Description: Relationship between surreal division and multiplication. Weak version that does not assume reciprocals. (Contributed by Scott Fenton, 12-Mar-2025)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | divsmulwd.1 | |- ( ph -> A e. No ) |
|
| divsmulwd.2 | |- ( ph -> B e. No ) |
||
| divsmulwd.3 | |- ( ph -> C e. No ) |
||
| divsmulwd.4 | |- ( ph -> C =/= 0s ) |
||
| divsmulwd.5 | |- ( ph -> E. x e. No ( C x.s x ) = 1s ) |
||
| Assertion | divsmulwd | |- ( ph -> ( ( A /su C ) = B <-> ( C x.s B ) = A ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | divsmulwd.1 | |- ( ph -> A e. No ) |
|
| 2 | divsmulwd.2 | |- ( ph -> B e. No ) |
|
| 3 | divsmulwd.3 | |- ( ph -> C e. No ) |
|
| 4 | divsmulwd.4 | |- ( ph -> C =/= 0s ) |
|
| 5 | divsmulwd.5 | |- ( ph -> E. x e. No ( C x.s x ) = 1s ) |
|
| 6 | 3 4 | jca | |- ( ph -> ( C e. No /\ C =/= 0s ) ) |
| 7 | divsmulw | |- ( ( ( A e. No /\ B e. No /\ ( C e. No /\ C =/= 0s ) ) /\ E. x e. No ( C x.s x ) = 1s ) -> ( ( A /su C ) = B <-> ( C x.s B ) = A ) ) |
|
| 8 | 1 2 6 5 7 | syl31anc | |- ( ph -> ( ( A /su C ) = B <-> ( C x.s B ) = A ) ) |