Step |
Hyp |
Ref |
Expression |
1 |
|
dgraaval |
⊢ ( 𝐴 ∈ 𝔸 → ( degAA ‘ 𝐴 ) = inf ( { 𝑎 ∈ ℕ ∣ ∃ 𝑝 ∈ ( ( Poly ‘ ℚ ) ∖ { 0𝑝 } ) ( ( deg ‘ 𝑝 ) = 𝑎 ∧ ( 𝑝 ‘ 𝐴 ) = 0 ) } , ℝ , < ) ) |
2 |
|
ssrab2 |
⊢ { 𝑎 ∈ ℕ ∣ ∃ 𝑝 ∈ ( ( Poly ‘ ℚ ) ∖ { 0𝑝 } ) ( ( deg ‘ 𝑝 ) = 𝑎 ∧ ( 𝑝 ‘ 𝐴 ) = 0 ) } ⊆ ℕ |
3 |
|
nnuz |
⊢ ℕ = ( ℤ≥ ‘ 1 ) |
4 |
2 3
|
sseqtri |
⊢ { 𝑎 ∈ ℕ ∣ ∃ 𝑝 ∈ ( ( Poly ‘ ℚ ) ∖ { 0𝑝 } ) ( ( deg ‘ 𝑝 ) = 𝑎 ∧ ( 𝑝 ‘ 𝐴 ) = 0 ) } ⊆ ( ℤ≥ ‘ 1 ) |
5 |
|
eldifsn |
⊢ ( 𝑏 ∈ ( ( Poly ‘ ℚ ) ∖ { 0𝑝 } ) ↔ ( 𝑏 ∈ ( Poly ‘ ℚ ) ∧ 𝑏 ≠ 0𝑝 ) ) |
6 |
5
|
biimpi |
⊢ ( 𝑏 ∈ ( ( Poly ‘ ℚ ) ∖ { 0𝑝 } ) → ( 𝑏 ∈ ( Poly ‘ ℚ ) ∧ 𝑏 ≠ 0𝑝 ) ) |
7 |
6
|
ad2antrr |
⊢ ( ( ( 𝑏 ∈ ( ( Poly ‘ ℚ ) ∖ { 0𝑝 } ) ∧ ( 𝑏 ‘ 𝐴 ) = 0 ) ∧ 𝐴 ∈ ℂ ) → ( 𝑏 ∈ ( Poly ‘ ℚ ) ∧ 𝑏 ≠ 0𝑝 ) ) |
8 |
|
simpr |
⊢ ( ( ( 𝑏 ∈ ( ( Poly ‘ ℚ ) ∖ { 0𝑝 } ) ∧ ( 𝑏 ‘ 𝐴 ) = 0 ) ∧ 𝐴 ∈ ℂ ) → 𝐴 ∈ ℂ ) |
9 |
|
simplr |
⊢ ( ( ( 𝑏 ∈ ( ( Poly ‘ ℚ ) ∖ { 0𝑝 } ) ∧ ( 𝑏 ‘ 𝐴 ) = 0 ) ∧ 𝐴 ∈ ℂ ) → ( 𝑏 ‘ 𝐴 ) = 0 ) |
10 |
|
dgrnznn |
⊢ ( ( ( 𝑏 ∈ ( Poly ‘ ℚ ) ∧ 𝑏 ≠ 0𝑝 ) ∧ ( 𝐴 ∈ ℂ ∧ ( 𝑏 ‘ 𝐴 ) = 0 ) ) → ( deg ‘ 𝑏 ) ∈ ℕ ) |
11 |
7 8 9 10
|
syl12anc |
⊢ ( ( ( 𝑏 ∈ ( ( Poly ‘ ℚ ) ∖ { 0𝑝 } ) ∧ ( 𝑏 ‘ 𝐴 ) = 0 ) ∧ 𝐴 ∈ ℂ ) → ( deg ‘ 𝑏 ) ∈ ℕ ) |
12 |
|
simpll |
⊢ ( ( ( 𝑏 ∈ ( ( Poly ‘ ℚ ) ∖ { 0𝑝 } ) ∧ ( 𝑏 ‘ 𝐴 ) = 0 ) ∧ 𝐴 ∈ ℂ ) → 𝑏 ∈ ( ( Poly ‘ ℚ ) ∖ { 0𝑝 } ) ) |
13 |
|
eqid |
⊢ ( deg ‘ 𝑏 ) = ( deg ‘ 𝑏 ) |
14 |
9 13
|
jctil |
⊢ ( ( ( 𝑏 ∈ ( ( Poly ‘ ℚ ) ∖ { 0𝑝 } ) ∧ ( 𝑏 ‘ 𝐴 ) = 0 ) ∧ 𝐴 ∈ ℂ ) → ( ( deg ‘ 𝑏 ) = ( deg ‘ 𝑏 ) ∧ ( 𝑏 ‘ 𝐴 ) = 0 ) ) |
15 |
|
eqeq2 |
⊢ ( 𝑎 = ( deg ‘ 𝑏 ) → ( ( deg ‘ 𝑝 ) = 𝑎 ↔ ( deg ‘ 𝑝 ) = ( deg ‘ 𝑏 ) ) ) |
16 |
15
|
anbi1d |
⊢ ( 𝑎 = ( deg ‘ 𝑏 ) → ( ( ( deg ‘ 𝑝 ) = 𝑎 ∧ ( 𝑝 ‘ 𝐴 ) = 0 ) ↔ ( ( deg ‘ 𝑝 ) = ( deg ‘ 𝑏 ) ∧ ( 𝑝 ‘ 𝐴 ) = 0 ) ) ) |
17 |
|
fveqeq2 |
⊢ ( 𝑝 = 𝑏 → ( ( deg ‘ 𝑝 ) = ( deg ‘ 𝑏 ) ↔ ( deg ‘ 𝑏 ) = ( deg ‘ 𝑏 ) ) ) |
18 |
|
fveq1 |
⊢ ( 𝑝 = 𝑏 → ( 𝑝 ‘ 𝐴 ) = ( 𝑏 ‘ 𝐴 ) ) |
19 |
18
|
eqeq1d |
⊢ ( 𝑝 = 𝑏 → ( ( 𝑝 ‘ 𝐴 ) = 0 ↔ ( 𝑏 ‘ 𝐴 ) = 0 ) ) |
20 |
17 19
|
anbi12d |
⊢ ( 𝑝 = 𝑏 → ( ( ( deg ‘ 𝑝 ) = ( deg ‘ 𝑏 ) ∧ ( 𝑝 ‘ 𝐴 ) = 0 ) ↔ ( ( deg ‘ 𝑏 ) = ( deg ‘ 𝑏 ) ∧ ( 𝑏 ‘ 𝐴 ) = 0 ) ) ) |
21 |
16 20
|
rspc2ev |
⊢ ( ( ( deg ‘ 𝑏 ) ∈ ℕ ∧ 𝑏 ∈ ( ( Poly ‘ ℚ ) ∖ { 0𝑝 } ) ∧ ( ( deg ‘ 𝑏 ) = ( deg ‘ 𝑏 ) ∧ ( 𝑏 ‘ 𝐴 ) = 0 ) ) → ∃ 𝑎 ∈ ℕ ∃ 𝑝 ∈ ( ( Poly ‘ ℚ ) ∖ { 0𝑝 } ) ( ( deg ‘ 𝑝 ) = 𝑎 ∧ ( 𝑝 ‘ 𝐴 ) = 0 ) ) |
22 |
11 12 14 21
|
syl3anc |
⊢ ( ( ( 𝑏 ∈ ( ( Poly ‘ ℚ ) ∖ { 0𝑝 } ) ∧ ( 𝑏 ‘ 𝐴 ) = 0 ) ∧ 𝐴 ∈ ℂ ) → ∃ 𝑎 ∈ ℕ ∃ 𝑝 ∈ ( ( Poly ‘ ℚ ) ∖ { 0𝑝 } ) ( ( deg ‘ 𝑝 ) = 𝑎 ∧ ( 𝑝 ‘ 𝐴 ) = 0 ) ) |
23 |
22
|
ex |
⊢ ( ( 𝑏 ∈ ( ( Poly ‘ ℚ ) ∖ { 0𝑝 } ) ∧ ( 𝑏 ‘ 𝐴 ) = 0 ) → ( 𝐴 ∈ ℂ → ∃ 𝑎 ∈ ℕ ∃ 𝑝 ∈ ( ( Poly ‘ ℚ ) ∖ { 0𝑝 } ) ( ( deg ‘ 𝑝 ) = 𝑎 ∧ ( 𝑝 ‘ 𝐴 ) = 0 ) ) ) |
24 |
23
|
rexlimiva |
⊢ ( ∃ 𝑏 ∈ ( ( Poly ‘ ℚ ) ∖ { 0𝑝 } ) ( 𝑏 ‘ 𝐴 ) = 0 → ( 𝐴 ∈ ℂ → ∃ 𝑎 ∈ ℕ ∃ 𝑝 ∈ ( ( Poly ‘ ℚ ) ∖ { 0𝑝 } ) ( ( deg ‘ 𝑝 ) = 𝑎 ∧ ( 𝑝 ‘ 𝐴 ) = 0 ) ) ) |
25 |
24
|
impcom |
⊢ ( ( 𝐴 ∈ ℂ ∧ ∃ 𝑏 ∈ ( ( Poly ‘ ℚ ) ∖ { 0𝑝 } ) ( 𝑏 ‘ 𝐴 ) = 0 ) → ∃ 𝑎 ∈ ℕ ∃ 𝑝 ∈ ( ( Poly ‘ ℚ ) ∖ { 0𝑝 } ) ( ( deg ‘ 𝑝 ) = 𝑎 ∧ ( 𝑝 ‘ 𝐴 ) = 0 ) ) |
26 |
|
elqaa |
⊢ ( 𝐴 ∈ 𝔸 ↔ ( 𝐴 ∈ ℂ ∧ ∃ 𝑏 ∈ ( ( Poly ‘ ℚ ) ∖ { 0𝑝 } ) ( 𝑏 ‘ 𝐴 ) = 0 ) ) |
27 |
|
rabn0 |
⊢ ( { 𝑎 ∈ ℕ ∣ ∃ 𝑝 ∈ ( ( Poly ‘ ℚ ) ∖ { 0𝑝 } ) ( ( deg ‘ 𝑝 ) = 𝑎 ∧ ( 𝑝 ‘ 𝐴 ) = 0 ) } ≠ ∅ ↔ ∃ 𝑎 ∈ ℕ ∃ 𝑝 ∈ ( ( Poly ‘ ℚ ) ∖ { 0𝑝 } ) ( ( deg ‘ 𝑝 ) = 𝑎 ∧ ( 𝑝 ‘ 𝐴 ) = 0 ) ) |
28 |
25 26 27
|
3imtr4i |
⊢ ( 𝐴 ∈ 𝔸 → { 𝑎 ∈ ℕ ∣ ∃ 𝑝 ∈ ( ( Poly ‘ ℚ ) ∖ { 0𝑝 } ) ( ( deg ‘ 𝑝 ) = 𝑎 ∧ ( 𝑝 ‘ 𝐴 ) = 0 ) } ≠ ∅ ) |
29 |
|
infssuzcl |
⊢ ( ( { 𝑎 ∈ ℕ ∣ ∃ 𝑝 ∈ ( ( Poly ‘ ℚ ) ∖ { 0𝑝 } ) ( ( deg ‘ 𝑝 ) = 𝑎 ∧ ( 𝑝 ‘ 𝐴 ) = 0 ) } ⊆ ( ℤ≥ ‘ 1 ) ∧ { 𝑎 ∈ ℕ ∣ ∃ 𝑝 ∈ ( ( Poly ‘ ℚ ) ∖ { 0𝑝 } ) ( ( deg ‘ 𝑝 ) = 𝑎 ∧ ( 𝑝 ‘ 𝐴 ) = 0 ) } ≠ ∅ ) → inf ( { 𝑎 ∈ ℕ ∣ ∃ 𝑝 ∈ ( ( Poly ‘ ℚ ) ∖ { 0𝑝 } ) ( ( deg ‘ 𝑝 ) = 𝑎 ∧ ( 𝑝 ‘ 𝐴 ) = 0 ) } , ℝ , < ) ∈ { 𝑎 ∈ ℕ ∣ ∃ 𝑝 ∈ ( ( Poly ‘ ℚ ) ∖ { 0𝑝 } ) ( ( deg ‘ 𝑝 ) = 𝑎 ∧ ( 𝑝 ‘ 𝐴 ) = 0 ) } ) |
30 |
4 28 29
|
sylancr |
⊢ ( 𝐴 ∈ 𝔸 → inf ( { 𝑎 ∈ ℕ ∣ ∃ 𝑝 ∈ ( ( Poly ‘ ℚ ) ∖ { 0𝑝 } ) ( ( deg ‘ 𝑝 ) = 𝑎 ∧ ( 𝑝 ‘ 𝐴 ) = 0 ) } , ℝ , < ) ∈ { 𝑎 ∈ ℕ ∣ ∃ 𝑝 ∈ ( ( Poly ‘ ℚ ) ∖ { 0𝑝 } ) ( ( deg ‘ 𝑝 ) = 𝑎 ∧ ( 𝑝 ‘ 𝐴 ) = 0 ) } ) |
31 |
1 30
|
eqeltrd |
⊢ ( 𝐴 ∈ 𝔸 → ( degAA ‘ 𝐴 ) ∈ { 𝑎 ∈ ℕ ∣ ∃ 𝑝 ∈ ( ( Poly ‘ ℚ ) ∖ { 0𝑝 } ) ( ( deg ‘ 𝑝 ) = 𝑎 ∧ ( 𝑝 ‘ 𝐴 ) = 0 ) } ) |
32 |
|
eqeq2 |
⊢ ( 𝑎 = ( degAA ‘ 𝐴 ) → ( ( deg ‘ 𝑝 ) = 𝑎 ↔ ( deg ‘ 𝑝 ) = ( degAA ‘ 𝐴 ) ) ) |
33 |
32
|
anbi1d |
⊢ ( 𝑎 = ( degAA ‘ 𝐴 ) → ( ( ( deg ‘ 𝑝 ) = 𝑎 ∧ ( 𝑝 ‘ 𝐴 ) = 0 ) ↔ ( ( deg ‘ 𝑝 ) = ( degAA ‘ 𝐴 ) ∧ ( 𝑝 ‘ 𝐴 ) = 0 ) ) ) |
34 |
33
|
rexbidv |
⊢ ( 𝑎 = ( degAA ‘ 𝐴 ) → ( ∃ 𝑝 ∈ ( ( Poly ‘ ℚ ) ∖ { 0𝑝 } ) ( ( deg ‘ 𝑝 ) = 𝑎 ∧ ( 𝑝 ‘ 𝐴 ) = 0 ) ↔ ∃ 𝑝 ∈ ( ( Poly ‘ ℚ ) ∖ { 0𝑝 } ) ( ( deg ‘ 𝑝 ) = ( degAA ‘ 𝐴 ) ∧ ( 𝑝 ‘ 𝐴 ) = 0 ) ) ) |
35 |
34
|
elrab |
⊢ ( ( degAA ‘ 𝐴 ) ∈ { 𝑎 ∈ ℕ ∣ ∃ 𝑝 ∈ ( ( Poly ‘ ℚ ) ∖ { 0𝑝 } ) ( ( deg ‘ 𝑝 ) = 𝑎 ∧ ( 𝑝 ‘ 𝐴 ) = 0 ) } ↔ ( ( degAA ‘ 𝐴 ) ∈ ℕ ∧ ∃ 𝑝 ∈ ( ( Poly ‘ ℚ ) ∖ { 0𝑝 } ) ( ( deg ‘ 𝑝 ) = ( degAA ‘ 𝐴 ) ∧ ( 𝑝 ‘ 𝐴 ) = 0 ) ) ) |
36 |
31 35
|
sylib |
⊢ ( 𝐴 ∈ 𝔸 → ( ( degAA ‘ 𝐴 ) ∈ ℕ ∧ ∃ 𝑝 ∈ ( ( Poly ‘ ℚ ) ∖ { 0𝑝 } ) ( ( deg ‘ 𝑝 ) = ( degAA ‘ 𝐴 ) ∧ ( 𝑝 ‘ 𝐴 ) = 0 ) ) ) |