MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  addsproplem4 Structured version   Visualization version   GIF version

Theorem addsproplem4 27709
Description: Lemma for surreal addition properties. Show the second half of the inductive hypothesis when 𝑌 is older than 𝑍. (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)
addsproplem4.6 (𝜑 → ( bday 𝑌) ∈ ( bday 𝑍))
Assertion
Ref Expression
addsproplem4 (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 addsproplem4
Dummy variables 𝑎 𝑏 𝑐 𝑑 𝑒 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 addsproplem.1 . . . . . 6 (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 uncom 4153 . . . . . . . . . 10 ((( bday 𝑋) +no ( bday 𝑌)) ∪ (( bday 𝑋) +no ( bday 𝑍))) = ((( bday 𝑋) +no ( bday 𝑍)) ∪ (( bday 𝑋) +no ( bday 𝑌)))
32eleq2i 2824 . . . . . . . . 9 (((( 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))))
43imbi1i 349 . . . . . . . 8 ((((( 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)))))
54ralbii 3092 . . . . . . 7 (∀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)))))
652ralbii 3127 . . . . . 6 (∀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)))))
71, 6sylib 217 . . . . 5 (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)))))
8 addspropord.2 . . . . 5 (𝜑𝑋 No )
9 addspropord.4 . . . . 5 (𝜑𝑍 No )
107, 8, 9addsproplem3 27708 . . . 4 (𝜑 → ((𝑋 +s 𝑍) ∈ No ∧ ({𝑎 ∣ ∃𝑐 ∈ ( L ‘𝑋)𝑎 = (𝑐 +s 𝑍)} ∪ {𝑏 ∣ ∃𝑑 ∈ ( L ‘𝑍)𝑏 = (𝑋 +s 𝑑)}) <<s {(𝑋 +s 𝑍)} ∧ {(𝑋 +s 𝑍)} <<s ({𝑒 ∣ ∃𝑔 ∈ ( R ‘𝑋)𝑒 = (𝑔 +s 𝑍)} ∪ {𝑓 ∣ ∃ ∈ ( R ‘𝑍)𝑓 = (𝑋 +s )})))
1110simp2d 1142 . . 3 (𝜑 → ({𝑎 ∣ ∃𝑐 ∈ ( L ‘𝑋)𝑎 = (𝑐 +s 𝑍)} ∪ {𝑏 ∣ ∃𝑑 ∈ ( L ‘𝑍)𝑏 = (𝑋 +s 𝑑)}) <<s {(𝑋 +s 𝑍)})
12 addsproplem4.6 . . . . . . . 8 (𝜑 → ( bday 𝑌) ∈ ( bday 𝑍))
13 bdayelon 27529 . . . . . . . . 9 ( bday 𝑍) ∈ On
14 addspropord.3 . . . . . . . . 9 (𝜑𝑌 No )
15 oldbday 27647 . . . . . . . . 9 ((( bday 𝑍) ∈ On ∧ 𝑌 No ) → (𝑌 ∈ ( O ‘( bday 𝑍)) ↔ ( bday 𝑌) ∈ ( bday 𝑍)))
1613, 14, 15sylancr 586 . . . . . . . 8 (𝜑 → (𝑌 ∈ ( O ‘( bday 𝑍)) ↔ ( bday 𝑌) ∈ ( bday 𝑍)))
1712, 16mpbird 257 . . . . . . 7 (𝜑𝑌 ∈ ( O ‘( bday 𝑍)))
18 addspropord.5 . . . . . . 7 (𝜑𝑌 <s 𝑍)
19 breq1 5151 . . . . . . . 8 (𝑦 = 𝑌 → (𝑦 <s 𝑍𝑌 <s 𝑍))
20 leftval 27610 . . . . . . . 8 ( L ‘𝑍) = {𝑦 ∈ ( O ‘( bday 𝑍)) ∣ 𝑦 <s 𝑍}
2119, 20elrab2 3686 . . . . . . 7 (𝑌 ∈ ( L ‘𝑍) ↔ (𝑌 ∈ ( O ‘( bday 𝑍)) ∧ 𝑌 <s 𝑍))
2217, 18, 21sylanbrc 582 . . . . . 6 (𝜑𝑌 ∈ ( L ‘𝑍))
23 eqid 2731 . . . . . 6 (𝑋 +s 𝑌) = (𝑋 +s 𝑌)
24 oveq2 7420 . . . . . . 7 (𝑑 = 𝑌 → (𝑋 +s 𝑑) = (𝑋 +s 𝑌))
2524rspceeqv 3633 . . . . . 6 ((𝑌 ∈ ( L ‘𝑍) ∧ (𝑋 +s 𝑌) = (𝑋 +s 𝑌)) → ∃𝑑 ∈ ( L ‘𝑍)(𝑋 +s 𝑌) = (𝑋 +s 𝑑))
2622, 23, 25sylancl 585 . . . . 5 (𝜑 → ∃𝑑 ∈ ( L ‘𝑍)(𝑋 +s 𝑌) = (𝑋 +s 𝑑))
27 ovex 7445 . . . . . 6 (𝑋 +s 𝑌) ∈ V
28 eqeq1 2735 . . . . . . 7 (𝑏 = (𝑋 +s 𝑌) → (𝑏 = (𝑋 +s 𝑑) ↔ (𝑋 +s 𝑌) = (𝑋 +s 𝑑)))
2928rexbidv 3177 . . . . . 6 (𝑏 = (𝑋 +s 𝑌) → (∃𝑑 ∈ ( L ‘𝑍)𝑏 = (𝑋 +s 𝑑) ↔ ∃𝑑 ∈ ( L ‘𝑍)(𝑋 +s 𝑌) = (𝑋 +s 𝑑)))
3027, 29elab 3668 . . . . 5 ((𝑋 +s 𝑌) ∈ {𝑏 ∣ ∃𝑑 ∈ ( L ‘𝑍)𝑏 = (𝑋 +s 𝑑)} ↔ ∃𝑑 ∈ ( L ‘𝑍)(𝑋 +s 𝑌) = (𝑋 +s 𝑑))
3126, 30sylibr 233 . . . 4 (𝜑 → (𝑋 +s 𝑌) ∈ {𝑏 ∣ ∃𝑑 ∈ ( L ‘𝑍)𝑏 = (𝑋 +s 𝑑)})
32 elun2 4177 . . . 4 ((𝑋 +s 𝑌) ∈ {𝑏 ∣ ∃𝑑 ∈ ( L ‘𝑍)𝑏 = (𝑋 +s 𝑑)} → (𝑋 +s 𝑌) ∈ ({𝑎 ∣ ∃𝑐 ∈ ( L ‘𝑋)𝑎 = (𝑐 +s 𝑍)} ∪ {𝑏 ∣ ∃𝑑 ∈ ( L ‘𝑍)𝑏 = (𝑋 +s 𝑑)}))
3331, 32syl 17 . . 3 (𝜑 → (𝑋 +s 𝑌) ∈ ({𝑎 ∣ ∃𝑐 ∈ ( L ‘𝑋)𝑎 = (𝑐 +s 𝑍)} ∪ {𝑏 ∣ ∃𝑑 ∈ ( L ‘𝑍)𝑏 = (𝑋 +s 𝑑)}))
34 ovex 7445 . . . . 5 (𝑋 +s 𝑍) ∈ V
3534snid 4664 . . . 4 (𝑋 +s 𝑍) ∈ {(𝑋 +s 𝑍)}
3635a1i 11 . . 3 (𝜑 → (𝑋 +s 𝑍) ∈ {(𝑋 +s 𝑍)})
3711, 33, 36ssltsepcd 27547 . 2 (𝜑 → (𝑋 +s 𝑌) <s (𝑋 +s 𝑍))
3814, 8addscomd 27704 . 2 (𝜑 → (𝑌 +s 𝑋) = (𝑋 +s 𝑌))
399, 8addscomd 27704 . 2 (𝜑 → (𝑍 +s 𝑋) = (𝑋 +s 𝑍))
4037, 38, 393brtr4d 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
  Copyright terms: Public domain W3C validator