝑧 <s 𝑧𝑅)
196195adantl 485 . . . . . . . . . . . . . 14 ((((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 28056 . . . . . . . . . . . . . 14 ((𝑧 ∈ No ∧ 𝑧𝑅 ∈ No ∧ 𝑥 ∈ No ) → (𝑧 <s 𝑧𝑅 → (𝑥 +s 𝑧) <s (𝑥 +s 𝑧𝑅)))
198193, 196, 197sylc 65 . . . . . . . . . . . . 13 ((((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))
199187, 190, 191, 192, 198leltstrd 27806 . . . . . . . . . . . 12 ((((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 7400 . . . . . . . . . . . . . . 15 (𝑧𝑂 = 𝑧𝑅 → (𝑦 +s 𝑧𝑂) = (𝑦 +s 𝑧𝑅))
201 oveq2 7400 . . . . . . . . . . . . . . 15 (𝑧𝑂 = 𝑧𝑅 → (𝑥 +s 𝑧𝑂) = (𝑥 +s 𝑧𝑅))
202200, 201breq12d 5112 . . . . . . . . . . . . . 14 (𝑧𝑂 = 𝑧𝑅 → ((𝑦 +s 𝑧𝑂) <s (𝑥 +s 𝑧𝑂) ↔ (𝑦 +s 𝑧𝑅) <s (𝑥 +s 𝑧𝑅)))
203202imbi1d 343 . . . . . . . . . . . . 13 (𝑧𝑂 = 𝑧𝑅 → (((𝑦 +s 𝑧𝑂) <s (𝑥 +s 𝑧𝑂) → 𝑦 <s 𝑥) ↔ ((𝑦 +s 𝑧𝑅) <s (𝑥 +s 𝑧𝑅) → 𝑦 <s 𝑥)))
204 simplr3 1230 . . . . . . . . . . . . 13 ((((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 780 . . . . . . . . . . . . . 14 ((((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 4135 . . . . . . . . . . . . . 14 (𝑧𝑅 ∈ ( R ‘𝑧) → 𝑧𝑅 ∈ (( L ‘𝑧) ∪ ( R ‘𝑧)))
207205, 206syl 17 . . . . . . . . . . . . 13 ((((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)))
208203, 204, 207rspcdva 3582 . . . . . . . . . . . 12 ((((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))
209199, 208mpd 15 . . . . . . . . . . 11 ((((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)
210209rexlimdvaa 3163 . . . . . . . . . 10 (((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))
211182, 210jaod 870 . . . . . . . . 9 (((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))
212167, 211biimtrid 244 . . . . . . . 8 (((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))
213138, 212jaod 870 . . . . . . 7 (((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))
21464, 213sylbid 242 . . . . . 6 (((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))
215214ex 416 . . . . 5 ((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)))
2164, 8, 12, 16, 20, 22, 25, 29, 33, 37, 215no3inds 28028 . . . 4 ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ((𝐵 +s 𝐶) <s (𝐴 +s 𝐶) → 𝐵 <s 𝐴))
217 addscl 28051 . . . . . 6 ((𝐵 ∈ No ∧ 𝐶 ∈ No ) → (𝐵 +s 𝐶) ∈ No )
2182173adant1 1142 . . . . 5 ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → (𝐵 +s 𝐶) ∈ No )
219 addscl 28051 . . . . . 6 ((𝐴 ∈ No ∧ 𝐶 ∈ No ) → (𝐴 +s 𝐶) ∈ No )
2202193adant2 1143 . . . . 5 ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → (𝐴 +s 𝐶) ∈ No )
221 ltnles 27794 . . . . 5 (((𝐵 +s 𝐶) ∈ No ∧ (𝐴 +s 𝐶) ∈ No ) → ((𝐵 +s 𝐶) <s (𝐴 +s 𝐶) ↔ ¬ (𝐴 +s 𝐶) ≤s (𝐵 +s 𝐶)))
222218, 220, 221syl2anc 593 . . . 4 ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ((𝐵 +s 𝐶) <s (𝐴 +s 𝐶) ↔ ¬ (𝐴 +s 𝐶) ≤s (𝐵 +s 𝐶)))
223 ltnles 27794 . . . . . 6 ((𝐵 ∈ No ∧ 𝐴 ∈ No ) → (𝐵 <s 𝐴 ↔ ¬ 𝐴 ≤s 𝐵))
224223ancoms 462 . . . . 5 ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐵 <s 𝐴 ↔ ¬ 𝐴 ≤s 𝐵))
2252243adant3 1144 . . . 4 ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → (𝐵 <s 𝐴 ↔ ¬ 𝐴 ≤s 𝐵))
226216, 222, 2253imtr3d 295 . . 3 ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → (¬ (𝐴 +s 𝐶) ≤s (𝐵 +s 𝐶) → ¬ 𝐴 ≤s 𝐵))
227226con4d 115 . 2 ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → (𝐴 ≤s 𝐵 → (𝐴 +s 𝐶) ≤s (𝐵 +s 𝐶)))
228 leadds1im 28057 . 2 ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ((𝐴 +s 𝐶) ≤s (𝐵 +s 𝐶) → 𝐴 ≤s 𝐵))
229227, 228impbid 214 1 ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → (𝐴 ≤s 𝐵 ↔ (𝐴 +s 𝐶) ≤s (𝐵 +s 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 399   ∨ wo 858   ∧ w3a 1097   = wceq 1559  ∃wex 1798   ∈ wcel 2141  {cab 2739   ≠ wne 2956  ∀wral 3075  ∃wrex 3085   ∪ cun 3902  ∅c0 4285  {csn 4581   class class class wbr 5099  ‘cfv 6517  (class class class)co 7392   No csur 27681   <s clts 27682   ≤s cles 27785   <<s cslts 27827   |s ccuts 27829   L cleft 27895   R cright 27896   +s cadds 28029
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714
This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-tp 4586  df-op 4588  df-ot 4590  df-uni 4865  df-int 4905  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5540  df-eprel 5545  df-po 5553  df-so 5554  df-fr 5598  df-se 5599  df-we 5600  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-pred 6284  df-ord 6345  df-on 6346  df-suc 6348  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-riota 7349  df-ov 7395  df-oprab 7396  df-mpo 7397  df-1st 7966  df-2nd 7967  df-frecs 8257  df-wrecs 8288  df-recs 8337  df-1o 8432  df-2o 8433  df-nadd 8631  df-no 27684  df-lts 27685  df-bday 27686  df-les 27786  df-slts 27828  df-cuts 27830  df-0s 27877  df-made 27897  df-old 27898  df-left 27900  df-right 27901  df-norec2 28019  df-adds 28030
This theorem is referenced by:  leadds2  28060  addscan2  28063  leadds1d  28065  nnsge1  28413  zsoring  28479
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