Theorem no2indlesm 27946

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 27935	. 2 ⊢ 𝑅 Fr No
3	1	lrrecpo 27933	. 2 ⊢ 𝑅 Po No
4	1	lrrecse 27934	. 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 27936	. . . . . 6 ⊢ (𝑥 ∈ No → Pred(𝑅, No , 𝑥) = (( L ‘𝑥) ∪ ( R ‘𝑥)))
11	10	adantr 480	. . . . 5 ⊢ ((𝑥 ∈ No ∧ 𝑦 ∈ No ) → Pred(𝑅, No , 𝑥) = (( L ‘𝑥) ∪ ( R ‘𝑥)))
12	1	lrrecpred 27936	. . . . . . 7 ⊢ (𝑦 ∈ No → Pred(𝑅, No , 𝑦) = (( L ‘𝑦) ∪ ( R ‘𝑦)))
13	12	adantl 481	. . . . . 6 ⊢ ((𝑥 ∈ No ∧ 𝑦 ∈ No ) → Pred(𝑅, No , 𝑦) = (( L ‘𝑦) ∪ ( R ‘𝑦)))
14	13	raleqdv 3295	. . . . 5 ⊢ ((𝑥 ∈ No ∧ 𝑦 ∈ No ) → (∀𝑤 ∈ Pred (𝑅, No , 𝑦)𝜒 ↔ ∀𝑤 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))𝜒))
15	11, 14	raleqbidv 3311	. . . 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 3295	. . . 4 ⊢ ((𝑥 ∈ No ∧ 𝑦 ∈ No ) → (∀𝑧 ∈ Pred (𝑅, No , 𝑥)𝜓 ↔ ∀𝑧 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))𝜓))
17	13	raleqdv 3295	. . . 4 ⊢ ((𝑥 ∈ No ∧ 𝑦 ∈ No ) → (∀𝑤 ∈ Pred (𝑅, No , 𝑦)𝜃 ↔ ∀𝑤 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))𝜃))
18	15, 16, 17	3anbi123d 1439	. . 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 240	. 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 8098	1 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → 𝜂)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3051   ∪ cun 3887  {copab 5147  Predcpred 6264  ‘cfv 6498   No csur 27603   L cleft 27817   R cright 27818

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-1st 7942  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-1o 8405  df-2o 8406  df-no 27606  df-lts 27607  df-bday 27608  df-slts 27750  df-cuts 27752  df-made 27819  df-old 27820  df-left 27822  df-right 27823

This theorem is referenced by:  no2inds  27947