Step |
Hyp |
Ref |
Expression |
1 |
|
grprinvlem.c |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) → ( 𝑥 + 𝑦 ) ∈ 𝐵 ) |
2 |
|
grprinvlem.o |
⊢ ( 𝜑 → 𝑂 ∈ 𝐵 ) |
3 |
|
grprinvlem.i |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → ( 𝑂 + 𝑥 ) = 𝑥 ) |
4 |
|
grprinvlem.a |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ) → ( ( 𝑥 + 𝑦 ) + 𝑧 ) = ( 𝑥 + ( 𝑦 + 𝑧 ) ) ) |
5 |
|
grprinvlem.n |
⊢ ( ( 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
|
grprinvd |
⊢ ( ( 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 ) |