Step |
Hyp |
Ref |
Expression |
1 |
|
2sq.1 |
⊢ 𝑆 = ran ( 𝑤 ∈ ℤ[i] ↦ ( ( abs ‘ 𝑤 ) ↑ 2 ) ) |
2 |
|
2sqlem7.2 |
⊢ 𝑌 = { 𝑧 ∣ ∃ 𝑥 ∈ ℤ ∃ 𝑦 ∈ ℤ ( ( 𝑥 gcd 𝑦 ) = 1 ∧ 𝑧 = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ) } |
3 |
|
breq1 |
⊢ ( 𝑏 = 𝐵 → ( 𝑏 ∥ 𝑎 ↔ 𝐵 ∥ 𝑎 ) ) |
4 |
|
eleq1 |
⊢ ( 𝑏 = 𝐵 → ( 𝑏 ∈ 𝑆 ↔ 𝐵 ∈ 𝑆 ) ) |
5 |
3 4
|
imbi12d |
⊢ ( 𝑏 = 𝐵 → ( ( 𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆 ) ↔ ( 𝐵 ∥ 𝑎 → 𝐵 ∈ 𝑆 ) ) ) |
6 |
5
|
ralbidv |
⊢ ( 𝑏 = 𝐵 → ( ∀ 𝑎 ∈ 𝑌 ( 𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆 ) ↔ ∀ 𝑎 ∈ 𝑌 ( 𝐵 ∥ 𝑎 → 𝐵 ∈ 𝑆 ) ) ) |
7 |
|
oveq2 |
⊢ ( 𝑚 = 1 → ( 1 ... 𝑚 ) = ( 1 ... 1 ) ) |
8 |
7
|
raleqdv |
⊢ ( 𝑚 = 1 → ( ∀ 𝑏 ∈ ( 1 ... 𝑚 ) ∀ 𝑎 ∈ 𝑌 ( 𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆 ) ↔ ∀ 𝑏 ∈ ( 1 ... 1 ) ∀ 𝑎 ∈ 𝑌 ( 𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆 ) ) ) |
9 |
|
oveq2 |
⊢ ( 𝑚 = 𝑛 → ( 1 ... 𝑚 ) = ( 1 ... 𝑛 ) ) |
10 |
9
|
raleqdv |
⊢ ( 𝑚 = 𝑛 → ( ∀ 𝑏 ∈ ( 1 ... 𝑚 ) ∀ 𝑎 ∈ 𝑌 ( 𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆 ) ↔ ∀ 𝑏 ∈ ( 1 ... 𝑛 ) ∀ 𝑎 ∈ 𝑌 ( 𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆 ) ) ) |
11 |
|
oveq2 |
⊢ ( 𝑚 = ( 𝑛 + 1 ) → ( 1 ... 𝑚 ) = ( 1 ... ( 𝑛 + 1 ) ) ) |
12 |
11
|
raleqdv |
⊢ ( 𝑚 = ( 𝑛 + 1 ) → ( ∀ 𝑏 ∈ ( 1 ... 𝑚 ) ∀ 𝑎 ∈ 𝑌 ( 𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆 ) ↔ ∀ 𝑏 ∈ ( 1 ... ( 𝑛 + 1 ) ) ∀ 𝑎 ∈ 𝑌 ( 𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆 ) ) ) |
13 |
|
oveq2 |
⊢ ( 𝑚 = 𝐵 → ( 1 ... 𝑚 ) = ( 1 ... 𝐵 ) ) |
14 |
13
|
raleqdv |
⊢ ( 𝑚 = 𝐵 → ( ∀ 𝑏 ∈ ( 1 ... 𝑚 ) ∀ 𝑎 ∈ 𝑌 ( 𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆 ) ↔ ∀ 𝑏 ∈ ( 1 ... 𝐵 ) ∀ 𝑎 ∈ 𝑌 ( 𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆 ) ) ) |
15 |
|
elfz1eq |
⊢ ( 𝑏 ∈ ( 1 ... 1 ) → 𝑏 = 1 ) |
16 |
|
1z |
⊢ 1 ∈ ℤ |
17 |
|
zgz |
⊢ ( 1 ∈ ℤ → 1 ∈ ℤ[i] ) |
18 |
16 17
|
ax-mp |
⊢ 1 ∈ ℤ[i] |
19 |
|
sq1 |
⊢ ( 1 ↑ 2 ) = 1 |
20 |
19
|
eqcomi |
⊢ 1 = ( 1 ↑ 2 ) |
21 |
|
fveq2 |
⊢ ( 𝑥 = 1 → ( abs ‘ 𝑥 ) = ( abs ‘ 1 ) ) |
22 |
|
abs1 |
⊢ ( abs ‘ 1 ) = 1 |
23 |
21 22
|
eqtrdi |
⊢ ( 𝑥 = 1 → ( abs ‘ 𝑥 ) = 1 ) |
24 |
23
|
oveq1d |
⊢ ( 𝑥 = 1 → ( ( abs ‘ 𝑥 ) ↑ 2 ) = ( 1 ↑ 2 ) ) |
25 |
24
|
rspceeqv |
⊢ ( ( 1 ∈ ℤ[i] ∧ 1 = ( 1 ↑ 2 ) ) → ∃ 𝑥 ∈ ℤ[i] 1 = ( ( abs ‘ 𝑥 ) ↑ 2 ) ) |
26 |
18 20 25
|
mp2an |
⊢ ∃ 𝑥 ∈ ℤ[i] 1 = ( ( abs ‘ 𝑥 ) ↑ 2 ) |
27 |
1
|
2sqlem1 |
⊢ ( 1 ∈ 𝑆 ↔ ∃ 𝑥 ∈ ℤ[i] 1 = ( ( abs ‘ 𝑥 ) ↑ 2 ) ) |
28 |
26 27
|
mpbir |
⊢ 1 ∈ 𝑆 |
29 |
15 28
|
eqeltrdi |
⊢ ( 𝑏 ∈ ( 1 ... 1 ) → 𝑏 ∈ 𝑆 ) |
30 |
29
|
a1d |
⊢ ( 𝑏 ∈ ( 1 ... 1 ) → ( 𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆 ) ) |
31 |
30
|
ralrimivw |
⊢ ( 𝑏 ∈ ( 1 ... 1 ) → ∀ 𝑎 ∈ 𝑌 ( 𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆 ) ) |
32 |
31
|
rgen |
⊢ ∀ 𝑏 ∈ ( 1 ... 1 ) ∀ 𝑎 ∈ 𝑌 ( 𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆 ) |
33 |
|
simplr |
⊢ ( ( ( 𝑛 ∈ ℕ ∧ ∀ 𝑏 ∈ ( 1 ... 𝑛 ) ∀ 𝑎 ∈ 𝑌 ( 𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆 ) ) ∧ ( 𝑚 ∈ 𝑌 ∧ ( 𝑛 + 1 ) ∥ 𝑚 ) ) → ∀ 𝑏 ∈ ( 1 ... 𝑛 ) ∀ 𝑎 ∈ 𝑌 ( 𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆 ) ) |
34 |
|
nncn |
⊢ ( 𝑛 ∈ ℕ → 𝑛 ∈ ℂ ) |
35 |
34
|
ad2antrr |
⊢ ( ( ( 𝑛 ∈ ℕ ∧ ∀ 𝑏 ∈ ( 1 ... 𝑛 ) ∀ 𝑎 ∈ 𝑌 ( 𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆 ) ) ∧ ( 𝑚 ∈ 𝑌 ∧ ( 𝑛 + 1 ) ∥ 𝑚 ) ) → 𝑛 ∈ ℂ ) |
36 |
|
ax-1cn |
⊢ 1 ∈ ℂ |
37 |
|
pncan |
⊢ ( ( 𝑛 ∈ ℂ ∧ 1 ∈ ℂ ) → ( ( 𝑛 + 1 ) − 1 ) = 𝑛 ) |
38 |
35 36 37
|
sylancl |
⊢ ( ( ( 𝑛 ∈ ℕ ∧ ∀ 𝑏 ∈ ( 1 ... 𝑛 ) ∀ 𝑎 ∈ 𝑌 ( 𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆 ) ) ∧ ( 𝑚 ∈ 𝑌 ∧ ( 𝑛 + 1 ) ∥ 𝑚 ) ) → ( ( 𝑛 + 1 ) − 1 ) = 𝑛 ) |
39 |
38
|
oveq2d |
⊢ ( ( ( 𝑛 ∈ ℕ ∧ ∀ 𝑏 ∈ ( 1 ... 𝑛 ) ∀ 𝑎 ∈ 𝑌 ( 𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆 ) ) ∧ ( 𝑚 ∈ 𝑌 ∧ ( 𝑛 + 1 ) ∥ 𝑚 ) ) → ( 1 ... ( ( 𝑛 + 1 ) − 1 ) ) = ( 1 ... 𝑛 ) ) |
40 |
39
|
raleqdv |
⊢ ( ( ( 𝑛 ∈ ℕ ∧ ∀ 𝑏 ∈ ( 1 ... 𝑛 ) ∀ 𝑎 ∈ 𝑌 ( 𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆 ) ) ∧ ( 𝑚 ∈ 𝑌 ∧ ( 𝑛 + 1 ) ∥ 𝑚 ) ) → ( ∀ 𝑏 ∈ ( 1 ... ( ( 𝑛 + 1 ) − 1 ) ) ∀ 𝑎 ∈ 𝑌 ( 𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆 ) ↔ ∀ 𝑏 ∈ ( 1 ... 𝑛 ) ∀ 𝑎 ∈ 𝑌 ( 𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆 ) ) ) |
41 |
33 40
|
mpbird |
⊢ ( ( ( 𝑛 ∈ ℕ ∧ ∀ 𝑏 ∈ ( 1 ... 𝑛 ) ∀ 𝑎 ∈ 𝑌 ( 𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆 ) ) ∧ ( 𝑚 ∈ 𝑌 ∧ ( 𝑛 + 1 ) ∥ 𝑚 ) ) → ∀ 𝑏 ∈ ( 1 ... ( ( 𝑛 + 1 ) − 1 ) ) ∀ 𝑎 ∈ 𝑌 ( 𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆 ) ) |
42 |
|
simprr |
⊢ ( ( ( 𝑛 ∈ ℕ ∧ ∀ 𝑏 ∈ ( 1 ... 𝑛 ) ∀ 𝑎 ∈ 𝑌 ( 𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆 ) ) ∧ ( 𝑚 ∈ 𝑌 ∧ ( 𝑛 + 1 ) ∥ 𝑚 ) ) → ( 𝑛 + 1 ) ∥ 𝑚 ) |
43 |
|
peano2nn |
⊢ ( 𝑛 ∈ ℕ → ( 𝑛 + 1 ) ∈ ℕ ) |
44 |
43
|
ad2antrr |
⊢ ( ( ( 𝑛 ∈ ℕ ∧ ∀ 𝑏 ∈ ( 1 ... 𝑛 ) ∀ 𝑎 ∈ 𝑌 ( 𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆 ) ) ∧ ( 𝑚 ∈ 𝑌 ∧ ( 𝑛 + 1 ) ∥ 𝑚 ) ) → ( 𝑛 + 1 ) ∈ ℕ ) |
45 |
|
simprl |
⊢ ( ( ( 𝑛 ∈ ℕ ∧ ∀ 𝑏 ∈ ( 1 ... 𝑛 ) ∀ 𝑎 ∈ 𝑌 ( 𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆 ) ) ∧ ( 𝑚 ∈ 𝑌 ∧ ( 𝑛 + 1 ) ∥ 𝑚 ) ) → 𝑚 ∈ 𝑌 ) |
46 |
1 2 41 42 44 45
|
2sqlem9 |
⊢ ( ( ( 𝑛 ∈ ℕ ∧ ∀ 𝑏 ∈ ( 1 ... 𝑛 ) ∀ 𝑎 ∈ 𝑌 ( 𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆 ) ) ∧ ( 𝑚 ∈ 𝑌 ∧ ( 𝑛 + 1 ) ∥ 𝑚 ) ) → ( 𝑛 + 1 ) ∈ 𝑆 ) |
47 |
46
|
expr |
⊢ ( ( ( 𝑛 ∈ ℕ ∧ ∀ 𝑏 ∈ ( 1 ... 𝑛 ) ∀ 𝑎 ∈ 𝑌 ( 𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆 ) ) ∧ 𝑚 ∈ 𝑌 ) → ( ( 𝑛 + 1 ) ∥ 𝑚 → ( 𝑛 + 1 ) ∈ 𝑆 ) ) |
48 |
47
|
ralrimiva |
⊢ ( ( 𝑛 ∈ ℕ ∧ ∀ 𝑏 ∈ ( 1 ... 𝑛 ) ∀ 𝑎 ∈ 𝑌 ( 𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆 ) ) → ∀ 𝑚 ∈ 𝑌 ( ( 𝑛 + 1 ) ∥ 𝑚 → ( 𝑛 + 1 ) ∈ 𝑆 ) ) |
49 |
48
|
ex |
⊢ ( 𝑛 ∈ ℕ → ( ∀ 𝑏 ∈ ( 1 ... 𝑛 ) ∀ 𝑎 ∈ 𝑌 ( 𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆 ) → ∀ 𝑚 ∈ 𝑌 ( ( 𝑛 + 1 ) ∥ 𝑚 → ( 𝑛 + 1 ) ∈ 𝑆 ) ) ) |
50 |
|
breq2 |
⊢ ( 𝑎 = 𝑚 → ( ( 𝑛 + 1 ) ∥ 𝑎 ↔ ( 𝑛 + 1 ) ∥ 𝑚 ) ) |
51 |
50
|
imbi1d |
⊢ ( 𝑎 = 𝑚 → ( ( ( 𝑛 + 1 ) ∥ 𝑎 → ( 𝑛 + 1 ) ∈ 𝑆 ) ↔ ( ( 𝑛 + 1 ) ∥ 𝑚 → ( 𝑛 + 1 ) ∈ 𝑆 ) ) ) |
52 |
51
|
cbvralvw |
⊢ ( ∀ 𝑎 ∈ 𝑌 ( ( 𝑛 + 1 ) ∥ 𝑎 → ( 𝑛 + 1 ) ∈ 𝑆 ) ↔ ∀ 𝑚 ∈ 𝑌 ( ( 𝑛 + 1 ) ∥ 𝑚 → ( 𝑛 + 1 ) ∈ 𝑆 ) ) |
53 |
49 52
|
syl6ibr |
⊢ ( 𝑛 ∈ ℕ → ( ∀ 𝑏 ∈ ( 1 ... 𝑛 ) ∀ 𝑎 ∈ 𝑌 ( 𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆 ) → ∀ 𝑎 ∈ 𝑌 ( ( 𝑛 + 1 ) ∥ 𝑎 → ( 𝑛 + 1 ) ∈ 𝑆 ) ) ) |
54 |
|
ovex |
⊢ ( 𝑛 + 1 ) ∈ V |
55 |
|
breq1 |
⊢ ( 𝑏 = ( 𝑛 + 1 ) → ( 𝑏 ∥ 𝑎 ↔ ( 𝑛 + 1 ) ∥ 𝑎 ) ) |
56 |
|
eleq1 |
⊢ ( 𝑏 = ( 𝑛 + 1 ) → ( 𝑏 ∈ 𝑆 ↔ ( 𝑛 + 1 ) ∈ 𝑆 ) ) |
57 |
55 56
|
imbi12d |
⊢ ( 𝑏 = ( 𝑛 + 1 ) → ( ( 𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆 ) ↔ ( ( 𝑛 + 1 ) ∥ 𝑎 → ( 𝑛 + 1 ) ∈ 𝑆 ) ) ) |
58 |
57
|
ralbidv |
⊢ ( 𝑏 = ( 𝑛 + 1 ) → ( ∀ 𝑎 ∈ 𝑌 ( 𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆 ) ↔ ∀ 𝑎 ∈ 𝑌 ( ( 𝑛 + 1 ) ∥ 𝑎 → ( 𝑛 + 1 ) ∈ 𝑆 ) ) ) |
59 |
54 58
|
ralsn |
⊢ ( ∀ 𝑏 ∈ { ( 𝑛 + 1 ) } ∀ 𝑎 ∈ 𝑌 ( 𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆 ) ↔ ∀ 𝑎 ∈ 𝑌 ( ( 𝑛 + 1 ) ∥ 𝑎 → ( 𝑛 + 1 ) ∈ 𝑆 ) ) |
60 |
53 59
|
syl6ibr |
⊢ ( 𝑛 ∈ ℕ → ( ∀ 𝑏 ∈ ( 1 ... 𝑛 ) ∀ 𝑎 ∈ 𝑌 ( 𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆 ) → ∀ 𝑏 ∈ { ( 𝑛 + 1 ) } ∀ 𝑎 ∈ 𝑌 ( 𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆 ) ) ) |
61 |
60
|
ancld |
⊢ ( 𝑛 ∈ ℕ → ( ∀ 𝑏 ∈ ( 1 ... 𝑛 ) ∀ 𝑎 ∈ 𝑌 ( 𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆 ) → ( ∀ 𝑏 ∈ ( 1 ... 𝑛 ) ∀ 𝑎 ∈ 𝑌 ( 𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆 ) ∧ ∀ 𝑏 ∈ { ( 𝑛 + 1 ) } ∀ 𝑎 ∈ 𝑌 ( 𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆 ) ) ) ) |
62 |
|
elnnuz |
⊢ ( 𝑛 ∈ ℕ ↔ 𝑛 ∈ ( ℤ≥ ‘ 1 ) ) |
63 |
|
fzsuc |
⊢ ( 𝑛 ∈ ( ℤ≥ ‘ 1 ) → ( 1 ... ( 𝑛 + 1 ) ) = ( ( 1 ... 𝑛 ) ∪ { ( 𝑛 + 1 ) } ) ) |
64 |
62 63
|
sylbi |
⊢ ( 𝑛 ∈ ℕ → ( 1 ... ( 𝑛 + 1 ) ) = ( ( 1 ... 𝑛 ) ∪ { ( 𝑛 + 1 ) } ) ) |
65 |
64
|
raleqdv |
⊢ ( 𝑛 ∈ ℕ → ( ∀ 𝑏 ∈ ( 1 ... ( 𝑛 + 1 ) ) ∀ 𝑎 ∈ 𝑌 ( 𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆 ) ↔ ∀ 𝑏 ∈ ( ( 1 ... 𝑛 ) ∪ { ( 𝑛 + 1 ) } ) ∀ 𝑎 ∈ 𝑌 ( 𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆 ) ) ) |
66 |
|
ralunb |
⊢ ( ∀ 𝑏 ∈ ( ( 1 ... 𝑛 ) ∪ { ( 𝑛 + 1 ) } ) ∀ 𝑎 ∈ 𝑌 ( 𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆 ) ↔ ( ∀ 𝑏 ∈ ( 1 ... 𝑛 ) ∀ 𝑎 ∈ 𝑌 ( 𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆 ) ∧ ∀ 𝑏 ∈ { ( 𝑛 + 1 ) } ∀ 𝑎 ∈ 𝑌 ( 𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆 ) ) ) |
67 |
65 66
|
bitrdi |
⊢ ( 𝑛 ∈ ℕ → ( ∀ 𝑏 ∈ ( 1 ... ( 𝑛 + 1 ) ) ∀ 𝑎 ∈ 𝑌 ( 𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆 ) ↔ ( ∀ 𝑏 ∈ ( 1 ... 𝑛 ) ∀ 𝑎 ∈ 𝑌 ( 𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆 ) ∧ ∀ 𝑏 ∈ { ( 𝑛 + 1 ) } ∀ 𝑎 ∈ 𝑌 ( 𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆 ) ) ) ) |
68 |
61 67
|
sylibrd |
⊢ ( 𝑛 ∈ ℕ → ( ∀ 𝑏 ∈ ( 1 ... 𝑛 ) ∀ 𝑎 ∈ 𝑌 ( 𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆 ) → ∀ 𝑏 ∈ ( 1 ... ( 𝑛 + 1 ) ) ∀ 𝑎 ∈ 𝑌 ( 𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆 ) ) ) |
69 |
8 10 12 14 32 68
|
nnind |
⊢ ( 𝐵 ∈ ℕ → ∀ 𝑏 ∈ ( 1 ... 𝐵 ) ∀ 𝑎 ∈ 𝑌 ( 𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆 ) ) |
70 |
|
elfz1end |
⊢ ( 𝐵 ∈ ℕ ↔ 𝐵 ∈ ( 1 ... 𝐵 ) ) |
71 |
70
|
biimpi |
⊢ ( 𝐵 ∈ ℕ → 𝐵 ∈ ( 1 ... 𝐵 ) ) |
72 |
6 69 71
|
rspcdva |
⊢ ( 𝐵 ∈ ℕ → ∀ 𝑎 ∈ 𝑌 ( 𝐵 ∥ 𝑎 → 𝐵 ∈ 𝑆 ) ) |
73 |
|
breq2 |
⊢ ( 𝑎 = 𝐴 → ( 𝐵 ∥ 𝑎 ↔ 𝐵 ∥ 𝐴 ) ) |
74 |
73
|
imbi1d |
⊢ ( 𝑎 = 𝐴 → ( ( 𝐵 ∥ 𝑎 → 𝐵 ∈ 𝑆 ) ↔ ( 𝐵 ∥ 𝐴 → 𝐵 ∈ 𝑆 ) ) ) |
75 |
74
|
rspcv |
⊢ ( 𝐴 ∈ 𝑌 → ( ∀ 𝑎 ∈ 𝑌 ( 𝐵 ∥ 𝑎 → 𝐵 ∈ 𝑆 ) → ( 𝐵 ∥ 𝐴 → 𝐵 ∈ 𝑆 ) ) ) |
76 |
72 75
|
syl5 |
⊢ ( 𝐴 ∈ 𝑌 → ( 𝐵 ∈ ℕ → ( 𝐵 ∥ 𝐴 → 𝐵 ∈ 𝑆 ) ) ) |
77 |
76
|
3imp |
⊢ ( ( 𝐴 ∈ 𝑌 ∧ 𝐵 ∈ ℕ ∧ 𝐵 ∥ 𝐴 ) → 𝐵 ∈ 𝑆 ) |