Metamath Proof Explorer

Theorem grprida

Description: Deduce right identity from left inverse and left identity in an associative structure (such as a group). (Contributed by NM, 10-Aug-2013) (Proof shortened by Mario Carneiro, 6-Jan-2015)

Ref Expression
Hypotheses grpinva.c ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) → ( 𝑥 + 𝑦 ) ∈ 𝐵 )
grpinva.o ⊢ ( 𝜑 → 𝑂 ∈ 𝐵 )
grpinva.i ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → ( 𝑂 + 𝑥 ) = 𝑥 )
grpinva.a ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) → ( ( 𝑥 + 𝑦 ) + 𝑧 ) = ( 𝑥 + ( 𝑦 + 𝑧 ) ) )
grpinva.r ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → ∃ 𝑦 ∈ 𝐵 ( 𝑦 + 𝑥 ) = 𝑂 )
Assertion grprida ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → ( 𝑥 + 𝑂 ) = 𝑥 )

Proof

Step Hyp Ref Expression
1 grpinva.c ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) → ( 𝑥 + 𝑦 ) ∈ 𝐵 )
2 grpinva.o ⊢ ( 𝜑 → 𝑂 ∈ 𝐵 )
3 grpinva.i ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → ( 𝑂 + 𝑥 ) = 𝑥 )
4 grpinva.a ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) → ( ( 𝑥 + 𝑦 ) + 𝑧 ) = ( 𝑥 + ( 𝑦 + 𝑧 ) ) )
5 grpinva.r ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → ∃ 𝑦 ∈ 𝐵 ( 𝑦 + 𝑥 ) = 𝑂 )
6 oveq1 ⊢ ( 𝑦 = 𝑛 → ( 𝑦 + 𝑥 ) = ( 𝑛 + 𝑥 ) )
7 6 eqeq1d ⊢ ( 𝑦 = 𝑛 → ( ( 𝑦 + 𝑥 ) = 𝑂 ↔ ( 𝑛 + 𝑥 ) = 𝑂 ) )
8 7 cbvrexvw ⊢ ( ∃ 𝑦 ∈ 𝐵 ( 𝑦 + 𝑥 ) = 𝑂 ↔ ∃ 𝑛 ∈ 𝐵 ( 𝑛 + 𝑥 ) = 𝑂 )
9 5 8 sylib ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → ∃ 𝑛 ∈ 𝐵 ( 𝑛 + 𝑥 ) = 𝑂 )
10 4 caovassg ⊢ ( ( 𝜑 ∧ ( 𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ) → ( ( 𝑢 + 𝑣 ) + 𝑤 ) = ( 𝑢 + ( 𝑣 + 𝑤 ) ) )
11 10 adantlr ⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ ( 𝑛 ∈ 𝐵 ∧ ( 𝑛 + 𝑥 ) = 𝑂 ) ) ) ∧ ( 𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ) → ( ( 𝑢 + 𝑣 ) + 𝑤 ) = ( 𝑢 + ( 𝑣 + 𝑤 ) ) )
12 simprl ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ ( 𝑛 ∈ 𝐵 ∧ ( 𝑛 + 𝑥 ) = 𝑂 ) ) ) → 𝑥 ∈ 𝐵 )
13 simprrl ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ ( 𝑛 ∈ 𝐵 ∧ ( 𝑛 + 𝑥 ) = 𝑂 ) ) ) → 𝑛 ∈ 𝐵 )
14 11 12 13 12 caovassd ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ ( 𝑛 ∈ 𝐵 ∧ ( 𝑛 + 𝑥 ) = 𝑂 ) ) ) → ( ( 𝑥 + 𝑛 ) + 𝑥 ) = ( 𝑥 + ( 𝑛 + 𝑥 ) ) )
15 simprrr ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ ( 𝑛 ∈ 𝐵 ∧ ( 𝑛 + 𝑥 ) = 𝑂 ) ) ) → ( 𝑛 + 𝑥 ) = 𝑂 )
16 1 2 3 4 5 12 13 15 grpinva ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ ( 𝑛 ∈ 𝐵 ∧ ( 𝑛 + 𝑥 ) = 𝑂 ) ) ) → ( 𝑥 + 𝑛 ) = 𝑂 )
17 16 oveq1d ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ ( 𝑛 ∈ 𝐵 ∧ ( 𝑛 + 𝑥 ) = 𝑂 ) ) ) → ( ( 𝑥 + 𝑛 ) + 𝑥 ) = ( 𝑂 + 𝑥 ) )
18 15 oveq2d ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ ( 𝑛 ∈ 𝐵 ∧ ( 𝑛 + 𝑥 ) = 𝑂 ) ) ) → ( 𝑥 + ( 𝑛 + 𝑥 ) ) = ( 𝑥 + 𝑂 ) )
19 14 17 18 3eqtr3d ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ ( 𝑛 ∈ 𝐵 ∧ ( 𝑛 + 𝑥 ) = 𝑂 ) ) ) → ( 𝑂 + 𝑥 ) = ( 𝑥 + 𝑂 ) )
20 19 anassrs ⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝑛 ∈ 𝐵 ∧ ( 𝑛 + 𝑥 ) = 𝑂 ) ) → ( 𝑂 + 𝑥 ) = ( 𝑥 + 𝑂 ) )
21 9 20 rexlimddv ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → ( 𝑂 + 𝑥 ) = ( 𝑥 + 𝑂 ) )
22 21 3 eqtr3d ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → ( 𝑥 + 𝑂 ) = 𝑥 )