Step |
Hyp |
Ref |
Expression |
1 |
|
remulinvcom.1 |
⊢ ( 𝜑 → 𝐴 ∈ ℝ ) |
2 |
|
remulinvcom.2 |
⊢ ( 𝜑 → 𝐵 ∈ ℝ ) |
3 |
|
remulinvcom.3 |
⊢ ( 𝜑 → ( 𝐴 · 𝐵 ) = 1 ) |
4 |
|
ax-1ne0 |
⊢ 1 ≠ 0 |
5 |
4
|
a1i |
⊢ ( 𝜑 → 1 ≠ 0 ) |
6 |
3 5
|
eqnetrd |
⊢ ( 𝜑 → ( 𝐴 · 𝐵 ) ≠ 0 ) |
7 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝐵 = 0 ) → 𝐵 = 0 ) |
8 |
7
|
oveq2d |
⊢ ( ( 𝜑 ∧ 𝐵 = 0 ) → ( 𝐴 · 𝐵 ) = ( 𝐴 · 0 ) ) |
9 |
1
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐵 = 0 ) → 𝐴 ∈ ℝ ) |
10 |
|
remul01 |
⊢ ( 𝐴 ∈ ℝ → ( 𝐴 · 0 ) = 0 ) |
11 |
9 10
|
syl |
⊢ ( ( 𝜑 ∧ 𝐵 = 0 ) → ( 𝐴 · 0 ) = 0 ) |
12 |
8 11
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝐵 = 0 ) → ( 𝐴 · 𝐵 ) = 0 ) |
13 |
6 12
|
mteqand |
⊢ ( 𝜑 → 𝐵 ≠ 0 ) |
14 |
|
ax-rrecex |
⊢ ( ( 𝐵 ∈ ℝ ∧ 𝐵 ≠ 0 ) → ∃ 𝑥 ∈ ℝ ( 𝐵 · 𝑥 ) = 1 ) |
15 |
2 13 14
|
syl2anc |
⊢ ( 𝜑 → ∃ 𝑥 ∈ ℝ ( 𝐵 · 𝑥 ) = 1 ) |
16 |
|
simprl |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ ∧ ( 𝐵 · 𝑥 ) = 1 ) ) → 𝑥 ∈ ℝ ) |
17 |
|
simprr |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ ∧ ( 𝐵 · 𝑥 ) = 1 ) ) → ( 𝐵 · 𝑥 ) = 1 ) |
18 |
4
|
a1i |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ ∧ ( 𝐵 · 𝑥 ) = 1 ) ) → 1 ≠ 0 ) |
19 |
17 18
|
eqnetrd |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ ∧ ( 𝐵 · 𝑥 ) = 1 ) ) → ( 𝐵 · 𝑥 ) ≠ 0 ) |
20 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ ∧ ( 𝐵 · 𝑥 ) = 1 ) ) ∧ 𝑥 = 0 ) → 𝑥 = 0 ) |
21 |
20
|
oveq2d |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ ∧ ( 𝐵 · 𝑥 ) = 1 ) ) ∧ 𝑥 = 0 ) → ( 𝐵 · 𝑥 ) = ( 𝐵 · 0 ) ) |
22 |
2
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ ∧ ( 𝐵 · 𝑥 ) = 1 ) ) ∧ 𝑥 = 0 ) → 𝐵 ∈ ℝ ) |
23 |
|
remul01 |
⊢ ( 𝐵 ∈ ℝ → ( 𝐵 · 0 ) = 0 ) |
24 |
22 23
|
syl |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ ∧ ( 𝐵 · 𝑥 ) = 1 ) ) ∧ 𝑥 = 0 ) → ( 𝐵 · 0 ) = 0 ) |
25 |
21 24
|
eqtrd |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ ∧ ( 𝐵 · 𝑥 ) = 1 ) ) ∧ 𝑥 = 0 ) → ( 𝐵 · 𝑥 ) = 0 ) |
26 |
19 25
|
mteqand |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ ∧ ( 𝐵 · 𝑥 ) = 1 ) ) → 𝑥 ≠ 0 ) |
27 |
|
ax-rrecex |
⊢ ( ( 𝑥 ∈ ℝ ∧ 𝑥 ≠ 0 ) → ∃ 𝑦 ∈ ℝ ( 𝑥 · 𝑦 ) = 1 ) |
28 |
16 26 27
|
syl2anc |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ ∧ ( 𝐵 · 𝑥 ) = 1 ) ) → ∃ 𝑦 ∈ ℝ ( 𝑥 · 𝑦 ) = 1 ) |
29 |
|
simplrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ ∧ ( 𝐵 · 𝑥 ) = 1 ) ) ∧ ( 𝑦 ∈ ℝ ∧ ( 𝑥 · 𝑦 ) = 1 ) ) → ( 𝐵 · 𝑥 ) = 1 ) |
30 |
29
|
oveq2d |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ ∧ ( 𝐵 · 𝑥 ) = 1 ) ) ∧ ( 𝑦 ∈ ℝ ∧ ( 𝑥 · 𝑦 ) = 1 ) ) → ( 𝐴 · ( 𝐵 · 𝑥 ) ) = ( 𝐴 · 1 ) ) |
31 |
30
|
oveq1d |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ ∧ ( 𝐵 · 𝑥 ) = 1 ) ) ∧ ( 𝑦 ∈ ℝ ∧ ( 𝑥 · 𝑦 ) = 1 ) ) → ( ( 𝐴 · ( 𝐵 · 𝑥 ) ) · 𝑦 ) = ( ( 𝐴 · 1 ) · 𝑦 ) ) |
32 |
1
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ ∧ ( 𝐵 · 𝑥 ) = 1 ) ) ∧ ( 𝑦 ∈ ℝ ∧ ( 𝑥 · 𝑦 ) = 1 ) ) → 𝐴 ∈ ℝ ) |
33 |
2
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ ∧ ( 𝐵 · 𝑥 ) = 1 ) ) ∧ ( 𝑦 ∈ ℝ ∧ ( 𝑥 · 𝑦 ) = 1 ) ) → 𝐵 ∈ ℝ ) |
34 |
32 33
|
remulcld |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ ∧ ( 𝐵 · 𝑥 ) = 1 ) ) ∧ ( 𝑦 ∈ ℝ ∧ ( 𝑥 · 𝑦 ) = 1 ) ) → ( 𝐴 · 𝐵 ) ∈ ℝ ) |
35 |
34
|
recnd |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ ∧ ( 𝐵 · 𝑥 ) = 1 ) ) ∧ ( 𝑦 ∈ ℝ ∧ ( 𝑥 · 𝑦 ) = 1 ) ) → ( 𝐴 · 𝐵 ) ∈ ℂ ) |
36 |
|
simplrl |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ ∧ ( 𝐵 · 𝑥 ) = 1 ) ) ∧ ( 𝑦 ∈ ℝ ∧ ( 𝑥 · 𝑦 ) = 1 ) ) → 𝑥 ∈ ℝ ) |
37 |
36
|
recnd |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ ∧ ( 𝐵 · 𝑥 ) = 1 ) ) ∧ ( 𝑦 ∈ ℝ ∧ ( 𝑥 · 𝑦 ) = 1 ) ) → 𝑥 ∈ ℂ ) |
38 |
|
simprl |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ ∧ ( 𝐵 · 𝑥 ) = 1 ) ) ∧ ( 𝑦 ∈ ℝ ∧ ( 𝑥 · 𝑦 ) = 1 ) ) → 𝑦 ∈ ℝ ) |
39 |
38
|
recnd |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ ∧ ( 𝐵 · 𝑥 ) = 1 ) ) ∧ ( 𝑦 ∈ ℝ ∧ ( 𝑥 · 𝑦 ) = 1 ) ) → 𝑦 ∈ ℂ ) |
40 |
35 37 39
|
mulassd |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ ∧ ( 𝐵 · 𝑥 ) = 1 ) ) ∧ ( 𝑦 ∈ ℝ ∧ ( 𝑥 · 𝑦 ) = 1 ) ) → ( ( ( 𝐴 · 𝐵 ) · 𝑥 ) · 𝑦 ) = ( ( 𝐴 · 𝐵 ) · ( 𝑥 · 𝑦 ) ) ) |
41 |
32
|
recnd |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ ∧ ( 𝐵 · 𝑥 ) = 1 ) ) ∧ ( 𝑦 ∈ ℝ ∧ ( 𝑥 · 𝑦 ) = 1 ) ) → 𝐴 ∈ ℂ ) |
42 |
33
|
recnd |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ ∧ ( 𝐵 · 𝑥 ) = 1 ) ) ∧ ( 𝑦 ∈ ℝ ∧ ( 𝑥 · 𝑦 ) = 1 ) ) → 𝐵 ∈ ℂ ) |
43 |
41 42 37
|
mulassd |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ ∧ ( 𝐵 · 𝑥 ) = 1 ) ) ∧ ( 𝑦 ∈ ℝ ∧ ( 𝑥 · 𝑦 ) = 1 ) ) → ( ( 𝐴 · 𝐵 ) · 𝑥 ) = ( 𝐴 · ( 𝐵 · 𝑥 ) ) ) |
44 |
43
|
oveq1d |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ ∧ ( 𝐵 · 𝑥 ) = 1 ) ) ∧ ( 𝑦 ∈ ℝ ∧ ( 𝑥 · 𝑦 ) = 1 ) ) → ( ( ( 𝐴 · 𝐵 ) · 𝑥 ) · 𝑦 ) = ( ( 𝐴 · ( 𝐵 · 𝑥 ) ) · 𝑦 ) ) |
45 |
3
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ ∧ ( 𝐵 · 𝑥 ) = 1 ) ) ∧ ( 𝑦 ∈ ℝ ∧ ( 𝑥 · 𝑦 ) = 1 ) ) → ( 𝐴 · 𝐵 ) = 1 ) |
46 |
|
simprr |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ ∧ ( 𝐵 · 𝑥 ) = 1 ) ) ∧ ( 𝑦 ∈ ℝ ∧ ( 𝑥 · 𝑦 ) = 1 ) ) → ( 𝑥 · 𝑦 ) = 1 ) |
47 |
45 46
|
oveq12d |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ ∧ ( 𝐵 · 𝑥 ) = 1 ) ) ∧ ( 𝑦 ∈ ℝ ∧ ( 𝑥 · 𝑦 ) = 1 ) ) → ( ( 𝐴 · 𝐵 ) · ( 𝑥 · 𝑦 ) ) = ( 1 · 1 ) ) |
48 |
|
1t1e1ALT |
⊢ ( 1 · 1 ) = 1 |
49 |
47 48
|
eqtrdi |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ ∧ ( 𝐵 · 𝑥 ) = 1 ) ) ∧ ( 𝑦 ∈ ℝ ∧ ( 𝑥 · 𝑦 ) = 1 ) ) → ( ( 𝐴 · 𝐵 ) · ( 𝑥 · 𝑦 ) ) = 1 ) |
50 |
40 44 49
|
3eqtr3d |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ ∧ ( 𝐵 · 𝑥 ) = 1 ) ) ∧ ( 𝑦 ∈ ℝ ∧ ( 𝑥 · 𝑦 ) = 1 ) ) → ( ( 𝐴 · ( 𝐵 · 𝑥 ) ) · 𝑦 ) = 1 ) |
51 |
|
ax-1rid |
⊢ ( 𝐴 ∈ ℝ → ( 𝐴 · 1 ) = 𝐴 ) |
52 |
32 51
|
syl |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ ∧ ( 𝐵 · 𝑥 ) = 1 ) ) ∧ ( 𝑦 ∈ ℝ ∧ ( 𝑥 · 𝑦 ) = 1 ) ) → ( 𝐴 · 1 ) = 𝐴 ) |
53 |
52
|
oveq1d |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ ∧ ( 𝐵 · 𝑥 ) = 1 ) ) ∧ ( 𝑦 ∈ ℝ ∧ ( 𝑥 · 𝑦 ) = 1 ) ) → ( ( 𝐴 · 1 ) · 𝑦 ) = ( 𝐴 · 𝑦 ) ) |
54 |
31 50 53
|
3eqtr3rd |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ ∧ ( 𝐵 · 𝑥 ) = 1 ) ) ∧ ( 𝑦 ∈ ℝ ∧ ( 𝑥 · 𝑦 ) = 1 ) ) → ( 𝐴 · 𝑦 ) = 1 ) |
55 |
54 46
|
eqtr4d |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ ∧ ( 𝐵 · 𝑥 ) = 1 ) ) ∧ ( 𝑦 ∈ ℝ ∧ ( 𝑥 · 𝑦 ) = 1 ) ) → ( 𝐴 · 𝑦 ) = ( 𝑥 · 𝑦 ) ) |
56 |
4
|
a1i |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ ∧ ( 𝐵 · 𝑥 ) = 1 ) ) ∧ ( 𝑦 ∈ ℝ ∧ ( 𝑥 · 𝑦 ) = 1 ) ) → 1 ≠ 0 ) |
57 |
46 56
|
eqnetrd |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ ∧ ( 𝐵 · 𝑥 ) = 1 ) ) ∧ ( 𝑦 ∈ ℝ ∧ ( 𝑥 · 𝑦 ) = 1 ) ) → ( 𝑥 · 𝑦 ) ≠ 0 ) |
58 |
|
simpr |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ ∧ ( 𝐵 · 𝑥 ) = 1 ) ) ∧ ( 𝑦 ∈ ℝ ∧ ( 𝑥 · 𝑦 ) = 1 ) ) ∧ 𝑦 = 0 ) → 𝑦 = 0 ) |
59 |
58
|
oveq2d |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ ∧ ( 𝐵 · 𝑥 ) = 1 ) ) ∧ ( 𝑦 ∈ ℝ ∧ ( 𝑥 · 𝑦 ) = 1 ) ) ∧ 𝑦 = 0 ) → ( 𝑥 · 𝑦 ) = ( 𝑥 · 0 ) ) |
60 |
36
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ ∧ ( 𝐵 · 𝑥 ) = 1 ) ) ∧ ( 𝑦 ∈ ℝ ∧ ( 𝑥 · 𝑦 ) = 1 ) ) ∧ 𝑦 = 0 ) → 𝑥 ∈ ℝ ) |
61 |
|
remul01 |
⊢ ( 𝑥 ∈ ℝ → ( 𝑥 · 0 ) = 0 ) |
62 |
60 61
|
syl |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ ∧ ( 𝐵 · 𝑥 ) = 1 ) ) ∧ ( 𝑦 ∈ ℝ ∧ ( 𝑥 · 𝑦 ) = 1 ) ) ∧ 𝑦 = 0 ) → ( 𝑥 · 0 ) = 0 ) |
63 |
59 62
|
eqtrd |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ ∧ ( 𝐵 · 𝑥 ) = 1 ) ) ∧ ( 𝑦 ∈ ℝ ∧ ( 𝑥 · 𝑦 ) = 1 ) ) ∧ 𝑦 = 0 ) → ( 𝑥 · 𝑦 ) = 0 ) |
64 |
57 63
|
mteqand |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ ∧ ( 𝐵 · 𝑥 ) = 1 ) ) ∧ ( 𝑦 ∈ ℝ ∧ ( 𝑥 · 𝑦 ) = 1 ) ) → 𝑦 ≠ 0 ) |
65 |
32 36 38 64
|
remulcan2d |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ ∧ ( 𝐵 · 𝑥 ) = 1 ) ) ∧ ( 𝑦 ∈ ℝ ∧ ( 𝑥 · 𝑦 ) = 1 ) ) → ( ( 𝐴 · 𝑦 ) = ( 𝑥 · 𝑦 ) ↔ 𝐴 = 𝑥 ) ) |
66 |
55 65
|
mpbid |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ ∧ ( 𝐵 · 𝑥 ) = 1 ) ) ∧ ( 𝑦 ∈ ℝ ∧ ( 𝑥 · 𝑦 ) = 1 ) ) → 𝐴 = 𝑥 ) |
67 |
|
simpr |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ ∧ ( 𝐵 · 𝑥 ) = 1 ) ) ∧ ( 𝑦 ∈ ℝ ∧ ( 𝑥 · 𝑦 ) = 1 ) ) ∧ 𝐴 = 𝑥 ) → 𝐴 = 𝑥 ) |
68 |
67
|
oveq2d |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ ∧ ( 𝐵 · 𝑥 ) = 1 ) ) ∧ ( 𝑦 ∈ ℝ ∧ ( 𝑥 · 𝑦 ) = 1 ) ) ∧ 𝐴 = 𝑥 ) → ( 𝐵 · 𝐴 ) = ( 𝐵 · 𝑥 ) ) |
69 |
17
|
ad2antrr |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ ∧ ( 𝐵 · 𝑥 ) = 1 ) ) ∧ ( 𝑦 ∈ ℝ ∧ ( 𝑥 · 𝑦 ) = 1 ) ) ∧ 𝐴 = 𝑥 ) → ( 𝐵 · 𝑥 ) = 1 ) |
70 |
68 69
|
eqtrd |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ ∧ ( 𝐵 · 𝑥 ) = 1 ) ) ∧ ( 𝑦 ∈ ℝ ∧ ( 𝑥 · 𝑦 ) = 1 ) ) ∧ 𝐴 = 𝑥 ) → ( 𝐵 · 𝐴 ) = 1 ) |
71 |
66 70
|
mpdan |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ ∧ ( 𝐵 · 𝑥 ) = 1 ) ) ∧ ( 𝑦 ∈ ℝ ∧ ( 𝑥 · 𝑦 ) = 1 ) ) → ( 𝐵 · 𝐴 ) = 1 ) |
72 |
28 71
|
rexlimddv |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℝ ∧ ( 𝐵 · 𝑥 ) = 1 ) ) → ( 𝐵 · 𝐴 ) = 1 ) |
73 |
15 72
|
rexlimddv |
⊢ ( 𝜑 → ( 𝐵 · 𝐴 ) = 1 ) |