Metamath Proof Explorer

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 
|- ( ph -> A e. No )
mulscut.2 
|- ( ph -> B e. No )
Assertion mulscut2 
|- ( 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 
 |-  ( ph -> A e. No )
2 mulscut.2 
 |-  ( ph -> B e. No )
3 1 2 mulscut 
 |-  ( 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 ) ) } ) <
4 3anass 
 |-  ( ( ( 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 
 |-  ( 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 ) ) } ) <
6 5 simprd 
 |-  ( 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 ) ) } ) <
7 ovex 
 |-  ( A x.s B ) e. _V
8 7 snnz 
 |-  { ( A x.s B ) } =/= (/)
9 sslttr 
 |-  ( ( ( { 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 
 |-  ( ( ( { 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 
 |-  ( 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 ) ) } ) <