Theorem no2indlesm 28184

Description: Double induction on surreals with explicit notation for the relationships. (Contributed by Scott Fenton, 22-Aug-2024.)

Hypotheses

Ref	Expression
no2indlesm.a	⊢ 𝑅 = {⟨𝑎, 𝑏⟩ ∣ 𝑎 ∈ (( L ‘𝑏) ∪ ( R ‘𝑏))}
no2indlesm.1	⊢ (𝑥 = 𝑧 → (𝜑 ↔ 𝜓))
no2indlesm.2	⊢ (𝑦 = 𝑤 → (𝜓 ↔ 𝜒))
no2indlesm.3	⊢ (𝑥 = 𝑧 → (𝜃 ↔ 𝜒))
no2indlesm.4	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏))
no2indlesm.5	⊢ (𝑦 = 𝐵 → (𝜏 ↔ 𝜂))
no2indlesm.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
no2indlesm	⊢ ((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 no2indlesm

Step	Hyp	Ref	Expression
1		no2indlesm.a	. . 3 ⊢ 𝑅 = {⟨𝑎, 𝑏⟩ ∣ 𝑎 ∈ (( L ‘𝑏) ∪ ( R ‘𝑏))}
2	1	lrrecfr 28173	. 2 ⊢ 𝑅 Fr No
3	1	lrrecpo 28171	. 2 ⊢ 𝑅 Po No
4	1	lrrecse 28172	. 2 ⊢ 𝑅 Se No
5		no2indlesm.1	. 2 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ 𝜓))
6		no2indlesm.2	. 2 ⊢ (𝑦 = 𝑤 → (𝜓 ↔ 𝜒))
7		no2indlesm.3	. 2 ⊢ (𝑥 = 𝑧 → (𝜃 ↔ 𝜒))
8		no2indlesm.4	. 2 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏))
9		no2indlesm.5	. 2 ⊢ (𝑦 = 𝐵 → (𝜏 ↔ 𝜂))
10	1	lrrecpred 28174	. . . . . 6 ⊢ (𝑥 ∈ No → Pred(𝑅, No , 𝑥) = (( L ‘𝑥) ∪ ( R ‘𝑥)))
11	10	adantr 486	. . . . 5 ⊢ ((𝑥 ∈ No ∧ 𝑦 ∈ No ) → Pred(𝑅, No , 𝑥) = (( L ‘𝑥) ∪ ( R ‘𝑥)))
12	1	lrrecpred 28174	. . . . . . 7 ⊢ (𝑦 ∈ No → Pred(𝑅, No , 𝑦) = (( L ‘𝑦) ∪ ( R ‘𝑦)))
13	12	adantl 487	. . . . . 6 ⊢ ((𝑥 ∈ No ∧ 𝑦 ∈ No ) → Pred(𝑅, No , 𝑦) = (( L ‘𝑦) ∪ ( R ‘𝑦)))
14	13	raleqdv 3326	. . . . 5 ⊢ ((𝑥 ∈ No ∧ 𝑦 ∈ No ) → (∀𝑤 ∈ Pred (𝑅, No , 𝑦)𝜒 ↔ ∀𝑤 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))𝜒))
15	11, 14	raleqbidv 3341	. . . 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))
16	11	raleqdv 3326	. . . 4 ⊢ ((𝑥 ∈ No ∧ 𝑦 ∈ No ) → (∀𝑧 ∈ Pred (𝑅, No , 𝑥)𝜓 ↔ ∀𝑧 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))𝜓))
17	13	raleqdv 3326	. . . 4 ⊢ ((𝑥 ∈ No ∧ 𝑦 ∈ No ) → (∀𝑤 ∈ Pred (𝑅, No , 𝑦)𝜃 ↔ ∀𝑤 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))𝜃))
18	15, 16, 17	3anbi123d 1464	. . 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		no2indlesm.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))
20	18, 19	sylbid 243	. 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))
21	2, 3, 4, 2, 3, 4, 5, 6, 7, 8, 9, 20	xpord2ind 8153	1 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → 𝜂)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146  ∀wral 3082   ∪ cun 3906  {copab 5178  Predcpred 6308  ‘cfv 6543   No csur 27841   L cleft 28055   R cright 28056

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-rep 5243  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409  ax-un 7745

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-rmo 3372  df-reu 3373  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3928  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-tp 4599  df-op 4601  df-uni 4878  df-int 4918  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-tr 5224  df-id 5561  df-eprel 5566  df-po 5574  df-so 5575  df-fr 5619  df-se 5620  df-we 5621  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6309  df-ord 6370  df-on 6371  df-suc 6373  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7380  df-ov 7426  df-oprab 7427  df-mpo 7428  df-1st 7995  df-2nd 7996  df-frecs 8287  df-wrecs 8318  df-recs 8367  df-1o 8462  df-2o 8463  df-no 27844  df-lts 27845  df-bday 27846  df-slts 27988  df-cuts 27990  df-made 28057  df-old 28058  df-left 28060  df-right 28061

This theorem is used by:  no2inds  28185