Theorem no2indslem 34090

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.b	⊢ 𝑆 = {⟨𝑐, 𝑑⟩ ∣ (𝑐 ∈ ( No × No ) ∧ 𝑑 ∈ ( No × No ) ∧ (((1st ‘𝑐)𝑅(1st ‘𝑑) ∨ (1st ‘𝑐) = (1st ‘𝑑)) ∧ ((2nd ‘𝑐)𝑅(2nd ‘𝑑) ∨ (2nd ‘𝑐) = (2nd ‘𝑑)) ∧ 𝑐 ≠ 𝑑))}
no2indslem.1	⊢ (𝑥 = 𝑧 → (𝜑 ↔ 𝜓))
no2indslem.2	⊢ (𝑦 = 𝑤 → (𝜓 ↔ 𝜒))
no2indslem.3	⊢ (𝑥 = 𝑧 → (𝜃 ↔ 𝜒))
no2indslem.4	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏))
no2indslem.5	⊢ (𝑦 = 𝐵 → (𝜏 ↔ 𝜂))
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   𝑐,𝑑   𝜒,𝑦   𝜂,𝑦   𝜑,𝑧   𝜓,𝑤,𝑥   𝑅,𝑐,𝑑   𝑤,𝑅,𝑧   𝑤,𝑆,𝑥,𝑦,𝑧   𝜏,𝑥   𝜃,𝑧   𝑥,𝑤,𝑦,𝑧

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑤,𝑎,𝑏,𝑐,𝑑)   𝜓(𝑦,𝑧,𝑎,𝑏,𝑐,𝑑)   𝜒(𝑥,𝑧,𝑤,𝑎,𝑏,𝑐,𝑑)   𝜃(𝑥,𝑦,𝑤,𝑎,𝑏,𝑐,𝑑)   𝜏(𝑦,𝑧,𝑤,𝑎,𝑏,𝑐,𝑑)   𝜂(𝑥,𝑧,𝑤,𝑎,𝑏,𝑐,𝑑)   𝐴(𝑧,𝑤,𝑎,𝑏,𝑐,𝑑)   𝐵(𝑥,𝑧,𝑤,𝑎,𝑏,𝑐,𝑑)   𝑅(𝑥,𝑦,𝑎,𝑏)   𝑆(𝑎,𝑏,𝑐,𝑑)

Proof of Theorem no2indslem

Step	Hyp	Ref	Expression
1		no2indslem.b	. 2 ⊢ 𝑆 = {⟨𝑐, 𝑑⟩ ∣ (𝑐 ∈ ( No × No ) ∧ 𝑑 ∈ ( No × No ) ∧ (((1st ‘𝑐)𝑅(1st ‘𝑑) ∨ (1st ‘𝑐) = (1st ‘𝑑)) ∧ ((2nd ‘𝑐)𝑅(2nd ‘𝑑) ∨ (2nd ‘𝑐) = (2nd ‘𝑑)) ∧ 𝑐 ≠ 𝑑))}
2		no2indslem.a	. . 3 ⊢ 𝑅 = {⟨𝑎, 𝑏⟩ ∣ 𝑎 ∈ (( L ‘𝑏) ∪ ( R ‘𝑏))}
3	2	lrrecfr 34079	. 2 ⊢ 𝑅 Fr No
4	2	lrrecpo 34077	. 2 ⊢ 𝑅 Po No
5	2	lrrecse 34078	. 2 ⊢ 𝑅 Se No
6		no2indslem.1	. 2 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ 𝜓))
7		no2indslem.2	. 2 ⊢ (𝑦 = 𝑤 → (𝜓 ↔ 𝜒))
8		no2indslem.3	. 2 ⊢ (𝑥 = 𝑧 → (𝜃 ↔ 𝜒))
9		no2indslem.4	. 2 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏))
10		no2indslem.5	. 2 ⊢ (𝑦 = 𝐵 → (𝜏 ↔ 𝜂))
11	2	lrrecpred 34080	. . . . . 6 ⊢ (𝑥 ∈ No → Pred(𝑅, No , 𝑥) = (( L ‘𝑥) ∪ ( R ‘𝑥)))
12	11	adantr 480	. . . . 5 ⊢ ((𝑥 ∈ No ∧ 𝑦 ∈ No ) → Pred(𝑅, No , 𝑥) = (( L ‘𝑥) ∪ ( R ‘𝑥)))
13	2	lrrecpred 34080	. . . . . . 7 ⊢ (𝑦 ∈ No → Pred(𝑅, No , 𝑦) = (( L ‘𝑦) ∪ ( R ‘𝑦)))
14	13	adantl 481	. . . . . 6 ⊢ ((𝑥 ∈ No ∧ 𝑦 ∈ No ) → Pred(𝑅, No , 𝑦) = (( L ‘𝑦) ∪ ( R ‘𝑦)))
15	14	raleqdv 3346	. . . . 5 ⊢ ((𝑥 ∈ No ∧ 𝑦 ∈ No ) → (∀𝑤 ∈ Pred (𝑅, No , 𝑦)𝜒 ↔ ∀𝑤 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))𝜒))
16	12, 15	raleqbidv 3334	. . . 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))
17	12	raleqdv 3346	. . . 4 ⊢ ((𝑥 ∈ No ∧ 𝑦 ∈ No ) → (∀𝑧 ∈ Pred (𝑅, No , 𝑥)𝜓 ↔ ∀𝑧 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))𝜓))
18	14	raleqdv 3346	. . . 4 ⊢ ((𝑥 ∈ No ∧ 𝑦 ∈ No ) → (∀𝑤 ∈ Pred (𝑅, No , 𝑦)𝜃 ↔ ∀𝑤 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))𝜃))
19	16, 17, 18	3anbi123d 1434	. . 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)))
20		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))
21	19, 20	sylbid 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))
22	1, 3, 4, 5, 3, 4, 5, 6, 7, 8, 9, 10, 21	xpord2ind 33773	1 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → 𝜂)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∨ wo 843   ∧ w3a 1085   = wceq 1541   ∈ wcel 2109   ≠ wne 2944  ∀wral 3065   ∪ cun 3889   class class class wbr 5078  {copab 5140   × cxp 5586  Predcpred 6198  ‘cfv 6430  1^st c1st 7815  2^nd c2nd 7816   No csur 33822   L cleft 34008   R cright 34009

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710  ax-rep 5213  ax-sep 5226  ax-nul 5233  ax-pow 5291  ax-pr 5355  ax-un 7579

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-ral 3070  df-rex 3071  df-reu 3072  df-rmo 3073  df-rab 3074  df-v 3432  df-sbc 3720  df-csb 3837  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-pss 3910  df-nul 4262  df-if 4465  df-pw 4540  df-sn 4567  df-pr 4569  df-tp 4571  df-op 4573  df-uni 4845  df-int 4885  df-iun 4931  df-br 5079  df-opab 5141  df-mpt 5162  df-tr 5196  df-id 5488  df-eprel 5494  df-po 5502  df-so 5503  df-fr 5543  df-se 5544  df-we 5545  df-xp 5594  df-rel 5595  df-cnv 5596  df-co 5597  df-dm 5598  df-rn 5599  df-res 5600  df-ima 5601  df-pred 6199  df-ord 6266  df-on 6267  df-suc 6269  df-iota 6388  df-fun 6432  df-fn 6433  df-f 6434  df-f1 6435  df-fo 6436  df-f1o 6437  df-fv 6438  df-riota 7225  df-ov 7271  df-oprab 7272  df-mpo 7273  df-1st 7817  df-2nd 7818  df-frecs 8081  df-wrecs 8112  df-recs 8186  df-1o 8281  df-2o 8282  df-no 33825  df-slt 33826  df-bday 33827  df-sslt 33955  df-scut 33957  df-made 34010  df-old 34011  df-left 34013  df-right 34014

This theorem is referenced by:  no2inds  34091