Theorem no2indlesm 28373

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 28362	. 2 ⊢ 𝑅 Fr No
3	1	lrrecpo 28360	. 2 ⊢ 𝑅 Po No
4	1	lrrecse 28361	. 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 28363	. . . . . 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 28363	. . . . . . 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 3323	. . . . 5 ⊢ ((𝑥 ∈ No ∧ 𝑦 ∈ No) → (∀𝑤 ∈ Pred (𝑅, No, 𝑦)𝜒 ↔ ∀𝑤 ∈ ((L‘𝑦) ∪ (R‘𝑦))𝜒))
15	11, 14	raleqbidv 3338	. . . 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 3323	. . . 4 ⊢ ((𝑥 ∈ No ∧ 𝑦 ∈ No) → (∀𝑧 ∈ Pred (𝑅, No, 𝑥)𝜓 ↔ ∀𝑧 ∈ ((L‘𝑥) ∪ (R‘𝑥))𝜓))
17	13	raleqdv 3323	. . . 4 ⊢ ((𝑥 ∈ No ∧ 𝑦 ∈ No) → (∀𝑤 ∈ Pred (𝑅, No, 𝑦)𝜃 ↔ ∀𝑤 ∈ ((L‘𝑦) ∪ (R‘𝑦))𝜃))
18	15, 16, 17	3anbi123d 1465	. . 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 8195	1 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → 𝜂)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  ∀wral 3080   ∪ cun 4152  {copab 5171  Predcpred 6324  ‘cfv 6561  Nocsur 28030  Lcleft 28244  Rcright 28245

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5236  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7778

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5551  df-eprel 5556  df-po 5564  df-so 5565  df-fr 5609  df-se 5610  df-we 5611  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-pred 6325  df-ord 6386  df-on 6387  df-suc 6389  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7403  df-ov 7449  df-oprab 7450  df-mpo 7451  df-1st 8028  df-2nd 8029  df-frecs 8330  df-wrecs 8361  df-recs 8410  df-1o 8507  df-2o 8508  df-no 28033  df-lts 28034  df-bday 28035  df-slts 28177  df-cuts 28179  df-made 28246  df-old 28247  df-left 28249  df-right 28250

This theorem is used by:  no2inds  28374