Theorem no2indslem 27892

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.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   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 no2indslem

Step	Hyp	Ref	Expression
1		no2indslem.a	. . 3 ⊢ 𝑅 = {⟨𝑎, 𝑏⟩ ∣ 𝑎 ∈ (( L ‘𝑏) ∪ ( R ‘𝑏))}
2	1	lrrecfr 27881	. 2 ⊢ 𝑅 Fr No
3	1	lrrecpo 27879	. 2 ⊢ 𝑅 Po No
4	1	lrrecse 27880	. 2 ⊢ 𝑅 Se No
5		no2indslem.1	. 2 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ 𝜓))
6		no2indslem.2	. 2 ⊢ (𝑦 = 𝑤 → (𝜓 ↔ 𝜒))
7		no2indslem.3	. 2 ⊢ (𝑥 = 𝑧 → (𝜃 ↔ 𝜒))
8		no2indslem.4	. 2 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏))
9		no2indslem.5	. 2 ⊢ (𝑦 = 𝐵 → (𝜏 ↔ 𝜂))
10	1	lrrecpred 27882	. . . . . 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 27882	. . . . . . 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 3292	. . . . 5 ⊢ ((𝑥 ∈ No ∧ 𝑦 ∈ No ) → (∀𝑤 ∈ Pred (𝑅, No , 𝑦)𝜒 ↔ ∀𝑤 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))𝜒))
15	11, 14	raleqbidv 3312	. . . 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 3292	. . . 4 ⊢ ((𝑥 ∈ No ∧ 𝑦 ∈ No ) → (∀𝑧 ∈ Pred (𝑅, No , 𝑥)𝜓 ↔ ∀𝑧 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))𝜓))
17	13	raleqdv 3292	. . . 4 ⊢ ((𝑥 ∈ No ∧ 𝑦 ∈ No ) → (∀𝑤 ∈ Pred (𝑅, No , 𝑦)𝜃 ↔ ∀𝑤 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))𝜃))
18	15, 16, 17	3anbi123d 1438	. . 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		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))
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 8073	1 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → 𝜂)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111  ∀wral 3047   ∪ cun 3895  {copab 5148  Predcpred 6242  ‘cfv 6476   No csur 27573   L cleft 27781   R cright 27782

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5212  ax-sep 5229  ax-nul 5239  ax-pow 5298  ax-pr 5365  ax-un 7663

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4279  df-if 4471  df-pw 4547  df-sn 4572  df-pr 4574  df-tp 4576  df-op 4578  df-uni 4855  df-int 4893  df-iun 4938  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5506  df-eprel 5511  df-po 5519  df-so 5520  df-fr 5564  df-se 5565  df-we 5566  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-res 5623  df-ima 5624  df-pred 6243  df-ord 6304  df-on 6305  df-suc 6307  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-f1 6481  df-fo 6482  df-f1o 6483  df-fv 6484  df-riota 7298  df-ov 7344  df-oprab 7345  df-mpo 7346  df-1st 7916  df-2nd 7917  df-frecs 8206  df-wrecs 8237  df-recs 8286  df-1o 8380  df-2o 8381  df-no 27576  df-slt 27577  df-bday 27578  df-sslt 27716  df-scut 27718  df-made 27783  df-old 27784  df-left 27786  df-right 27787

This theorem is referenced by:  no2inds  27893