Step |
Hyp |
Ref |
Expression |
1 |
|
qcn |
⊢ ( 𝐴 ∈ ℚ → 𝐴 ∈ ℂ ) |
2 |
|
qsscn |
⊢ ℚ ⊆ ℂ |
3 |
|
1z |
⊢ 1 ∈ ℤ |
4 |
|
zq |
⊢ ( 1 ∈ ℤ → 1 ∈ ℚ ) |
5 |
3 4
|
ax-mp |
⊢ 1 ∈ ℚ |
6 |
|
plyid |
⊢ ( ( ℚ ⊆ ℂ ∧ 1 ∈ ℚ ) → Xp ∈ ( Poly ‘ ℚ ) ) |
7 |
2 5 6
|
mp2an |
⊢ Xp ∈ ( Poly ‘ ℚ ) |
8 |
7
|
a1i |
⊢ ( 𝐴 ∈ ℚ → Xp ∈ ( Poly ‘ ℚ ) ) |
9 |
|
plyconst |
⊢ ( ( ℚ ⊆ ℂ ∧ 𝐴 ∈ ℚ ) → ( ℂ × { 𝐴 } ) ∈ ( Poly ‘ ℚ ) ) |
10 |
2 9
|
mpan |
⊢ ( 𝐴 ∈ ℚ → ( ℂ × { 𝐴 } ) ∈ ( Poly ‘ ℚ ) ) |
11 |
|
qaddcl |
⊢ ( ( 𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ ) → ( 𝑥 + 𝑦 ) ∈ ℚ ) |
12 |
11
|
adantl |
⊢ ( ( 𝐴 ∈ ℚ ∧ ( 𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ ) ) → ( 𝑥 + 𝑦 ) ∈ ℚ ) |
13 |
|
qmulcl |
⊢ ( ( 𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ ) → ( 𝑥 · 𝑦 ) ∈ ℚ ) |
14 |
13
|
adantl |
⊢ ( ( 𝐴 ∈ ℚ ∧ ( 𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ ) ) → ( 𝑥 · 𝑦 ) ∈ ℚ ) |
15 |
|
qnegcl |
⊢ ( 1 ∈ ℚ → - 1 ∈ ℚ ) |
16 |
5 15
|
ax-mp |
⊢ - 1 ∈ ℚ |
17 |
16
|
a1i |
⊢ ( 𝐴 ∈ ℚ → - 1 ∈ ℚ ) |
18 |
8 10 12 14 17
|
plysub |
⊢ ( 𝐴 ∈ ℚ → ( Xp ∘f − ( ℂ × { 𝐴 } ) ) ∈ ( Poly ‘ ℚ ) ) |
19 |
|
peano2cn |
⊢ ( 𝐴 ∈ ℂ → ( 𝐴 + 1 ) ∈ ℂ ) |
20 |
1 19
|
syl |
⊢ ( 𝐴 ∈ ℚ → ( 𝐴 + 1 ) ∈ ℂ ) |
21 |
|
fnresi |
⊢ ( I ↾ ℂ ) Fn ℂ |
22 |
|
df-idp |
⊢ Xp = ( I ↾ ℂ ) |
23 |
22
|
fneq1i |
⊢ ( Xp Fn ℂ ↔ ( I ↾ ℂ ) Fn ℂ ) |
24 |
21 23
|
mpbir |
⊢ Xp Fn ℂ |
25 |
24
|
a1i |
⊢ ( 𝐴 ∈ ℚ → Xp Fn ℂ ) |
26 |
|
fnconstg |
⊢ ( 𝐴 ∈ ℚ → ( ℂ × { 𝐴 } ) Fn ℂ ) |
27 |
|
cnex |
⊢ ℂ ∈ V |
28 |
27
|
a1i |
⊢ ( 𝐴 ∈ ℚ → ℂ ∈ V ) |
29 |
|
inidm |
⊢ ( ℂ ∩ ℂ ) = ℂ |
30 |
22
|
fveq1i |
⊢ ( Xp ‘ ( 𝐴 + 1 ) ) = ( ( I ↾ ℂ ) ‘ ( 𝐴 + 1 ) ) |
31 |
|
fvresi |
⊢ ( ( 𝐴 + 1 ) ∈ ℂ → ( ( I ↾ ℂ ) ‘ ( 𝐴 + 1 ) ) = ( 𝐴 + 1 ) ) |
32 |
30 31
|
syl5eq |
⊢ ( ( 𝐴 + 1 ) ∈ ℂ → ( Xp ‘ ( 𝐴 + 1 ) ) = ( 𝐴 + 1 ) ) |
33 |
32
|
adantl |
⊢ ( ( 𝐴 ∈ ℚ ∧ ( 𝐴 + 1 ) ∈ ℂ ) → ( Xp ‘ ( 𝐴 + 1 ) ) = ( 𝐴 + 1 ) ) |
34 |
|
fvconst2g |
⊢ ( ( 𝐴 ∈ ℚ ∧ ( 𝐴 + 1 ) ∈ ℂ ) → ( ( ℂ × { 𝐴 } ) ‘ ( 𝐴 + 1 ) ) = 𝐴 ) |
35 |
25 26 28 28 29 33 34
|
ofval |
⊢ ( ( 𝐴 ∈ ℚ ∧ ( 𝐴 + 1 ) ∈ ℂ ) → ( ( Xp ∘f − ( ℂ × { 𝐴 } ) ) ‘ ( 𝐴 + 1 ) ) = ( ( 𝐴 + 1 ) − 𝐴 ) ) |
36 |
20 35
|
mpdan |
⊢ ( 𝐴 ∈ ℚ → ( ( Xp ∘f − ( ℂ × { 𝐴 } ) ) ‘ ( 𝐴 + 1 ) ) = ( ( 𝐴 + 1 ) − 𝐴 ) ) |
37 |
|
ax-1cn |
⊢ 1 ∈ ℂ |
38 |
|
pncan2 |
⊢ ( ( 𝐴 ∈ ℂ ∧ 1 ∈ ℂ ) → ( ( 𝐴 + 1 ) − 𝐴 ) = 1 ) |
39 |
1 37 38
|
sylancl |
⊢ ( 𝐴 ∈ ℚ → ( ( 𝐴 + 1 ) − 𝐴 ) = 1 ) |
40 |
36 39
|
eqtrd |
⊢ ( 𝐴 ∈ ℚ → ( ( Xp ∘f − ( ℂ × { 𝐴 } ) ) ‘ ( 𝐴 + 1 ) ) = 1 ) |
41 |
|
ax-1ne0 |
⊢ 1 ≠ 0 |
42 |
41
|
a1i |
⊢ ( 𝐴 ∈ ℚ → 1 ≠ 0 ) |
43 |
40 42
|
eqnetrd |
⊢ ( 𝐴 ∈ ℚ → ( ( Xp ∘f − ( ℂ × { 𝐴 } ) ) ‘ ( 𝐴 + 1 ) ) ≠ 0 ) |
44 |
|
ne0p |
⊢ ( ( ( 𝐴 + 1 ) ∈ ℂ ∧ ( ( Xp ∘f − ( ℂ × { 𝐴 } ) ) ‘ ( 𝐴 + 1 ) ) ≠ 0 ) → ( Xp ∘f − ( ℂ × { 𝐴 } ) ) ≠ 0𝑝 ) |
45 |
20 43 44
|
syl2anc |
⊢ ( 𝐴 ∈ ℚ → ( Xp ∘f − ( ℂ × { 𝐴 } ) ) ≠ 0𝑝 ) |
46 |
|
eldifsn |
⊢ ( ( Xp ∘f − ( ℂ × { 𝐴 } ) ) ∈ ( ( Poly ‘ ℚ ) ∖ { 0𝑝 } ) ↔ ( ( Xp ∘f − ( ℂ × { 𝐴 } ) ) ∈ ( Poly ‘ ℚ ) ∧ ( Xp ∘f − ( ℂ × { 𝐴 } ) ) ≠ 0𝑝 ) ) |
47 |
18 45 46
|
sylanbrc |
⊢ ( 𝐴 ∈ ℚ → ( Xp ∘f − ( ℂ × { 𝐴 } ) ) ∈ ( ( Poly ‘ ℚ ) ∖ { 0𝑝 } ) ) |
48 |
22
|
fveq1i |
⊢ ( Xp ‘ 𝐴 ) = ( ( I ↾ ℂ ) ‘ 𝐴 ) |
49 |
|
fvresi |
⊢ ( 𝐴 ∈ ℂ → ( ( I ↾ ℂ ) ‘ 𝐴 ) = 𝐴 ) |
50 |
48 49
|
syl5eq |
⊢ ( 𝐴 ∈ ℂ → ( Xp ‘ 𝐴 ) = 𝐴 ) |
51 |
50
|
adantl |
⊢ ( ( 𝐴 ∈ ℚ ∧ 𝐴 ∈ ℂ ) → ( Xp ‘ 𝐴 ) = 𝐴 ) |
52 |
|
fvconst2g |
⊢ ( ( 𝐴 ∈ ℚ ∧ 𝐴 ∈ ℂ ) → ( ( ℂ × { 𝐴 } ) ‘ 𝐴 ) = 𝐴 ) |
53 |
25 26 28 28 29 51 52
|
ofval |
⊢ ( ( 𝐴 ∈ ℚ ∧ 𝐴 ∈ ℂ ) → ( ( Xp ∘f − ( ℂ × { 𝐴 } ) ) ‘ 𝐴 ) = ( 𝐴 − 𝐴 ) ) |
54 |
1 53
|
mpdan |
⊢ ( 𝐴 ∈ ℚ → ( ( Xp ∘f − ( ℂ × { 𝐴 } ) ) ‘ 𝐴 ) = ( 𝐴 − 𝐴 ) ) |
55 |
1
|
subidd |
⊢ ( 𝐴 ∈ ℚ → ( 𝐴 − 𝐴 ) = 0 ) |
56 |
54 55
|
eqtrd |
⊢ ( 𝐴 ∈ ℚ → ( ( Xp ∘f − ( ℂ × { 𝐴 } ) ) ‘ 𝐴 ) = 0 ) |
57 |
|
fveq1 |
⊢ ( 𝑓 = ( Xp ∘f − ( ℂ × { 𝐴 } ) ) → ( 𝑓 ‘ 𝐴 ) = ( ( Xp ∘f − ( ℂ × { 𝐴 } ) ) ‘ 𝐴 ) ) |
58 |
57
|
eqeq1d |
⊢ ( 𝑓 = ( Xp ∘f − ( ℂ × { 𝐴 } ) ) → ( ( 𝑓 ‘ 𝐴 ) = 0 ↔ ( ( Xp ∘f − ( ℂ × { 𝐴 } ) ) ‘ 𝐴 ) = 0 ) ) |
59 |
58
|
rspcev |
⊢ ( ( ( Xp ∘f − ( ℂ × { 𝐴 } ) ) ∈ ( ( Poly ‘ ℚ ) ∖ { 0𝑝 } ) ∧ ( ( Xp ∘f − ( ℂ × { 𝐴 } ) ) ‘ 𝐴 ) = 0 ) → ∃ 𝑓 ∈ ( ( Poly ‘ ℚ ) ∖ { 0𝑝 } ) ( 𝑓 ‘ 𝐴 ) = 0 ) |
60 |
47 56 59
|
syl2anc |
⊢ ( 𝐴 ∈ ℚ → ∃ 𝑓 ∈ ( ( Poly ‘ ℚ ) ∖ { 0𝑝 } ) ( 𝑓 ‘ 𝐴 ) = 0 ) |
61 |
|
elqaa |
⊢ ( 𝐴 ∈ 𝔸 ↔ ( 𝐴 ∈ ℂ ∧ ∃ 𝑓 ∈ ( ( Poly ‘ ℚ ) ∖ { 0𝑝 } ) ( 𝑓 ‘ 𝐴 ) = 0 ) ) |
62 |
1 60 61
|
sylanbrc |
⊢ ( 𝐴 ∈ ℚ → 𝐴 ∈ 𝔸 ) |