Step |
Hyp |
Ref |
Expression |
1 |
|
cnring |
⊢ ℂfld ∈ Ring |
2 |
|
cnfldbas |
⊢ ℂ = ( Base ‘ ℂfld ) |
3 |
|
cnfld0 |
⊢ 0 = ( 0g ‘ ℂfld ) |
4 |
|
cndrng |
⊢ ℂfld ∈ DivRing |
5 |
2 3 4
|
drngui |
⊢ ( ℂ ∖ { 0 } ) = ( Unit ‘ ℂfld ) |
6 |
|
eqid |
⊢ ( /r ‘ ℂfld ) = ( /r ‘ ℂfld ) |
7 |
2 5 6
|
dvrcl |
⊢ ( ( ℂfld ∈ Ring ∧ 𝑥 ∈ ℂ ∧ 𝑦 ∈ ( ℂ ∖ { 0 } ) ) → ( 𝑥 ( /r ‘ ℂfld ) 𝑦 ) ∈ ℂ ) |
8 |
1 7
|
mp3an1 |
⊢ ( ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ ( ℂ ∖ { 0 } ) ) → ( 𝑥 ( /r ‘ ℂfld ) 𝑦 ) ∈ ℂ ) |
9 |
|
difssd |
⊢ ( 𝑥 ∈ ℂ → ( ℂ ∖ { 0 } ) ⊆ ℂ ) |
10 |
9
|
sselda |
⊢ ( ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ ( ℂ ∖ { 0 } ) ) → 𝑦 ∈ ℂ ) |
11 |
|
ovmpot |
⊢ ( ( ( 𝑥 ( /r ‘ ℂfld ) 𝑦 ) ∈ ℂ ∧ 𝑦 ∈ ℂ ) → ( ( 𝑥 ( /r ‘ ℂfld ) 𝑦 ) ( 𝑢 ∈ ℂ , 𝑣 ∈ ℂ ↦ ( 𝑢 · 𝑣 ) ) 𝑦 ) = ( ( 𝑥 ( /r ‘ ℂfld ) 𝑦 ) · 𝑦 ) ) |
12 |
8 10 11
|
syl2anc |
⊢ ( ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ ( ℂ ∖ { 0 } ) ) → ( ( 𝑥 ( /r ‘ ℂfld ) 𝑦 ) ( 𝑢 ∈ ℂ , 𝑣 ∈ ℂ ↦ ( 𝑢 · 𝑣 ) ) 𝑦 ) = ( ( 𝑥 ( /r ‘ ℂfld ) 𝑦 ) · 𝑦 ) ) |
13 |
|
mpocnfldmul |
⊢ ( 𝑢 ∈ ℂ , 𝑣 ∈ ℂ ↦ ( 𝑢 · 𝑣 ) ) = ( .r ‘ ℂfld ) |
14 |
2 5 6 13
|
dvrcan1 |
⊢ ( ( ℂfld ∈ Ring ∧ 𝑥 ∈ ℂ ∧ 𝑦 ∈ ( ℂ ∖ { 0 } ) ) → ( ( 𝑥 ( /r ‘ ℂfld ) 𝑦 ) ( 𝑢 ∈ ℂ , 𝑣 ∈ ℂ ↦ ( 𝑢 · 𝑣 ) ) 𝑦 ) = 𝑥 ) |
15 |
1 14
|
mp3an1 |
⊢ ( ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ ( ℂ ∖ { 0 } ) ) → ( ( 𝑥 ( /r ‘ ℂfld ) 𝑦 ) ( 𝑢 ∈ ℂ , 𝑣 ∈ ℂ ↦ ( 𝑢 · 𝑣 ) ) 𝑦 ) = 𝑥 ) |
16 |
12 15
|
eqtr3d |
⊢ ( ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ ( ℂ ∖ { 0 } ) ) → ( ( 𝑥 ( /r ‘ ℂfld ) 𝑦 ) · 𝑦 ) = 𝑥 ) |
17 |
16
|
oveq1d |
⊢ ( ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ ( ℂ ∖ { 0 } ) ) → ( ( ( 𝑥 ( /r ‘ ℂfld ) 𝑦 ) · 𝑦 ) / 𝑦 ) = ( 𝑥 / 𝑦 ) ) |
18 |
|
eldifsni |
⊢ ( 𝑦 ∈ ( ℂ ∖ { 0 } ) → 𝑦 ≠ 0 ) |
19 |
18
|
adantl |
⊢ ( ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ ( ℂ ∖ { 0 } ) ) → 𝑦 ≠ 0 ) |
20 |
8 10 19
|
divcan4d |
⊢ ( ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ ( ℂ ∖ { 0 } ) ) → ( ( ( 𝑥 ( /r ‘ ℂfld ) 𝑦 ) · 𝑦 ) / 𝑦 ) = ( 𝑥 ( /r ‘ ℂfld ) 𝑦 ) ) |
21 |
17 20
|
eqtr3d |
⊢ ( ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ ( ℂ ∖ { 0 } ) ) → ( 𝑥 / 𝑦 ) = ( 𝑥 ( /r ‘ ℂfld ) 𝑦 ) ) |
22 |
|
simpl |
⊢ ( ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ ( ℂ ∖ { 0 } ) ) → 𝑥 ∈ ℂ ) |
23 |
|
divval |
⊢ ( ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑦 ≠ 0 ) → ( 𝑥 / 𝑦 ) = ( ℩ 𝑧 ∈ ℂ ( 𝑦 · 𝑧 ) = 𝑥 ) ) |
24 |
22 10 19 23
|
syl3anc |
⊢ ( ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ ( ℂ ∖ { 0 } ) ) → ( 𝑥 / 𝑦 ) = ( ℩ 𝑧 ∈ ℂ ( 𝑦 · 𝑧 ) = 𝑥 ) ) |
25 |
21 24
|
eqtr3d |
⊢ ( ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ ( ℂ ∖ { 0 } ) ) → ( 𝑥 ( /r ‘ ℂfld ) 𝑦 ) = ( ℩ 𝑧 ∈ ℂ ( 𝑦 · 𝑧 ) = 𝑥 ) ) |
26 |
|
eqid |
⊢ ( .r ‘ ℂfld ) = ( .r ‘ ℂfld ) |
27 |
|
eqid |
⊢ ( invr ‘ ℂfld ) = ( invr ‘ ℂfld ) |
28 |
2 26 5 27 6
|
dvrval |
⊢ ( ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ ( ℂ ∖ { 0 } ) ) → ( 𝑥 ( /r ‘ ℂfld ) 𝑦 ) = ( 𝑥 ( .r ‘ ℂfld ) ( ( invr ‘ ℂfld ) ‘ 𝑦 ) ) ) |
29 |
25 28
|
eqtr3d |
⊢ ( ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ ( ℂ ∖ { 0 } ) ) → ( ℩ 𝑧 ∈ ℂ ( 𝑦 · 𝑧 ) = 𝑥 ) = ( 𝑥 ( .r ‘ ℂfld ) ( ( invr ‘ ℂfld ) ‘ 𝑦 ) ) ) |
30 |
29
|
mpoeq3ia |
⊢ ( 𝑥 ∈ ℂ , 𝑦 ∈ ( ℂ ∖ { 0 } ) ↦ ( ℩ 𝑧 ∈ ℂ ( 𝑦 · 𝑧 ) = 𝑥 ) ) = ( 𝑥 ∈ ℂ , 𝑦 ∈ ( ℂ ∖ { 0 } ) ↦ ( 𝑥 ( .r ‘ ℂfld ) ( ( invr ‘ ℂfld ) ‘ 𝑦 ) ) ) |
31 |
|
df-div |
⊢ / = ( 𝑥 ∈ ℂ , 𝑦 ∈ ( ℂ ∖ { 0 } ) ↦ ( ℩ 𝑧 ∈ ℂ ( 𝑦 · 𝑧 ) = 𝑥 ) ) |
32 |
2 26 5 27 6
|
dvrfval |
⊢ ( /r ‘ ℂfld ) = ( 𝑥 ∈ ℂ , 𝑦 ∈ ( ℂ ∖ { 0 } ) ↦ ( 𝑥 ( .r ‘ ℂfld ) ( ( invr ‘ ℂfld ) ‘ 𝑦 ) ) ) |
33 |
30 31 32
|
3eqtr4i |
⊢ / = ( /r ‘ ℂfld ) |