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 |
|
grprinvlem.x |
⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝑋 ∈ 𝐵 ) |
7 |
|
grprinvlem.e |
⊢ ( ( 𝜑 ∧ 𝜓 ) → ( 𝑋 + 𝑋 ) = 𝑋 ) |
8 |
5
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐵 ∃ 𝑦 ∈ 𝐵 ( 𝑦 + 𝑥 ) = 𝑂 ) |
9 |
|
oveq2 |
⊢ ( 𝑥 = 𝑧 → ( 𝑦 + 𝑥 ) = ( 𝑦 + 𝑧 ) ) |
10 |
9
|
eqeq1d |
⊢ ( 𝑥 = 𝑧 → ( ( 𝑦 + 𝑥 ) = 𝑂 ↔ ( 𝑦 + 𝑧 ) = 𝑂 ) ) |
11 |
10
|
rexbidv |
⊢ ( 𝑥 = 𝑧 → ( ∃ 𝑦 ∈ 𝐵 ( 𝑦 + 𝑥 ) = 𝑂 ↔ ∃ 𝑦 ∈ 𝐵 ( 𝑦 + 𝑧 ) = 𝑂 ) ) |
12 |
11
|
cbvralvw |
⊢ ( ∀ 𝑥 ∈ 𝐵 ∃ 𝑦 ∈ 𝐵 ( 𝑦 + 𝑥 ) = 𝑂 ↔ ∀ 𝑧 ∈ 𝐵 ∃ 𝑦 ∈ 𝐵 ( 𝑦 + 𝑧 ) = 𝑂 ) |
13 |
8 12
|
sylib |
⊢ ( 𝜑 → ∀ 𝑧 ∈ 𝐵 ∃ 𝑦 ∈ 𝐵 ( 𝑦 + 𝑧 ) = 𝑂 ) |
14 |
|
oveq2 |
⊢ ( 𝑧 = 𝑋 → ( 𝑦 + 𝑧 ) = ( 𝑦 + 𝑋 ) ) |
15 |
14
|
eqeq1d |
⊢ ( 𝑧 = 𝑋 → ( ( 𝑦 + 𝑧 ) = 𝑂 ↔ ( 𝑦 + 𝑋 ) = 𝑂 ) ) |
16 |
15
|
rexbidv |
⊢ ( 𝑧 = 𝑋 → ( ∃ 𝑦 ∈ 𝐵 ( 𝑦 + 𝑧 ) = 𝑂 ↔ ∃ 𝑦 ∈ 𝐵 ( 𝑦 + 𝑋 ) = 𝑂 ) ) |
17 |
16
|
rspccva |
⊢ ( ( ∀ 𝑧 ∈ 𝐵 ∃ 𝑦 ∈ 𝐵 ( 𝑦 + 𝑧 ) = 𝑂 ∧ 𝑋 ∈ 𝐵 ) → ∃ 𝑦 ∈ 𝐵 ( 𝑦 + 𝑋 ) = 𝑂 ) |
18 |
13 6 17
|
syl2an2r |
⊢ ( ( 𝜑 ∧ 𝜓 ) → ∃ 𝑦 ∈ 𝐵 ( 𝑦 + 𝑋 ) = 𝑂 ) |
19 |
7
|
oveq2d |
⊢ ( ( 𝜑 ∧ 𝜓 ) → ( 𝑦 + ( 𝑋 + 𝑋 ) ) = ( 𝑦 + 𝑋 ) ) |
20 |
19
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝜓 ) ∧ ( 𝑦 ∈ 𝐵 ∧ ( 𝑦 + 𝑋 ) = 𝑂 ) ) → ( 𝑦 + ( 𝑋 + 𝑋 ) ) = ( 𝑦 + 𝑋 ) ) |
21 |
|
simprr |
⊢ ( ( ( 𝜑 ∧ 𝜓 ) ∧ ( 𝑦 ∈ 𝐵 ∧ ( 𝑦 + 𝑋 ) = 𝑂 ) ) → ( 𝑦 + 𝑋 ) = 𝑂 ) |
22 |
21
|
oveq1d |
⊢ ( ( ( 𝜑 ∧ 𝜓 ) ∧ ( 𝑦 ∈ 𝐵 ∧ ( 𝑦 + 𝑋 ) = 𝑂 ) ) → ( ( 𝑦 + 𝑋 ) + 𝑋 ) = ( 𝑂 + 𝑋 ) ) |
23 |
4
|
caovassg |
⊢ ( ( 𝜑 ∧ ( 𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ) → ( ( 𝑢 + 𝑣 ) + 𝑤 ) = ( 𝑢 + ( 𝑣 + 𝑤 ) ) ) |
24 |
23
|
ad4ant14 |
⊢ ( ( ( ( 𝜑 ∧ 𝜓 ) ∧ ( 𝑦 ∈ 𝐵 ∧ ( 𝑦 + 𝑋 ) = 𝑂 ) ) ∧ ( 𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵 ) ) → ( ( 𝑢 + 𝑣 ) + 𝑤 ) = ( 𝑢 + ( 𝑣 + 𝑤 ) ) ) |
25 |
|
simprl |
⊢ ( ( ( 𝜑 ∧ 𝜓 ) ∧ ( 𝑦 ∈ 𝐵 ∧ ( 𝑦 + 𝑋 ) = 𝑂 ) ) → 𝑦 ∈ 𝐵 ) |
26 |
6
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝜓 ) ∧ ( 𝑦 ∈ 𝐵 ∧ ( 𝑦 + 𝑋 ) = 𝑂 ) ) → 𝑋 ∈ 𝐵 ) |
27 |
24 25 26 26
|
caovassd |
⊢ ( ( ( 𝜑 ∧ 𝜓 ) ∧ ( 𝑦 ∈ 𝐵 ∧ ( 𝑦 + 𝑋 ) = 𝑂 ) ) → ( ( 𝑦 + 𝑋 ) + 𝑋 ) = ( 𝑦 + ( 𝑋 + 𝑋 ) ) ) |
28 |
|
oveq2 |
⊢ ( 𝑦 = 𝑋 → ( 𝑂 + 𝑦 ) = ( 𝑂 + 𝑋 ) ) |
29 |
|
id |
⊢ ( 𝑦 = 𝑋 → 𝑦 = 𝑋 ) |
30 |
28 29
|
eqeq12d |
⊢ ( 𝑦 = 𝑋 → ( ( 𝑂 + 𝑦 ) = 𝑦 ↔ ( 𝑂 + 𝑋 ) = 𝑋 ) ) |
31 |
3
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐵 ( 𝑂 + 𝑥 ) = 𝑥 ) |
32 |
|
oveq2 |
⊢ ( 𝑥 = 𝑦 → ( 𝑂 + 𝑥 ) = ( 𝑂 + 𝑦 ) ) |
33 |
|
id |
⊢ ( 𝑥 = 𝑦 → 𝑥 = 𝑦 ) |
34 |
32 33
|
eqeq12d |
⊢ ( 𝑥 = 𝑦 → ( ( 𝑂 + 𝑥 ) = 𝑥 ↔ ( 𝑂 + 𝑦 ) = 𝑦 ) ) |
35 |
34
|
cbvralvw |
⊢ ( ∀ 𝑥 ∈ 𝐵 ( 𝑂 + 𝑥 ) = 𝑥 ↔ ∀ 𝑦 ∈ 𝐵 ( 𝑂 + 𝑦 ) = 𝑦 ) |
36 |
31 35
|
sylib |
⊢ ( 𝜑 → ∀ 𝑦 ∈ 𝐵 ( 𝑂 + 𝑦 ) = 𝑦 ) |
37 |
36
|
adantr |
⊢ ( ( 𝜑 ∧ 𝜓 ) → ∀ 𝑦 ∈ 𝐵 ( 𝑂 + 𝑦 ) = 𝑦 ) |
38 |
30 37 6
|
rspcdva |
⊢ ( ( 𝜑 ∧ 𝜓 ) → ( 𝑂 + 𝑋 ) = 𝑋 ) |
39 |
38
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝜓 ) ∧ ( 𝑦 ∈ 𝐵 ∧ ( 𝑦 + 𝑋 ) = 𝑂 ) ) → ( 𝑂 + 𝑋 ) = 𝑋 ) |
40 |
22 27 39
|
3eqtr3d |
⊢ ( ( ( 𝜑 ∧ 𝜓 ) ∧ ( 𝑦 ∈ 𝐵 ∧ ( 𝑦 + 𝑋 ) = 𝑂 ) ) → ( 𝑦 + ( 𝑋 + 𝑋 ) ) = 𝑋 ) |
41 |
20 40 21
|
3eqtr3d |
⊢ ( ( ( 𝜑 ∧ 𝜓 ) ∧ ( 𝑦 ∈ 𝐵 ∧ ( 𝑦 + 𝑋 ) = 𝑂 ) ) → 𝑋 = 𝑂 ) |
42 |
18 41
|
rexlimddv |
⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝑋 = 𝑂 ) |