Theorem mulscut2

Description: Show that the cut involved in surreal multiplication is actually a cut. (Contributed by Scott Fenton, 7-Mar-2025)

	Ref	Expression
Hypotheses	mulscut.1	$$⊢\phi \to A\in \mathrm{No}$$
	mulscut.2	$$⊢\phi \to B\in \mathrm{No}$$
Assertion	mulscut2	Could not format assertion : No typesetting found for |- ( ph -> ( { a | E. p e. ( _Left ` A ) E. q e. ( _Left ` B ) a = ( ( ( p x.s B ) +s ( A x.s q ) ) -s ( p x.s q ) ) } u. { b | E. r e. ( _Right ` A ) E. s e. ( _Right ` B ) b = ( ( ( r x.s B ) +s ( A x.s s ) ) -s ( r x.s s ) ) } ) <

Proof

Step	Hyp	Ref	Expression
1		mulscut.1	$$⊢\phi \to A\in \mathrm{No}$$
2		mulscut.2	$$⊢\phi \to B\in \mathrm{No}$$
3	1 2	mulscut	Could not format ( ph -> ( ( A x.s B ) e. No /\ ( { a | E. p e. ( _Left ` A ) E. q e. ( _Left ` B ) a = ( ( ( p x.s B ) +s ( A x.s q ) ) -s ( p x.s q ) ) } u. { b | E. r e. ( _Right ` A ) E. s e. ( _Right ` B ) b = ( ( ( r x.s B ) +s ( A x.s s ) ) -s ( r x.s s ) ) } ) < ( ( A x.s B ) e. No /\ ( { a | E. p e. ( _Left ` A ) E. q e. ( _Left ` B ) a = ( ( ( p x.s B ) +s ( A x.s q ) ) -s ( p x.s q ) ) } u. { b | E. r e. ( _Right ` A ) E. s e. ( _Right ` B ) b = ( ( ( r x.s B ) +s ( A x.s s ) ) -s ( r x.s s ) ) } ) <
4		3anass	Could not format ( ( ( A x.s B ) e. No /\ ( { a | E. p e. ( _Left ` A ) E. q e. ( _Left ` B ) a = ( ( ( p x.s B ) +s ( A x.s q ) ) -s ( p x.s q ) ) } u. { b | E. r e. ( _Right ` A ) E. s e. ( _Right ` B ) b = ( ( ( r x.s B ) +s ( A x.s s ) ) -s ( r x.s s ) ) } ) < ( ( A x.s B ) e. No /\ ( ( { a | E. p e. ( _Left ` A ) E. q e. ( _Left ` B ) a = ( ( ( p x.s B ) +s ( A x.s q ) ) -s ( p x.s q ) ) } u. { b | E. r e. ( _Right ` A ) E. s e. ( _Right ` B ) b = ( ( ( r x.s B ) +s ( A x.s s ) ) -s ( r x.s s ) ) } ) < ( ( A x.s B ) e. No /\ ( ( { a | E. p e. ( _Left ` A ) E. q e. ( _Left ` B ) a = ( ( ( p x.s B ) +s ( A x.s q ) ) -s ( p x.s q ) ) } u. { b | E. r e. ( _Right ` A ) E. s e. ( _Right ` B ) b = ( ( ( r x.s B ) +s ( A x.s s ) ) -s ( r x.s s ) ) } ) <
5	3 4	sylib	Could not format ( ph -> ( ( A x.s B ) e. No /\ ( ( { a | E. p e. ( _Left ` A ) E. q e. ( _Left ` B ) a = ( ( ( p x.s B ) +s ( A x.s q ) ) -s ( p x.s q ) ) } u. { b | E. r e. ( _Right ` A ) E. s e. ( _Right ` B ) b = ( ( ( r x.s B ) +s ( A x.s s ) ) -s ( r x.s s ) ) } ) < ( ( A x.s B ) e. No /\ ( ( { a | E. p e. ( _Left ` A ) E. q e. ( _Left ` B ) a = ( ( ( p x.s B ) +s ( A x.s q ) ) -s ( p x.s q ) ) } u. { b | E. r e. ( _Right ` A ) E. s e. ( _Right ` B ) b = ( ( ( r x.s B ) +s ( A x.s s ) ) -s ( r x.s s ) ) } ) <
6	5	simprd	Could not format ( ph -> ( ( { a | E. p e. ( _Left ` A ) E. q e. ( _Left ` B ) a = ( ( ( p x.s B ) +s ( A x.s q ) ) -s ( p x.s q ) ) } u. { b | E. r e. ( _Right ` A ) E. s e. ( _Right ` B ) b = ( ( ( r x.s B ) +s ( A x.s s ) ) -s ( r x.s s ) ) } ) < ( ( { a | E. p e. ( _Left ` A ) E. q e. ( _Left ` B ) a = ( ( ( p x.s B ) +s ( A x.s q ) ) -s ( p x.s q ) ) } u. { b | E. r e. ( _Right ` A ) E. s e. ( _Right ` B ) b = ( ( ( r x.s B ) +s ( A x.s s ) ) -s ( r x.s s ) ) } ) <
7		ovex	Could not format ( A x.s B ) e. _V : No typesetting found for |- ( A x.s B ) e. _V with typecode |-
8	7	snnz	Could not format { ( A x.s B ) } =/= (/) : No typesetting found for |- { ( A x.s B ) } =/= (/) with typecode |-
9		sslttr	Could not format ( ( ( { a | E. p e. ( _Left ` A ) E. q e. ( _Left ` B ) a = ( ( ( p x.s B ) +s ( A x.s q ) ) -s ( p x.s q ) ) } u. { b | E. r e. ( _Right ` A ) E. s e. ( _Right ` B ) b = ( ( ( r x.s B ) +s ( A x.s s ) ) -s ( r x.s s ) ) } ) < ( { a | E. p e. ( _Left ` A ) E. q e. ( _Left ` B ) a = ( ( ( p x.s B ) +s ( A x.s q ) ) -s ( p x.s q ) ) } u. { b | E. r e. ( _Right ` A ) E. s e. ( _Right ` B ) b = ( ( ( r x.s B ) +s ( A x.s s ) ) -s ( r x.s s ) ) } ) < ( { a | E. p e. ( _Left ` A ) E. q e. ( _Left ` B ) a = ( ( ( p x.s B ) +s ( A x.s q ) ) -s ( p x.s q ) ) } u. { b | E. r e. ( _Right ` A ) E. s e. ( _Right ` B ) b = ( ( ( r x.s B ) +s ( A x.s s ) ) -s ( r x.s s ) ) } ) <
10	8 9	mp3an3	Could not format ( ( ( { a | E. p e. ( _Left ` A ) E. q e. ( _Left ` B ) a = ( ( ( p x.s B ) +s ( A x.s q ) ) -s ( p x.s q ) ) } u. { b | E. r e. ( _Right ` A ) E. s e. ( _Right ` B ) b = ( ( ( r x.s B ) +s ( A x.s s ) ) -s ( r x.s s ) ) } ) < ( { a | E. p e. ( _Left ` A ) E. q e. ( _Left ` B ) a = ( ( ( p x.s B ) +s ( A x.s q ) ) -s ( p x.s q ) ) } u. { b | E. r e. ( _Right ` A ) E. s e. ( _Right ` B ) b = ( ( ( r x.s B ) +s ( A x.s s ) ) -s ( r x.s s ) ) } ) < ( { a | E. p e. ( _Left ` A ) E. q e. ( _Left ` B ) a = ( ( ( p x.s B ) +s ( A x.s q ) ) -s ( p x.s q ) ) } u. { b | E. r e. ( _Right ` A ) E. s e. ( _Right ` B ) b = ( ( ( r x.s B ) +s ( A x.s s ) ) -s ( r x.s s ) ) } ) <
11	6 10	syl	Could not format ( ph -> ( { a | E. p e. ( _Left ` A ) E. q e. ( _Left ` B ) a = ( ( ( p x.s B ) +s ( A x.s q ) ) -s ( p x.s q ) ) } u. { b | E. r e. ( _Right ` A ) E. s e. ( _Right ` B ) b = ( ( ( r x.s B ) +s ( A x.s s ) ) -s ( r x.s s ) ) } ) < ( { a | E. p e. ( _Left ` A ) E. q e. ( _Left ` B ) a = ( ( ( p x.s B ) +s ( A x.s q ) ) -s ( p x.s q ) ) } u. { b | E. r e. ( _Right ` A ) E. s e. ( _Right ` B ) b = ( ( ( r x.s B ) +s ( A x.s s ) ) -s ( r x.s s ) ) } ) <