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Theorem addsproplem5 27710
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 27708 . . . 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 1143 . . 3 (𝜑 → {(𝑋 +s 𝑌)} <<s ({𝑎 ∣ ∃𝑐 ∈ ( R ‘𝑋)𝑎 = (𝑐 +s 𝑌)} ∪ {𝑏 ∣ ∃𝑑 ∈ ( R ‘𝑌)𝑏 = (𝑋 +s 𝑑)}))
6 ovex 7445 . . . . 5 (𝑋 +s 𝑌) ∈ V
76snid 4664 . . . 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 27529 . . . . . . . . 9 ( bday 𝑌) ∈ On
11 addspropord.4 . . . . . . . . 9 (𝜑𝑍 No )
12 oldbday 27647 . . . . . . . . 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 5152 . . . . . . . 8 (𝑧 = 𝑍 → (𝑌 <s 𝑧𝑌 <s 𝑍))
17 rightval 27611 . . . . . . . 8 ( R ‘𝑌) = {𝑧 ∈ ( O ‘( bday 𝑌)) ∣ 𝑌 <s 𝑧}
1816, 17elrab2 3686 . . . . . . 7 (𝑍 ∈ ( R ‘𝑌) ↔ (𝑍 ∈ ( O ‘( bday 𝑌)) ∧ 𝑌 <s 𝑍))
1914, 15, 18sylanbrc 582 . . . . . 6 (𝜑𝑍 ∈ ( R ‘𝑌))
20 eqid 2731 . . . . . 6 (𝑋 +s 𝑍) = (𝑋 +s 𝑍)
21 oveq2 7420 . . . . . . 7 (𝑑 = 𝑍 → (𝑋 +s 𝑑) = (𝑋 +s 𝑍))
2221rspceeqv 3633 . . . . . 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 7445 . . . . . 6 (𝑋 +s 𝑍) ∈ V
25 eqeq1 2735 . . . . . . 7 (𝑏 = (𝑋 +s 𝑍) → (𝑏 = (𝑋 +s 𝑑) ↔ (𝑋 +s 𝑍) = (𝑋 +s 𝑑)))
2625rexbidv 3177 . . . . . 6 (𝑏 = (𝑋 +s 𝑍) → (∃𝑑 ∈ ( R ‘𝑌)𝑏 = (𝑋 +s 𝑑) ↔ ∃𝑑 ∈ ( R ‘𝑌)(𝑋 +s 𝑍) = (𝑋 +s 𝑑)))
2724, 26elab 3668 . . . . 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 4177 . . . 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 27547 . 2 (𝜑 → (𝑋 +s 𝑌) <s (𝑋 +s 𝑍))
323, 2addscomd 27704 . 2 (𝜑 → (𝑌 +s 𝑋) = (𝑋 +s 𝑌))
3311, 2addscomd 27704 . 2 (𝜑 → (𝑍 +s 𝑋) = (𝑋 +s 𝑍))
3431, 32, 333brtr4d 5180 1 (𝜑 → (𝑌 +s 𝑋) <s (𝑍 +s 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1540  wcel 2105  {cab 2708  wral 3060  wrex 3069  cun 3946  {csn 4628   class class class wbr 5148  Oncon0 6364  cfv 6543  (class class class)co 7412   +no cnadd 8670   No csur 27394   <s cslt 27395   bday cbday 27396   <<s csslt 27533   O cold 27590   L cleft 27592   R cright 27593   +s cadds 27696
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  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-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 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7979  df-2nd 7980  df-frecs 8272  df-wrecs 8303  df-recs 8377  df-1o 8472  df-2o 8473  df-nadd 8671  df-no 27397  df-slt 27398  df-bday 27399  df-sslt 27534  df-scut 27536  df-0s 27577  df-made 27594  df-old 27595  df-left 27597  df-right 27598  df-norec2 27686  df-adds 27697
This theorem is referenced by:  addsproplem7  27712
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