Theorem ltmuls

Description: An ordering relationship for surreal multiplication. Compare theorem 8(iii) of Conway p. 19. (Contributed by Scott Fenton, 5-Mar-2025)

	Ref	Expression
Assertion	ltmuls	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝐶 ∈ No ∧ 𝐷 ∈ No ) ) → ( ( 𝐴 <s 𝐵 ∧ 𝐶 <s 𝐷 ) → ( ( 𝐴 ·s 𝐷 ) -s ( 𝐴 ·s 𝐶 ) ) <s ( ( 𝐵 ·s 𝐷 ) -s ( 𝐵 ·s 𝐶 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		0no	⊢ 0s ∈ No
2	1 1	pm3.2i	⊢ ( 0s ∈ No ∧ 0s ∈ No )
3		mulsprop	⊢ ( ( ( 0s ∈ No ∧ 0s ∈ No ) ∧ ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝐶 ∈ No ∧ 𝐷 ∈ No ) ) → ( ( 0s ·s 0s ) ∈ No ∧ ( ( 𝐴 <s 𝐵 ∧ 𝐶 <s 𝐷 ) → ( ( 𝐴 ·s 𝐷 ) -s ( 𝐴 ·s 𝐶 ) ) <s ( ( 𝐵 ·s 𝐷 ) -s ( 𝐵 ·s 𝐶 ) ) ) ) )
4	2 3	mp3an1	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝐶 ∈ No ∧ 𝐷 ∈ No ) ) → ( ( 0s ·s 0s ) ∈ No ∧ ( ( 𝐴 <s 𝐵 ∧ 𝐶 <s 𝐷 ) → ( ( 𝐴 ·s 𝐷 ) -s ( 𝐴 ·s 𝐶 ) ) <s ( ( 𝐵 ·s 𝐷 ) -s ( 𝐵 ·s 𝐶 ) ) ) ) )
5	4	simprd	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝐶 ∈ No ∧ 𝐷 ∈ No ) ) → ( ( 𝐴 <s 𝐵 ∧ 𝐶 <s 𝐷 ) → ( ( 𝐴 ·s 𝐷 ) -s ( 𝐴 ·s 𝐶 ) ) <s ( ( 𝐵 ·s 𝐷 ) -s ( 𝐵 ·s 𝐶 ) ) ) )