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Theorem addsproplem5 27903
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 27901 . . . 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 1141 . . 3 (𝜑 → {(𝑋 +s 𝑌)} <<s ({𝑎 ∣ ∃𝑐 ∈ ( R ‘𝑋)𝑎 = (𝑐 +s 𝑌)} ∪ {𝑏 ∣ ∃𝑑 ∈ ( R ‘𝑌)𝑏 = (𝑋 +s 𝑑)}))
6 ovex 7446 . . . . 5 (𝑋 +s 𝑌) ∈ V
76snid 4661 . . . 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 27722 . . . . . . . . 9 ( bday 𝑌) ∈ On
11 addspropord.4 . . . . . . . . 9 (𝜑𝑍 No )
12 oldbday 27840 . . . . . . . . 9 ((( bday 𝑌) ∈ On ∧ 𝑍 No ) → (𝑍 ∈ ( O ‘( bday 𝑌)) ↔ ( bday 𝑍) ∈ ( bday 𝑌)))
1310, 11, 12sylancr 585 . . . . . . . 8 (𝜑 → (𝑍 ∈ ( O ‘( bday 𝑌)) ↔ ( bday 𝑍) ∈ ( bday 𝑌)))
149, 13mpbird 256 . . . . . . 7 (𝜑𝑍 ∈ ( O ‘( bday 𝑌)))
15 addspropord.5 . . . . . . 7 (𝜑𝑌 <s 𝑍)
16 breq2 5148 . . . . . . . 8 (𝑧 = 𝑍 → (𝑌 <s 𝑧𝑌 <s 𝑍))
17 rightval 27804 . . . . . . . 8 ( R ‘𝑌) = {𝑧 ∈ ( O ‘( bday 𝑌)) ∣ 𝑌 <s 𝑧}
1816, 17elrab2 3679 . . . . . . 7 (𝑍 ∈ ( R ‘𝑌) ↔ (𝑍 ∈ ( O ‘( bday 𝑌)) ∧ 𝑌 <s 𝑍))
1914, 15, 18sylanbrc 581 . . . . . 6 (𝜑𝑍 ∈ ( R ‘𝑌))
20 eqid 2725 . . . . . 6 (𝑋 +s 𝑍) = (𝑋 +s 𝑍)
21 oveq2 7421 . . . . . . 7 (𝑑 = 𝑍 → (𝑋 +s 𝑑) = (𝑋 +s 𝑍))
2221rspceeqv 3625 . . . . . 6 ((𝑍 ∈ ( R ‘𝑌) ∧ (𝑋 +s 𝑍) = (𝑋 +s 𝑍)) → ∃𝑑 ∈ ( R ‘𝑌)(𝑋 +s 𝑍) = (𝑋 +s 𝑑))
2319, 20, 22sylancl 584 . . . . 5 (𝜑 → ∃𝑑 ∈ ( R ‘𝑌)(𝑋 +s 𝑍) = (𝑋 +s 𝑑))
24 ovex 7446 . . . . . 6 (𝑋 +s 𝑍) ∈ V
25 eqeq1 2729 . . . . . . 7 (𝑏 = (𝑋 +s 𝑍) → (𝑏 = (𝑋 +s 𝑑) ↔ (𝑋 +s 𝑍) = (𝑋 +s 𝑑)))
2625rexbidv 3169 . . . . . 6 (𝑏 = (𝑋 +s 𝑍) → (∃𝑑 ∈ ( R ‘𝑌)𝑏 = (𝑋 +s 𝑑) ↔ ∃𝑑 ∈ ( R ‘𝑌)(𝑋 +s 𝑍) = (𝑋 +s 𝑑)))
2724, 26elab 3661 . . . . 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 4172 . . . 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 27740 . 2 (𝜑 → (𝑋 +s 𝑌) <s (𝑋 +s 𝑍))
323, 2addscomd 27897 . 2 (𝜑 → (𝑌 +s 𝑋) = (𝑋 +s 𝑌))
3311, 2addscomd 27897 . 2 (𝜑 → (𝑍 +s 𝑋) = (𝑋 +s 𝑍))
3431, 32, 333brtr4d 5176 1 (𝜑 → (𝑌 +s 𝑋) <s (𝑍 +s 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1533  wcel 2098  {cab 2702  wral 3051  wrex 3060  cun 3939  {csn 4625   class class class wbr 5144  Oncon0 6365  cfv 6543  (class class class)co 7413   +no cnadd 8679   No csur 27586   <s cslt 27587   bday cbday 27588   <<s csslt 27726   O cold 27783   L cleft 27785   R cright 27786   +s cadds 27889
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5281  ax-sep 5295  ax-nul 5302  ax-pow 5360  ax-pr 5424  ax-un 7735
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3961  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-tp 4630  df-op 4632  df-uni 4905  df-int 4946  df-iun 4994  df-br 5145  df-opab 5207  df-mpt 5228  df-tr 5262  df-id 5571  df-eprel 5577  df-po 5585  df-so 5586  df-fr 5628  df-se 5629  df-we 5630  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-pred 6301  df-ord 6368  df-on 6369  df-suc 6371  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 7369  df-ov 7416  df-oprab 7417  df-mpo 7418  df-1st 7987  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-1o 8480  df-2o 8481  df-nadd 8680  df-no 27589  df-slt 27590  df-bday 27591  df-sslt 27727  df-scut 27729  df-0s 27770  df-made 27787  df-old 27788  df-left 27790  df-right 27791  df-norec2 27879  df-adds 27890
This theorem is referenced by:  addsproplem7  27905
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