Step |
Hyp |
Ref |
Expression |
1 |
|
ressxr |
⊢ ℝ ⊆ ℝ* |
2 |
|
xrecex |
⊢ ( ( 𝐵 ∈ ℝ ∧ 𝐵 ≠ 0 ) → ∃ 𝑦 ∈ ℝ ( 𝐵 ·e 𝑦 ) = 1 ) |
3 |
2
|
3adant1 |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ ∧ 𝐵 ≠ 0 ) → ∃ 𝑦 ∈ ℝ ( 𝐵 ·e 𝑦 ) = 1 ) |
4 |
|
ssrexv |
⊢ ( ℝ ⊆ ℝ* → ( ∃ 𝑦 ∈ ℝ ( 𝐵 ·e 𝑦 ) = 1 → ∃ 𝑦 ∈ ℝ* ( 𝐵 ·e 𝑦 ) = 1 ) ) |
5 |
1 3 4
|
mpsyl |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ ∧ 𝐵 ≠ 0 ) → ∃ 𝑦 ∈ ℝ* ( 𝐵 ·e 𝑦 ) = 1 ) |
6 |
|
simprl |
⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ ) ∧ ( 𝑦 ∈ ℝ* ∧ ( 𝐵 ·e 𝑦 ) = 1 ) ) → 𝑦 ∈ ℝ* ) |
7 |
|
simpll |
⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ ) ∧ ( 𝑦 ∈ ℝ* ∧ ( 𝐵 ·e 𝑦 ) = 1 ) ) → 𝐴 ∈ ℝ* ) |
8 |
6 7
|
xmulcld |
⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ ) ∧ ( 𝑦 ∈ ℝ* ∧ ( 𝐵 ·e 𝑦 ) = 1 ) ) → ( 𝑦 ·e 𝐴 ) ∈ ℝ* ) |
9 |
|
oveq1 |
⊢ ( ( 𝐵 ·e 𝑦 ) = 1 → ( ( 𝐵 ·e 𝑦 ) ·e 𝐴 ) = ( 1 ·e 𝐴 ) ) |
10 |
9
|
ad2antll |
⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ ) ∧ ( 𝑦 ∈ ℝ* ∧ ( 𝐵 ·e 𝑦 ) = 1 ) ) → ( ( 𝐵 ·e 𝑦 ) ·e 𝐴 ) = ( 1 ·e 𝐴 ) ) |
11 |
|
simplr |
⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ ) ∧ ( 𝑦 ∈ ℝ* ∧ ( 𝐵 ·e 𝑦 ) = 1 ) ) → 𝐵 ∈ ℝ ) |
12 |
11
|
rexrd |
⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ ) ∧ ( 𝑦 ∈ ℝ* ∧ ( 𝐵 ·e 𝑦 ) = 1 ) ) → 𝐵 ∈ ℝ* ) |
13 |
|
xmulass |
⊢ ( ( 𝐵 ∈ ℝ* ∧ 𝑦 ∈ ℝ* ∧ 𝐴 ∈ ℝ* ) → ( ( 𝐵 ·e 𝑦 ) ·e 𝐴 ) = ( 𝐵 ·e ( 𝑦 ·e 𝐴 ) ) ) |
14 |
12 6 7 13
|
syl3anc |
⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ ) ∧ ( 𝑦 ∈ ℝ* ∧ ( 𝐵 ·e 𝑦 ) = 1 ) ) → ( ( 𝐵 ·e 𝑦 ) ·e 𝐴 ) = ( 𝐵 ·e ( 𝑦 ·e 𝐴 ) ) ) |
15 |
|
xmulid2 |
⊢ ( 𝐴 ∈ ℝ* → ( 1 ·e 𝐴 ) = 𝐴 ) |
16 |
7 15
|
syl |
⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ ) ∧ ( 𝑦 ∈ ℝ* ∧ ( 𝐵 ·e 𝑦 ) = 1 ) ) → ( 1 ·e 𝐴 ) = 𝐴 ) |
17 |
10 14 16
|
3eqtr3d |
⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ ) ∧ ( 𝑦 ∈ ℝ* ∧ ( 𝐵 ·e 𝑦 ) = 1 ) ) → ( 𝐵 ·e ( 𝑦 ·e 𝐴 ) ) = 𝐴 ) |
18 |
|
oveq2 |
⊢ ( 𝑥 = ( 𝑦 ·e 𝐴 ) → ( 𝐵 ·e 𝑥 ) = ( 𝐵 ·e ( 𝑦 ·e 𝐴 ) ) ) |
19 |
18
|
eqeq1d |
⊢ ( 𝑥 = ( 𝑦 ·e 𝐴 ) → ( ( 𝐵 ·e 𝑥 ) = 𝐴 ↔ ( 𝐵 ·e ( 𝑦 ·e 𝐴 ) ) = 𝐴 ) ) |
20 |
19
|
rspcev |
⊢ ( ( ( 𝑦 ·e 𝐴 ) ∈ ℝ* ∧ ( 𝐵 ·e ( 𝑦 ·e 𝐴 ) ) = 𝐴 ) → ∃ 𝑥 ∈ ℝ* ( 𝐵 ·e 𝑥 ) = 𝐴 ) |
21 |
8 17 20
|
syl2anc |
⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ ) ∧ ( 𝑦 ∈ ℝ* ∧ ( 𝐵 ·e 𝑦 ) = 1 ) ) → ∃ 𝑥 ∈ ℝ* ( 𝐵 ·e 𝑥 ) = 𝐴 ) |
22 |
21
|
rexlimdvaa |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ ) → ( ∃ 𝑦 ∈ ℝ* ( 𝐵 ·e 𝑦 ) = 1 → ∃ 𝑥 ∈ ℝ* ( 𝐵 ·e 𝑥 ) = 𝐴 ) ) |
23 |
22
|
3adant3 |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ ∧ 𝐵 ≠ 0 ) → ( ∃ 𝑦 ∈ ℝ* ( 𝐵 ·e 𝑦 ) = 1 → ∃ 𝑥 ∈ ℝ* ( 𝐵 ·e 𝑥 ) = 𝐴 ) ) |
24 |
5 23
|
mpd |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ ∧ 𝐵 ≠ 0 ) → ∃ 𝑥 ∈ ℝ* ( 𝐵 ·e 𝑥 ) = 𝐴 ) |
25 |
|
eqtr3 |
⊢ ( ( ( 𝐵 ·e 𝑥 ) = 𝐴 ∧ ( 𝐵 ·e 𝑦 ) = 𝐴 ) → ( 𝐵 ·e 𝑥 ) = ( 𝐵 ·e 𝑦 ) ) |
26 |
|
simp1 |
⊢ ( ( 𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ* ∧ ( 𝐵 ∈ ℝ ∧ 𝐵 ≠ 0 ) ) → 𝑥 ∈ ℝ* ) |
27 |
|
simp2 |
⊢ ( ( 𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ* ∧ ( 𝐵 ∈ ℝ ∧ 𝐵 ≠ 0 ) ) → 𝑦 ∈ ℝ* ) |
28 |
|
simp3l |
⊢ ( ( 𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ* ∧ ( 𝐵 ∈ ℝ ∧ 𝐵 ≠ 0 ) ) → 𝐵 ∈ ℝ ) |
29 |
|
simp3r |
⊢ ( ( 𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ* ∧ ( 𝐵 ∈ ℝ ∧ 𝐵 ≠ 0 ) ) → 𝐵 ≠ 0 ) |
30 |
26 27 28 29
|
xmulcand |
⊢ ( ( 𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ* ∧ ( 𝐵 ∈ ℝ ∧ 𝐵 ≠ 0 ) ) → ( ( 𝐵 ·e 𝑥 ) = ( 𝐵 ·e 𝑦 ) ↔ 𝑥 = 𝑦 ) ) |
31 |
25 30
|
syl5ib |
⊢ ( ( 𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ* ∧ ( 𝐵 ∈ ℝ ∧ 𝐵 ≠ 0 ) ) → ( ( ( 𝐵 ·e 𝑥 ) = 𝐴 ∧ ( 𝐵 ·e 𝑦 ) = 𝐴 ) → 𝑥 = 𝑦 ) ) |
32 |
31
|
3expa |
⊢ ( ( ( 𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) ∧ ( 𝐵 ∈ ℝ ∧ 𝐵 ≠ 0 ) ) → ( ( ( 𝐵 ·e 𝑥 ) = 𝐴 ∧ ( 𝐵 ·e 𝑦 ) = 𝐴 ) → 𝑥 = 𝑦 ) ) |
33 |
32
|
expcom |
⊢ ( ( 𝐵 ∈ ℝ ∧ 𝐵 ≠ 0 ) → ( ( 𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) → ( ( ( 𝐵 ·e 𝑥 ) = 𝐴 ∧ ( 𝐵 ·e 𝑦 ) = 𝐴 ) → 𝑥 = 𝑦 ) ) ) |
34 |
33
|
3adant1 |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ ∧ 𝐵 ≠ 0 ) → ( ( 𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) → ( ( ( 𝐵 ·e 𝑥 ) = 𝐴 ∧ ( 𝐵 ·e 𝑦 ) = 𝐴 ) → 𝑥 = 𝑦 ) ) ) |
35 |
34
|
ralrimivv |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ ∧ 𝐵 ≠ 0 ) → ∀ 𝑥 ∈ ℝ* ∀ 𝑦 ∈ ℝ* ( ( ( 𝐵 ·e 𝑥 ) = 𝐴 ∧ ( 𝐵 ·e 𝑦 ) = 𝐴 ) → 𝑥 = 𝑦 ) ) |
36 |
|
oveq2 |
⊢ ( 𝑥 = 𝑦 → ( 𝐵 ·e 𝑥 ) = ( 𝐵 ·e 𝑦 ) ) |
37 |
36
|
eqeq1d |
⊢ ( 𝑥 = 𝑦 → ( ( 𝐵 ·e 𝑥 ) = 𝐴 ↔ ( 𝐵 ·e 𝑦 ) = 𝐴 ) ) |
38 |
37
|
reu4 |
⊢ ( ∃! 𝑥 ∈ ℝ* ( 𝐵 ·e 𝑥 ) = 𝐴 ↔ ( ∃ 𝑥 ∈ ℝ* ( 𝐵 ·e 𝑥 ) = 𝐴 ∧ ∀ 𝑥 ∈ ℝ* ∀ 𝑦 ∈ ℝ* ( ( ( 𝐵 ·e 𝑥 ) = 𝐴 ∧ ( 𝐵 ·e 𝑦 ) = 𝐴 ) → 𝑥 = 𝑦 ) ) ) |
39 |
24 35 38
|
sylanbrc |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ ∧ 𝐵 ≠ 0 ) → ∃! 𝑥 ∈ ℝ* ( 𝐵 ·e 𝑥 ) = 𝐴 ) |