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Theorem sleadd1 27699
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
sleadd1 ((ðī ∈ No ∧ ðĩ ∈ No ∧ ðķ ∈ No ) → (ðī â‰Īs ðĩ ↔ (ðī +s ðķ) â‰Īs (ðĩ +s ðķ)))
Proof of Theorem sleadd1
Dummy variables ð‘Ĩ ð‘Ķ 𝑧 𝑎 𝑏 𝑐 𝑑 𝑝 𝑞 ð‘Ĩðŋ ð‘Ķðŋ 𝑧ðŋ ð‘Ĩ𝑅 ð‘Ķ𝑅 𝑧𝑅 ð‘Ĩ𝑂 ð‘Ķ𝑂 𝑧𝑂 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq1 7418 . . . . . . 7 (ð‘Ĩ = ð‘Ĩ𝑂 → (ð‘Ĩ +s 𝑧) = (ð‘Ĩ𝑂 +s 𝑧))
21breq2d 5160 . . . . . 6 (ð‘Ĩ = ð‘Ĩ𝑂 → ((ð‘Ķ +s 𝑧) <s (ð‘Ĩ +s 𝑧) ↔ (ð‘Ķ +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧)))
3 breq2 5152 . . . . . 6 (ð‘Ĩ = ð‘Ĩ𝑂 → (ð‘Ķ <s ð‘Ĩ ↔ ð‘Ķ <s ð‘Ĩ𝑂))
42, 3imbi12d 344 . . . . 5 (ð‘Ĩ = ð‘Ĩ𝑂 → (((ð‘Ķ +s 𝑧) <s (ð‘Ĩ +s 𝑧) → ð‘Ķ <s ð‘Ĩ) ↔ ((ð‘Ķ +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ <s ð‘Ĩ𝑂)))
5 oveq1 7418 . . . . . . 7 (ð‘Ķ = ð‘Ķ𝑂 → (ð‘Ķ +s 𝑧) = (ð‘Ķ𝑂 +s 𝑧))
65breq1d 5158 . . . . . 6 (ð‘Ķ = ð‘Ķ𝑂 → ((ð‘Ķ +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) ↔ (ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧)))
7 breq1 5151 . . . . . 6 (ð‘Ķ = ð‘Ķ𝑂 → (ð‘Ķ <s ð‘Ĩ𝑂 ↔ ð‘Ķ𝑂 <s ð‘Ĩ𝑂))
86, 7imbi12d 344 . . . . 5 (ð‘Ķ = ð‘Ķ𝑂 → (((ð‘Ķ +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ <s ð‘Ĩ𝑂) ↔ ((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂)))
9 oveq2 7419 . . . . . . 7 (𝑧 = 𝑧𝑂 → (ð‘Ķ𝑂 +s 𝑧) = (ð‘Ķ𝑂 +s 𝑧𝑂))
10 oveq2 7419 . . . . . . 7 (𝑧 = 𝑧𝑂 → (ð‘Ĩ𝑂 +s 𝑧) = (ð‘Ĩ𝑂 +s 𝑧𝑂))
119, 10breq12d 5161 . . . . . 6 (𝑧 = 𝑧𝑂 → ((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) ↔ (ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂)))
1211imbi1d 341 . . . . 5 (𝑧 = 𝑧𝑂 → (((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ↔ ((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂)))
13 oveq1 7418 . . . . . . 7 (ð‘Ĩ = ð‘Ĩ𝑂 → (ð‘Ĩ +s 𝑧𝑂) = (ð‘Ĩ𝑂 +s 𝑧𝑂))
1413breq2d 5160 . . . . . 6 (ð‘Ĩ = ð‘Ĩ𝑂 → ((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) ↔ (ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂)))
15 breq2 5152 . . . . . 6 (ð‘Ĩ = ð‘Ĩ𝑂 → (ð‘Ķ𝑂 <s ð‘Ĩ ↔ ð‘Ķ𝑂 <s ð‘Ĩ𝑂))
1614, 15imbi12d 344 . . . . 5 (ð‘Ĩ = ð‘Ĩ𝑂 → (((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ) ↔ ((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂)))
17 oveq1 7418 . . . . . . 7 (ð‘Ķ = ð‘Ķ𝑂 → (ð‘Ķ +s 𝑧𝑂) = (ð‘Ķ𝑂 +s 𝑧𝑂))
1817breq1d 5158 . . . . . 6 (ð‘Ķ = ð‘Ķ𝑂 → ((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) ↔ (ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂)))
19 breq1 5151 . . . . . 6 (ð‘Ķ = ð‘Ķ𝑂 → (ð‘Ķ <s ð‘Ĩ ↔ ð‘Ķ𝑂 <s ð‘Ĩ))
2018, 19imbi12d 344 . . . . 5 (ð‘Ķ = ð‘Ķ𝑂 → (((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ) ↔ ((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ)))
2117breq1d 5158 . . . . . 6 (ð‘Ķ = ð‘Ķ𝑂 → ((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) ↔ (ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂)))
2221, 7imbi12d 344 . . . . 5 (ð‘Ķ = ð‘Ķ𝑂 → (((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ𝑂) ↔ ((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂)))
23 oveq2 7419 . . . . . . 7 (𝑧 = 𝑧𝑂 → (ð‘Ĩ +s 𝑧) = (ð‘Ĩ +s 𝑧𝑂))
249, 23breq12d 5161 . . . . . 6 (𝑧 = 𝑧𝑂 → ((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ +s 𝑧) ↔ (ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂)))
2524imbi1d 341 . . . . 5 (𝑧 = 𝑧𝑂 → (((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ) ↔ ((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ)))
26 oveq1 7418 . . . . . . 7 (ð‘Ĩ = ðī → (ð‘Ĩ +s 𝑧) = (ðī +s 𝑧))
2726breq2d 5160 . . . . . 6 (ð‘Ĩ = ðī → ((ð‘Ķ +s 𝑧) <s (ð‘Ĩ +s 𝑧) ↔ (ð‘Ķ +s 𝑧) <s (ðī +s 𝑧)))
28 breq2 5152 . . . . . 6 (ð‘Ĩ = ðī → (ð‘Ķ <s ð‘Ĩ ↔ ð‘Ķ <s ðī))
2927, 28imbi12d 344 . . . . 5 (ð‘Ĩ = ðī → (((ð‘Ķ +s 𝑧) <s (ð‘Ĩ +s 𝑧) → ð‘Ķ <s ð‘Ĩ) ↔ ((ð‘Ķ +s 𝑧) <s (ðī +s 𝑧) → ð‘Ķ <s ðī)))
30 oveq1 7418 . . . . . . 7 (ð‘Ķ = ðĩ → (ð‘Ķ +s 𝑧) = (ðĩ +s 𝑧))
3130breq1d 5158 . . . . . 6 (ð‘Ķ = ðĩ → ((ð‘Ķ +s 𝑧) <s (ðī +s 𝑧) ↔ (ðĩ +s 𝑧) <s (ðī +s 𝑧)))
32 breq1 5151 . . . . . 6 (ð‘Ķ = ðĩ → (ð‘Ķ <s ðī ↔ ðĩ <s ðī))
3331, 32imbi12d 344 . . . . 5 (ð‘Ķ = ðĩ → (((ð‘Ķ +s 𝑧) <s (ðī +s 𝑧) → ð‘Ķ <s ðī) ↔ ((ðĩ +s 𝑧) <s (ðī +s 𝑧) → ðĩ <s ðī)))
34 oveq2 7419 . . . . . . 7 (𝑧 = ðķ → (ðĩ +s 𝑧) = (ðĩ +s ðķ))
35 oveq2 7419 . . . . . . 7 (𝑧 = ðķ → (ðī +s 𝑧) = (ðī +s ðķ))
3634, 35breq12d 5161 . . . . . 6 (𝑧 = ðķ → ((ðĩ +s 𝑧) <s (ðī +s 𝑧) ↔ (ðĩ +s ðķ) <s (ðī +s ðķ)))
3736imbi1d 341 . . . . 5 (𝑧 = ðķ → (((ðĩ +s 𝑧) <s (ðī +s 𝑧) → ðĩ <s ðī) ↔ ((ðĩ +s ðķ) <s (ðī +s ðķ) → ðĩ <s ðī)))
38 simp2 1137 . . . . . . . . . . . 12 ((ð‘Ĩ ∈ No ∧ ð‘Ķ ∈ No ∧ 𝑧 ∈ No ) → ð‘Ķ ∈ No )
39 simp3 1138 . . . . . . . . . . . 12 ((ð‘Ĩ ∈ No ∧ ð‘Ķ ∈ No ∧ 𝑧 ∈ No ) → 𝑧 ∈ No )
4038, 39addscut 27688 . . . . . . . . . . 11 ((ð‘Ĩ ∈ No ∧ ð‘Ķ ∈ No ∧ 𝑧 ∈ No ) → ((ð‘Ķ +s 𝑧) ∈ No ∧ ({𝑎 âˆĢ ∃ð‘Ķðŋ ∈ ( L ‘ð‘Ķ)𝑎 = (ð‘Ķðŋ +s 𝑧)} ∊ {𝑏 âˆĢ ∃𝑧ðŋ ∈ ( L ‘𝑧)𝑏 = (ð‘Ķ +s 𝑧ðŋ)}) <<s {(ð‘Ķ +s 𝑧)} ∧ {(ð‘Ķ +s 𝑧)} <<s ({𝑐 âˆĢ ∃ð‘Ķ𝑅 ∈ ( R ‘ð‘Ķ)𝑐 = (ð‘Ķ𝑅 +s 𝑧)} ∊ {𝑑 âˆĢ ∃𝑧𝑅 ∈ ( R ‘𝑧)𝑑 = (ð‘Ķ +s 𝑧𝑅)})))
41 simp2 1137 . . . . . . . . . . 11 (((ð‘Ķ +s 𝑧) ∈ No ∧ ({𝑎 âˆĢ ∃ð‘Ķðŋ ∈ ( L ‘ð‘Ķ)𝑎 = (ð‘Ķðŋ +s 𝑧)} ∊ {𝑏 âˆĢ ∃𝑧ðŋ ∈ ( L ‘𝑧)𝑏 = (ð‘Ķ +s 𝑧ðŋ)}) <<s {(ð‘Ķ +s 𝑧)} ∧ {(ð‘Ķ +s 𝑧)} <<s ({𝑐 âˆĢ ∃ð‘Ķ𝑅 ∈ ( R ‘ð‘Ķ)𝑐 = (ð‘Ķ𝑅 +s 𝑧)} ∊ {𝑑 âˆĢ ∃𝑧𝑅 ∈ ( R ‘𝑧)𝑑 = (ð‘Ķ +s 𝑧𝑅)})) → ({𝑎 âˆĢ ∃ð‘Ķðŋ ∈ ( L ‘ð‘Ķ)𝑎 = (ð‘Ķðŋ +s 𝑧)} ∊ {𝑏 âˆĢ ∃𝑧ðŋ ∈ ( L ‘𝑧)𝑏 = (ð‘Ķ +s 𝑧ðŋ)}) <<s {(ð‘Ķ +s 𝑧)})
4240, 41syl 17 . . . . . . . . . 10 ((ð‘Ĩ ∈ No ∧ ð‘Ķ ∈ No ∧ 𝑧 ∈ No ) → ({𝑎 âˆĢ ∃ð‘Ķðŋ ∈ ( L ‘ð‘Ķ)𝑎 = (ð‘Ķðŋ +s 𝑧)} ∊ {𝑏 âˆĢ ∃𝑧ðŋ ∈ ( L ‘𝑧)𝑏 = (ð‘Ķ +s 𝑧ðŋ)}) <<s {(ð‘Ķ +s 𝑧)})
4340simp3d 1144 . . . . . . . . . 10 ((ð‘Ĩ ∈ No ∧ ð‘Ķ ∈ No ∧ 𝑧 ∈ No ) → {(ð‘Ķ +s 𝑧)} <<s ({𝑐 âˆĢ ∃ð‘Ķ𝑅 ∈ ( R ‘ð‘Ķ)𝑐 = (ð‘Ķ𝑅 +s 𝑧)} ∊ {𝑑 âˆĢ ∃𝑧𝑅 ∈ ( R ‘𝑧)𝑑 = (ð‘Ķ +s 𝑧𝑅)}))
44 ovex 7444 . . . . . . . . . . . 12 (ð‘Ķ +s 𝑧) ∈ V
4544snnz 4780 . . . . . . . . . . 11 {(ð‘Ķ +s 𝑧)} ≠ ∅
46 sslttr 27533 . . . . . . . . . . 11 ((({𝑎 âˆĢ ∃ð‘Ķðŋ ∈ ( L ‘ð‘Ķ)𝑎 = (ð‘Ķðŋ +s 𝑧)} ∊ {𝑏 âˆĢ ∃𝑧ðŋ ∈ ( L ‘𝑧)𝑏 = (ð‘Ķ +s 𝑧ðŋ)}) <<s {(ð‘Ķ +s 𝑧)} ∧ {(ð‘Ķ +s 𝑧)} <<s ({𝑐 âˆĢ ∃ð‘Ķ𝑅 ∈ ( R ‘ð‘Ķ)𝑐 = (ð‘Ķ𝑅 +s 𝑧)} ∊ {𝑑 âˆĢ ∃𝑧𝑅 ∈ ( R ‘𝑧)𝑑 = (ð‘Ķ +s 𝑧𝑅)}) ∧ {(ð‘Ķ +s 𝑧)} ≠ ∅) → ({𝑎 âˆĢ ∃ð‘Ķðŋ ∈ ( L ‘ð‘Ķ)𝑎 = (ð‘Ķðŋ +s 𝑧)} ∊ {𝑏 âˆĢ ∃𝑧ðŋ ∈ ( L ‘𝑧)𝑏 = (ð‘Ķ +s 𝑧ðŋ)}) <<s ({𝑐 âˆĢ ∃ð‘Ķ𝑅 ∈ ( R ‘ð‘Ķ)𝑐 = (ð‘Ķ𝑅 +s 𝑧)} ∊ {𝑑 âˆĢ ∃𝑧𝑅 ∈ ( R ‘𝑧)𝑑 = (ð‘Ķ +s 𝑧𝑅)}))
4745, 46mp3an3 1450 . . . . . . . . . 10 ((({𝑎 âˆĢ ∃ð‘Ķðŋ ∈ ( L ‘ð‘Ķ)𝑎 = (ð‘Ķðŋ +s 𝑧)} ∊ {𝑏 âˆĢ ∃𝑧ðŋ ∈ ( L ‘𝑧)𝑏 = (ð‘Ķ +s 𝑧ðŋ)}) <<s {(ð‘Ķ +s 𝑧)} ∧ {(ð‘Ķ +s 𝑧)} <<s ({𝑐 âˆĢ ∃ð‘Ķ𝑅 ∈ ( R ‘ð‘Ķ)𝑐 = (ð‘Ķ𝑅 +s 𝑧)} ∊ {𝑑 âˆĢ ∃𝑧𝑅 ∈ ( R ‘𝑧)𝑑 = (ð‘Ķ +s 𝑧𝑅)})) → ({𝑎 âˆĢ ∃ð‘Ķðŋ ∈ ( L ‘ð‘Ķ)𝑎 = (ð‘Ķðŋ +s 𝑧)} ∊ {𝑏 âˆĢ ∃𝑧ðŋ ∈ ( L ‘𝑧)𝑏 = (ð‘Ķ +s 𝑧ðŋ)}) <<s ({𝑐 âˆĢ ∃ð‘Ķ𝑅 ∈ ( R ‘ð‘Ķ)𝑐 = (ð‘Ķ𝑅 +s 𝑧)} ∊ {𝑑 âˆĢ ∃𝑧𝑅 ∈ ( R ‘𝑧)𝑑 = (ð‘Ķ +s 𝑧𝑅)}))
4842, 43, 47syl2anc 584 . . . . . . . . 9 ((ð‘Ĩ ∈ No ∧ ð‘Ķ ∈ No ∧ 𝑧 ∈ No ) → ({𝑎 âˆĢ ∃ð‘Ķðŋ ∈ ( L ‘ð‘Ķ)𝑎 = (ð‘Ķðŋ +s 𝑧)} ∊ {𝑏 âˆĢ ∃𝑧ðŋ ∈ ( L ‘𝑧)𝑏 = (ð‘Ķ +s 𝑧ðŋ)}) <<s ({𝑐 âˆĢ ∃ð‘Ķ𝑅 ∈ ( R ‘ð‘Ķ)𝑐 = (ð‘Ķ𝑅 +s 𝑧)} ∊ {𝑑 âˆĢ ∃𝑧𝑅 ∈ ( R ‘𝑧)𝑑 = (ð‘Ķ +s 𝑧𝑅)}))
49 simp1 1136 . . . . . . . . . . . 12 ((ð‘Ĩ ∈ No ∧ ð‘Ķ ∈ No ∧ 𝑧 ∈ No ) → ð‘Ĩ ∈ No )
5049, 39addscut 27688 . . . . . . . . . . 11 ((ð‘Ĩ ∈ No ∧ ð‘Ķ ∈ No ∧ 𝑧 ∈ No ) → ((ð‘Ĩ +s 𝑧) ∈ No ∧ ({𝑎 âˆĢ ∃ð‘Ĩðŋ ∈ ( L ‘ð‘Ĩ)𝑎 = (ð‘Ĩðŋ +s 𝑧)} ∊ {𝑏 âˆĢ ∃𝑧ðŋ ∈ ( L ‘𝑧)𝑏 = (ð‘Ĩ +s 𝑧ðŋ)}) <<s {(ð‘Ĩ +s 𝑧)} ∧ {(ð‘Ĩ +s 𝑧)} <<s ({𝑐 âˆĢ ∃ð‘Ĩ𝑅 ∈ ( R ‘ð‘Ĩ)𝑐 = (ð‘Ĩ𝑅 +s 𝑧)} ∊ {𝑑 âˆĢ ∃𝑧𝑅 ∈ ( R ‘𝑧)𝑑 = (ð‘Ĩ +s 𝑧𝑅)})))
51 simp2 1137 . . . . . . . . . . 11 (((ð‘Ĩ +s 𝑧) ∈ No ∧ ({𝑎 âˆĢ ∃ð‘Ĩðŋ ∈ ( L ‘ð‘Ĩ)𝑎 = (ð‘Ĩðŋ +s 𝑧)} ∊ {𝑏 âˆĢ ∃𝑧ðŋ ∈ ( L ‘𝑧)𝑏 = (ð‘Ĩ +s 𝑧ðŋ)}) <<s {(ð‘Ĩ +s 𝑧)} ∧ {(ð‘Ĩ +s 𝑧)} <<s ({𝑐 âˆĢ ∃ð‘Ĩ𝑅 ∈ ( R ‘ð‘Ĩ)𝑐 = (ð‘Ĩ𝑅 +s 𝑧)} ∊ {𝑑 âˆĢ ∃𝑧𝑅 ∈ ( R ‘𝑧)𝑑 = (ð‘Ĩ +s 𝑧𝑅)})) → ({𝑎 âˆĢ ∃ð‘Ĩðŋ ∈ ( L ‘ð‘Ĩ)𝑎 = (ð‘Ĩðŋ +s 𝑧)} ∊ {𝑏 âˆĢ ∃𝑧ðŋ ∈ ( L ‘𝑧)𝑏 = (ð‘Ĩ +s 𝑧ðŋ)}) <<s {(ð‘Ĩ +s 𝑧)})
5250, 51syl 17 . . . . . . . . . 10 ((ð‘Ĩ ∈ No ∧ ð‘Ķ ∈ No ∧ 𝑧 ∈ No ) → ({𝑎 âˆĢ ∃ð‘Ĩðŋ ∈ ( L ‘ð‘Ĩ)𝑎 = (ð‘Ĩðŋ +s 𝑧)} ∊ {𝑏 âˆĢ ∃𝑧ðŋ ∈ ( L ‘𝑧)𝑏 = (ð‘Ĩ +s 𝑧ðŋ)}) <<s {(ð‘Ĩ +s 𝑧)})
5350simp3d 1144 . . . . . . . . . 10 ((ð‘Ĩ ∈ No ∧ ð‘Ķ ∈ No ∧ 𝑧 ∈ No ) → {(ð‘Ĩ +s 𝑧)} <<s ({𝑐 âˆĢ ∃ð‘Ĩ𝑅 ∈ ( R ‘ð‘Ĩ)𝑐 = (ð‘Ĩ𝑅 +s 𝑧)} ∊ {𝑑 âˆĢ ∃𝑧𝑅 ∈ ( R ‘𝑧)𝑑 = (ð‘Ĩ +s 𝑧𝑅)}))
54 ovex 7444 . . . . . . . . . . . 12 (ð‘Ĩ +s 𝑧) ∈ V
5554snnz 4780 . . . . . . . . . . 11 {(ð‘Ĩ +s 𝑧)} ≠ ∅
56 sslttr 27533 . . . . . . . . . . 11 ((({𝑎 âˆĢ ∃ð‘Ĩðŋ ∈ ( L ‘ð‘Ĩ)𝑎 = (ð‘Ĩðŋ +s 𝑧)} ∊ {𝑏 âˆĢ ∃𝑧ðŋ ∈ ( L ‘𝑧)𝑏 = (ð‘Ĩ +s 𝑧ðŋ)}) <<s {(ð‘Ĩ +s 𝑧)} ∧ {(ð‘Ĩ +s 𝑧)} <<s ({𝑐 âˆĢ ∃ð‘Ĩ𝑅 ∈ ( R ‘ð‘Ĩ)𝑐 = (ð‘Ĩ𝑅 +s 𝑧)} ∊ {𝑑 âˆĢ ∃𝑧𝑅 ∈ ( R ‘𝑧)𝑑 = (ð‘Ĩ +s 𝑧𝑅)}) ∧ {(ð‘Ĩ +s 𝑧)} ≠ ∅) → ({𝑎 âˆĢ ∃ð‘Ĩðŋ ∈ ( L ‘ð‘Ĩ)𝑎 = (ð‘Ĩðŋ +s 𝑧)} ∊ {𝑏 âˆĢ ∃𝑧ðŋ ∈ ( L ‘𝑧)𝑏 = (ð‘Ĩ +s 𝑧ðŋ)}) <<s ({𝑐 âˆĢ ∃ð‘Ĩ𝑅 ∈ ( R ‘ð‘Ĩ)𝑐 = (ð‘Ĩ𝑅 +s 𝑧)} ∊ {𝑑 âˆĢ ∃𝑧𝑅 ∈ ( R ‘𝑧)𝑑 = (ð‘Ĩ +s 𝑧𝑅)}))
5755, 56mp3an3 1450 . . . . . . . . . 10 ((({𝑎 âˆĢ ∃ð‘Ĩðŋ ∈ ( L ‘ð‘Ĩ)𝑎 = (ð‘Ĩðŋ +s 𝑧)} ∊ {𝑏 âˆĢ ∃𝑧ðŋ ∈ ( L ‘𝑧)𝑏 = (ð‘Ĩ +s 𝑧ðŋ)}) <<s {(ð‘Ĩ +s 𝑧)} ∧ {(ð‘Ĩ +s 𝑧)} <<s ({𝑐 âˆĢ ∃ð‘Ĩ𝑅 ∈ ( R ‘ð‘Ĩ)𝑐 = (ð‘Ĩ𝑅 +s 𝑧)} ∊ {𝑑 âˆĢ ∃𝑧𝑅 ∈ ( R ‘𝑧)𝑑 = (ð‘Ĩ +s 𝑧𝑅)})) → ({𝑎 âˆĢ ∃ð‘Ĩðŋ ∈ ( L ‘ð‘Ĩ)𝑎 = (ð‘Ĩðŋ +s 𝑧)} ∊ {𝑏 âˆĢ ∃𝑧ðŋ ∈ ( L ‘𝑧)𝑏 = (ð‘Ĩ +s 𝑧ðŋ)}) <<s ({𝑐 âˆĢ ∃ð‘Ĩ𝑅 ∈ ( R ‘ð‘Ĩ)𝑐 = (ð‘Ĩ𝑅 +s 𝑧)} ∊ {𝑑 âˆĢ ∃𝑧𝑅 ∈ ( R ‘𝑧)𝑑 = (ð‘Ĩ +s 𝑧𝑅)}))
5852, 53, 57syl2anc 584 . . . . . . . . 9 ((ð‘Ĩ ∈ No ∧ ð‘Ķ ∈ No ∧ 𝑧 ∈ No ) → ({𝑎 âˆĢ ∃ð‘Ĩðŋ ∈ ( L ‘ð‘Ĩ)𝑎 = (ð‘Ĩðŋ +s 𝑧)} ∊ {𝑏 âˆĢ ∃𝑧ðŋ ∈ ( L ‘𝑧)𝑏 = (ð‘Ĩ +s 𝑧ðŋ)}) <<s ({𝑐 âˆĢ ∃ð‘Ĩ𝑅 ∈ ( R ‘ð‘Ĩ)𝑐 = (ð‘Ĩ𝑅 +s 𝑧)} ∊ {𝑑 âˆĢ ∃𝑧𝑅 ∈ ( R ‘𝑧)𝑑 = (ð‘Ĩ +s 𝑧𝑅)}))
59 addsval2 27673 . . . . . . . . . 10 ((ð‘Ķ ∈ No ∧ 𝑧 ∈ No ) → (ð‘Ķ +s 𝑧) = (({𝑎 âˆĢ ∃ð‘Ķðŋ ∈ ( L ‘ð‘Ķ)𝑎 = (ð‘Ķðŋ +s 𝑧)} ∊ {𝑏 âˆĢ ∃𝑧ðŋ ∈ ( L ‘𝑧)𝑏 = (ð‘Ķ +s 𝑧ðŋ)}) |s ({𝑐 âˆĢ ∃ð‘Ķ𝑅 ∈ ( R ‘ð‘Ķ)𝑐 = (ð‘Ķ𝑅 +s 𝑧)} ∊ {𝑑 âˆĢ ∃𝑧𝑅 ∈ ( R ‘𝑧)𝑑 = (ð‘Ķ +s 𝑧𝑅)})))
60593adant1 1130 . . . . . . . . 9 ((ð‘Ĩ ∈ No ∧ ð‘Ķ ∈ No ∧ 𝑧 ∈ No ) → (ð‘Ķ +s 𝑧) = (({𝑎 âˆĢ ∃ð‘Ķðŋ ∈ ( L ‘ð‘Ķ)𝑎 = (ð‘Ķðŋ +s 𝑧)} ∊ {𝑏 âˆĢ ∃𝑧ðŋ ∈ ( L ‘𝑧)𝑏 = (ð‘Ķ +s 𝑧ðŋ)}) |s ({𝑐 âˆĢ ∃ð‘Ķ𝑅 ∈ ( R ‘ð‘Ķ)𝑐 = (ð‘Ķ𝑅 +s 𝑧)} ∊ {𝑑 âˆĢ ∃𝑧𝑅 ∈ ( R ‘𝑧)𝑑 = (ð‘Ķ +s 𝑧𝑅)})))
61 addsval2 27673 . . . . . . . . . 10 ((ð‘Ĩ ∈ No ∧ 𝑧 ∈ No ) → (ð‘Ĩ +s 𝑧) = (({𝑎 âˆĢ ∃ð‘Ĩðŋ ∈ ( L ‘ð‘Ĩ)𝑎 = (ð‘Ĩðŋ +s 𝑧)} ∊ {𝑏 âˆĢ ∃𝑧ðŋ ∈ ( L ‘𝑧)𝑏 = (ð‘Ĩ +s 𝑧ðŋ)}) |s ({𝑐 âˆĢ ∃ð‘Ĩ𝑅 ∈ ( R ‘ð‘Ĩ)𝑐 = (ð‘Ĩ𝑅 +s 𝑧)} ∊ {𝑑 âˆĢ ∃𝑧𝑅 ∈ ( R ‘𝑧)𝑑 = (ð‘Ĩ +s 𝑧𝑅)})))
62613adant2 1131 . . . . . . . . 9 ((ð‘Ĩ ∈ No ∧ ð‘Ķ ∈ No ∧ 𝑧 ∈ No ) → (ð‘Ĩ +s 𝑧) = (({𝑎 âˆĢ ∃ð‘Ĩðŋ ∈ ( L ‘ð‘Ĩ)𝑎 = (ð‘Ĩðŋ +s 𝑧)} ∊ {𝑏 âˆĢ ∃𝑧ðŋ ∈ ( L ‘𝑧)𝑏 = (ð‘Ĩ +s 𝑧ðŋ)}) |s ({𝑐 âˆĢ ∃ð‘Ĩ𝑅 ∈ ( R ‘ð‘Ĩ)𝑐 = (ð‘Ĩ𝑅 +s 𝑧)} ∊ {𝑑 âˆĢ ∃𝑧𝑅 ∈ ( R ‘𝑧)𝑑 = (ð‘Ĩ +s 𝑧𝑅)})))
63 sltrec 27546 . . . . . . . . 9 (((({𝑎 âˆĢ ∃ð‘Ķðŋ ∈ ( L ‘ð‘Ķ)𝑎 = (ð‘Ķðŋ +s 𝑧)} ∊ {𝑏 âˆĢ ∃𝑧ðŋ ∈ ( L ‘𝑧)𝑏 = (ð‘Ķ +s 𝑧ðŋ)}) <<s ({𝑐 âˆĢ ∃ð‘Ķ𝑅 ∈ ( R ‘ð‘Ķ)𝑐 = (ð‘Ķ𝑅 +s 𝑧)} ∊ {𝑑 âˆĢ ∃𝑧𝑅 ∈ ( R ‘𝑧)𝑑 = (ð‘Ķ +s 𝑧𝑅)}) ∧ ({𝑎 âˆĢ ∃ð‘Ĩðŋ ∈ ( L ‘ð‘Ĩ)𝑎 = (ð‘Ĩðŋ +s 𝑧)} ∊ {𝑏 âˆĢ ∃𝑧ðŋ ∈ ( L ‘𝑧)𝑏 = (ð‘Ĩ +s 𝑧ðŋ)}) <<s ({𝑐 âˆĢ ∃ð‘Ĩ𝑅 ∈ ( R ‘ð‘Ĩ)𝑐 = (ð‘Ĩ𝑅 +s 𝑧)} ∊ {𝑑 âˆĢ ∃𝑧𝑅 ∈ ( R ‘𝑧)𝑑 = (ð‘Ĩ +s 𝑧𝑅)})) ∧ ((ð‘Ķ +s 𝑧) = (({𝑎 âˆĢ ∃ð‘Ķðŋ ∈ ( L ‘ð‘Ķ)𝑎 = (ð‘Ķðŋ +s 𝑧)} ∊ {𝑏 âˆĢ ∃𝑧ðŋ ∈ ( L ‘𝑧)𝑏 = (ð‘Ķ +s 𝑧ðŋ)}) |s ({𝑐 âˆĢ ∃ð‘Ķ𝑅 ∈ ( R ‘ð‘Ķ)𝑐 = (ð‘Ķ𝑅 +s 𝑧)} ∊ {𝑑 âˆĢ ∃𝑧𝑅 ∈ ( R ‘𝑧)𝑑 = (ð‘Ķ +s 𝑧𝑅)})) ∧ (ð‘Ĩ +s 𝑧) = (({𝑎 âˆĢ ∃ð‘Ĩðŋ ∈ ( L ‘ð‘Ĩ)𝑎 = (ð‘Ĩðŋ +s 𝑧)} ∊ {𝑏 âˆĢ ∃𝑧ðŋ ∈ ( L ‘𝑧)𝑏 = (ð‘Ĩ +s 𝑧ðŋ)}) |s ({𝑐 âˆĢ ∃ð‘Ĩ𝑅 ∈ ( R ‘ð‘Ĩ)𝑐 = (ð‘Ĩ𝑅 +s 𝑧)} ∊ {𝑑 âˆĢ ∃𝑧𝑅 ∈ ( R ‘𝑧)𝑑 = (ð‘Ĩ +s 𝑧𝑅)})))) → ((ð‘Ķ +s 𝑧) <s (ð‘Ĩ +s 𝑧) ↔ (∃𝑝 ∈ ({𝑎 âˆĢ ∃ð‘Ĩðŋ ∈ ( L ‘ð‘Ĩ)𝑎 = (ð‘Ĩðŋ +s 𝑧)} ∊ {𝑏 âˆĢ ∃𝑧ðŋ ∈ ( L ‘𝑧)𝑏 = (ð‘Ĩ +s 𝑧ðŋ)})(ð‘Ķ +s 𝑧) â‰Īs 𝑝 âˆĻ ∃𝑞 ∈ ({𝑐 âˆĢ ∃ð‘Ķ𝑅 ∈ ( R ‘ð‘Ķ)𝑐 = (ð‘Ķ𝑅 +s 𝑧)} ∊ {𝑑 âˆĢ ∃𝑧𝑅 ∈ ( R ‘𝑧)𝑑 = (ð‘Ķ +s 𝑧𝑅)})𝑞 â‰Īs (ð‘Ĩ +s 𝑧))))
6448, 58, 60, 62, 63syl22anc 837 . . . . . . . 8 ((ð‘Ĩ ∈ No ∧ ð‘Ķ ∈ No ∧ 𝑧 ∈ No ) → ((ð‘Ķ +s 𝑧) <s (ð‘Ĩ +s 𝑧) ↔ (∃𝑝 ∈ ({𝑎 âˆĢ ∃ð‘Ĩðŋ ∈ ( L ‘ð‘Ĩ)𝑎 = (ð‘Ĩðŋ +s 𝑧)} ∊ {𝑏 âˆĢ ∃𝑧ðŋ ∈ ( L ‘𝑧)𝑏 = (ð‘Ĩ +s 𝑧ðŋ)})(ð‘Ķ +s 𝑧) â‰Īs 𝑝 âˆĻ ∃𝑞 ∈ ({𝑐 âˆĢ ∃ð‘Ķ𝑅 ∈ ( R ‘ð‘Ķ)𝑐 = (ð‘Ķ𝑅 +s 𝑧)} ∊ {𝑑 âˆĢ ∃𝑧𝑅 ∈ ( R ‘𝑧)𝑑 = (ð‘Ķ +s 𝑧𝑅)})𝑞 â‰Īs (ð‘Ĩ +s 𝑧))))
6564adantr 481 . . . . . . 7 (((ð‘Ĩ ∈ No ∧ ð‘Ķ ∈ No ∧ 𝑧 ∈ No ) ∧ ((∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ𝑂)) ∧ (∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))((ð‘Ķ +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ <s ð‘Ĩ𝑂) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ))) → ((ð‘Ķ +s 𝑧) <s (ð‘Ĩ +s 𝑧) ↔ (∃𝑝 ∈ ({𝑎 âˆĢ ∃ð‘Ĩðŋ ∈ ( L ‘ð‘Ĩ)𝑎 = (ð‘Ĩðŋ +s 𝑧)} ∊ {𝑏 âˆĢ ∃𝑧ðŋ ∈ ( L ‘𝑧)𝑏 = (ð‘Ĩ +s 𝑧ðŋ)})(ð‘Ķ +s 𝑧) â‰Īs 𝑝 âˆĻ ∃𝑞 ∈ ({𝑐 âˆĢ ∃ð‘Ķ𝑅 ∈ ( R ‘ð‘Ķ)𝑐 = (ð‘Ķ𝑅 +s 𝑧)} ∊ {𝑑 âˆĢ ∃𝑧𝑅 ∈ ( R ‘𝑧)𝑑 = (ð‘Ķ +s 𝑧𝑅)})𝑞 â‰Īs (ð‘Ĩ +s 𝑧))))
66 rexun 4190 . . . . . . . . . 10 (∃𝑝 ∈ ({𝑎 âˆĢ ∃ð‘Ĩðŋ ∈ ( L ‘ð‘Ĩ)𝑎 = (ð‘Ĩðŋ +s 𝑧)} ∊ {𝑏 âˆĢ ∃𝑧ðŋ ∈ ( L ‘𝑧)𝑏 = (ð‘Ĩ +s 𝑧ðŋ)})(ð‘Ķ +s 𝑧) â‰Īs 𝑝 ↔ (∃𝑝 ∈ {𝑎 âˆĢ ∃ð‘Ĩðŋ ∈ ( L ‘ð‘Ĩ)𝑎 = (ð‘Ĩðŋ +s 𝑧)} (ð‘Ķ +s 𝑧) â‰Īs 𝑝 âˆĻ ∃𝑝 ∈ {𝑏 âˆĢ ∃𝑧ðŋ ∈ ( L ‘𝑧)𝑏 = (ð‘Ĩ +s 𝑧ðŋ)} (ð‘Ķ +s 𝑧) â‰Īs 𝑝))
67 eqeq1 2736 . . . . . . . . . . . . . 14 (𝑎 = 𝑝 → (𝑎 = (ð‘Ĩðŋ +s 𝑧) ↔ 𝑝 = (ð‘Ĩðŋ +s 𝑧)))
6867rexbidv 3178 . . . . . . . . . . . . 13 (𝑎 = 𝑝 → (∃ð‘Ĩðŋ ∈ ( L ‘ð‘Ĩ)𝑎 = (ð‘Ĩðŋ +s 𝑧) ↔ ∃ð‘Ĩðŋ ∈ ( L ‘ð‘Ĩ)𝑝 = (ð‘Ĩðŋ +s 𝑧)))
6968rexab 3690 . . . . . . . . . . . 12 (∃𝑝 ∈ {𝑎 âˆĢ ∃ð‘Ĩðŋ ∈ ( L ‘ð‘Ĩ)𝑎 = (ð‘Ĩðŋ +s 𝑧)} (ð‘Ķ +s 𝑧) â‰Īs 𝑝 ↔ ∃𝑝(∃ð‘Ĩðŋ ∈ ( L ‘ð‘Ĩ)𝑝 = (ð‘Ĩðŋ +s 𝑧) ∧ (ð‘Ķ +s 𝑧) â‰Īs 𝑝))
70 rexcom4 3285 . . . . . . . . . . . . . 14 (∃ð‘Ĩðŋ ∈ ( L ‘ð‘Ĩ)∃𝑝(𝑝 = (ð‘Ĩðŋ +s 𝑧) ∧ (ð‘Ķ +s 𝑧) â‰Īs 𝑝) ↔ ∃𝑝∃ð‘Ĩðŋ ∈ ( L ‘ð‘Ĩ)(𝑝 = (ð‘Ĩðŋ +s 𝑧) ∧ (ð‘Ķ +s 𝑧) â‰Īs 𝑝))
71 r19.41v 3188 . . . . . . . . . . . . . . 15 (∃ð‘Ĩðŋ ∈ ( L ‘ð‘Ĩ)(𝑝 = (ð‘Ĩðŋ +s 𝑧) ∧ (ð‘Ķ +s 𝑧) â‰Īs 𝑝) ↔ (∃ð‘Ĩðŋ ∈ ( L ‘ð‘Ĩ)𝑝 = (ð‘Ĩðŋ +s 𝑧) ∧ (ð‘Ķ +s 𝑧) â‰Īs 𝑝))
7271exbii 1850 . . . . . . . . . . . . . 14 (∃𝑝∃ð‘Ĩðŋ ∈ ( L ‘ð‘Ĩ)(𝑝 = (ð‘Ĩðŋ +s 𝑧) ∧ (ð‘Ķ +s 𝑧) â‰Īs 𝑝) ↔ ∃𝑝(∃ð‘Ĩðŋ ∈ ( L ‘ð‘Ĩ)𝑝 = (ð‘Ĩðŋ +s 𝑧) ∧ (ð‘Ķ +s 𝑧) â‰Īs 𝑝))
7370, 72bitri 274 . . . . . . . . . . . . 13 (∃ð‘Ĩðŋ ∈ ( L ‘ð‘Ĩ)∃𝑝(𝑝 = (ð‘Ĩðŋ +s 𝑧) ∧ (ð‘Ķ +s 𝑧) â‰Īs 𝑝) ↔ ∃𝑝(∃ð‘Ĩðŋ ∈ ( L ‘ð‘Ĩ)𝑝 = (ð‘Ĩðŋ +s 𝑧) ∧ (ð‘Ķ +s 𝑧) â‰Īs 𝑝))
74 ovex 7444 . . . . . . . . . . . . . . 15 (ð‘Ĩðŋ +s 𝑧) ∈ V
75 breq2 5152 . . . . . . . . . . . . . . 15 (𝑝 = (ð‘Ĩðŋ +s 𝑧) → ((ð‘Ķ +s 𝑧) â‰Īs 𝑝 ↔ (ð‘Ķ +s 𝑧) â‰Īs (ð‘Ĩðŋ +s 𝑧)))
7674, 75ceqsexv 3525 . . . . . . . . . . . . . 14 (∃𝑝(𝑝 = (ð‘Ĩðŋ +s 𝑧) ∧ (ð‘Ķ +s 𝑧) â‰Īs 𝑝) ↔ (ð‘Ķ +s 𝑧) â‰Īs (ð‘Ĩðŋ +s 𝑧))
7776rexbii 3094 . . . . . . . . . . . . 13 (∃ð‘Ĩðŋ ∈ ( L ‘ð‘Ĩ)∃𝑝(𝑝 = (ð‘Ĩðŋ +s 𝑧) ∧ (ð‘Ķ +s 𝑧) â‰Īs 𝑝) ↔ ∃ð‘Ĩðŋ ∈ ( L ‘ð‘Ĩ)(ð‘Ķ +s 𝑧) â‰Īs (ð‘Ĩðŋ +s 𝑧))
7873, 77bitr3i 276 . . . . . . . . . . . 12 (∃𝑝(∃ð‘Ĩðŋ ∈ ( L ‘ð‘Ĩ)𝑝 = (ð‘Ĩðŋ +s 𝑧) ∧ (ð‘Ķ +s 𝑧) â‰Īs 𝑝) ↔ ∃ð‘Ĩðŋ ∈ ( L ‘ð‘Ĩ)(ð‘Ķ +s 𝑧) â‰Īs (ð‘Ĩðŋ +s 𝑧))
7969, 78bitri 274 . . . . . . . . . . 11 (∃𝑝 ∈ {𝑎 âˆĢ ∃ð‘Ĩðŋ ∈ ( L ‘ð‘Ĩ)𝑎 = (ð‘Ĩðŋ +s 𝑧)} (ð‘Ķ +s 𝑧) â‰Īs 𝑝 ↔ ∃ð‘Ĩðŋ ∈ ( L ‘ð‘Ĩ)(ð‘Ķ +s 𝑧) â‰Īs (ð‘Ĩðŋ +s 𝑧))
80 eqeq1 2736 . . . . . . . . . . . . . 14 (𝑏 = 𝑝 → (𝑏 = (ð‘Ĩ +s 𝑧ðŋ) ↔ 𝑝 = (ð‘Ĩ +s 𝑧ðŋ)))
8180rexbidv 3178 . . . . . . . . . . . . 13 (𝑏 = 𝑝 → (∃𝑧ðŋ ∈ ( L ‘𝑧)𝑏 = (ð‘Ĩ +s 𝑧ðŋ) ↔ ∃𝑧ðŋ ∈ ( L ‘𝑧)𝑝 = (ð‘Ĩ +s 𝑧ðŋ)))
8281rexab 3690 . . . . . . . . . . . 12 (∃𝑝 ∈ {𝑏 âˆĢ ∃𝑧ðŋ ∈ ( L ‘𝑧)𝑏 = (ð‘Ĩ +s 𝑧ðŋ)} (ð‘Ķ +s 𝑧) â‰Īs 𝑝 ↔ ∃𝑝(∃𝑧ðŋ ∈ ( L ‘𝑧)𝑝 = (ð‘Ĩ +s 𝑧ðŋ) ∧ (ð‘Ķ +s 𝑧) â‰Īs 𝑝))
83 rexcom4 3285 . . . . . . . . . . . . . 14 (∃𝑧ðŋ ∈ ( L ‘𝑧)∃𝑝(𝑝 = (ð‘Ĩ +s 𝑧ðŋ) ∧ (ð‘Ķ +s 𝑧) â‰Īs 𝑝) ↔ ∃𝑝∃𝑧ðŋ ∈ ( L ‘𝑧)(𝑝 = (ð‘Ĩ +s 𝑧ðŋ) ∧ (ð‘Ķ +s 𝑧) â‰Īs 𝑝))
84 r19.41v 3188 . . . . . . . . . . . . . . 15 (∃𝑧ðŋ ∈ ( L ‘𝑧)(𝑝 = (ð‘Ĩ +s 𝑧ðŋ) ∧ (ð‘Ķ +s 𝑧) â‰Īs 𝑝) ↔ (∃𝑧ðŋ ∈ ( L ‘𝑧)𝑝 = (ð‘Ĩ +s 𝑧ðŋ) ∧ (ð‘Ķ +s 𝑧) â‰Īs 𝑝))
8584exbii 1850 . . . . . . . . . . . . . 14 (∃𝑝∃𝑧ðŋ ∈ ( L ‘𝑧)(𝑝 = (ð‘Ĩ +s 𝑧ðŋ) ∧ (ð‘Ķ +s 𝑧) â‰Īs 𝑝) ↔ ∃𝑝(∃𝑧ðŋ ∈ ( L ‘𝑧)𝑝 = (ð‘Ĩ +s 𝑧ðŋ) ∧ (ð‘Ķ +s 𝑧) â‰Īs 𝑝))
8683, 85bitri 274 . . . . . . . . . . . . 13 (∃𝑧ðŋ ∈ ( L ‘𝑧)∃𝑝(𝑝 = (ð‘Ĩ +s 𝑧ðŋ) ∧ (ð‘Ķ +s 𝑧) â‰Īs 𝑝) ↔ ∃𝑝(∃𝑧ðŋ ∈ ( L ‘𝑧)𝑝 = (ð‘Ĩ +s 𝑧ðŋ) ∧ (ð‘Ķ +s 𝑧) â‰Īs 𝑝))
87 ovex 7444 . . . . . . . . . . . . . . 15 (ð‘Ĩ +s 𝑧ðŋ) ∈ V
88 breq2 5152 . . . . . . . . . . . . . . 15 (𝑝 = (ð‘Ĩ +s 𝑧ðŋ) → ((ð‘Ķ +s 𝑧) â‰Īs 𝑝 ↔ (ð‘Ķ +s 𝑧) â‰Īs (ð‘Ĩ +s 𝑧ðŋ)))
8987, 88ceqsexv 3525 . . . . . . . . . . . . . 14 (∃𝑝(𝑝 = (ð‘Ĩ +s 𝑧ðŋ) ∧ (ð‘Ķ +s 𝑧) â‰Īs 𝑝) ↔ (ð‘Ķ +s 𝑧) â‰Īs (ð‘Ĩ +s 𝑧ðŋ))
9089rexbii 3094 . . . . . . . . . . . . 13 (∃𝑧ðŋ ∈ ( L ‘𝑧)∃𝑝(𝑝 = (ð‘Ĩ +s 𝑧ðŋ) ∧ (ð‘Ķ +s 𝑧) â‰Īs 𝑝) ↔ ∃𝑧ðŋ ∈ ( L ‘𝑧)(ð‘Ķ +s 𝑧) â‰Īs (ð‘Ĩ +s 𝑧ðŋ))
9186, 90bitr3i 276 . . . . . . . . . . . 12 (∃𝑝(∃𝑧ðŋ ∈ ( L ‘𝑧)𝑝 = (ð‘Ĩ +s 𝑧ðŋ) ∧ (ð‘Ķ +s 𝑧) â‰Īs 𝑝) ↔ ∃𝑧ðŋ ∈ ( L ‘𝑧)(ð‘Ķ +s 𝑧) â‰Īs (ð‘Ĩ +s 𝑧ðŋ))
9282, 91bitri 274 . . . . . . . . . . 11 (∃𝑝 ∈ {𝑏 âˆĢ ∃𝑧ðŋ ∈ ( L ‘𝑧)𝑏 = (ð‘Ĩ +s 𝑧ðŋ)} (ð‘Ķ +s 𝑧) â‰Īs 𝑝 ↔ ∃𝑧ðŋ ∈ ( L ‘𝑧)(ð‘Ķ +s 𝑧) â‰Īs (ð‘Ĩ +s 𝑧ðŋ))
9379, 92orbi12i 913 . . . . . . . . . 10 ((∃𝑝 ∈ {𝑎 âˆĢ ∃ð‘Ĩðŋ ∈ ( L ‘ð‘Ĩ)𝑎 = (ð‘Ĩðŋ +s 𝑧)} (ð‘Ķ +s 𝑧) â‰Īs 𝑝 âˆĻ ∃𝑝 ∈ {𝑏 âˆĢ ∃𝑧ðŋ ∈ ( L ‘𝑧)𝑏 = (ð‘Ĩ +s 𝑧ðŋ)} (ð‘Ķ +s 𝑧) â‰Īs 𝑝) ↔ (∃ð‘Ĩðŋ ∈ ( L ‘ð‘Ĩ)(ð‘Ķ +s 𝑧) â‰Īs (ð‘Ĩðŋ +s 𝑧) âˆĻ ∃𝑧ðŋ ∈ ( L ‘𝑧)(ð‘Ķ +s 𝑧) â‰Īs (ð‘Ĩ +s 𝑧ðŋ)))
9466, 93bitri 274 . . . . . . . . 9 (∃𝑝 ∈ ({𝑎 âˆĢ ∃ð‘Ĩðŋ ∈ ( L ‘ð‘Ĩ)𝑎 = (ð‘Ĩðŋ +s 𝑧)} ∊ {𝑏 âˆĢ ∃𝑧ðŋ ∈ ( L ‘𝑧)𝑏 = (ð‘Ĩ +s 𝑧ðŋ)})(ð‘Ķ +s 𝑧) â‰Īs 𝑝 ↔ (∃ð‘Ĩðŋ ∈ ( L ‘ð‘Ĩ)(ð‘Ķ +s 𝑧) â‰Īs (ð‘Ĩðŋ +s 𝑧) âˆĻ ∃𝑧ðŋ ∈ ( L ‘𝑧)(ð‘Ķ +s 𝑧) â‰Īs (ð‘Ĩ +s 𝑧ðŋ)))
95 simpll2 1213 . . . . . . . . . . . 12 ((((ð‘Ĩ ∈ No ∧ ð‘Ķ ∈ No ∧ 𝑧 ∈ No ) ∧ ((∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ𝑂)) ∧ (∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))((ð‘Ķ +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ <s ð‘Ĩ𝑂) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ))) ∧ (ð‘Ĩðŋ ∈ ( L ‘ð‘Ĩ) ∧ (ð‘Ķ +s 𝑧) â‰Īs (ð‘Ĩðŋ +s 𝑧))) → ð‘Ķ ∈ No )
96 leftssno 27600 . . . . . . . . . . . . . . 15 ( L ‘ð‘Ĩ) ⊆ No
9796sseli 3978 . . . . . . . . . . . . . 14 (ð‘Ĩðŋ ∈ ( L ‘ð‘Ĩ) → ð‘Ĩðŋ ∈ No )
9897adantr 481 . . . . . . . . . . . . 13 ((ð‘Ĩðŋ ∈ ( L ‘ð‘Ĩ) ∧ (ð‘Ķ +s 𝑧) â‰Īs (ð‘Ĩðŋ +s 𝑧)) → ð‘Ĩðŋ ∈ No )
9998adantl 482 . . . . . . . . . . . 12 ((((ð‘Ĩ ∈ No ∧ ð‘Ķ ∈ No ∧ 𝑧 ∈ No ) ∧ ((∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ𝑂)) ∧ (∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))((ð‘Ķ +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ <s ð‘Ĩ𝑂) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ))) ∧ (ð‘Ĩðŋ ∈ ( L ‘ð‘Ĩ) ∧ (ð‘Ķ +s 𝑧) â‰Īs (ð‘Ĩðŋ +s 𝑧))) → ð‘Ĩðŋ ∈ No )
100 simpll1 1212 . . . . . . . . . . . 12 ((((ð‘Ĩ ∈ No ∧ ð‘Ķ ∈ No ∧ 𝑧 ∈ No ) ∧ ((∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ𝑂)) ∧ (∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))((ð‘Ķ +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ <s ð‘Ĩ𝑂) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ))) ∧ (ð‘Ĩðŋ ∈ ( L ‘ð‘Ĩ) ∧ (ð‘Ķ +s 𝑧) â‰Īs (ð‘Ĩðŋ +s 𝑧))) → ð‘Ĩ ∈ No )
101 simprr 771 . . . . . . . . . . . . 13 ((((ð‘Ĩ ∈ No ∧ ð‘Ķ ∈ No ∧ 𝑧 ∈ No ) ∧ ((∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ𝑂)) ∧ (∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))((ð‘Ķ +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ <s ð‘Ĩ𝑂) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ))) ∧ (ð‘Ĩðŋ ∈ ( L ‘ð‘Ĩ) ∧ (ð‘Ķ +s 𝑧) â‰Īs (ð‘Ĩðŋ +s 𝑧))) → (ð‘Ķ +s 𝑧) â‰Īs (ð‘Ĩðŋ +s 𝑧))
102 simpll3 1214 . . . . . . . . . . . . . 14 ((((ð‘Ĩ ∈ No ∧ ð‘Ķ ∈ No ∧ 𝑧 ∈ No ) ∧ ((∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ𝑂)) ∧ (∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))((ð‘Ķ +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ <s ð‘Ĩ𝑂) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ))) ∧ (ð‘Ĩðŋ ∈ ( L ‘ð‘Ĩ) ∧ (ð‘Ķ +s 𝑧) â‰Īs (ð‘Ĩðŋ +s 𝑧))) → 𝑧 ∈ No )
103 sleadd1im 27697 . . . . . . . . . . . . . 14 ((ð‘Ķ ∈ No ∧ ð‘Ĩðŋ ∈ No ∧ 𝑧 ∈ No ) → ((ð‘Ķ +s 𝑧) â‰Īs (ð‘Ĩðŋ +s 𝑧) → ð‘Ķ â‰Īs ð‘Ĩðŋ))
10495, 99, 102, 103syl3anc 1371 . . . . . . . . . . . . 13 ((((ð‘Ĩ ∈ No ∧ ð‘Ķ ∈ No ∧ 𝑧 ∈ No ) ∧ ((∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ𝑂)) ∧ (∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))((ð‘Ķ +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ <s ð‘Ĩ𝑂) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ))) ∧ (ð‘Ĩðŋ ∈ ( L ‘ð‘Ĩ) ∧ (ð‘Ķ +s 𝑧) â‰Īs (ð‘Ĩðŋ +s 𝑧))) → ((ð‘Ķ +s 𝑧) â‰Īs (ð‘Ĩðŋ +s 𝑧) → ð‘Ķ â‰Īs ð‘Ĩðŋ))
105101, 104mpd 15 . . . . . . . . . . . 12 ((((ð‘Ĩ ∈ No ∧ ð‘Ķ ∈ No ∧ 𝑧 ∈ No ) ∧ ((∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ𝑂)) ∧ (∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))((ð‘Ķ +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ <s ð‘Ĩ𝑂) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ))) ∧ (ð‘Ĩðŋ ∈ ( L ‘ð‘Ĩ) ∧ (ð‘Ķ +s 𝑧) â‰Īs (ð‘Ĩðŋ +s 𝑧))) → ð‘Ķ â‰Īs ð‘Ĩðŋ)
106 leftval 27583 . . . . . . . . . . . . . . . 16 ( L ‘ð‘Ĩ) = {ð‘Ĩðŋ ∈ ( O ‘( bday ‘ð‘Ĩ)) âˆĢ ð‘Ĩðŋ <s ð‘Ĩ}
107106reqabi 3454 . . . . . . . . . . . . . . 15 (ð‘Ĩðŋ ∈ ( L ‘ð‘Ĩ) ↔ (ð‘Ĩðŋ ∈ ( O ‘( bday ‘ð‘Ĩ)) ∧ ð‘Ĩðŋ <s ð‘Ĩ))
108107simprbi 497 . . . . . . . . . . . . . 14 (ð‘Ĩðŋ ∈ ( L ‘ð‘Ĩ) → ð‘Ĩðŋ <s ð‘Ĩ)
109108adantr 481 . . . . . . . . . . . . 13 ((ð‘Ĩðŋ ∈ ( L ‘ð‘Ĩ) ∧ (ð‘Ķ +s 𝑧) â‰Īs (ð‘Ĩðŋ +s 𝑧)) → ð‘Ĩðŋ <s ð‘Ĩ)
110109adantl 482 . . . . . . . . . . . 12 ((((ð‘Ĩ ∈ No ∧ ð‘Ķ ∈ No ∧ 𝑧 ∈ No ) ∧ ((∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ𝑂)) ∧ (∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))((ð‘Ķ +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ <s ð‘Ĩ𝑂) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ))) ∧ (ð‘Ĩðŋ ∈ ( L ‘ð‘Ĩ) ∧ (ð‘Ķ +s 𝑧) â‰Īs (ð‘Ĩðŋ +s 𝑧))) → ð‘Ĩðŋ <s ð‘Ĩ)
11195, 99, 100, 105, 110slelttrd 27488 . . . . . . . . . . 11 ((((ð‘Ĩ ∈ No ∧ ð‘Ķ ∈ No ∧ 𝑧 ∈ No ) ∧ ((∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ𝑂)) ∧ (∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))((ð‘Ķ +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ <s ð‘Ĩ𝑂) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ))) ∧ (ð‘Ĩðŋ ∈ ( L ‘ð‘Ĩ) ∧ (ð‘Ķ +s 𝑧) â‰Īs (ð‘Ĩðŋ +s 𝑧))) → ð‘Ķ <s ð‘Ĩ)
112111rexlimdvaa 3156 . . . . . . . . . 10 (((ð‘Ĩ ∈ No ∧ ð‘Ķ ∈ No ∧ 𝑧 ∈ No ) ∧ ((∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ𝑂)) ∧ (∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))((ð‘Ķ +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ <s ð‘Ĩ𝑂) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ))) → (∃ð‘Ĩðŋ ∈ ( L ‘ð‘Ĩ)(ð‘Ķ +s 𝑧) â‰Īs (ð‘Ĩðŋ +s 𝑧) → ð‘Ķ <s ð‘Ĩ))
113 simpll2 1213 . . . . . . . . . . . . . 14 ((((ð‘Ĩ ∈ No ∧ ð‘Ķ ∈ No ∧ 𝑧 ∈ No ) ∧ ((∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ𝑂)) ∧ (∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))((ð‘Ķ +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ <s ð‘Ĩ𝑂) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ))) ∧ (𝑧ðŋ ∈ ( L ‘𝑧) ∧ (ð‘Ķ +s 𝑧) â‰Īs (ð‘Ĩ +s 𝑧ðŋ))) → ð‘Ķ ∈ No )
114 leftssno 27600 . . . . . . . . . . . . . . . . 17 ( L ‘𝑧) ⊆ No
115114sseli 3978 . . . . . . . . . . . . . . . 16 (𝑧ðŋ ∈ ( L ‘𝑧) → 𝑧ðŋ ∈ No )
116115adantr 481 . . . . . . . . . . . . . . 15 ((𝑧ðŋ ∈ ( L ‘𝑧) ∧ (ð‘Ķ +s 𝑧) â‰Īs (ð‘Ĩ +s 𝑧ðŋ)) → 𝑧ðŋ ∈ No )
117116adantl 482 . . . . . . . . . . . . . 14 ((((ð‘Ĩ ∈ No ∧ ð‘Ķ ∈ No ∧ 𝑧 ∈ No ) ∧ ((∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ𝑂)) ∧ (∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))((ð‘Ķ +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ <s ð‘Ĩ𝑂) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ))) ∧ (𝑧ðŋ ∈ ( L ‘𝑧) ∧ (ð‘Ķ +s 𝑧) â‰Īs (ð‘Ĩ +s 𝑧ðŋ))) → 𝑧ðŋ ∈ No )
118113, 117addscld 27690 . . . . . . . . . . . . 13 ((((ð‘Ĩ ∈ No ∧ ð‘Ķ ∈ No ∧ 𝑧 ∈ No ) ∧ ((∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ𝑂)) ∧ (∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))((ð‘Ķ +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ <s ð‘Ĩ𝑂) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ))) ∧ (𝑧ðŋ ∈ ( L ‘𝑧) ∧ (ð‘Ķ +s 𝑧) â‰Īs (ð‘Ĩ +s 𝑧ðŋ))) → (ð‘Ķ +s 𝑧ðŋ) ∈ No )
119 simpll3 1214 . . . . . . . . . . . . . 14 ((((ð‘Ĩ ∈ No ∧ ð‘Ķ ∈ No ∧ 𝑧 ∈ No ) ∧ ((∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ𝑂)) ∧ (∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))((ð‘Ķ +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ <s ð‘Ĩ𝑂) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ))) ∧ (𝑧ðŋ ∈ ( L ‘𝑧) ∧ (ð‘Ķ +s 𝑧) â‰Īs (ð‘Ĩ +s 𝑧ðŋ))) → 𝑧 ∈ No )
120113, 119addscld 27690 . . . . . . . . . . . . 13 ((((ð‘Ĩ ∈ No ∧ ð‘Ķ ∈ No ∧ 𝑧 ∈ No ) ∧ ((∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ𝑂)) ∧ (∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))((ð‘Ķ +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ <s ð‘Ĩ𝑂) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ))) ∧ (𝑧ðŋ ∈ ( L ‘𝑧) ∧ (ð‘Ķ +s 𝑧) â‰Īs (ð‘Ĩ +s 𝑧ðŋ))) → (ð‘Ķ +s 𝑧) ∈ No )
121 simpll1 1212 . . . . . . . . . . . . . 14 ((((ð‘Ĩ ∈ No ∧ ð‘Ķ ∈ No ∧ 𝑧 ∈ No ) ∧ ((∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ𝑂)) ∧ (∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))((ð‘Ķ +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ <s ð‘Ĩ𝑂) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ))) ∧ (𝑧ðŋ ∈ ( L ‘𝑧) ∧ (ð‘Ķ +s 𝑧) â‰Īs (ð‘Ĩ +s 𝑧ðŋ))) → ð‘Ĩ ∈ No )
122121, 117addscld 27690 . . . . . . . . . . . . 13 ((((ð‘Ĩ ∈ No ∧ ð‘Ķ ∈ No ∧ 𝑧 ∈ No ) ∧ ((∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ𝑂)) ∧ (∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))((ð‘Ķ +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ <s ð‘Ĩ𝑂) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ))) ∧ (𝑧ðŋ ∈ ( L ‘𝑧) ∧ (ð‘Ķ +s 𝑧) â‰Īs (ð‘Ĩ +s 𝑧ðŋ))) → (ð‘Ĩ +s 𝑧ðŋ) ∈ No )
123 leftval 27583 . . . . . . . . . . . . . . . . . 18 ( L ‘𝑧) = {𝑧ðŋ ∈ ( O ‘( bday ‘𝑧)) âˆĢ 𝑧ðŋ <s 𝑧}
124123reqabi 3454 . . . . . . . . . . . . . . . . 17 (𝑧ðŋ ∈ ( L ‘𝑧) ↔ (𝑧ðŋ ∈ ( O ‘( bday ‘𝑧)) ∧ 𝑧ðŋ <s 𝑧))
125124simprbi 497 . . . . . . . . . . . . . . . 16 (𝑧ðŋ ∈ ( L ‘𝑧) → 𝑧ðŋ <s 𝑧)
126125adantr 481 . . . . . . . . . . . . . . 15 ((𝑧ðŋ ∈ ( L ‘𝑧) ∧ (ð‘Ķ +s 𝑧) â‰Īs (ð‘Ĩ +s 𝑧ðŋ)) → 𝑧ðŋ <s 𝑧)
127126adantl 482 . . . . . . . . . . . . . 14 ((((ð‘Ĩ ∈ No ∧ ð‘Ķ ∈ No ∧ 𝑧 ∈ No ) ∧ ((∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ𝑂)) ∧ (∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))((ð‘Ķ +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ <s ð‘Ĩ𝑂) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ))) ∧ (𝑧ðŋ ∈ ( L ‘𝑧) ∧ (ð‘Ķ +s 𝑧) â‰Īs (ð‘Ĩ +s 𝑧ðŋ))) → 𝑧ðŋ <s 𝑧)
128 sltadd2im 27696 . . . . . . . . . . . . . . 15 ((𝑧ðŋ ∈ No ∧ 𝑧 ∈ No ∧ ð‘Ķ ∈ No ) → (𝑧ðŋ <s 𝑧 → (ð‘Ķ +s 𝑧ðŋ) <s (ð‘Ķ +s 𝑧)))
129117, 119, 113, 128syl3anc 1371 . . . . . . . . . . . . . 14 ((((ð‘Ĩ ∈ No ∧ ð‘Ķ ∈ No ∧ 𝑧 ∈ No ) ∧ ((∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ𝑂)) ∧ (∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))((ð‘Ķ +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ <s ð‘Ĩ𝑂) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ))) ∧ (𝑧ðŋ ∈ ( L ‘𝑧) ∧ (ð‘Ķ +s 𝑧) â‰Īs (ð‘Ĩ +s 𝑧ðŋ))) → (𝑧ðŋ <s 𝑧 → (ð‘Ķ +s 𝑧ðŋ) <s (ð‘Ķ +s 𝑧)))
130127, 129mpd 15 . . . . . . . . . . . . 13 ((((ð‘Ĩ ∈ No ∧ ð‘Ķ ∈ No ∧ 𝑧 ∈ No ) ∧ ((∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ𝑂)) ∧ (∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))((ð‘Ķ +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ <s ð‘Ĩ𝑂) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ))) ∧ (𝑧ðŋ ∈ ( L ‘𝑧) ∧ (ð‘Ķ +s 𝑧) â‰Īs (ð‘Ĩ +s 𝑧ðŋ))) → (ð‘Ķ +s 𝑧ðŋ) <s (ð‘Ķ +s 𝑧))
131 simprr 771 . . . . . . . . . . . . 13 ((((ð‘Ĩ ∈ No ∧ ð‘Ķ ∈ No ∧ 𝑧 ∈ No ) ∧ ((∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ𝑂)) ∧ (∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))((ð‘Ķ +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ <s ð‘Ĩ𝑂) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ))) ∧ (𝑧ðŋ ∈ ( L ‘𝑧) ∧ (ð‘Ķ +s 𝑧) â‰Īs (ð‘Ĩ +s 𝑧ðŋ))) → (ð‘Ķ +s 𝑧) â‰Īs (ð‘Ĩ +s 𝑧ðŋ))
132118, 120, 122, 130, 131sltletrd 27487 . . . . . . . . . . . 12 ((((ð‘Ĩ ∈ No ∧ ð‘Ķ ∈ No ∧ 𝑧 ∈ No ) ∧ ((∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ𝑂)) ∧ (∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))((ð‘Ķ +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ <s ð‘Ĩ𝑂) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ))) ∧ (𝑧ðŋ ∈ ( L ‘𝑧) ∧ (ð‘Ķ +s 𝑧) â‰Īs (ð‘Ĩ +s 𝑧ðŋ))) → (ð‘Ķ +s 𝑧ðŋ) <s (ð‘Ĩ +s 𝑧ðŋ))
133 oveq2 7419 . . . . . . . . . . . . . . 15 (𝑧𝑂 = 𝑧ðŋ → (ð‘Ķ +s 𝑧𝑂) = (ð‘Ķ +s 𝑧ðŋ))
134 oveq2 7419 . . . . . . . . . . . . . . 15 (𝑧𝑂 = 𝑧ðŋ → (ð‘Ĩ +s 𝑧𝑂) = (ð‘Ĩ +s 𝑧ðŋ))
135133, 134breq12d 5161 . . . . . . . . . . . . . 14 (𝑧𝑂 = 𝑧ðŋ → ((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) ↔ (ð‘Ķ +s 𝑧ðŋ) <s (ð‘Ĩ +s 𝑧ðŋ)))
136135imbi1d 341 . . . . . . . . . . . . 13 (𝑧𝑂 = 𝑧ðŋ → (((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ) ↔ ((ð‘Ķ +s 𝑧ðŋ) <s (ð‘Ĩ +s 𝑧ðŋ) → ð‘Ķ <s ð‘Ĩ)))
137 simplr3 1217 . . . . . . . . . . . . 13 ((((ð‘Ĩ ∈ No ∧ ð‘Ķ ∈ No ∧ 𝑧 ∈ No ) ∧ ((∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ𝑂)) ∧ (∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))((ð‘Ķ +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ <s ð‘Ĩ𝑂) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ))) ∧ (𝑧ðŋ ∈ ( L ‘𝑧) ∧ (ð‘Ķ +s 𝑧) â‰Īs (ð‘Ĩ +s 𝑧ðŋ))) → ∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ))
138 simprl 769 . . . . . . . . . . . . . 14 ((((ð‘Ĩ ∈ No ∧ ð‘Ķ ∈ No ∧ 𝑧 ∈ No ) ∧ ((∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ𝑂)) ∧ (∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))((ð‘Ķ +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ <s ð‘Ĩ𝑂) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ))) ∧ (𝑧ðŋ ∈ ( L ‘𝑧) ∧ (ð‘Ķ +s 𝑧) â‰Īs (ð‘Ĩ +s 𝑧ðŋ))) → 𝑧ðŋ ∈ ( L ‘𝑧))
139 elun1 4176 . . . . . . . . . . . . . 14 (𝑧ðŋ ∈ ( L ‘𝑧) → 𝑧ðŋ ∈ (( L ‘𝑧) ∊ ( R ‘𝑧)))
140138, 139syl 17 . . . . . . . . . . . . 13 ((((ð‘Ĩ ∈ No ∧ ð‘Ķ ∈ No ∧ 𝑧 ∈ No ) ∧ ((∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ𝑂)) ∧ (∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))((ð‘Ķ +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ <s ð‘Ĩ𝑂) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ))) ∧ (𝑧ðŋ ∈ ( L ‘𝑧) ∧ (ð‘Ķ +s 𝑧) â‰Īs (ð‘Ĩ +s 𝑧ðŋ))) → 𝑧ðŋ ∈ (( L ‘𝑧) ∊ ( R ‘𝑧)))
141136, 137, 140rspcdva 3613 . . . . . . . . . . . 12 ((((ð‘Ĩ ∈ No ∧ ð‘Ķ ∈ No ∧ 𝑧 ∈ No ) ∧ ((∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ𝑂)) ∧ (∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))((ð‘Ķ +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ <s ð‘Ĩ𝑂) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ))) ∧ (𝑧ðŋ ∈ ( L ‘𝑧) ∧ (ð‘Ķ +s 𝑧) â‰Īs (ð‘Ĩ +s 𝑧ðŋ))) → ((ð‘Ķ +s 𝑧ðŋ) <s (ð‘Ĩ +s 𝑧ðŋ) → ð‘Ķ <s ð‘Ĩ))
142132, 141mpd 15 . . . . . . . . . . 11 ((((ð‘Ĩ ∈ No ∧ ð‘Ķ ∈ No ∧ 𝑧 ∈ No ) ∧ ((∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ𝑂)) ∧ (∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))((ð‘Ķ +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ <s ð‘Ĩ𝑂) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ))) ∧ (𝑧ðŋ ∈ ( L ‘𝑧) ∧ (ð‘Ķ +s 𝑧) â‰Īs (ð‘Ĩ +s 𝑧ðŋ))) → ð‘Ķ <s ð‘Ĩ)
143142rexlimdvaa 3156 . . . . . . . . . 10 (((ð‘Ĩ ∈ No ∧ ð‘Ķ ∈ No ∧ 𝑧 ∈ No ) ∧ ((∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ𝑂)) ∧ (∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))((ð‘Ķ +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ <s ð‘Ĩ𝑂) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ))) → (∃𝑧ðŋ ∈ ( L ‘𝑧)(ð‘Ķ +s 𝑧) â‰Īs (ð‘Ĩ +s 𝑧ðŋ) → ð‘Ķ <s ð‘Ĩ))
144112, 143jaod 857 . . . . . . . . 9 (((ð‘Ĩ ∈ No ∧ ð‘Ķ ∈ No ∧ 𝑧 ∈ No ) ∧ ((∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ𝑂)) ∧ (∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))((ð‘Ķ +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ <s ð‘Ĩ𝑂) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ))) → ((∃ð‘Ĩðŋ ∈ ( L ‘ð‘Ĩ)(ð‘Ķ +s 𝑧) â‰Īs (ð‘Ĩðŋ +s 𝑧) âˆĻ ∃𝑧ðŋ ∈ ( L ‘𝑧)(ð‘Ķ +s 𝑧) â‰Īs (ð‘Ĩ +s 𝑧ðŋ)) → ð‘Ķ <s ð‘Ĩ))
14594, 144biimtrid 241 . . . . . . . 8 (((ð‘Ĩ ∈ No ∧ ð‘Ķ ∈ No ∧ 𝑧 ∈ No ) ∧ ((∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ𝑂)) ∧ (∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))((ð‘Ķ +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ <s ð‘Ĩ𝑂) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ))) → (∃𝑝 ∈ ({𝑎 âˆĢ ∃ð‘Ĩðŋ ∈ ( L ‘ð‘Ĩ)𝑎 = (ð‘Ĩðŋ +s 𝑧)} ∊ {𝑏 âˆĢ ∃𝑧ðŋ ∈ ( L ‘𝑧)𝑏 = (ð‘Ĩ +s 𝑧ðŋ)})(ð‘Ķ +s 𝑧) â‰Īs 𝑝 → ð‘Ķ <s ð‘Ĩ))
146 rexun 4190 . . . . . . . . . 10 (∃𝑞 ∈ ({𝑐 âˆĢ ∃ð‘Ķ𝑅 ∈ ( R ‘ð‘Ķ)𝑐 = (ð‘Ķ𝑅 +s 𝑧)} ∊ {𝑑 âˆĢ ∃𝑧𝑅 ∈ ( R ‘𝑧)𝑑 = (ð‘Ķ +s 𝑧𝑅)})𝑞 â‰Īs (ð‘Ĩ +s 𝑧) ↔ (∃𝑞 ∈ {𝑐 âˆĢ ∃ð‘Ķ𝑅 ∈ ( R ‘ð‘Ķ)𝑐 = (ð‘Ķ𝑅 +s 𝑧)}𝑞 â‰Īs (ð‘Ĩ +s 𝑧) âˆĻ ∃𝑞 ∈ {𝑑 âˆĢ ∃𝑧𝑅 ∈ ( R ‘𝑧)𝑑 = (ð‘Ķ +s 𝑧𝑅)}𝑞 â‰Īs (ð‘Ĩ +s 𝑧)))
147 eqeq1 2736 . . . . . . . . . . . . . 14 (𝑐 = 𝑞 → (𝑐 = (ð‘Ķ𝑅 +s 𝑧) ↔ 𝑞 = (ð‘Ķ𝑅 +s 𝑧)))
148147rexbidv 3178 . . . . . . . . . . . . 13 (𝑐 = 𝑞 → (∃ð‘Ķ𝑅 ∈ ( R ‘ð‘Ķ)𝑐 = (ð‘Ķ𝑅 +s 𝑧) ↔ ∃ð‘Ķ𝑅 ∈ ( R ‘ð‘Ķ)𝑞 = (ð‘Ķ𝑅 +s 𝑧)))
149148rexab 3690 . . . . . . . . . . . 12 (∃𝑞 ∈ {𝑐 âˆĢ ∃ð‘Ķ𝑅 ∈ ( R ‘ð‘Ķ)𝑐 = (ð‘Ķ𝑅 +s 𝑧)}𝑞 â‰Īs (ð‘Ĩ +s 𝑧) ↔ ∃𝑞(∃ð‘Ķ𝑅 ∈ ( R ‘ð‘Ķ)𝑞 = (ð‘Ķ𝑅 +s 𝑧) ∧ 𝑞 â‰Īs (ð‘Ĩ +s 𝑧)))
150 rexcom4 3285 . . . . . . . . . . . . . 14 (∃ð‘Ķ𝑅 ∈ ( R ‘ð‘Ķ)∃𝑞(𝑞 = (ð‘Ķ𝑅 +s 𝑧) ∧ 𝑞 â‰Īs (ð‘Ĩ +s 𝑧)) ↔ ∃𝑞∃ð‘Ķ𝑅 ∈ ( R ‘ð‘Ķ)(𝑞 = (ð‘Ķ𝑅 +s 𝑧) ∧ 𝑞 â‰Īs (ð‘Ĩ +s 𝑧)))
151 r19.41v 3188 . . . . . . . . . . . . . . 15 (∃ð‘Ķ𝑅 ∈ ( R ‘ð‘Ķ)(𝑞 = (ð‘Ķ𝑅 +s 𝑧) ∧ 𝑞 â‰Īs (ð‘Ĩ +s 𝑧)) ↔ (∃ð‘Ķ𝑅 ∈ ( R ‘ð‘Ķ)𝑞 = (ð‘Ķ𝑅 +s 𝑧) ∧ 𝑞 â‰Īs (ð‘Ĩ +s 𝑧)))
152151exbii 1850 . . . . . . . . . . . . . 14 (∃𝑞∃ð‘Ķ𝑅 ∈ ( R ‘ð‘Ķ)(𝑞 = (ð‘Ķ𝑅 +s 𝑧) ∧ 𝑞 â‰Īs (ð‘Ĩ +s 𝑧)) ↔ ∃𝑞(∃ð‘Ķ𝑅 ∈ ( R ‘ð‘Ķ)𝑞 = (ð‘Ķ𝑅 +s 𝑧) ∧ 𝑞 â‰Īs (ð‘Ĩ +s 𝑧)))
153150, 152bitri 274 . . . . . . . . . . . . 13 (∃ð‘Ķ𝑅 ∈ ( R ‘ð‘Ķ)∃𝑞(𝑞 = (ð‘Ķ𝑅 +s 𝑧) ∧ 𝑞 â‰Īs (ð‘Ĩ +s 𝑧)) ↔ ∃𝑞(∃ð‘Ķ𝑅 ∈ ( R ‘ð‘Ķ)𝑞 = (ð‘Ķ𝑅 +s 𝑧) ∧ 𝑞 â‰Īs (ð‘Ĩ +s 𝑧)))
154 ovex 7444 . . . . . . . . . . . . . . 15 (ð‘Ķ𝑅 +s 𝑧) ∈ V
155 breq1 5151 . . . . . . . . . . . . . . 15 (𝑞 = (ð‘Ķ𝑅 +s 𝑧) → (𝑞 â‰Īs (ð‘Ĩ +s 𝑧) ↔ (ð‘Ķ𝑅 +s 𝑧) â‰Īs (ð‘Ĩ +s 𝑧)))
156154, 155ceqsexv 3525 . . . . . . . . . . . . . 14 (∃𝑞(𝑞 = (ð‘Ķ𝑅 +s 𝑧) ∧ 𝑞 â‰Īs (ð‘Ĩ +s 𝑧)) ↔ (ð‘Ķ𝑅 +s 𝑧) â‰Īs (ð‘Ĩ +s 𝑧))
157156rexbii 3094 . . . . . . . . . . . . 13 (∃ð‘Ķ𝑅 ∈ ( R ‘ð‘Ķ)∃𝑞(𝑞 = (ð‘Ķ𝑅 +s 𝑧) ∧ 𝑞 â‰Īs (ð‘Ĩ +s 𝑧)) ↔ ∃ð‘Ķ𝑅 ∈ ( R ‘ð‘Ķ)(ð‘Ķ𝑅 +s 𝑧) â‰Īs (ð‘Ĩ +s 𝑧))
158153, 157bitr3i 276 . . . . . . . . . . . 12 (∃𝑞(∃ð‘Ķ𝑅 ∈ ( R ‘ð‘Ķ)𝑞 = (ð‘Ķ𝑅 +s 𝑧) ∧ 𝑞 â‰Īs (ð‘Ĩ +s 𝑧)) ↔ ∃ð‘Ķ𝑅 ∈ ( R ‘ð‘Ķ)(ð‘Ķ𝑅 +s 𝑧) â‰Īs (ð‘Ĩ +s 𝑧))
159149, 158bitri 274 . . . . . . . . . . 11 (∃𝑞 ∈ {𝑐 âˆĢ ∃ð‘Ķ𝑅 ∈ ( R ‘ð‘Ķ)𝑐 = (ð‘Ķ𝑅 +s 𝑧)}𝑞 â‰Īs (ð‘Ĩ +s 𝑧) ↔ ∃ð‘Ķ𝑅 ∈ ( R ‘ð‘Ķ)(ð‘Ķ𝑅 +s 𝑧) â‰Īs (ð‘Ĩ +s 𝑧))
160 eqeq1 2736 . . . . . . . . . . . . . 14 (𝑑 = 𝑞 → (𝑑 = (ð‘Ķ +s 𝑧𝑅) ↔ 𝑞 = (ð‘Ķ +s 𝑧𝑅)))
161160rexbidv 3178 . . . . . . . . . . . . 13 (𝑑 = 𝑞 → (∃𝑧𝑅 ∈ ( R ‘𝑧)𝑑 = (ð‘Ķ +s 𝑧𝑅) ↔ ∃𝑧𝑅 ∈ ( R ‘𝑧)𝑞 = (ð‘Ķ +s 𝑧𝑅)))
162161rexab 3690 . . . . . . . . . . . 12 (∃𝑞 ∈ {𝑑 âˆĢ ∃𝑧𝑅 ∈ ( R ‘𝑧)𝑑 = (ð‘Ķ +s 𝑧𝑅)}𝑞 â‰Īs (ð‘Ĩ +s 𝑧) ↔ ∃𝑞(∃𝑧𝑅 ∈ ( R ‘𝑧)𝑞 = (ð‘Ķ +s 𝑧𝑅) ∧ 𝑞 â‰Īs (ð‘Ĩ +s 𝑧)))
163 rexcom4 3285 . . . . . . . . . . . . . 14 (∃𝑧𝑅 ∈ ( R ‘𝑧)∃𝑞(𝑞 = (ð‘Ķ +s 𝑧𝑅) ∧ 𝑞 â‰Īs (ð‘Ĩ +s 𝑧)) ↔ ∃𝑞∃𝑧𝑅 ∈ ( R ‘𝑧)(𝑞 = (ð‘Ķ +s 𝑧𝑅) ∧ 𝑞 â‰Īs (ð‘Ĩ +s 𝑧)))
164 r19.41v 3188 . . . . . . . . . . . . . . 15 (∃𝑧𝑅 ∈ ( R ‘𝑧)(𝑞 = (ð‘Ķ +s 𝑧𝑅) ∧ 𝑞 â‰Īs (ð‘Ĩ +s 𝑧)) ↔ (∃𝑧𝑅 ∈ ( R ‘𝑧)𝑞 = (ð‘Ķ +s 𝑧𝑅) ∧ 𝑞 â‰Īs (ð‘Ĩ +s 𝑧)))
165164exbii 1850 . . . . . . . . . . . . . 14 (∃𝑞∃𝑧𝑅 ∈ ( R ‘𝑧)(𝑞 = (ð‘Ķ +s 𝑧𝑅) ∧ 𝑞 â‰Īs (ð‘Ĩ +s 𝑧)) ↔ ∃𝑞(∃𝑧𝑅 ∈ ( R ‘𝑧)𝑞 = (ð‘Ķ +s 𝑧𝑅) ∧ 𝑞 â‰Īs (ð‘Ĩ +s 𝑧)))
166163, 165bitri 274 . . . . . . . . . . . . 13 (∃𝑧𝑅 ∈ ( R ‘𝑧)∃𝑞(𝑞 = (ð‘Ķ +s 𝑧𝑅) ∧ 𝑞 â‰Īs (ð‘Ĩ +s 𝑧)) ↔ ∃𝑞(∃𝑧𝑅 ∈ ( R ‘𝑧)𝑞 = (ð‘Ķ +s 𝑧𝑅) ∧ 𝑞 â‰Īs (ð‘Ĩ +s 𝑧)))
167 ovex 7444 . . . . . . . . . . . . . . 15 (ð‘Ķ +s 𝑧𝑅) ∈ V
168 breq1 5151 . . . . . . . . . . . . . . 15 (𝑞 = (ð‘Ķ +s 𝑧𝑅) → (𝑞 â‰Īs (ð‘Ĩ +s 𝑧) ↔ (ð‘Ķ +s 𝑧𝑅) â‰Īs (ð‘Ĩ +s 𝑧)))
169167, 168ceqsexv 3525 . . . . . . . . . . . . . 14 (∃𝑞(𝑞 = (ð‘Ķ +s 𝑧𝑅) ∧ 𝑞 â‰Īs (ð‘Ĩ +s 𝑧)) ↔ (ð‘Ķ +s 𝑧𝑅) â‰Īs (ð‘Ĩ +s 𝑧))
170169rexbii 3094 . . . . . . . . . . . . 13 (∃𝑧𝑅 ∈ ( R ‘𝑧)∃𝑞(𝑞 = (ð‘Ķ +s 𝑧𝑅) ∧ 𝑞 â‰Īs (ð‘Ĩ +s 𝑧)) ↔ ∃𝑧𝑅 ∈ ( R ‘𝑧)(ð‘Ķ +s 𝑧𝑅) â‰Īs (ð‘Ĩ +s 𝑧))
171166, 170bitr3i 276 . . . . . . . . . . . 12 (∃𝑞(∃𝑧𝑅 ∈ ( R ‘𝑧)𝑞 = (ð‘Ķ +s 𝑧𝑅) ∧ 𝑞 â‰Īs (ð‘Ĩ +s 𝑧)) ↔ ∃𝑧𝑅 ∈ ( R ‘𝑧)(ð‘Ķ +s 𝑧𝑅) â‰Īs (ð‘Ĩ +s 𝑧))
172162, 171bitri 274 . . . . . . . . . . 11 (∃𝑞 ∈ {𝑑 âˆĢ ∃𝑧𝑅 ∈ ( R ‘𝑧)𝑑 = (ð‘Ķ +s 𝑧𝑅)}𝑞 â‰Īs (ð‘Ĩ +s 𝑧) ↔ ∃𝑧𝑅 ∈ ( R ‘𝑧)(ð‘Ķ +s 𝑧𝑅) â‰Īs (ð‘Ĩ +s 𝑧))
173159, 172orbi12i 913 . . . . . . . . . 10 ((∃𝑞 ∈ {𝑐 âˆĢ ∃ð‘Ķ𝑅 ∈ ( R ‘ð‘Ķ)𝑐 = (ð‘Ķ𝑅 +s 𝑧)}𝑞 â‰Īs (ð‘Ĩ +s 𝑧) âˆĻ ∃𝑞 ∈ {𝑑 âˆĢ ∃𝑧𝑅 ∈ ( R ‘𝑧)𝑑 = (ð‘Ķ +s 𝑧𝑅)}𝑞 â‰Īs (ð‘Ĩ +s 𝑧)) ↔ (∃ð‘Ķ𝑅 ∈ ( R ‘ð‘Ķ)(ð‘Ķ𝑅 +s 𝑧) â‰Īs (ð‘Ĩ +s 𝑧) âˆĻ ∃𝑧𝑅 ∈ ( R ‘𝑧)(ð‘Ķ +s 𝑧𝑅) â‰Īs (ð‘Ĩ +s 𝑧)))
174146, 173bitri 274 . . . . . . . . 9 (∃𝑞 ∈ ({𝑐 âˆĢ ∃ð‘Ķ𝑅 ∈ ( R ‘ð‘Ķ)𝑐 = (ð‘Ķ𝑅 +s 𝑧)} ∊ {𝑑 âˆĢ ∃𝑧𝑅 ∈ ( R ‘𝑧)𝑑 = (ð‘Ķ +s 𝑧𝑅)})𝑞 â‰Īs (ð‘Ĩ +s 𝑧) ↔ (∃ð‘Ķ𝑅 ∈ ( R ‘ð‘Ķ)(ð‘Ķ𝑅 +s 𝑧) â‰Īs (ð‘Ĩ +s 𝑧) âˆĻ ∃𝑧𝑅 ∈ ( R ‘𝑧)(ð‘Ķ +s 𝑧𝑅) â‰Īs (ð‘Ĩ +s 𝑧)))
175 simpll2 1213 . . . . . . . . . . . 12 ((((ð‘Ĩ ∈ No ∧ ð‘Ķ ∈ No ∧ 𝑧 ∈ No ) ∧ ((∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ𝑂)) ∧ (∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))((ð‘Ķ +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ <s ð‘Ĩ𝑂) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ))) ∧ (ð‘Ķ𝑅 ∈ ( R ‘ð‘Ķ) ∧ (ð‘Ķ𝑅 +s 𝑧) â‰Īs (ð‘Ĩ +s 𝑧))) → ð‘Ķ ∈ No )
176 rightssno 27601 . . . . . . . . . . . . . . 15 ( R ‘ð‘Ķ) ⊆ No
177176sseli 3978 . . . . . . . . . . . . . 14 (ð‘Ķ𝑅 ∈ ( R ‘ð‘Ķ) → ð‘Ķ𝑅 ∈ No )
178177adantr 481 . . . . . . . . . . . . 13 ((ð‘Ķ𝑅 ∈ ( R ‘ð‘Ķ) ∧ (ð‘Ķ𝑅 +s 𝑧) â‰Īs (ð‘Ĩ +s 𝑧)) → ð‘Ķ𝑅 ∈ No )
179178adantl 482 . . . . . . . . . . . 12 ((((ð‘Ĩ ∈ No ∧ ð‘Ķ ∈ No ∧ 𝑧 ∈ No ) ∧ ((∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ𝑂)) ∧ (∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))((ð‘Ķ +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ <s ð‘Ĩ𝑂) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ))) ∧ (ð‘Ķ𝑅 ∈ ( R ‘ð‘Ķ) ∧ (ð‘Ķ𝑅 +s 𝑧) â‰Īs (ð‘Ĩ +s 𝑧))) → ð‘Ķ𝑅 ∈ No )
180 simpll1 1212 . . . . . . . . . . . 12 ((((ð‘Ĩ ∈ No ∧ ð‘Ķ ∈ No ∧ 𝑧 ∈ No ) ∧ ((∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ𝑂)) ∧ (∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))((ð‘Ķ +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ <s ð‘Ĩ𝑂) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ))) ∧ (ð‘Ķ𝑅 ∈ ( R ‘ð‘Ķ) ∧ (ð‘Ķ𝑅 +s 𝑧) â‰Īs (ð‘Ĩ +s 𝑧))) → ð‘Ĩ ∈ No )
181 rightval 27584 . . . . . . . . . . . . . . . 16 ( R ‘ð‘Ķ) = {ð‘Ķ𝑅 ∈ ( O ‘( bday ‘ð‘Ķ)) âˆĢ ð‘Ķ <s ð‘Ķ𝑅}
182181reqabi 3454 . . . . . . . . . . . . . . 15 (ð‘Ķ𝑅 ∈ ( R ‘ð‘Ķ) ↔ (ð‘Ķ𝑅 ∈ ( O ‘( bday ‘ð‘Ķ)) ∧ ð‘Ķ <s ð‘Ķ𝑅))
183182simprbi 497 . . . . . . . . . . . . . 14 (ð‘Ķ𝑅 ∈ ( R ‘ð‘Ķ) → ð‘Ķ <s ð‘Ķ𝑅)
184183adantr 481 . . . . . . . . . . . . 13 ((ð‘Ķ𝑅 ∈ ( R ‘ð‘Ķ) ∧ (ð‘Ķ𝑅 +s 𝑧) â‰Īs (ð‘Ĩ +s 𝑧)) → ð‘Ķ <s ð‘Ķ𝑅)
185184adantl 482 . . . . . . . . . . . 12 ((((ð‘Ĩ ∈ No ∧ ð‘Ķ ∈ No ∧ 𝑧 ∈ No ) ∧ ((∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ𝑂)) ∧ (∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))((ð‘Ķ +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ <s ð‘Ĩ𝑂) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ))) ∧ (ð‘Ķ𝑅 ∈ ( R ‘ð‘Ķ) ∧ (ð‘Ķ𝑅 +s 𝑧) â‰Īs (ð‘Ĩ +s 𝑧))) → ð‘Ķ <s ð‘Ķ𝑅)
186 simprr 771 . . . . . . . . . . . . 13 ((((ð‘Ĩ ∈ No ∧ ð‘Ķ ∈ No ∧ 𝑧 ∈ No ) ∧ ((∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ𝑂)) ∧ (∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))((ð‘Ķ +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ <s ð‘Ĩ𝑂) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ))) ∧ (ð‘Ķ𝑅 ∈ ( R ‘ð‘Ķ) ∧ (ð‘Ķ𝑅 +s 𝑧) â‰Īs (ð‘Ĩ +s 𝑧))) → (ð‘Ķ𝑅 +s 𝑧) â‰Īs (ð‘Ĩ +s 𝑧))
187 simpll3 1214 . . . . . . . . . . . . . 14 ((((ð‘Ĩ ∈ No ∧ ð‘Ķ ∈ No ∧ 𝑧 ∈ No ) ∧ ((∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ𝑂)) ∧ (∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))((ð‘Ķ +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ <s ð‘Ĩ𝑂) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ))) ∧ (ð‘Ķ𝑅 ∈ ( R ‘ð‘Ķ) ∧ (ð‘Ķ𝑅 +s 𝑧) â‰Īs (ð‘Ĩ +s 𝑧))) → 𝑧 ∈ No )
188 sleadd1im 27697 . . . . . . . . . . . . . 14 ((ð‘Ķ𝑅 ∈ No ∧ ð‘Ĩ ∈ No ∧ 𝑧 ∈ No ) → ((ð‘Ķ𝑅 +s 𝑧) â‰Īs (ð‘Ĩ +s 𝑧) → ð‘Ķ𝑅 â‰Īs ð‘Ĩ))
189179, 180, 187, 188syl3anc 1371 . . . . . . . . . . . . 13 ((((ð‘Ĩ ∈ No ∧ ð‘Ķ ∈ No ∧ 𝑧 ∈ No ) ∧ ((∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ𝑂)) ∧ (∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))((ð‘Ķ +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ <s ð‘Ĩ𝑂) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ))) ∧ (ð‘Ķ𝑅 ∈ ( R ‘ð‘Ķ) ∧ (ð‘Ķ𝑅 +s 𝑧) â‰Īs (ð‘Ĩ +s 𝑧))) → ((ð‘Ķ𝑅 +s 𝑧) â‰Īs (ð‘Ĩ +s 𝑧) → ð‘Ķ𝑅 â‰Īs ð‘Ĩ))
190186, 189mpd 15 . . . . . . . . . . . 12 ((((ð‘Ĩ ∈ No ∧ ð‘Ķ ∈ No ∧ 𝑧 ∈ No ) ∧ ((∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ𝑂)) ∧ (∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))((ð‘Ķ +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ <s ð‘Ĩ𝑂) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ))) ∧ (ð‘Ķ𝑅 ∈ ( R ‘ð‘Ķ) ∧ (ð‘Ķ𝑅 +s 𝑧) â‰Īs (ð‘Ĩ +s 𝑧))) → ð‘Ķ𝑅 â‰Īs ð‘Ĩ)
191175, 179, 180, 185, 190sltletrd 27487 . . . . . . . . . . 11 ((((ð‘Ĩ ∈ No ∧ ð‘Ķ ∈ No ∧ 𝑧 ∈ No ) ∧ ((∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ𝑂)) ∧ (∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))((ð‘Ķ +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ <s ð‘Ĩ𝑂) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ))) ∧ (ð‘Ķ𝑅 ∈ ( R ‘ð‘Ķ) ∧ (ð‘Ķ𝑅 +s 𝑧) â‰Īs (ð‘Ĩ +s 𝑧))) → ð‘Ķ <s ð‘Ĩ)
192191rexlimdvaa 3156 . . . . . . . . . 10 (((ð‘Ĩ ∈ No ∧ ð‘Ķ ∈ No ∧ 𝑧 ∈ No ) ∧ ((∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ𝑂)) ∧ (∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))((ð‘Ķ +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ <s ð‘Ĩ𝑂) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ))) → (∃ð‘Ķ𝑅 ∈ ( R ‘ð‘Ķ)(ð‘Ķ𝑅 +s 𝑧) â‰Īs (ð‘Ĩ +s 𝑧) → ð‘Ķ <s ð‘Ĩ))
193 simpll2 1213 . . . . . . . . . . . . . 14 ((((ð‘Ĩ ∈ No ∧ ð‘Ķ ∈ No ∧ 𝑧 ∈ No ) ∧ ((∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ𝑂)) ∧ (∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))((ð‘Ķ +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ <s ð‘Ĩ𝑂) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ))) ∧ (𝑧𝑅 ∈ ( R ‘𝑧) ∧ (ð‘Ķ +s 𝑧𝑅) â‰Īs (ð‘Ĩ +s 𝑧))) → ð‘Ķ ∈ No )
194 rightssno 27601 . . . . . . . . . . . . . . . . 17 ( R ‘𝑧) ⊆ No
195194sseli 3978 . . . . . . . . . . . . . . . 16 (𝑧𝑅 ∈ ( R ‘𝑧) → 𝑧𝑅 ∈ No )
196195adantr 481 . . . . . . . . . . . . . . 15 ((𝑧𝑅 ∈ ( R ‘𝑧) ∧ (ð‘Ķ +s 𝑧𝑅) â‰Īs (ð‘Ĩ +s 𝑧)) → 𝑧𝑅 ∈ No )
197196adantl 482 . . . . . . . . . . . . . 14 ((((ð‘Ĩ ∈ No ∧ ð‘Ķ ∈ No ∧ 𝑧 ∈ No ) ∧ ((∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ𝑂)) ∧ (∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))((ð‘Ķ +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ <s ð‘Ĩ𝑂) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ))) ∧ (𝑧𝑅 ∈ ( R ‘𝑧) ∧ (ð‘Ķ +s 𝑧𝑅) â‰Īs (ð‘Ĩ +s 𝑧))) → 𝑧𝑅 ∈ No )
198193, 197addscld 27690 . . . . . . . . . . . . 13 ((((ð‘Ĩ ∈ No ∧ ð‘Ķ ∈ No ∧ 𝑧 ∈ No ) ∧ ((∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ𝑂)) ∧ (∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))((ð‘Ķ +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ <s ð‘Ĩ𝑂) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ))) ∧ (𝑧𝑅 ∈ ( R ‘𝑧) ∧ (ð‘Ķ +s 𝑧𝑅) â‰Īs (ð‘Ĩ +s 𝑧))) → (ð‘Ķ +s 𝑧𝑅) ∈ No )
199 simpll1 1212 . . . . . . . . . . . . . 14 ((((ð‘Ĩ ∈ No ∧ ð‘Ķ ∈ No ∧ 𝑧 ∈ No ) ∧ ((∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ𝑂)) ∧ (∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))((ð‘Ķ +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ <s ð‘Ĩ𝑂) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ))) ∧ (𝑧𝑅 ∈ ( R ‘𝑧) ∧ (ð‘Ķ +s 𝑧𝑅) â‰Īs (ð‘Ĩ +s 𝑧))) → ð‘Ĩ ∈ No )
200 simpll3 1214 . . . . . . . . . . . . . 14 ((((ð‘Ĩ ∈ No ∧ ð‘Ķ ∈ No ∧ 𝑧 ∈ No ) ∧ ((∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ𝑂)) ∧ (∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))((ð‘Ķ +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ <s ð‘Ĩ𝑂) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ))) ∧ (𝑧𝑅 ∈ ( R ‘𝑧) ∧ (ð‘Ķ +s 𝑧𝑅) â‰Īs (ð‘Ĩ +s 𝑧))) → 𝑧 ∈ No )
201199, 200addscld 27690 . . . . . . . . . . . . 13 ((((ð‘Ĩ ∈ No ∧ ð‘Ķ ∈ No ∧ 𝑧 ∈ No ) ∧ ((∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ𝑂)) ∧ (∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))((ð‘Ķ +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ <s ð‘Ĩ𝑂) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ))) ∧ (𝑧𝑅 ∈ ( R ‘𝑧) ∧ (ð‘Ķ +s 𝑧𝑅) â‰Īs (ð‘Ĩ +s 𝑧))) → (ð‘Ĩ +s 𝑧) ∈ No )
202199, 197addscld 27690 . . . . . . . . . . . . 13 ((((ð‘Ĩ ∈ No ∧ ð‘Ķ ∈ No ∧ 𝑧 ∈ No ) ∧ ((∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ𝑂)) ∧ (∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))((ð‘Ķ +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ <s ð‘Ĩ𝑂) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ))) ∧ (𝑧𝑅 ∈ ( R ‘𝑧) ∧ (ð‘Ķ +s 𝑧𝑅) â‰Īs (ð‘Ĩ +s 𝑧))) → (ð‘Ĩ +s 𝑧𝑅) ∈ No )
203 simprr 771 . . . . . . . . . . . . 13 ((((ð‘Ĩ ∈ No ∧ ð‘Ķ ∈ No ∧ 𝑧 ∈ No ) ∧ ((∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ𝑂)) ∧ (∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))((ð‘Ķ +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ <s ð‘Ĩ𝑂) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ))) ∧ (𝑧𝑅 ∈ ( R ‘𝑧) ∧ (ð‘Ķ +s 𝑧𝑅) â‰Īs (ð‘Ĩ +s 𝑧))) → (ð‘Ķ +s 𝑧𝑅) â‰Īs (ð‘Ĩ +s 𝑧))
204200, 197, 1993jca 1128 . . . . . . . . . . . . . 14 ((((ð‘Ĩ ∈ No ∧ ð‘Ķ ∈ No ∧ 𝑧 ∈ No ) ∧ ((∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ𝑂)) ∧ (∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))((ð‘Ķ +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ <s ð‘Ĩ𝑂) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ))) ∧ (𝑧𝑅 ∈ ( R ‘𝑧) ∧ (ð‘Ķ +s 𝑧𝑅) â‰Īs (ð‘Ĩ +s 𝑧))) → (𝑧 ∈ No ∧ 𝑧𝑅 ∈ No ∧ ð‘Ĩ ∈ No ))
205 rightval 27584 . . . . . . . . . . . . . . . . . 18 ( R ‘𝑧) = {𝑧𝑅 ∈ ( O ‘( bday ‘𝑧)) âˆĢ 𝑧 <s 𝑧𝑅}
206205reqabi 3454 . . . . . . . . . . . . . . . . 17 (𝑧𝑅 ∈ ( R ‘𝑧) ↔ (𝑧𝑅 ∈ ( O ‘( bday ‘𝑧)) ∧ 𝑧 <s 𝑧𝑅))
207206simprbi 497 . . . . . . . . . . . . . . . 16 (𝑧𝑅 ∈ ( R ‘𝑧) → 𝑧 <s 𝑧𝑅)
208207adantr 481 . . . . . . . . . . . . . . 15 ((𝑧𝑅 ∈ ( R ‘𝑧) ∧ (ð‘Ķ +s 𝑧𝑅) â‰Īs (ð‘Ĩ +s 𝑧)) → 𝑧 <s 𝑧𝑅)
209208adantl 482 . . . . . . . . . . . . . 14 ((((ð‘Ĩ ∈ No ∧ ð‘Ķ ∈ No ∧ 𝑧 ∈ No ) ∧ ((∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ𝑂)) ∧ (∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))((ð‘Ķ +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ <s ð‘Ĩ𝑂) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ))) ∧ (𝑧𝑅 ∈ ( R ‘𝑧) ∧ (ð‘Ķ +s 𝑧𝑅) â‰Īs (ð‘Ĩ +s 𝑧))) → 𝑧 <s 𝑧𝑅)
210 sltadd2im 27696 . . . . . . . . . . . . . 14 ((𝑧 ∈ No ∧ 𝑧𝑅 ∈ No ∧ ð‘Ĩ ∈ No ) → (𝑧 <s 𝑧𝑅 → (ð‘Ĩ +s 𝑧) <s (ð‘Ĩ +s 𝑧𝑅)))
211204, 209, 210sylc 65 . . . . . . . . . . . . 13 ((((ð‘Ĩ ∈ No ∧ ð‘Ķ ∈ No ∧ 𝑧 ∈ No ) ∧ ((∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ𝑂)) ∧ (∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))((ð‘Ķ +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ <s ð‘Ĩ𝑂) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ))) ∧ (𝑧𝑅 ∈ ( R ‘𝑧) ∧ (ð‘Ķ +s 𝑧𝑅) â‰Īs (ð‘Ĩ +s 𝑧))) → (ð‘Ĩ +s 𝑧) <s (ð‘Ĩ +s 𝑧𝑅))
212198, 201, 202, 203, 211slelttrd 27488 . . . . . . . . . . . 12 ((((ð‘Ĩ ∈ No ∧ ð‘Ķ ∈ No ∧ 𝑧 ∈ No ) ∧ ((∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ𝑂)) ∧ (∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))((ð‘Ķ +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ <s ð‘Ĩ𝑂) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ))) ∧ (𝑧𝑅 ∈ ( R ‘𝑧) ∧ (ð‘Ķ +s 𝑧𝑅) â‰Īs (ð‘Ĩ +s 𝑧))) → (ð‘Ķ +s 𝑧𝑅) <s (ð‘Ĩ +s 𝑧𝑅))
213 oveq2 7419 . . . . . . . . . . . . . . 15 (𝑧𝑂 = 𝑧𝑅 → (ð‘Ķ +s 𝑧𝑂) = (ð‘Ķ +s 𝑧𝑅))
214 oveq2 7419 . . . . . . . . . . . . . . 15 (𝑧𝑂 = 𝑧𝑅 → (ð‘Ĩ +s 𝑧𝑂) = (ð‘Ĩ +s 𝑧𝑅))
215213, 214breq12d 5161 . . . . . . . . . . . . . 14 (𝑧𝑂 = 𝑧𝑅 → ((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) ↔ (ð‘Ķ +s 𝑧𝑅) <s (ð‘Ĩ +s 𝑧𝑅)))
216215imbi1d 341 . . . . . . . . . . . . 13 (𝑧𝑂 = 𝑧𝑅 → (((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ) ↔ ((ð‘Ķ +s 𝑧𝑅) <s (ð‘Ĩ +s 𝑧𝑅) → ð‘Ķ <s ð‘Ĩ)))
217 simplr3 1217 . . . . . . . . . . . . 13 ((((ð‘Ĩ ∈ No ∧ ð‘Ķ ∈ No ∧ 𝑧 ∈ No ) ∧ ((∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ𝑂)) ∧ (∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))((ð‘Ķ +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ <s ð‘Ĩ𝑂) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ))) ∧ (𝑧𝑅 ∈ ( R ‘𝑧) ∧ (ð‘Ķ +s 𝑧𝑅) â‰Īs (ð‘Ĩ +s 𝑧))) → ∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ))
218 simprl 769 . . . . . . . . . . . . . 14 ((((ð‘Ĩ ∈ No ∧ ð‘Ķ ∈ No ∧ 𝑧 ∈ No ) ∧ ((∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ𝑂)) ∧ (∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))((ð‘Ķ +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ <s ð‘Ĩ𝑂) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ))) ∧ (𝑧𝑅 ∈ ( R ‘𝑧) ∧ (ð‘Ķ +s 𝑧𝑅) â‰Īs (ð‘Ĩ +s 𝑧))) → 𝑧𝑅 ∈ ( R ‘𝑧))
219 elun2 4177 . . . . . . . . . . . . . 14 (𝑧𝑅 ∈ ( R ‘𝑧) → 𝑧𝑅 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧)))
220218, 219syl 17 . . . . . . . . . . . . 13 ((((ð‘Ĩ ∈ No ∧ ð‘Ķ ∈ No ∧ 𝑧 ∈ No ) ∧ ((∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ𝑂)) ∧ (∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))((ð‘Ķ +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ <s ð‘Ĩ𝑂) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ))) ∧ (𝑧𝑅 ∈ ( R ‘𝑧) ∧ (ð‘Ķ +s 𝑧𝑅) â‰Īs (ð‘Ĩ +s 𝑧))) → 𝑧𝑅 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧)))
221216, 217, 220rspcdva 3613 . . . . . . . . . . . 12 ((((ð‘Ĩ ∈ No ∧ ð‘Ķ ∈ No ∧ 𝑧 ∈ No ) ∧ ((∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ𝑂)) ∧ (∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))((ð‘Ķ +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ <s ð‘Ĩ𝑂) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ))) ∧ (𝑧𝑅 ∈ ( R ‘𝑧) ∧ (ð‘Ķ +s 𝑧𝑅) â‰Īs (ð‘Ĩ +s 𝑧))) → ((ð‘Ķ +s 𝑧𝑅) <s (ð‘Ĩ +s 𝑧𝑅) → ð‘Ķ <s ð‘Ĩ))
222212, 221mpd 15 . . . . . . . . . . 11 ((((ð‘Ĩ ∈ No ∧ ð‘Ķ ∈ No ∧ 𝑧 ∈ No ) ∧ ((∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ𝑂)) ∧ (∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))((ð‘Ķ +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ <s ð‘Ĩ𝑂) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ))) ∧ (𝑧𝑅 ∈ ( R ‘𝑧) ∧ (ð‘Ķ +s 𝑧𝑅) â‰Īs (ð‘Ĩ +s 𝑧))) → ð‘Ķ <s ð‘Ĩ)
223222rexlimdvaa 3156 . . . . . . . . . 10 (((ð‘Ĩ ∈ No ∧ ð‘Ķ ∈ No ∧ 𝑧 ∈ No ) ∧ ((∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ𝑂)) ∧ (∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))((ð‘Ķ +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ <s ð‘Ĩ𝑂) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ))) → (∃𝑧𝑅 ∈ ( R ‘𝑧)(ð‘Ķ +s 𝑧𝑅) â‰Īs (ð‘Ĩ +s 𝑧) → ð‘Ķ <s ð‘Ĩ))
224192, 223jaod 857 . . . . . . . . 9 (((ð‘Ĩ ∈ No ∧ ð‘Ķ ∈ No ∧ 𝑧 ∈ No ) ∧ ((∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ𝑂)) ∧ (∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))((ð‘Ķ +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ <s ð‘Ĩ𝑂) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ))) → ((∃ð‘Ķ𝑅 ∈ ( R ‘ð‘Ķ)(ð‘Ķ𝑅 +s 𝑧) â‰Īs (ð‘Ĩ +s 𝑧) âˆĻ ∃𝑧𝑅 ∈ ( R ‘𝑧)(ð‘Ķ +s 𝑧𝑅) â‰Īs (ð‘Ĩ +s 𝑧)) → ð‘Ķ <s ð‘Ĩ))
225174, 224biimtrid 241 . . . . . . . 8 (((ð‘Ĩ ∈ No ∧ ð‘Ķ ∈ No ∧ 𝑧 ∈ No ) ∧ ((∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ𝑂)) ∧ (∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))((ð‘Ķ +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ <s ð‘Ĩ𝑂) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ))) → (∃𝑞 ∈ ({𝑐 âˆĢ ∃ð‘Ķ𝑅 ∈ ( R ‘ð‘Ķ)𝑐 = (ð‘Ķ𝑅 +s 𝑧)} ∊ {𝑑 âˆĢ ∃𝑧𝑅 ∈ ( R ‘𝑧)𝑑 = (ð‘Ķ +s 𝑧𝑅)})𝑞 â‰Īs (ð‘Ĩ +s 𝑧) → ð‘Ķ <s ð‘Ĩ))
226145, 225jaod 857 . . . . . . 7 (((ð‘Ĩ ∈ No ∧ ð‘Ķ ∈ No ∧ 𝑧 ∈ No ) ∧ ((∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ𝑂)) ∧ (∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))((ð‘Ķ +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ <s ð‘Ĩ𝑂) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ))) → ((∃𝑝 ∈ ({𝑎 âˆĢ ∃ð‘Ĩðŋ ∈ ( L ‘ð‘Ĩ)𝑎 = (ð‘Ĩðŋ +s 𝑧)} ∊ {𝑏 âˆĢ ∃𝑧ðŋ ∈ ( L ‘𝑧)𝑏 = (ð‘Ĩ +s 𝑧ðŋ)})(ð‘Ķ +s 𝑧) â‰Īs 𝑝 âˆĻ ∃𝑞 ∈ ({𝑐 âˆĢ ∃ð‘Ķ𝑅 ∈ ( R ‘ð‘Ķ)𝑐 = (ð‘Ķ𝑅 +s 𝑧)} ∊ {𝑑 âˆĢ ∃𝑧𝑅 ∈ ( R ‘𝑧)𝑑 = (ð‘Ķ +s 𝑧𝑅)})𝑞 â‰Īs (ð‘Ĩ +s 𝑧)) → ð‘Ķ <s ð‘Ĩ))
22765, 226sylbid 239 . . . . . 6 (((ð‘Ĩ ∈ No ∧ ð‘Ķ ∈ No ∧ 𝑧 ∈ No ) ∧ ((∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ𝑂)) ∧ (∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))((ð‘Ķ +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ <s ð‘Ĩ𝑂) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ))) → ((ð‘Ķ +s 𝑧) <s (ð‘Ĩ +s 𝑧) → ð‘Ķ <s ð‘Ĩ))
228227ex 413 . . . . 5 ((ð‘Ĩ ∈ No ∧ ð‘Ķ ∈ No ∧ 𝑧 ∈ No ) → (((∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ𝑂) ∧ ∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ𝑂 +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ𝑂)) ∧ (∀ð‘Ĩ𝑂 ∈ (( L ‘ð‘Ĩ) ∊ ( R ‘ð‘Ĩ))((ð‘Ķ +s 𝑧) <s (ð‘Ĩ𝑂 +s 𝑧) → ð‘Ķ <s ð‘Ĩ𝑂) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ𝑂 +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ𝑂 <s ð‘Ĩ) ∧ ∀ð‘Ķ𝑂 ∈ (( L ‘ð‘Ķ) ∊ ( R ‘ð‘Ķ))((ð‘Ķ𝑂 +s 𝑧) <s (ð‘Ĩ +s 𝑧) → ð‘Ķ𝑂 <s ð‘Ĩ)) ∧ ∀𝑧𝑂 ∈ (( L ‘𝑧) ∊ ( R ‘𝑧))((ð‘Ķ +s 𝑧𝑂) <s (ð‘Ĩ +s 𝑧𝑂) → ð‘Ķ <s ð‘Ĩ)) → ((ð‘Ķ +s 𝑧) <s (ð‘Ĩ +s 𝑧) → ð‘Ķ <s ð‘Ĩ)))
2294, 8, 12, 16, 20, 22, 25, 29, 33, 37, 228no3inds 27668 . . . 4 ((ðī ∈ No ∧ ðĩ ∈ No ∧ ðķ ∈ No ) → ((ðĩ +s ðķ) <s (ðī +s ðķ) → ðĩ <s ðī))
230 addscl 27691 . . . . . 6 ((ðĩ ∈ No ∧ ðķ ∈ No ) → (ðĩ +s ðķ) ∈ No )
2312303adant1 1130 . . . . 5 ((ðī ∈ No ∧ ðĩ ∈ No ∧ ðķ ∈ No ) → (ðĩ +s ðķ) ∈ No )
232 addscl 27691 . . . . . 6 ((ðī ∈ No ∧ ðķ ∈ No ) → (ðī +s ðķ) ∈ No )
2332323adant2 1131 . . . . 5 ((ðī ∈ No ∧ ðĩ ∈ No ∧ ðķ ∈ No ) → (ðī +s ðķ) ∈ No )
234 sltnle 27480 . . . . 5 (((ðĩ +s ðķ) ∈ No ∧ (ðī +s ðķ) ∈ No ) → ((ðĩ +s ðķ) <s (ðī +s ðķ) ↔ ÂŽ (ðī +s ðķ) â‰Īs (ðĩ +s ðķ)))
235231, 233, 234syl2anc 584 . . . 4 ((ðī ∈ No ∧ ðĩ ∈ No ∧ ðķ ∈ No ) → ((ðĩ +s ðķ) <s (ðī +s ðķ) ↔ ÂŽ (ðī +s ðķ) â‰Īs (ðĩ +s ðķ)))
236 sltnle 27480 . . . . . 6 ((ðĩ ∈ No ∧ ðī ∈ No ) → (ðĩ <s ðī ↔ ÂŽ ðī â‰Īs ðĩ))
237236ancoms 459 . . . . 5 ((ðī ∈ No ∧ ðĩ ∈ No ) → (ðĩ <s ðī ↔ ÂŽ ðī â‰Īs ðĩ))
2382373adant3 1132 . . . 4 ((ðī ∈ No ∧ ðĩ ∈ No ∧ ðķ ∈ No ) → (ðĩ <s ðī ↔ ÂŽ ðī â‰Īs ðĩ))
239229, 235, 2383imtr3d 292 . . 3 ((ðī ∈ No ∧ ðĩ ∈ No ∧ ðķ ∈ No ) → (ÂŽ (ðī +s ðķ) â‰Īs (ðĩ +s ðķ) → ÂŽ ðī â‰Īs ðĩ))
240239con4d 115 . 2 ((ðī ∈ No ∧ ðĩ ∈ No ∧ ðķ ∈ No ) → (ðī â‰Īs ðĩ → (ðī +s ðķ) â‰Īs (ðĩ +s ðķ)))
241 sleadd1im 27697 . 2 ((ðī ∈ No ∧ ðĩ ∈ No ∧ ðķ ∈ No ) → ((ðī +s ðķ) â‰Īs (ðĩ +s ðķ) → ðī â‰Īs ðĩ))
242240, 241impbid 211 1 ((ðī ∈ No ∧ ðĩ ∈ No ∧ ðķ ∈ No ) → (ðī â‰Īs ðĩ ↔ (ðī +s ðķ) â‰Īs (ðĩ +s ðķ)))
Colors of variables: wff setvar class
Syntax hints:  ÂŽ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   âˆĻ wo 845   ∧ w3a 1087   = wceq 1541  âˆƒwex 1781   ∈ wcel 2106  {cab 2709   ≠ wne 2940  âˆ€wral 3061  âˆƒwrex 3070   ∊ cun 3946  âˆ…c0 4322  {csn 4628   class class class wbr 5148  â€˜cfv 6543  (class class class)co 7411   No csur 27367   <s cslt 27368   bday cbday 27369   â‰Īs csle 27471   <<s csslt 27506   |s cscut 27508   O cold 27563   L cleft 27565   R cright 27566   +s cadds 27669
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-tp 4633  df-op 4635  df-ot 4637  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-se 5632  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-1o 8468  df-2o 8469  df-nadd 8667  df-no 27370  df-slt 27371  df-bday 27372  df-sle 27472  df-sslt 27507  df-scut 27509  df-0s 27550  df-made 27567  df-old 27568  df-left 27570  df-right 27571  df-norec2 27659  df-adds 27670
This theorem is referenced by:  sleadd2  27700  addscan2  27703  sleadd1d  27705
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