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Theorem addsproplem5 27864
Description: Lemma for surreal addition properties. Show the second half of the inductive hypothesis when 𝑍 is older than 𝑌. (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)
addsproplem5.6 (𝜑 → ( bday 𝑍) ∈ ( bday 𝑌))
Assertion
Ref Expression
addsproplem5 (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 addsproplem5
Dummy variables 𝑎 𝑏 𝑐 𝑑 𝑒 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 addsproplem.1 . . . . 5 (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)))))
2 addspropord.2 . . . . 5 (𝜑𝑋 No )
3 addspropord.3 . . . . 5 (𝜑𝑌 No )
41, 2, 3addsproplem3 27862 . . . 4 (𝜑 → ((𝑋 +s 𝑌) ∈ No ∧ ({𝑒 ∣ ∃𝑓 ∈ ( L ‘𝑋)𝑒 = (𝑓 +s 𝑌)} ∪ {𝑔 ∣ ∃ ∈ ( L ‘𝑌)𝑔 = (𝑋 +s )}) <<s {(𝑋 +s 𝑌)} ∧ {(𝑋 +s 𝑌)} <<s ({𝑎 ∣ ∃𝑐 ∈ ( R ‘𝑋)𝑎 = (𝑐 +s 𝑌)} ∪ {𝑏 ∣ ∃𝑑 ∈ ( R ‘𝑌)𝑏 = (𝑋 +s 𝑑)})))
54simp3d 1142 . . 3 (𝜑 → {(𝑋 +s 𝑌)} <<s ({𝑎 ∣ ∃𝑐 ∈ ( R ‘𝑋)𝑎 = (𝑐 +s 𝑌)} ∪ {𝑏 ∣ ∃𝑑 ∈ ( R ‘𝑌)𝑏 = (𝑋 +s 𝑑)}))
6 ovex 7447 . . . . 5 (𝑋 +s 𝑌) ∈ V
76snid 4660 . . . 4 (𝑋 +s 𝑌) ∈ {(𝑋 +s 𝑌)}
87a1i 11 . . 3 (𝜑 → (𝑋 +s 𝑌) ∈ {(𝑋 +s 𝑌)})
9 addsproplem5.6 . . . . . . . 8 (𝜑 → ( bday 𝑍) ∈ ( bday 𝑌))
10 bdayelon 27683 . . . . . . . . 9 ( bday 𝑌) ∈ On
11 addspropord.4 . . . . . . . . 9 (𝜑𝑍 No )
12 oldbday 27801 . . . . . . . . 9 ((( bday 𝑌) ∈ On ∧ 𝑍 No ) → (𝑍 ∈ ( O ‘( bday 𝑌)) ↔ ( bday 𝑍) ∈ ( bday 𝑌)))
1310, 11, 12sylancr 586 . . . . . . . 8 (𝜑 → (𝑍 ∈ ( O ‘( bday 𝑌)) ↔ ( bday 𝑍) ∈ ( bday 𝑌)))
149, 13mpbird 257 . . . . . . 7 (𝜑𝑍 ∈ ( O ‘( bday 𝑌)))
15 addspropord.5 . . . . . . 7 (𝜑𝑌 <s 𝑍)
16 breq2 5146 . . . . . . . 8 (𝑧 = 𝑍 → (𝑌 <s 𝑧𝑌 <s 𝑍))
17 rightval 27765 . . . . . . . 8 ( R ‘𝑌) = {𝑧 ∈ ( O ‘( bday 𝑌)) ∣ 𝑌 <s 𝑧}
1816, 17elrab2 3683 . . . . . . 7 (𝑍 ∈ ( R ‘𝑌) ↔ (𝑍 ∈ ( O ‘( bday 𝑌)) ∧ 𝑌 <s 𝑍))
1914, 15, 18sylanbrc 582 . . . . . 6 (𝜑𝑍 ∈ ( R ‘𝑌))
20 eqid 2727 . . . . . 6 (𝑋 +s 𝑍) = (𝑋 +s 𝑍)
21 oveq2 7422 . . . . . . 7 (𝑑 = 𝑍 → (𝑋 +s 𝑑) = (𝑋 +s 𝑍))
2221rspceeqv 3629 . . . . . 6 ((𝑍 ∈ ( R ‘𝑌) ∧ (𝑋 +s 𝑍) = (𝑋 +s 𝑍)) → ∃𝑑 ∈ ( R ‘𝑌)(𝑋 +s 𝑍) = (𝑋 +s 𝑑))
2319, 20, 22sylancl 585 . . . . 5 (𝜑 → ∃𝑑 ∈ ( R ‘𝑌)(𝑋 +s 𝑍) = (𝑋 +s 𝑑))
24 ovex 7447 . . . . . 6 (𝑋 +s 𝑍) ∈ V
25 eqeq1 2731 . . . . . . 7 (𝑏 = (𝑋 +s 𝑍) → (𝑏 = (𝑋 +s 𝑑) ↔ (𝑋 +s 𝑍) = (𝑋 +s 𝑑)))
2625rexbidv 3173 . . . . . 6 (𝑏 = (𝑋 +s 𝑍) → (∃𝑑 ∈ ( R ‘𝑌)𝑏 = (𝑋 +s 𝑑) ↔ ∃𝑑 ∈ ( R ‘𝑌)(𝑋 +s 𝑍) = (𝑋 +s 𝑑)))
2724, 26elab 3665 . . . . 5 ((𝑋 +s 𝑍) ∈ {𝑏 ∣ ∃𝑑 ∈ ( R ‘𝑌)𝑏 = (𝑋 +s 𝑑)} ↔ ∃𝑑 ∈ ( R ‘𝑌)(𝑋 +s 𝑍) = (𝑋 +s 𝑑))
2823, 27sylibr 233 . . . 4 (𝜑 → (𝑋 +s 𝑍) ∈ {𝑏 ∣ ∃𝑑 ∈ ( R ‘𝑌)𝑏 = (𝑋 +s 𝑑)})
29 elun2 4173 . . . 4 ((𝑋 +s 𝑍) ∈ {𝑏 ∣ ∃𝑑 ∈ ( R ‘𝑌)𝑏 = (𝑋 +s 𝑑)} → (𝑋 +s 𝑍) ∈ ({𝑎 ∣ ∃𝑐 ∈ ( R ‘𝑋)𝑎 = (𝑐 +s 𝑌)} ∪ {𝑏 ∣ ∃𝑑 ∈ ( R ‘𝑌)𝑏 = (𝑋 +s 𝑑)}))
3028, 29syl 17 . . 3 (𝜑 → (𝑋 +s 𝑍) ∈ ({𝑎 ∣ ∃𝑐 ∈ ( R ‘𝑋)𝑎 = (𝑐 +s 𝑌)} ∪ {𝑏 ∣ ∃𝑑 ∈ ( R ‘𝑌)𝑏 = (𝑋 +s 𝑑)}))
315, 8, 30ssltsepcd 27701 . 2 (𝜑 → (𝑋 +s 𝑌) <s (𝑋 +s 𝑍))
323, 2addscomd 27858 . 2 (𝜑 → (𝑌 +s 𝑋) = (𝑋 +s 𝑌))
3311, 2addscomd 27858 . 2 (𝜑 → (𝑍 +s 𝑋) = (𝑋 +s 𝑍))
3431, 32, 333brtr4d 5174 1 (𝜑 → (𝑌 +s 𝑋) <s (𝑍 +s 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1534  wcel 2099  {cab 2704  wral 3056  wrex 3065  cun 3942  {csn 4624   class class class wbr 5142  Oncon0 6363  cfv 6542  (class class class)co 7414   +no cnadd 8677   No csur 27547   <s cslt 27548   bday cbday 27549   <<s csslt 27687   O cold 27744   L cleft 27746   R cright 27747   +s cadds 27850
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7732
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-rmo 3371  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-tp 4629  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-se 5628  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-1st 7985  df-2nd 7986  df-frecs 8278  df-wrecs 8309  df-recs 8383  df-1o 8478  df-2o 8479  df-nadd 8678  df-no 27550  df-slt 27551  df-bday 27552  df-sslt 27688  df-scut 27690  df-0s 27731  df-made 27748  df-old 27749  df-left 27751  df-right 27752  df-norec2 27840  df-adds 27851
This theorem is referenced by:  addsproplem7  27866
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