Description: Base set of an opposite group. (Contributed by Stefan O'Rear, 26-Aug-2015)
Ref | Expression | ||
---|---|---|---|
Hypotheses | oppgbas.1 | ⊢ 𝑂 = ( oppg ‘ 𝑅 ) | |
oppgbas.2 | ⊢ 𝐵 = ( Base ‘ 𝑅 ) | ||
Assertion | oppgbas | ⊢ 𝐵 = ( Base ‘ 𝑂 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oppgbas.1 | ⊢ 𝑂 = ( oppg ‘ 𝑅 ) | |
2 | oppgbas.2 | ⊢ 𝐵 = ( Base ‘ 𝑅 ) | |
3 | eqid | ⊢ ( +g ‘ 𝑅 ) = ( +g ‘ 𝑅 ) | |
4 | 3 1 | oppgval | ⊢ 𝑂 = ( 𝑅 sSet 〈 ( +g ‘ ndx ) , tpos ( +g ‘ 𝑅 ) 〉 ) |
5 | baseid | ⊢ Base = Slot ( Base ‘ ndx ) | |
6 | basendxnplusgndx | ⊢ ( Base ‘ ndx ) ≠ ( +g ‘ ndx ) | |
7 | 4 5 6 | setsplusg | ⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑂 ) |
8 | 2 7 | eqtri | ⊢ 𝐵 = ( Base ‘ 𝑂 ) |