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Theorem addsproplem5 27288
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 27286 . . . 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 1145 . . 3 (𝜑 → {(𝑋 +s 𝑌)} <<s ({𝑎 ∣ ∃𝑐 ∈ ( R ‘𝑋)𝑎 = (𝑐 +s 𝑌)} ∪ {𝑏 ∣ ∃𝑑 ∈ ( R ‘𝑌)𝑏 = (𝑋 +s 𝑑)}))
6 ovex 7391 . . . . 5 (𝑋 +s 𝑌) ∈ V
76snid 4623 . . . 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 27119 . . . . . . . . 9 ( bday 𝑌) ∈ On
11 addspropord.4 . . . . . . . . 9 (𝜑𝑍 No )
12 oldbday 27233 . . . . . . . . 9 ((( bday 𝑌) ∈ On ∧ 𝑍 No ) → (𝑍 ∈ ( O ‘( bday 𝑌)) ↔ ( bday 𝑍) ∈ ( bday 𝑌)))
1310, 11, 12sylancr 588 . . . . . . . 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 5110 . . . . . . . 8 (𝑧 = 𝑍 → (𝑌 <s 𝑧𝑌 <s 𝑍))
17 rightval 27197 . . . . . . . 8 ( R ‘𝑌) = {𝑧 ∈ ( O ‘( bday 𝑌)) ∣ 𝑌 <s 𝑧}
1816, 17elrab2 3649 . . . . . . 7 (𝑍 ∈ ( R ‘𝑌) ↔ (𝑍 ∈ ( O ‘( bday 𝑌)) ∧ 𝑌 <s 𝑍))
1914, 15, 18sylanbrc 584 . . . . . 6 (𝜑𝑍 ∈ ( R ‘𝑌))
20 eqid 2737 . . . . . 6 (𝑋 +s 𝑍) = (𝑋 +s 𝑍)
21 oveq2 7366 . . . . . . 7 (𝑑 = 𝑍 → (𝑋 +s 𝑑) = (𝑋 +s 𝑍))
2221rspceeqv 3596 . . . . . 6 ((𝑍 ∈ ( R ‘𝑌) ∧ (𝑋 +s 𝑍) = (𝑋 +s 𝑍)) → ∃𝑑 ∈ ( R ‘𝑌)(𝑋 +s 𝑍) = (𝑋 +s 𝑑))
2319, 20, 22sylancl 587 . . . . 5 (𝜑 → ∃𝑑 ∈ ( R ‘𝑌)(𝑋 +s 𝑍) = (𝑋 +s 𝑑))
24 ovex 7391 . . . . . 6 (𝑋 +s 𝑍) ∈ V
25 eqeq1 2741 . . . . . . 7 (𝑏 = (𝑋 +s 𝑍) → (𝑏 = (𝑋 +s 𝑑) ↔ (𝑋 +s 𝑍) = (𝑋 +s 𝑑)))
2625rexbidv 3176 . . . . . 6 (𝑏 = (𝑋 +s 𝑍) → (∃𝑑 ∈ ( R ‘𝑌)𝑏 = (𝑋 +s 𝑑) ↔ ∃𝑑 ∈ ( R ‘𝑌)(𝑋 +s 𝑍) = (𝑋 +s 𝑑)))
2724, 26elab 3631 . . . . 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 4138 . . . 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 27136 . 2 (𝜑 → (𝑋 +s 𝑌) <s (𝑋 +s 𝑍))
323, 2addscomd 27282 . 2 (𝜑 → (𝑌 +s 𝑋) = (𝑋 +s 𝑌))
3311, 2addscomd 27282 . 2 (𝜑 → (𝑍 +s 𝑋) = (𝑋 +s 𝑍))
3431, 32, 333brtr4d 5138 1 (𝜑 → (𝑌 +s 𝑋) <s (𝑍 +s 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = 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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