Description: The group addition 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 ) | |
| tngplusg.2 | ⊢ + = ( +g ‘ 𝐺 ) | ||
| Assertion | tngplusg | ⊢ ( 𝑁 ∈ 𝑉 → + = ( +g ‘ 𝑇 ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | tngbas.t | ⊢ 𝑇 = ( 𝐺 toNrmGrp 𝑁 ) | |
| 2 | tngplusg.2 | ⊢ + = ( +g ‘ 𝐺 ) | |
| 3 | plusgid | ⊢ +g = Slot ( +g ‘ ndx ) | |
| 4 | tsetndxnplusgndx | ⊢ ( TopSet ‘ ndx ) ≠ ( +g ‘ ndx ) | |
| 5 | 4 | necomi | ⊢ ( +g ‘ ndx ) ≠ ( TopSet ‘ ndx ) |
| 6 | dsndxnplusgndx | ⊢ ( dist ‘ ndx ) ≠ ( +g ‘ ndx ) | |
| 7 | 6 | necomi | ⊢ ( +g ‘ ndx ) ≠ ( dist ‘ ndx ) |
| 8 | 1 3 5 7 | tnglem | ⊢ ( 𝑁 ∈ 𝑉 → ( +g ‘ 𝐺 ) = ( +g ‘ 𝑇 ) ) |
| 9 | 2 8 | syl5eq | ⊢ ( 𝑁 ∈ 𝑉 → + = ( +g ‘ 𝑇 ) ) |