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Theorem no2indslem 27826
Description: Double induction on surreals with explicit notation for the relationships. (Contributed by Scott Fenton, 22-Aug-2024.)
Hypotheses
Ref Expression
no2indslem.a 𝑅 = {⟨𝑎, 𝑏⟩ ∣ 𝑎 ∈ (( L ‘𝑏) ∪ ( R ‘𝑏))}
no2indslem.1 (𝑥 = 𝑧 → (𝜑𝜓))
no2indslem.2 (𝑦 = 𝑤 → (𝜓𝜒))
no2indslem.3 (𝑥 = 𝑧 → (𝜃𝜒))
no2indslem.4 (𝑥 = 𝐴 → (𝜑𝜏))
no2indslem.5 (𝑦 = 𝐵 → (𝜏𝜂))
no2indslem.i ((𝑥 No 𝑦 No ) → ((∀𝑧 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑤 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))𝜒 ∧ ∀𝑧 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))𝜓 ∧ ∀𝑤 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))𝜃) → 𝜑))
Assertion
Ref Expression
no2indslem ((𝐴 No 𝐵 No ) → 𝜂)
Distinct variable groups:   𝑎,𝑏,𝑥   𝑥,𝐴   𝑦,𝑎   𝑦,𝐴   𝑥,𝑏,𝑦   𝑦,𝐵   𝜒,𝑦   𝜂,𝑦   𝜑,𝑧   𝜓,𝑤,𝑥   𝑥,𝑅,𝑦,𝑤,𝑧   𝜏,𝑥   𝜃,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑤,𝑎,𝑏)   𝜓(𝑦,𝑧,𝑎,𝑏)   𝜒(𝑥,𝑧,𝑤,𝑎,𝑏)   𝜃(𝑥,𝑦,𝑤,𝑎,𝑏)   𝜏(𝑦,𝑧,𝑤,𝑎,𝑏)   𝜂(𝑥,𝑧,𝑤,𝑎,𝑏)   𝐴(𝑧,𝑤,𝑎,𝑏)   𝐵(𝑥,𝑧,𝑤,𝑎,𝑏)   𝑅(𝑎,𝑏)

Proof of Theorem no2indslem
StepHypRef Expression
1 no2indslem.a . . 3 𝑅 = {⟨𝑎, 𝑏⟩ ∣ 𝑎 ∈ (( L ‘𝑏) ∪ ( R ‘𝑏))}
21lrrecfr 27815 . 2 𝑅 Fr No
31lrrecpo 27813 . 2 𝑅 Po No
41lrrecse 27814 . 2 𝑅 Se No
5 no2indslem.1 . 2 (𝑥 = 𝑧 → (𝜑𝜓))
6 no2indslem.2 . 2 (𝑦 = 𝑤 → (𝜓𝜒))
7 no2indslem.3 . 2 (𝑥 = 𝑧 → (𝜃𝜒))
8 no2indslem.4 . 2 (𝑥 = 𝐴 → (𝜑𝜏))
9 no2indslem.5 . 2 (𝑦 = 𝐵 → (𝜏𝜂))
101lrrecpred 27816 . . . . . 6 (𝑥 No → Pred(𝑅, No , 𝑥) = (( L ‘𝑥) ∪ ( R ‘𝑥)))
1110adantr 480 . . . . 5 ((𝑥 No 𝑦 No ) → Pred(𝑅, No , 𝑥) = (( L ‘𝑥) ∪ ( R ‘𝑥)))
121lrrecpred 27816 . . . . . . 7 (𝑦 No → Pred(𝑅, No , 𝑦) = (( L ‘𝑦) ∪ ( R ‘𝑦)))
1312adantl 481 . . . . . 6 ((𝑥 No 𝑦 No ) → Pred(𝑅, No , 𝑦) = (( L ‘𝑦) ∪ ( R ‘𝑦)))
1413raleqdv 3319 . . . . 5 ((𝑥 No 𝑦 No ) → (∀𝑤 ∈ Pred (𝑅, No , 𝑦)𝜒 ↔ ∀𝑤 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))𝜒))
1511, 14raleqbidv 3336 . . . 4 ((𝑥 No 𝑦 No ) → (∀𝑧 ∈ Pred (𝑅, No , 𝑥)∀𝑤 ∈ Pred (𝑅, No , 𝑦)𝜒 ↔ ∀𝑧 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑤 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))𝜒))
1611raleqdv 3319 . . . 4 ((𝑥 No 𝑦 No ) → (∀𝑧 ∈ Pred (𝑅, No , 𝑥)𝜓 ↔ ∀𝑧 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))𝜓))
1713raleqdv 3319 . . . 4 ((𝑥 No 𝑦 No ) → (∀𝑤 ∈ Pred (𝑅, No , 𝑦)𝜃 ↔ ∀𝑤 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))𝜃))
1815, 16, 173anbi123d 1432 . . 3 ((𝑥 No 𝑦 No ) → ((∀𝑧 ∈ Pred (𝑅, No , 𝑥)∀𝑤 ∈ Pred (𝑅, No , 𝑦)𝜒 ∧ ∀𝑧 ∈ Pred (𝑅, No , 𝑥)𝜓 ∧ ∀𝑤 ∈ Pred (𝑅, No , 𝑦)𝜃) ↔ (∀𝑧 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑤 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))𝜒 ∧ ∀𝑧 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))𝜓 ∧ ∀𝑤 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))𝜃)))
19 no2indslem.i . . 3 ((𝑥 No 𝑦 No ) → ((∀𝑧 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑤 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))𝜒 ∧ ∀𝑧 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))𝜓 ∧ ∀𝑤 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))𝜃) → 𝜑))
2018, 19sylbid 239 . 2 ((𝑥 No 𝑦 No ) → ((∀𝑧 ∈ Pred (𝑅, No , 𝑥)∀𝑤 ∈ Pred (𝑅, No , 𝑦)𝜒 ∧ ∀𝑧 ∈ Pred (𝑅, No , 𝑥)𝜓 ∧ ∀𝑤 ∈ Pred (𝑅, No , 𝑦)𝜃) → 𝜑))
212, 3, 4, 2, 3, 4, 5, 6, 7, 8, 9, 20xpord2ind 8134 1 ((𝐴 No 𝐵 No ) → 𝜂)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  w3a 1084   = wceq 1533  wcel 2098  wral 3055  cun 3941  {copab 5203  Predcpred 6293  cfv 6537   No csur 27528   L cleft 27727   R cright 27728
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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  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 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-tp 4628  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-se 5625  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-ord 6361  df-on 6362  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7974  df-2nd 7975  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-1o 8467  df-2o 8468  df-no 27531  df-slt 27532  df-bday 27533  df-sslt 27669  df-scut 27671  df-made 27729  df-old 27730  df-left 27732  df-right 27733
This theorem is referenced by:  no2inds  27827
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