Description: The base set of a structure augmented with a norm. (Contributed by Mario Carneiro, 2-Oct-2015) (Revised by AV, 31-Oct-2024)
Ref | Expression | ||
---|---|---|---|
Hypotheses | tngbas.t | ⊢ 𝑇 = ( 𝐺 toNrmGrp 𝑁 ) | |
tngbas.2 | ⊢ 𝐵 = ( Base ‘ 𝐺 ) | ||
Assertion | tngbas | ⊢ ( 𝑁 ∈ 𝑉 → 𝐵 = ( Base ‘ 𝑇 ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tngbas.t | ⊢ 𝑇 = ( 𝐺 toNrmGrp 𝑁 ) | |
2 | tngbas.2 | ⊢ 𝐵 = ( Base ‘ 𝐺 ) | |
3 | baseid | ⊢ Base = Slot ( Base ‘ ndx ) | |
4 | tsetndxnbasendx | ⊢ ( TopSet ‘ ndx ) ≠ ( Base ‘ ndx ) | |
5 | 4 | necomi | ⊢ ( Base ‘ ndx ) ≠ ( TopSet ‘ ndx ) |
6 | dsndxnbasendx | ⊢ ( dist ‘ ndx ) ≠ ( Base ‘ ndx ) | |
7 | 6 | necomi | ⊢ ( Base ‘ ndx ) ≠ ( dist ‘ ndx ) |
8 | 1 3 5 7 | tnglem | ⊢ ( 𝑁 ∈ 𝑉 → ( Base ‘ 𝐺 ) = ( Base ‘ 𝑇 ) ) |
9 | 2 8 | eqtrid | ⊢ ( 𝑁 ∈ 𝑉 → 𝐵 = ( Base ‘ 𝑇 ) ) |