Description: Associative law for the multiplication operation of a non-unital ring. (Contributed by NM, 27-Aug-2011) (Revised by AV, 13-Feb-2025)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | rngass.b | ⊢ 𝐵 = ( Base ‘ 𝑅 ) | |
| rngass.t | ⊢ · = ( .r ‘ 𝑅 ) | ||
| Assertion | rngass | ⊢ ( ( 𝑅 ∈ Rng ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( 𝑋 · 𝑌 ) · 𝑍 ) = ( 𝑋 · ( 𝑌 · 𝑍 ) ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | rngass.b | ⊢ 𝐵 = ( Base ‘ 𝑅 ) | |
| 2 | rngass.t | ⊢ · = ( .r ‘ 𝑅 ) | |
| 3 | eqid | ⊢ ( mulGrp ‘ 𝑅 ) = ( mulGrp ‘ 𝑅 ) | |
| 4 | 3 | rngmgp | ⊢ ( 𝑅 ∈ Rng → ( mulGrp ‘ 𝑅 ) ∈ Smgrp ) |
| 5 | 3 1 | mgpbas | ⊢ 𝐵 = ( Base ‘ ( mulGrp ‘ 𝑅 ) ) |
| 6 | 3 2 | mgpplusg | ⊢ · = ( +g ‘ ( mulGrp ‘ 𝑅 ) ) |
| 7 | 5 6 | sgrpass | ⊢ ( ( ( mulGrp ‘ 𝑅 ) ∈ Smgrp ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( 𝑋 · 𝑌 ) · 𝑍 ) = ( 𝑋 · ( 𝑌 · 𝑍 ) ) ) |
| 8 | 4 7 | sylan | ⊢ ( ( 𝑅 ∈ Rng ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( 𝑋 · 𝑌 ) · 𝑍 ) = ( 𝑋 · ( 𝑌 · 𝑍 ) ) ) |