no2indslem

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

	Ref	Expression
Hypotheses	no2indslem.a	⊢ 𝑅 = { ⟨ 𝑎 , 𝑏 ⟩ ∣ 𝑎 ∈ ( ( L ‘ 𝑏 ) ∪ ( R ‘ 𝑏 ) ) }
	no2indslem.b	⊢ 𝑆 = { ⟨ 𝑐 , 𝑑 ⟩ ∣ ( 𝑐 ∈ ( No × No ) ∧ 𝑑 ∈ ( No × No ) ∧ ( ( ( 1^st ‘ 𝑐 ) 𝑅 ( 1^st ‘ 𝑑 ) ∨ ( 1^st ‘ 𝑐 ) = ( 1^st ‘ 𝑑 ) ) ∧ ( ( 2^nd ‘ 𝑐 ) 𝑅 ( 2^nd ‘ 𝑑 ) ∨ ( 2^nd ‘ 𝑐 ) = ( 2^nd ‘ 𝑑 ) ) ∧ 𝑐 ≠ 𝑑 ) ) }
	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	no2indslem	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → 𝜂 )

Proof

Step	Hyp	Ref	Expression
1		no2indslem.a	⊢ 𝑅 = { ⟨ 𝑎 , 𝑏 ⟩ ∣ 𝑎 ∈ ( ( L ‘ 𝑏 ) ∪ ( R ‘ 𝑏 ) ) }
2		no2indslem.b	⊢ 𝑆 = { ⟨ 𝑐 , 𝑑 ⟩ ∣ ( 𝑐 ∈ ( No × No ) ∧ 𝑑 ∈ ( No × No ) ∧ ( ( ( 1^st ‘ 𝑐 ) 𝑅 ( 1^st ‘ 𝑑 ) ∨ ( 1^st ‘ 𝑐 ) = ( 1^st ‘ 𝑑 ) ) ∧ ( ( 2^nd ‘ 𝑐 ) 𝑅 ( 2^nd ‘ 𝑑 ) ∨ ( 2^nd ‘ 𝑐 ) = ( 2^nd ‘ 𝑑 ) ) ∧ 𝑐 ≠ 𝑑 ) ) }
3		no2indslem.1	⊢ ( 𝑥 = 𝑧 → ( 𝜑 ↔ 𝜓 ) )
4		no2indslem.2	⊢ ( 𝑦 = 𝑤 → ( 𝜓 ↔ 𝜒 ) )
5		no2indslem.3	⊢ ( 𝑥 = 𝑧 → ( 𝜃 ↔ 𝜒 ) )
6		no2indslem.4	⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜏 ) )
7		no2indslem.5	⊢ ( 𝑦 = 𝐵 → ( 𝜏 ↔ 𝜂 ) )
8		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 ) )
9	1	lrrecfr	⊢ 𝑅 Fr No
10	1	lrrecpo	⊢ 𝑅 Po No
11	1	lrrecse	⊢ 𝑅 Se No
12	1	lrrecpred	⊢ ( 𝑥 ∈ No → Pred ( 𝑅 , No , 𝑥 ) = ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) )
13	12	adantr	⊢ ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ) → Pred ( 𝑅 , No , 𝑥 ) = ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) )
14	1	lrrecpred	⊢ ( 𝑦 ∈ No → Pred ( 𝑅 , No , 𝑦 ) = ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) )
15	14	adantl	⊢ ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ) → Pred ( 𝑅 , No , 𝑦 ) = ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) )
16	15	raleqdv	⊢ ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ) → ( ∀ 𝑤 ∈ Pred ( 𝑅 , No , 𝑦 ) 𝜒 ↔ ∀ 𝑤 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) 𝜒 ) )
17	13 16	raleqbidv	⊢ ( ( 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 ) )
18	13	raleqdv	⊢ ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ) → ( ∀ 𝑧 ∈ Pred ( 𝑅 , No , 𝑥 ) 𝜓 ↔ ∀ 𝑧 ∈ ( ( L ‘ 𝑥 ) ∪ ( R ‘ 𝑥 ) ) 𝜓 ) )
19	15	raleqdv	⊢ ( ( 𝑥 ∈ No ∧ 𝑦 ∈ No ) → ( ∀ 𝑤 ∈ Pred ( 𝑅 , No , 𝑦 ) 𝜃 ↔ ∀ 𝑤 ∈ ( ( L ‘ 𝑦 ) ∪ ( R ‘ 𝑦 ) ) 𝜃 ) )
20	17 18 19	3anbi123d	⊢ ( ( 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 ) ) )
21	20 8	sylbid	⊢ ( ( 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 ) )
22	2 9 10 11 9 10 11 3 4 5 6 7 21	xpord2ind	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → 𝜂 )

Metamath Proof Explorer

Theorem no2indslem

Proof