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 |
|
grprinvd.x |
⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝑋 ∈ 𝐵 ) |
7 |
|
grprinvd.n |
⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝑁 ∈ 𝐵 ) |
8 |
|
grprinvd.e |
⊢ ( ( 𝜑 ∧ 𝜓 ) → ( 𝑁 + 𝑋 ) = 𝑂 ) |
9 |
1
|
3expb |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝑥 + 𝑦 ) ∈ 𝐵 ) |
10 |
9
|
caovclg |
⊢ ( ( 𝜑 ∧ ( 𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ) ) → ( 𝑢 + 𝑣 ) ∈ 𝐵 ) |
11 |
10
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝜓 ) ∧ ( 𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ) ) → ( 𝑢 + 𝑣 ) ∈ 𝐵 ) |
12 |
11 6 7
|
caovcld |
⊢ ( ( 𝜑 ∧ 𝜓 ) → ( 𝑋 + 𝑁 ) ∈ 𝐵 ) |
13 |
4
|
caovassg |
⊢ ( ( 𝜑 ∧ ( 𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ) → ( ( 𝑢 + 𝑣 ) + 𝑤 ) = ( 𝑢 + ( 𝑣 + 𝑤 ) ) ) |
14 |
13
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝜓 ) ∧ ( 𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ) → ( ( 𝑢 + 𝑣 ) + 𝑤 ) = ( 𝑢 + ( 𝑣 + 𝑤 ) ) ) |
15 |
14 6 7 12
|
caovassd |
⊢ ( ( 𝜑 ∧ 𝜓 ) → ( ( 𝑋 + 𝑁 ) + ( 𝑋 + 𝑁 ) ) = ( 𝑋 + ( 𝑁 + ( 𝑋 + 𝑁 ) ) ) ) |
16 |
8
|
oveq1d |
⊢ ( ( 𝜑 ∧ 𝜓 ) → ( ( 𝑁 + 𝑋 ) + 𝑁 ) = ( 𝑂 + 𝑁 ) ) |
17 |
14 7 6 7
|
caovassd |
⊢ ( ( 𝜑 ∧ 𝜓 ) → ( ( 𝑁 + 𝑋 ) + 𝑁 ) = ( 𝑁 + ( 𝑋 + 𝑁 ) ) ) |
18 |
|
oveq2 |
⊢ ( 𝑦 = 𝑁 → ( 𝑂 + 𝑦 ) = ( 𝑂 + 𝑁 ) ) |
19 |
|
id |
⊢ ( 𝑦 = 𝑁 → 𝑦 = 𝑁 ) |
20 |
18 19
|
eqeq12d |
⊢ ( 𝑦 = 𝑁 → ( ( 𝑂 + 𝑦 ) = 𝑦 ↔ ( 𝑂 + 𝑁 ) = 𝑁 ) ) |
21 |
3
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐵 ( 𝑂 + 𝑥 ) = 𝑥 ) |
22 |
|
oveq2 |
⊢ ( 𝑥 = 𝑦 → ( 𝑂 + 𝑥 ) = ( 𝑂 + 𝑦 ) ) |
23 |
|
id |
⊢ ( 𝑥 = 𝑦 → 𝑥 = 𝑦 ) |
24 |
22 23
|
eqeq12d |
⊢ ( 𝑥 = 𝑦 → ( ( 𝑂 + 𝑥 ) = 𝑥 ↔ ( 𝑂 + 𝑦 ) = 𝑦 ) ) |
25 |
24
|
cbvralvw |
⊢ ( ∀ 𝑥 ∈ 𝐵 ( 𝑂 + 𝑥 ) = 𝑥 ↔ ∀ 𝑦 ∈ 𝐵 ( 𝑂 + 𝑦 ) = 𝑦 ) |
26 |
21 25
|
sylib |
⊢ ( 𝜑 → ∀ 𝑦 ∈ 𝐵 ( 𝑂 + 𝑦 ) = 𝑦 ) |
27 |
26
|
adantr |
⊢ ( ( 𝜑 ∧ 𝜓 ) → ∀ 𝑦 ∈ 𝐵 ( 𝑂 + 𝑦 ) = 𝑦 ) |
28 |
20 27 7
|
rspcdva |
⊢ ( ( 𝜑 ∧ 𝜓 ) → ( 𝑂 + 𝑁 ) = 𝑁 ) |
29 |
16 17 28
|
3eqtr3d |
⊢ ( ( 𝜑 ∧ 𝜓 ) → ( 𝑁 + ( 𝑋 + 𝑁 ) ) = 𝑁 ) |
30 |
29
|
oveq2d |
⊢ ( ( 𝜑 ∧ 𝜓 ) → ( 𝑋 + ( 𝑁 + ( 𝑋 + 𝑁 ) ) ) = ( 𝑋 + 𝑁 ) ) |
31 |
15 30
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝜓 ) → ( ( 𝑋 + 𝑁 ) + ( 𝑋 + 𝑁 ) ) = ( 𝑋 + 𝑁 ) ) |
32 |
1 2 3 4 5 12 31
|
grprinvlem |
⊢ ( ( 𝜑 ∧ 𝜓 ) → ( 𝑋 + 𝑁 ) = 𝑂 ) |