Theorem addcuts2

Description: Show that the cut involved in surreal addition is legitimate. (Contributed by Scott Fenton, 8-Mar-2025)

	Ref	Expression
Hypotheses	addcuts.1	⊢ ( 𝜑 → 𝑋 ∈ No )
	addcuts.2	⊢ ( 𝜑 → 𝑌 ∈ No )
Assertion	addcuts2	⊢ ( 𝜑 → ( { 𝑝 ∣ ∃ 𝑙 ∈ ( L ‘ 𝑋 ) 𝑝 = ( 𝑙 +s 𝑌 ) } ∪ { 𝑞 ∣ ∃ 𝑚 ∈ ( L ‘ 𝑌 ) 𝑞 = ( 𝑋 +s 𝑚 ) } ) <<s ( { 𝑤 ∣ ∃ 𝑟 ∈ ( R ‘ 𝑋 ) 𝑤 = ( 𝑟 +s 𝑌 ) } ∪ { 𝑡 ∣ ∃ 𝑠 ∈ ( R ‘ 𝑌 ) 𝑡 = ( 𝑋 +s 𝑠 ) } ) )

Proof

Step	Hyp	Ref	Expression
1		addcuts.1	⊢ ( 𝜑 → 𝑋 ∈ No )
2		addcuts.2	⊢ ( 𝜑 → 𝑌 ∈ No )
3	1 2	addcuts	⊢ ( 𝜑 → ( ( 𝑋 +s 𝑌 ) ∈ No ∧ ( { 𝑝 ∣ ∃ 𝑙 ∈ ( L ‘ 𝑋 ) 𝑝 = ( 𝑙 +s 𝑌 ) } ∪ { 𝑞 ∣ ∃ 𝑚 ∈ ( L ‘ 𝑌 ) 𝑞 = ( 𝑋 +s 𝑚 ) } ) <<s { ( 𝑋 +s 𝑌 ) } ∧ { ( 𝑋 +s 𝑌 ) } <<s ( { 𝑤 ∣ ∃ 𝑟 ∈ ( R ‘ 𝑋 ) 𝑤 = ( 𝑟 +s 𝑌 ) } ∪ { 𝑡 ∣ ∃ 𝑠 ∈ ( R ‘ 𝑌 ) 𝑡 = ( 𝑋 +s 𝑠 ) } ) ) )
4		3anass	⊢ ( ( ( 𝑋 +s 𝑌 ) ∈ No ∧ ( { 𝑝 ∣ ∃ 𝑙 ∈ ( L ‘ 𝑋 ) 𝑝 = ( 𝑙 +s 𝑌 ) } ∪ { 𝑞 ∣ ∃ 𝑚 ∈ ( L ‘ 𝑌 ) 𝑞 = ( 𝑋 +s 𝑚 ) } ) <<s { ( 𝑋 +s 𝑌 ) } ∧ { ( 𝑋 +s 𝑌 ) } <<s ( { 𝑤 ∣ ∃ 𝑟 ∈ ( R ‘ 𝑋 ) 𝑤 = ( 𝑟 +s 𝑌 ) } ∪ { 𝑡 ∣ ∃ 𝑠 ∈ ( R ‘ 𝑌 ) 𝑡 = ( 𝑋 +s 𝑠 ) } ) ) ↔ ( ( 𝑋 +s 𝑌 ) ∈ No ∧ ( ( { 𝑝 ∣ ∃ 𝑙 ∈ ( L ‘ 𝑋 ) 𝑝 = ( 𝑙 +s 𝑌 ) } ∪ { 𝑞 ∣ ∃ 𝑚 ∈ ( L ‘ 𝑌 ) 𝑞 = ( 𝑋 +s 𝑚 ) } ) <<s { ( 𝑋 +s 𝑌 ) } ∧ { ( 𝑋 +s 𝑌 ) } <<s ( { 𝑤 ∣ ∃ 𝑟 ∈ ( R ‘ 𝑋 ) 𝑤 = ( 𝑟 +s 𝑌 ) } ∪ { 𝑡 ∣ ∃ 𝑠 ∈ ( R ‘ 𝑌 ) 𝑡 = ( 𝑋 +s 𝑠 ) } ) ) ) )
5	3 4	sylib	⊢ ( 𝜑 → ( ( 𝑋 +s 𝑌 ) ∈ No ∧ ( ( { 𝑝 ∣ ∃ 𝑙 ∈ ( L ‘ 𝑋 ) 𝑝 = ( 𝑙 +s 𝑌 ) } ∪ { 𝑞 ∣ ∃ 𝑚 ∈ ( L ‘ 𝑌 ) 𝑞 = ( 𝑋 +s 𝑚 ) } ) <<s { ( 𝑋 +s 𝑌 ) } ∧ { ( 𝑋 +s 𝑌 ) } <<s ( { 𝑤 ∣ ∃ 𝑟 ∈ ( R ‘ 𝑋 ) 𝑤 = ( 𝑟 +s 𝑌 ) } ∪ { 𝑡 ∣ ∃ 𝑠 ∈ ( R ‘ 𝑌 ) 𝑡 = ( 𝑋 +s 𝑠 ) } ) ) ) )
6	5	simprd	⊢ ( 𝜑 → ( ( { 𝑝 ∣ ∃ 𝑙 ∈ ( L ‘ 𝑋 ) 𝑝 = ( 𝑙 +s 𝑌 ) } ∪ { 𝑞 ∣ ∃ 𝑚 ∈ ( L ‘ 𝑌 ) 𝑞 = ( 𝑋 +s 𝑚 ) } ) <<s { ( 𝑋 +s 𝑌 ) } ∧ { ( 𝑋 +s 𝑌 ) } <<s ( { 𝑤 ∣ ∃ 𝑟 ∈ ( R ‘ 𝑋 ) 𝑤 = ( 𝑟 +s 𝑌 ) } ∪ { 𝑡 ∣ ∃ 𝑠 ∈ ( R ‘ 𝑌 ) 𝑡 = ( 𝑋 +s 𝑠 ) } ) ) )
7		ovex	⊢ ( 𝑋 +s 𝑌 ) ∈ V
8	7	snnz	⊢ { ( 𝑋 +s 𝑌 ) } ≠ ∅
9		sltstr	⊢ ( ( ( { 𝑝 ∣ ∃ 𝑙 ∈ ( L ‘ 𝑋 ) 𝑝 = ( 𝑙 +s 𝑌 ) } ∪ { 𝑞 ∣ ∃ 𝑚 ∈ ( L ‘ 𝑌 ) 𝑞 = ( 𝑋 +s 𝑚 ) } ) <<s { ( 𝑋 +s 𝑌 ) } ∧ { ( 𝑋 +s 𝑌 ) } <<s ( { 𝑤 ∣ ∃ 𝑟 ∈ ( R ‘ 𝑋 ) 𝑤 = ( 𝑟 +s 𝑌 ) } ∪ { 𝑡 ∣ ∃ 𝑠 ∈ ( R ‘ 𝑌 ) 𝑡 = ( 𝑋 +s 𝑠 ) } ) ∧ { ( 𝑋 +s 𝑌 ) } ≠ ∅ ) → ( { 𝑝 ∣ ∃ 𝑙 ∈ ( L ‘ 𝑋 ) 𝑝 = ( 𝑙 +s 𝑌 ) } ∪ { 𝑞 ∣ ∃ 𝑚 ∈ ( L ‘ 𝑌 ) 𝑞 = ( 𝑋 +s 𝑚 ) } ) <<s ( { 𝑤 ∣ ∃ 𝑟 ∈ ( R ‘ 𝑋 ) 𝑤 = ( 𝑟 +s 𝑌 ) } ∪ { 𝑡 ∣ ∃ 𝑠 ∈ ( R ‘ 𝑌 ) 𝑡 = ( 𝑋 +s 𝑠 ) } ) )
10	8 9	mp3an3	⊢ ( ( ( { 𝑝 ∣ ∃ 𝑙 ∈ ( L ‘ 𝑋 ) 𝑝 = ( 𝑙 +s 𝑌 ) } ∪ { 𝑞 ∣ ∃ 𝑚 ∈ ( L ‘ 𝑌 ) 𝑞 = ( 𝑋 +s 𝑚 ) } ) <<s { ( 𝑋 +s 𝑌 ) } ∧ { ( 𝑋 +s 𝑌 ) } <<s ( { 𝑤 ∣ ∃ 𝑟 ∈ ( R ‘ 𝑋 ) 𝑤 = ( 𝑟 +s 𝑌 ) } ∪ { 𝑡 ∣ ∃ 𝑠 ∈ ( R ‘ 𝑌 ) 𝑡 = ( 𝑋 +s 𝑠 ) } ) ) → ( { 𝑝 ∣ ∃ 𝑙 ∈ ( L ‘ 𝑋 ) 𝑝 = ( 𝑙 +s 𝑌 ) } ∪ { 𝑞 ∣ ∃ 𝑚 ∈ ( L ‘ 𝑌 ) 𝑞 = ( 𝑋 +s 𝑚 ) } ) <<s ( { 𝑤 ∣ ∃ 𝑟 ∈ ( R ‘ 𝑋 ) 𝑤 = ( 𝑟 +s 𝑌 ) } ∪ { 𝑡 ∣ ∃ 𝑠 ∈ ( R ‘ 𝑌 ) 𝑡 = ( 𝑋 +s 𝑠 ) } ) )
11	6 10	syl	⊢ ( 𝜑 → ( { 𝑝 ∣ ∃ 𝑙 ∈ ( L ‘ 𝑋 ) 𝑝 = ( 𝑙 +s 𝑌 ) } ∪ { 𝑞 ∣ ∃ 𝑚 ∈ ( L ‘ 𝑌 ) 𝑞 = ( 𝑋 +s 𝑚 ) } ) <<s ( { 𝑤 ∣ ∃ 𝑟 ∈ ( R ‘ 𝑋 ) 𝑤 = ( 𝑟 +s 𝑌 ) } ∪ { 𝑡 ∣ ∃ 𝑠 ∈ ( R ‘ 𝑌 ) 𝑡 = ( 𝑋 +s 𝑠 ) } ) )