Theorem no2indlesm 28346

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 28335	. 2 ⊢ 𝑅 Fr No
3	1	lrrecpo 28333	. 2 ⊢ 𝑅 Po No
4	1	lrrecse 28334	. 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 28336	. . . . . 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 28336	. . . . . . 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 3319	. . . . 5 ⊢ ((𝑥 ∈ No ∧ 𝑦 ∈ No) → (∀𝑤 ∈ Pred (𝑅, No, 𝑦)𝜒 ↔ ∀𝑤 ∈ ((L‘𝑦) ∪ (R‘𝑦))𝜒))
15	11, 14	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))
16	11	raleqdv 3319	. . . 4 ⊢ ((𝑥 ∈ No ∧ 𝑦 ∈ No) → (∀𝑧 ∈ Pred (𝑅, No, 𝑥)𝜓 ↔ ∀𝑧 ∈ ((L‘𝑥) ∪ (R‘𝑥))𝜓))
17	13	raleqdv 3319	. . . 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 8168	1 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → 𝜂)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144  ∀wral 3076   ∪ cun 4149  {copab 5166  Predcpred 6303  ‘cfv 6539  Nocsur 28003  Lcleft 28217  Rcright 28218

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7752

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-se 5604  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-suc 6368  df-iota 6494  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-riota 7378  df-ov 7424  df-oprab 7425  df-mpo 7426  df-1st 8002  df-2nd 8003  df-frecs 8303  df-wrecs 8334  df-recs 8383  df-1o 8480  df-2o 8481  df-no 28006  df-lts 28007  df-bday 28008  df-slts 28150  df-cuts 28152  df-made 28219  df-old 28220  df-left 28222  df-right 28223

This theorem is used by:  no2inds  28347