Step |
Hyp |
Ref |
Expression |
1 |
|
xmulasslem.1 |
⊢ ( 𝑥 = 𝐷 → ( 𝜓 ↔ 𝑋 = 𝑌 ) ) |
2 |
|
xmulasslem.2 |
⊢ ( 𝑥 = -𝑒 𝐷 → ( 𝜓 ↔ 𝐸 = 𝐹 ) ) |
3 |
|
xmulasslem.x |
⊢ ( 𝜑 → 𝑋 ∈ ℝ* ) |
4 |
|
xmulasslem.y |
⊢ ( 𝜑 → 𝑌 ∈ ℝ* ) |
5 |
|
xmulasslem.d |
⊢ ( 𝜑 → 𝐷 ∈ ℝ* ) |
6 |
|
xmulasslem.ps |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ* ∧ 0 < 𝑥 ) ) → 𝜓 ) |
7 |
|
xmulasslem.0 |
⊢ ( 𝜑 → ( 𝑥 = 0 → 𝜓 ) ) |
8 |
|
xmulasslem.e |
⊢ ( 𝜑 → 𝐸 = -𝑒 𝑋 ) |
9 |
|
xmulasslem.f |
⊢ ( 𝜑 → 𝐹 = -𝑒 𝑌 ) |
10 |
|
0xr |
⊢ 0 ∈ ℝ* |
11 |
|
xrltso |
⊢ < Or ℝ* |
12 |
|
solin |
⊢ ( ( < Or ℝ* ∧ ( 𝐷 ∈ ℝ* ∧ 0 ∈ ℝ* ) ) → ( 𝐷 < 0 ∨ 𝐷 = 0 ∨ 0 < 𝐷 ) ) |
13 |
11 12
|
mpan |
⊢ ( ( 𝐷 ∈ ℝ* ∧ 0 ∈ ℝ* ) → ( 𝐷 < 0 ∨ 𝐷 = 0 ∨ 0 < 𝐷 ) ) |
14 |
5 10 13
|
sylancl |
⊢ ( 𝜑 → ( 𝐷 < 0 ∨ 𝐷 = 0 ∨ 0 < 𝐷 ) ) |
15 |
|
xlt0neg1 |
⊢ ( 𝐷 ∈ ℝ* → ( 𝐷 < 0 ↔ 0 < -𝑒 𝐷 ) ) |
16 |
5 15
|
syl |
⊢ ( 𝜑 → ( 𝐷 < 0 ↔ 0 < -𝑒 𝐷 ) ) |
17 |
|
xnegcl |
⊢ ( 𝐷 ∈ ℝ* → -𝑒 𝐷 ∈ ℝ* ) |
18 |
5 17
|
syl |
⊢ ( 𝜑 → -𝑒 𝐷 ∈ ℝ* ) |
19 |
|
breq2 |
⊢ ( 𝑥 = -𝑒 𝐷 → ( 0 < 𝑥 ↔ 0 < -𝑒 𝐷 ) ) |
20 |
19 2
|
imbi12d |
⊢ ( 𝑥 = -𝑒 𝐷 → ( ( 0 < 𝑥 → 𝜓 ) ↔ ( 0 < -𝑒 𝐷 → 𝐸 = 𝐹 ) ) ) |
21 |
20
|
imbi2d |
⊢ ( 𝑥 = -𝑒 𝐷 → ( ( 𝜑 → ( 0 < 𝑥 → 𝜓 ) ) ↔ ( 𝜑 → ( 0 < -𝑒 𝐷 → 𝐸 = 𝐹 ) ) ) ) |
22 |
6
|
exp32 |
⊢ ( 𝜑 → ( 𝑥 ∈ ℝ* → ( 0 < 𝑥 → 𝜓 ) ) ) |
23 |
22
|
com12 |
⊢ ( 𝑥 ∈ ℝ* → ( 𝜑 → ( 0 < 𝑥 → 𝜓 ) ) ) |
24 |
21 23
|
vtoclga |
⊢ ( -𝑒 𝐷 ∈ ℝ* → ( 𝜑 → ( 0 < -𝑒 𝐷 → 𝐸 = 𝐹 ) ) ) |
25 |
18 24
|
mpcom |
⊢ ( 𝜑 → ( 0 < -𝑒 𝐷 → 𝐸 = 𝐹 ) ) |
26 |
16 25
|
sylbid |
⊢ ( 𝜑 → ( 𝐷 < 0 → 𝐸 = 𝐹 ) ) |
27 |
8 9
|
eqeq12d |
⊢ ( 𝜑 → ( 𝐸 = 𝐹 ↔ -𝑒 𝑋 = -𝑒 𝑌 ) ) |
28 |
|
xneg11 |
⊢ ( ( 𝑋 ∈ ℝ* ∧ 𝑌 ∈ ℝ* ) → ( -𝑒 𝑋 = -𝑒 𝑌 ↔ 𝑋 = 𝑌 ) ) |
29 |
3 4 28
|
syl2anc |
⊢ ( 𝜑 → ( -𝑒 𝑋 = -𝑒 𝑌 ↔ 𝑋 = 𝑌 ) ) |
30 |
27 29
|
bitrd |
⊢ ( 𝜑 → ( 𝐸 = 𝐹 ↔ 𝑋 = 𝑌 ) ) |
31 |
26 30
|
sylibd |
⊢ ( 𝜑 → ( 𝐷 < 0 → 𝑋 = 𝑌 ) ) |
32 |
|
eqeq1 |
⊢ ( 𝑥 = 𝐷 → ( 𝑥 = 0 ↔ 𝐷 = 0 ) ) |
33 |
32 1
|
imbi12d |
⊢ ( 𝑥 = 𝐷 → ( ( 𝑥 = 0 → 𝜓 ) ↔ ( 𝐷 = 0 → 𝑋 = 𝑌 ) ) ) |
34 |
33
|
imbi2d |
⊢ ( 𝑥 = 𝐷 → ( ( 𝜑 → ( 𝑥 = 0 → 𝜓 ) ) ↔ ( 𝜑 → ( 𝐷 = 0 → 𝑋 = 𝑌 ) ) ) ) |
35 |
34 7
|
vtoclg |
⊢ ( 𝐷 ∈ ℝ* → ( 𝜑 → ( 𝐷 = 0 → 𝑋 = 𝑌 ) ) ) |
36 |
5 35
|
mpcom |
⊢ ( 𝜑 → ( 𝐷 = 0 → 𝑋 = 𝑌 ) ) |
37 |
|
breq2 |
⊢ ( 𝑥 = 𝐷 → ( 0 < 𝑥 ↔ 0 < 𝐷 ) ) |
38 |
37 1
|
imbi12d |
⊢ ( 𝑥 = 𝐷 → ( ( 0 < 𝑥 → 𝜓 ) ↔ ( 0 < 𝐷 → 𝑋 = 𝑌 ) ) ) |
39 |
38
|
imbi2d |
⊢ ( 𝑥 = 𝐷 → ( ( 𝜑 → ( 0 < 𝑥 → 𝜓 ) ) ↔ ( 𝜑 → ( 0 < 𝐷 → 𝑋 = 𝑌 ) ) ) ) |
40 |
39 23
|
vtoclga |
⊢ ( 𝐷 ∈ ℝ* → ( 𝜑 → ( 0 < 𝐷 → 𝑋 = 𝑌 ) ) ) |
41 |
5 40
|
mpcom |
⊢ ( 𝜑 → ( 0 < 𝐷 → 𝑋 = 𝑌 ) ) |
42 |
31 36 41
|
3jaod |
⊢ ( 𝜑 → ( ( 𝐷 < 0 ∨ 𝐷 = 0 ∨ 0 < 𝐷 ) → 𝑋 = 𝑌 ) ) |
43 |
14 42
|
mpd |
⊢ ( 𝜑 → 𝑋 = 𝑌 ) |