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Theorem addsproplem6 27289
Description: Lemma for surreal addition properties. Finally, we show the second half of the induction hypothesis when 𝑌 and 𝑍 are the same age. (Contributed by Scott Fenton, 21-Jan-2025.)
Hypotheses
Ref Expression
addsproplem.1 (𝜑 → ∀𝑥 No 𝑦 No 𝑧 No (((( bday 𝑥) +no ( bday 𝑦)) ∪ (( bday 𝑥) +no ( bday 𝑧))) ∈ ((( bday 𝑋) +no ( bday 𝑌)) ∪ (( bday 𝑋) +no ( bday 𝑍))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)))))
addspropord.2 (𝜑𝑋 No )
addspropord.3 (𝜑𝑌 No )
addspropord.4 (𝜑𝑍 No )
addspropord.5 (𝜑𝑌 <s 𝑍)
addsproplem6.6 (𝜑 → ( bday 𝑌) = ( bday 𝑍))
Assertion
Ref Expression
addsproplem6 (𝜑 → (𝑌 +s 𝑋) <s (𝑍 +s 𝑋))
Distinct variable groups:   𝑥,𝑋,𝑦,𝑧   𝑥,𝑌,𝑦,𝑧   𝑥,𝑍,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem addsproplem6
Dummy variables 𝑎 𝑏 𝑐 𝑑 𝑒 𝑓 𝑔 𝑚 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 addspropord.3 . . . 4 (𝜑𝑌 No )
2 addspropord.4 . . . 4 (𝜑𝑍 No )
3 addsproplem6.6 . . . 4 (𝜑 → ( bday 𝑌) = ( bday 𝑍))
4 addspropord.5 . . . 4 (𝜑𝑌 <s 𝑍)
5 nodense 27043 . . . 4 (((𝑌 No 𝑍 No ) ∧ (( bday 𝑌) = ( bday 𝑍) ∧ 𝑌 <s 𝑍)) → ∃𝑚 No (( bday 𝑚) ∈ ( bday 𝑌) ∧ 𝑌 <s 𝑚𝑚 <s 𝑍))
61, 2, 3, 4, 5syl22anc 838 . . 3 (𝜑 → ∃𝑚 No (( bday 𝑚) ∈ ( bday 𝑌) ∧ 𝑌 <s 𝑚𝑚 <s 𝑍))
7 addsproplem.1 . . . . . . 7 (𝜑 → ∀𝑥 No 𝑦 No 𝑧 No (((( bday 𝑥) +no ( bday 𝑦)) ∪ (( bday 𝑥) +no ( bday 𝑧))) ∈ ((( bday 𝑋) +no ( bday 𝑌)) ∪ (( bday 𝑋) +no ( bday 𝑍))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)))))
8 addspropord.2 . . . . . . 7 (𝜑𝑋 No )
97, 8, 1addsproplem3 27286 . . . . . 6 (𝜑 → ((𝑋 +s 𝑌) ∈ No ∧ ({𝑎 ∣ ∃𝑏 ∈ ( L ‘𝑋)𝑎 = (𝑏 +s 𝑌)} ∪ {𝑐 ∣ ∃𝑑 ∈ ( L ‘𝑌)𝑐 = (𝑋 +s 𝑑)}) <<s {(𝑋 +s 𝑌)} ∧ {(𝑋 +s 𝑌)} <<s ({𝑒 ∣ ∃𝑓 ∈ ( R ‘𝑋)𝑒 = (𝑓 +s 𝑌)} ∪ {𝑔 ∣ ∃ ∈ ( R ‘𝑌)𝑔 = (𝑋 +s )})))
109simp1d 1143 . . . . 5 (𝜑 → (𝑋 +s 𝑌) ∈ No )
1110adantr 482 . . . 4 ((𝜑 ∧ (𝑚 No ∧ (( bday 𝑚) ∈ ( bday 𝑌) ∧ 𝑌 <s 𝑚𝑚 <s 𝑍))) → (𝑋 +s 𝑌) ∈ No )
127adantr 482 . . . . . 6 ((𝜑 ∧ (𝑚 No ∧ (( bday 𝑚) ∈ ( bday 𝑌) ∧ 𝑌 <s 𝑚𝑚 <s 𝑍))) → ∀𝑥 No 𝑦 No 𝑧 No (((( bday 𝑥) +no ( bday 𝑦)) ∪ (( bday 𝑥) +no ( bday 𝑧))) ∈ ((( bday 𝑋) +no ( bday 𝑌)) ∪ (( bday 𝑋) +no ( bday 𝑍))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)))))
138adantr 482 . . . . . 6 ((𝜑 ∧ (𝑚 No ∧ (( bday 𝑚) ∈ ( bday 𝑌) ∧ 𝑌 <s 𝑚𝑚 <s 𝑍))) → 𝑋 No )
14 simprl 770 . . . . . 6 ((𝜑 ∧ (𝑚 No ∧ (( bday 𝑚) ∈ ( bday 𝑌) ∧ 𝑌 <s 𝑚𝑚 <s 𝑍))) → 𝑚 No )
15 unidm 4113 . . . . . . 7 ((( bday 𝑋) +no ( bday 𝑚)) ∪ (( bday 𝑋) +no ( bday 𝑚))) = (( bday 𝑋) +no ( bday 𝑚))
16 simprr1 1222 . . . . . . . . 9 ((𝜑 ∧ (𝑚 No ∧ (( bday 𝑚) ∈ ( bday 𝑌) ∧ 𝑌 <s 𝑚𝑚 <s 𝑍))) → ( bday 𝑚) ∈ ( bday 𝑌))
17 bdayelon 27119 . . . . . . . . . 10 ( bday 𝑚) ∈ On
18 bdayelon 27119 . . . . . . . . . 10 ( bday 𝑌) ∈ On
19 bdayelon 27119 . . . . . . . . . 10 ( bday 𝑋) ∈ On
20 naddel2 8634 . . . . . . . . . 10 ((( bday 𝑚) ∈ On ∧ ( bday 𝑌) ∈ On ∧ ( bday 𝑋) ∈ On) → (( bday 𝑚) ∈ ( bday 𝑌) ↔ (( bday 𝑋) +no ( bday 𝑚)) ∈ (( bday 𝑋) +no ( bday 𝑌))))
2117, 18, 19, 20mp3an 1462 . . . . . . . . 9 (( bday 𝑚) ∈ ( bday 𝑌) ↔ (( bday 𝑋) +no ( bday 𝑚)) ∈ (( bday 𝑋) +no ( bday 𝑌)))
2216, 21sylib 217 . . . . . . . 8 ((𝜑 ∧ (𝑚 No ∧ (( bday 𝑚) ∈ ( bday 𝑌) ∧ 𝑌 <s 𝑚𝑚 <s 𝑍))) → (( bday 𝑋) +no ( bday 𝑚)) ∈ (( bday 𝑋) +no ( bday 𝑌)))
23 elun1 4137 . . . . . . . 8 ((( bday 𝑋) +no ( bday 𝑚)) ∈ (( bday 𝑋) +no ( bday 𝑌)) → (( bday 𝑋) +no ( bday 𝑚)) ∈ ((( bday 𝑋) +no ( bday 𝑌)) ∪ (( bday 𝑋) +no ( bday 𝑍))))
2422, 23syl 17 . . . . . . 7 ((𝜑 ∧ (𝑚 No ∧ (( bday 𝑚) ∈ ( bday 𝑌) ∧ 𝑌 <s 𝑚𝑚 <s 𝑍))) → (( bday 𝑋) +no ( bday 𝑚)) ∈ ((( bday 𝑋) +no ( bday 𝑌)) ∪ (( bday 𝑋) +no ( bday 𝑍))))
2515, 24eqeltrid 2842 . . . . . 6 ((𝜑 ∧ (𝑚 No ∧ (( bday 𝑚) ∈ ( bday 𝑌) ∧ 𝑌 <s 𝑚𝑚 <s 𝑍))) → ((( bday 𝑋) +no ( bday 𝑚)) ∪ (( bday 𝑋) +no ( bday 𝑚))) ∈ ((( bday 𝑋) +no ( bday 𝑌)) ∪ (( bday 𝑋) +no ( bday 𝑍))))
2612, 13, 14, 14, 25addsproplem1 27284 . . . . 5 ((𝜑 ∧ (𝑚 No ∧ (( bday 𝑚) ∈ ( bday 𝑌) ∧ 𝑌 <s 𝑚𝑚 <s 𝑍))) → ((𝑋 +s 𝑚) ∈ No ∧ (𝑚 <s 𝑚 → (𝑚 +s 𝑋) <s (𝑚 +s 𝑋))))
2726simpld 496 . . . 4 ((𝜑 ∧ (𝑚 No ∧ (( bday 𝑚) ∈ ( bday 𝑌) ∧ 𝑌 <s 𝑚𝑚 <s 𝑍))) → (𝑋 +s 𝑚) ∈ No )
28 uncom 4114 . . . . . . . . . . . 12 ((( bday 𝑋) +no ( bday 𝑌)) ∪ (( bday 𝑋) +no ( bday 𝑍))) = ((( bday 𝑋) +no ( bday 𝑍)) ∪ (( bday 𝑋) +no ( bday 𝑌)))
2928eleq2i 2830 . . . . . . . . . . 11 (((( bday 𝑥) +no ( bday 𝑦)) ∪ (( bday 𝑥) +no ( bday 𝑧))) ∈ ((( bday 𝑋) +no ( bday 𝑌)) ∪ (( bday 𝑋) +no ( bday 𝑍))) ↔ ((( bday 𝑥) +no ( bday 𝑦)) ∪ (( bday 𝑥) +no ( bday 𝑧))) ∈ ((( bday 𝑋) +no ( bday 𝑍)) ∪ (( bday 𝑋) +no ( bday 𝑌))))
3029imbi1i 350 . . . . . . . . . 10 ((((( bday 𝑥) +no ( bday 𝑦)) ∪ (( bday 𝑥) +no ( bday 𝑧))) ∈ ((( bday 𝑋) +no ( bday 𝑌)) ∪ (( bday 𝑋) +no ( bday 𝑍))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)))) ↔ (((( bday 𝑥) +no ( bday 𝑦)) ∪ (( bday 𝑥) +no ( bday 𝑧))) ∈ ((( bday 𝑋) +no ( bday 𝑍)) ∪ (( bday 𝑋) +no ( bday 𝑌))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)))))
3130ralbii 3097 . . . . . . . . 9 (∀𝑧 No (((( bday 𝑥) +no ( bday 𝑦)) ∪ (( bday 𝑥) +no ( bday 𝑧))) ∈ ((( bday 𝑋) +no ( bday 𝑌)) ∪ (( bday 𝑋) +no ( bday 𝑍))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)))) ↔ ∀𝑧 No (((( bday 𝑥) +no ( bday 𝑦)) ∪ (( bday 𝑥) +no ( bday 𝑧))) ∈ ((( bday 𝑋) +no ( bday 𝑍)) ∪ (( bday 𝑋) +no ( bday 𝑌))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)))))
32312ralbii 3128 . . . . . . . 8 (∀𝑥 No 𝑦 No 𝑧 No (((( bday 𝑥) +no ( bday 𝑦)) ∪ (( bday 𝑥) +no ( bday 𝑧))) ∈ ((( bday 𝑋) +no ( bday 𝑌)) ∪ (( bday 𝑋) +no ( bday 𝑍))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)))) ↔ ∀𝑥 No 𝑦 No 𝑧 No (((( bday 𝑥) +no ( bday 𝑦)) ∪ (( bday 𝑥) +no ( bday 𝑧))) ∈ ((( bday 𝑋) +no ( bday 𝑍)) ∪ (( bday 𝑋) +no ( bday 𝑌))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)))))
337, 32sylib 217 . . . . . . 7 (𝜑 → ∀𝑥 No 𝑦 No 𝑧 No (((( bday 𝑥) +no ( bday 𝑦)) ∪ (( bday 𝑥) +no ( bday 𝑧))) ∈ ((( bday 𝑋) +no ( bday 𝑍)) ∪ (( bday 𝑋) +no ( bday 𝑌))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)))))
3433, 8, 2addsproplem3 27286 . . . . . 6 (𝜑 → ((𝑋 +s 𝑍) ∈ No ∧ ({𝑎 ∣ ∃𝑏 ∈ ( L ‘𝑋)𝑎 = (𝑏 +s 𝑍)} ∪ {𝑐 ∣ ∃𝑑 ∈ ( L ‘𝑍)𝑐 = (𝑋 +s 𝑑)}) <<s {(𝑋 +s 𝑍)} ∧ {(𝑋 +s 𝑍)} <<s ({𝑒 ∣ ∃𝑓 ∈ ( R ‘𝑋)𝑒 = (𝑓 +s 𝑍)} ∪ {𝑔 ∣ ∃ ∈ ( R ‘𝑍)𝑔 = (𝑋 +s )})))
3534simp1d 1143 . . . . 5 (𝜑 → (𝑋 +s 𝑍) ∈ No )
3635adantr 482 . . . 4 ((𝜑 ∧ (𝑚 No ∧ (( bday 𝑚) ∈ ( bday 𝑌) ∧ 𝑌 <s 𝑚𝑚 <s 𝑍))) → (𝑋 +s 𝑍) ∈ No )
379simp3d 1145 . . . . . 6 (𝜑 → {(𝑋 +s 𝑌)} <<s ({𝑒 ∣ ∃𝑓 ∈ ( R ‘𝑋)𝑒 = (𝑓 +s 𝑌)} ∪ {𝑔 ∣ ∃ ∈ ( R ‘𝑌)𝑔 = (𝑋 +s )}))
3837adantr 482 . . . . 5 ((𝜑 ∧ (𝑚 No ∧ (( bday 𝑚) ∈ ( bday 𝑌) ∧ 𝑌 <s 𝑚𝑚 <s 𝑍))) → {(𝑋 +s 𝑌)} <<s ({𝑒 ∣ ∃𝑓 ∈ ( R ‘𝑋)𝑒 = (𝑓 +s 𝑌)} ∪ {𝑔 ∣ ∃ ∈ ( R ‘𝑌)𝑔 = (𝑋 +s )}))
39 ovex 7391 . . . . . . 7 (𝑋 +s 𝑌) ∈ V
4039snid 4623 . . . . . 6 (𝑋 +s 𝑌) ∈ {(𝑋 +s 𝑌)}
4140a1i 11 . . . . 5 ((𝜑 ∧ (𝑚 No ∧ (( bday 𝑚) ∈ ( bday 𝑌) ∧ 𝑌 <s 𝑚𝑚 <s 𝑍))) → (𝑋 +s 𝑌) ∈ {(𝑋 +s 𝑌)})
42 oldbday 27233 . . . . . . . . . . 11 ((( bday 𝑌) ∈ On ∧ 𝑚 No ) → (𝑚 ∈ ( O ‘( bday 𝑌)) ↔ ( bday 𝑚) ∈ ( bday 𝑌)))
4318, 14, 42sylancr 588 . . . . . . . . . 10 ((𝜑 ∧ (𝑚 No ∧ (( bday 𝑚) ∈ ( bday 𝑌) ∧ 𝑌 <s 𝑚𝑚 <s 𝑍))) → (𝑚 ∈ ( O ‘( bday 𝑌)) ↔ ( bday 𝑚) ∈ ( bday 𝑌)))
4416, 43mpbird 257 . . . . . . . . 9 ((𝜑 ∧ (𝑚 No ∧ (( bday 𝑚) ∈ ( bday 𝑌) ∧ 𝑌 <s 𝑚𝑚 <s 𝑍))) → 𝑚 ∈ ( O ‘( bday 𝑌)))
45 simprr2 1223 . . . . . . . . 9 ((𝜑 ∧ (𝑚 No ∧ (( bday 𝑚) ∈ ( bday 𝑌) ∧ 𝑌 <s 𝑚𝑚 <s 𝑍))) → 𝑌 <s 𝑚)
46 breq2 5110 . . . . . . . . . 10 (𝑎 = 𝑚 → (𝑌 <s 𝑎𝑌 <s 𝑚))
47 rightval 27197 . . . . . . . . . 10 ( R ‘𝑌) = {𝑎 ∈ ( O ‘( bday 𝑌)) ∣ 𝑌 <s 𝑎}
4846, 47elrab2 3649 . . . . . . . . 9 (𝑚 ∈ ( R ‘𝑌) ↔ (𝑚 ∈ ( O ‘( bday 𝑌)) ∧ 𝑌 <s 𝑚))
4944, 45, 48sylanbrc 584 . . . . . . . 8 ((𝜑 ∧ (𝑚 No ∧ (( bday 𝑚) ∈ ( bday 𝑌) ∧ 𝑌 <s 𝑚𝑚 <s 𝑍))) → 𝑚 ∈ ( R ‘𝑌))
50 eqid 2737 . . . . . . . 8 (𝑋 +s 𝑚) = (𝑋 +s 𝑚)
51 oveq2 7366 . . . . . . . . 9 ( = 𝑚 → (𝑋 +s ) = (𝑋 +s 𝑚))
5251rspceeqv 3596 . . . . . . . 8 ((𝑚 ∈ ( R ‘𝑌) ∧ (𝑋 +s 𝑚) = (𝑋 +s 𝑚)) → ∃ ∈ ( R ‘𝑌)(𝑋 +s 𝑚) = (𝑋 +s ))
5349, 50, 52sylancl 587 . . . . . . 7 ((𝜑 ∧ (𝑚 No ∧ (( bday 𝑚) ∈ ( bday 𝑌) ∧ 𝑌 <s 𝑚𝑚 <s 𝑍))) → ∃ ∈ ( R ‘𝑌)(𝑋 +s 𝑚) = (𝑋 +s ))
54 ovex 7391 . . . . . . . 8 (𝑋 +s 𝑚) ∈ V
55 eqeq1 2741 . . . . . . . . 9 (𝑔 = (𝑋 +s 𝑚) → (𝑔 = (𝑋 +s ) ↔ (𝑋 +s 𝑚) = (𝑋 +s )))
5655rexbidv 3176 . . . . . . . 8 (𝑔 = (𝑋 +s 𝑚) → (∃ ∈ ( R ‘𝑌)𝑔 = (𝑋 +s ) ↔ ∃ ∈ ( R ‘𝑌)(𝑋 +s 𝑚) = (𝑋 +s )))
5754, 56elab 3631 . . . . . . 7 ((𝑋 +s 𝑚) ∈ {𝑔 ∣ ∃ ∈ ( R ‘𝑌)𝑔 = (𝑋 +s )} ↔ ∃ ∈ ( R ‘𝑌)(𝑋 +s 𝑚) = (𝑋 +s ))
5853, 57sylibr 233 . . . . . 6 ((𝜑 ∧ (𝑚 No ∧ (( bday 𝑚) ∈ ( bday 𝑌) ∧ 𝑌 <s 𝑚𝑚 <s 𝑍))) → (𝑋 +s 𝑚) ∈ {𝑔 ∣ ∃ ∈ ( R ‘𝑌)𝑔 = (𝑋 +s )})
59 elun2 4138 . . . . . 6 ((𝑋 +s 𝑚) ∈ {𝑔 ∣ ∃ ∈ ( R ‘𝑌)𝑔 = (𝑋 +s )} → (𝑋 +s 𝑚) ∈ ({𝑒 ∣ ∃𝑓 ∈ ( R ‘𝑋)𝑒 = (𝑓 +s 𝑌)} ∪ {𝑔 ∣ ∃ ∈ ( R ‘𝑌)𝑔 = (𝑋 +s )}))
6058, 59syl 17 . . . . 5 ((𝜑 ∧ (𝑚 No ∧ (( bday 𝑚) ∈ ( bday 𝑌) ∧ 𝑌 <s 𝑚𝑚 <s 𝑍))) → (𝑋 +s 𝑚) ∈ ({𝑒 ∣ ∃𝑓 ∈ ( R ‘𝑋)𝑒 = (𝑓 +s 𝑌)} ∪ {𝑔 ∣ ∃ ∈ ( R ‘𝑌)𝑔 = (𝑋 +s )}))
6138, 41, 60ssltsepcd 27136 . . . 4 ((𝜑 ∧ (𝑚 No ∧ (( bday 𝑚) ∈ ( bday 𝑌) ∧ 𝑌 <s 𝑚𝑚 <s 𝑍))) → (𝑋 +s 𝑌) <s (𝑋 +s 𝑚))
6233adantr 482 . . . . . . 7 ((𝜑 ∧ (𝑚 No ∧ (( bday 𝑚) ∈ ( bday 𝑌) ∧ 𝑌 <s 𝑚𝑚 <s 𝑍))) → ∀𝑥 No 𝑦 No 𝑧 No (((( bday 𝑥) +no ( bday 𝑦)) ∪ (( bday 𝑥) +no ( bday 𝑧))) ∈ ((( bday 𝑋) +no ( bday 𝑍)) ∪ (( bday 𝑋) +no ( bday 𝑌))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)))))
632adantr 482 . . . . . . 7 ((𝜑 ∧ (𝑚 No ∧ (( bday 𝑚) ∈ ( bday 𝑌) ∧ 𝑌 <s 𝑚𝑚 <s 𝑍))) → 𝑍 No )
6462, 13, 63addsproplem3 27286 . . . . . 6 ((𝜑 ∧ (𝑚 No ∧ (( bday 𝑚) ∈ ( bday 𝑌) ∧ 𝑌 <s 𝑚𝑚 <s 𝑍))) → ((𝑋 +s 𝑍) ∈ No ∧ ({𝑎 ∣ ∃𝑏 ∈ ( L ‘𝑋)𝑎 = (𝑏 +s 𝑍)} ∪ {𝑐 ∣ ∃𝑑 ∈ ( L ‘𝑍)𝑐 = (𝑋 +s 𝑑)}) <<s {(𝑋 +s 𝑍)} ∧ {(𝑋 +s 𝑍)} <<s ({𝑒 ∣ ∃𝑓 ∈ ( R ‘𝑋)𝑒 = (𝑓 +s 𝑍)} ∪ {𝑔 ∣ ∃ ∈ ( R ‘𝑍)𝑔 = (𝑋 +s )})))
6564simp2d 1144 . . . . 5 ((𝜑 ∧ (𝑚 No ∧ (( bday 𝑚) ∈ ( bday 𝑌) ∧ 𝑌 <s 𝑚𝑚 <s 𝑍))) → ({𝑎 ∣ ∃𝑏 ∈ ( L ‘𝑋)𝑎 = (𝑏 +s 𝑍)} ∪ {𝑐 ∣ ∃𝑑 ∈ ( L ‘𝑍)𝑐 = (𝑋 +s 𝑑)}) <<s {(𝑋 +s 𝑍)})
663adantr 482 . . . . . . . . . . 11 ((𝜑 ∧ (𝑚 No ∧ (( bday 𝑚) ∈ ( bday 𝑌) ∧ 𝑌 <s 𝑚𝑚 <s 𝑍))) → ( bday 𝑌) = ( bday 𝑍))
6716, 66eleqtrd 2840 . . . . . . . . . 10 ((𝜑 ∧ (𝑚 No ∧ (( bday 𝑚) ∈ ( bday 𝑌) ∧ 𝑌 <s 𝑚𝑚 <s 𝑍))) → ( bday 𝑚) ∈ ( bday 𝑍))
68 bdayelon 27119 . . . . . . . . . . 11 ( bday 𝑍) ∈ On
69 oldbday 27233 . . . . . . . . . . 11 ((( bday 𝑍) ∈ On ∧ 𝑚 No ) → (𝑚 ∈ ( O ‘( bday 𝑍)) ↔ ( bday 𝑚) ∈ ( bday 𝑍)))
7068, 14, 69sylancr 588 . . . . . . . . . 10 ((𝜑 ∧ (𝑚 No ∧ (( bday 𝑚) ∈ ( bday 𝑌) ∧ 𝑌 <s 𝑚𝑚 <s 𝑍))) → (𝑚 ∈ ( O ‘( bday 𝑍)) ↔ ( bday 𝑚) ∈ ( bday 𝑍)))
7167, 70mpbird 257 . . . . . . . . 9 ((𝜑 ∧ (𝑚 No ∧ (( bday 𝑚) ∈ ( bday 𝑌) ∧ 𝑌 <s 𝑚𝑚 <s 𝑍))) → 𝑚 ∈ ( O ‘( bday 𝑍)))
72 simprr3 1224 . . . . . . . . 9 ((𝜑 ∧ (𝑚 No ∧ (( bday 𝑚) ∈ ( bday 𝑌) ∧ 𝑌 <s 𝑚𝑚 <s 𝑍))) → 𝑚 <s 𝑍)
73 breq1 5109 . . . . . . . . . 10 (𝑎 = 𝑚 → (𝑎 <s 𝑍𝑚 <s 𝑍))
74 leftval 27196 . . . . . . . . . 10 ( L ‘𝑍) = {𝑎 ∈ ( O ‘( bday 𝑍)) ∣ 𝑎 <s 𝑍}
7573, 74elrab2 3649 . . . . . . . . 9 (𝑚 ∈ ( L ‘𝑍) ↔ (𝑚 ∈ ( O ‘( bday 𝑍)) ∧ 𝑚 <s 𝑍))
7671, 72, 75sylanbrc 584 . . . . . . . 8 ((𝜑 ∧ (𝑚 No ∧ (( bday 𝑚) ∈ ( bday 𝑌) ∧ 𝑌 <s 𝑚𝑚 <s 𝑍))) → 𝑚 ∈ ( L ‘𝑍))
77 oveq2 7366 . . . . . . . . 9 (𝑑 = 𝑚 → (𝑋 +s 𝑑) = (𝑋 +s 𝑚))
7877rspceeqv 3596 . . . . . . . 8 ((𝑚 ∈ ( L ‘𝑍) ∧ (𝑋 +s 𝑚) = (𝑋 +s 𝑚)) → ∃𝑑 ∈ ( L ‘𝑍)(𝑋 +s 𝑚) = (𝑋 +s 𝑑))
7976, 50, 78sylancl 587 . . . . . . 7 ((𝜑 ∧ (𝑚 No ∧ (( bday 𝑚) ∈ ( bday 𝑌) ∧ 𝑌 <s 𝑚𝑚 <s 𝑍))) → ∃𝑑 ∈ ( L ‘𝑍)(𝑋 +s 𝑚) = (𝑋 +s 𝑑))
80 eqeq1 2741 . . . . . . . . 9 (𝑐 = (𝑋 +s 𝑚) → (𝑐 = (𝑋 +s 𝑑) ↔ (𝑋 +s 𝑚) = (𝑋 +s 𝑑)))
8180rexbidv 3176 . . . . . . . 8 (𝑐 = (𝑋 +s 𝑚) → (∃𝑑 ∈ ( L ‘𝑍)𝑐 = (𝑋 +s 𝑑) ↔ ∃𝑑 ∈ ( L ‘𝑍)(𝑋 +s 𝑚) = (𝑋 +s 𝑑)))
8254, 81elab 3631 . . . . . . 7 ((𝑋 +s 𝑚) ∈ {𝑐 ∣ ∃𝑑 ∈ ( L ‘𝑍)𝑐 = (𝑋 +s 𝑑)} ↔ ∃𝑑 ∈ ( L ‘𝑍)(𝑋 +s 𝑚) = (𝑋 +s 𝑑))
8379, 82sylibr 233 . . . . . 6 ((𝜑 ∧ (𝑚 No ∧ (( bday 𝑚) ∈ ( bday 𝑌) ∧ 𝑌 <s 𝑚𝑚 <s 𝑍))) → (𝑋 +s 𝑚) ∈ {𝑐 ∣ ∃𝑑 ∈ ( L ‘𝑍)𝑐 = (𝑋 +s 𝑑)})
84 elun2 4138 . . . . . 6 ((𝑋 +s 𝑚) ∈ {𝑐 ∣ ∃𝑑 ∈ ( L ‘𝑍)𝑐 = (𝑋 +s 𝑑)} → (𝑋 +s 𝑚) ∈ ({𝑎 ∣ ∃𝑏 ∈ ( L ‘𝑋)𝑎 = (𝑏 +s 𝑍)} ∪ {𝑐 ∣ ∃𝑑 ∈ ( L ‘𝑍)𝑐 = (𝑋 +s 𝑑)}))
8583, 84syl 17 . . . . 5 ((𝜑 ∧ (𝑚 No ∧ (( bday 𝑚) ∈ ( bday 𝑌) ∧ 𝑌 <s 𝑚𝑚 <s 𝑍))) → (𝑋 +s 𝑚) ∈ ({𝑎 ∣ ∃𝑏 ∈ ( L ‘𝑋)𝑎 = (𝑏 +s 𝑍)} ∪ {𝑐 ∣ ∃𝑑 ∈ ( L ‘𝑍)𝑐 = (𝑋 +s 𝑑)}))
86 ovex 7391 . . . . . . 7 (𝑋 +s 𝑍) ∈ V
8786snid 4623 . . . . . 6 (𝑋 +s 𝑍) ∈ {(𝑋 +s 𝑍)}
8887a1i 11 . . . . 5 ((𝜑 ∧ (𝑚 No ∧ (( bday 𝑚) ∈ ( bday 𝑌) ∧ 𝑌 <s 𝑚𝑚 <s 𝑍))) → (𝑋 +s 𝑍) ∈ {(𝑋 +s 𝑍)})
8965, 85, 88ssltsepcd 27136 . . . 4 ((𝜑 ∧ (𝑚 No ∧ (( bday 𝑚) ∈ ( bday 𝑌) ∧ 𝑌 <s 𝑚𝑚 <s 𝑍))) → (𝑋 +s 𝑚) <s (𝑋 +s 𝑍))
9011, 27, 36, 61, 89slttrd 27110 . . 3 ((𝜑 ∧ (𝑚 No ∧ (( bday 𝑚) ∈ ( bday 𝑌) ∧ 𝑌 <s 𝑚𝑚 <s 𝑍))) → (𝑋 +s 𝑌) <s (𝑋 +s 𝑍))
916, 90rexlimddv 3159 . 2 (𝜑 → (𝑋 +s 𝑌) <s (𝑋 +s 𝑍))
921, 8addscomd 27282 . 2 (𝜑 → (𝑌 +s 𝑋) = (𝑋 +s 𝑌))
932, 8addscomd 27282 . 2 (𝜑 → (𝑍 +s 𝑋) = (𝑋 +s 𝑍))
9491, 92, 933brtr4d 5138 1 (𝜑 → (𝑌 +s 𝑋) <s (𝑍 +s 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  w3a 1088   = wceq 1542  wcel 2107  {cab 2714  wral 3065  wrex 3074  cun 3909  {csn 4587   class class class wbr 5106  Oncon0 6318  cfv 6497  (class class class)co 7358   +no cnadd 8612   No csur 26991   <s cslt 26992   bday cbday 26993   <<s csslt 27123   O cold 27176   L cleft 27178   R cright 27179   +s cadds 27274
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rmo 3354  df-reu 3355  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-uni 4867  df-int 4909  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-se 5590  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7922  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-1o 8413  df-2o 8414  df-nadd 8613  df-no 26994  df-slt 26995  df-bday 26996  df-sslt 27124  df-scut 27126  df-0s 27166  df-made 27180  df-old 27181  df-left 27183  df-right 27184  df-norec2 27264  df-adds 27275
This theorem is referenced by:  addsproplem7  27290
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