Step |
Hyp |
Ref |
Expression |
1 |
|
ip1i.1 |
⊢ 𝑋 = ( BaseSet ‘ 𝑈 ) |
2 |
|
ip1i.2 |
⊢ 𝐺 = ( +𝑣 ‘ 𝑈 ) |
3 |
|
ip1i.4 |
⊢ 𝑆 = ( ·𝑠OLD ‘ 𝑈 ) |
4 |
|
ip1i.7 |
⊢ 𝑃 = ( ·𝑖OLD ‘ 𝑈 ) |
5 |
|
ip1i.9 |
⊢ 𝑈 ∈ CPreHilOLD |
6 |
|
ipasslem1.b |
⊢ 𝐵 ∈ 𝑋 |
7 |
|
elq |
⊢ ( 𝐶 ∈ ℚ ↔ ∃ 𝑗 ∈ ℤ ∃ 𝑘 ∈ ℕ 𝐶 = ( 𝑗 / 𝑘 ) ) |
8 |
|
zcn |
⊢ ( 𝑗 ∈ ℤ → 𝑗 ∈ ℂ ) |
9 |
|
nnrecre |
⊢ ( 𝑘 ∈ ℕ → ( 1 / 𝑘 ) ∈ ℝ ) |
10 |
9
|
recnd |
⊢ ( 𝑘 ∈ ℕ → ( 1 / 𝑘 ) ∈ ℂ ) |
11 |
5
|
phnvi |
⊢ 𝑈 ∈ NrmCVec |
12 |
1 4
|
dipcl |
⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 𝑃 𝐵 ) ∈ ℂ ) |
13 |
11 6 12
|
mp3an13 |
⊢ ( 𝐴 ∈ 𝑋 → ( 𝐴 𝑃 𝐵 ) ∈ ℂ ) |
14 |
|
mulass |
⊢ ( ( 𝑗 ∈ ℂ ∧ ( 1 / 𝑘 ) ∈ ℂ ∧ ( 𝐴 𝑃 𝐵 ) ∈ ℂ ) → ( ( 𝑗 · ( 1 / 𝑘 ) ) · ( 𝐴 𝑃 𝐵 ) ) = ( 𝑗 · ( ( 1 / 𝑘 ) · ( 𝐴 𝑃 𝐵 ) ) ) ) |
15 |
8 10 13 14
|
syl3an |
⊢ ( ( 𝑗 ∈ ℤ ∧ 𝑘 ∈ ℕ ∧ 𝐴 ∈ 𝑋 ) → ( ( 𝑗 · ( 1 / 𝑘 ) ) · ( 𝐴 𝑃 𝐵 ) ) = ( 𝑗 · ( ( 1 / 𝑘 ) · ( 𝐴 𝑃 𝐵 ) ) ) ) |
16 |
8
|
adantr |
⊢ ( ( 𝑗 ∈ ℤ ∧ 𝑘 ∈ ℕ ) → 𝑗 ∈ ℂ ) |
17 |
|
nncn |
⊢ ( 𝑘 ∈ ℕ → 𝑘 ∈ ℂ ) |
18 |
17
|
adantl |
⊢ ( ( 𝑗 ∈ ℤ ∧ 𝑘 ∈ ℕ ) → 𝑘 ∈ ℂ ) |
19 |
|
nnne0 |
⊢ ( 𝑘 ∈ ℕ → 𝑘 ≠ 0 ) |
20 |
19
|
adantl |
⊢ ( ( 𝑗 ∈ ℤ ∧ 𝑘 ∈ ℕ ) → 𝑘 ≠ 0 ) |
21 |
16 18 20
|
divrecd |
⊢ ( ( 𝑗 ∈ ℤ ∧ 𝑘 ∈ ℕ ) → ( 𝑗 / 𝑘 ) = ( 𝑗 · ( 1 / 𝑘 ) ) ) |
22 |
21
|
3adant3 |
⊢ ( ( 𝑗 ∈ ℤ ∧ 𝑘 ∈ ℕ ∧ 𝐴 ∈ 𝑋 ) → ( 𝑗 / 𝑘 ) = ( 𝑗 · ( 1 / 𝑘 ) ) ) |
23 |
22
|
oveq1d |
⊢ ( ( 𝑗 ∈ ℤ ∧ 𝑘 ∈ ℕ ∧ 𝐴 ∈ 𝑋 ) → ( ( 𝑗 / 𝑘 ) · ( 𝐴 𝑃 𝐵 ) ) = ( ( 𝑗 · ( 1 / 𝑘 ) ) · ( 𝐴 𝑃 𝐵 ) ) ) |
24 |
22
|
oveq1d |
⊢ ( ( 𝑗 ∈ ℤ ∧ 𝑘 ∈ ℕ ∧ 𝐴 ∈ 𝑋 ) → ( ( 𝑗 / 𝑘 ) 𝑆 𝐴 ) = ( ( 𝑗 · ( 1 / 𝑘 ) ) 𝑆 𝐴 ) ) |
25 |
|
id |
⊢ ( 𝐴 ∈ 𝑋 → 𝐴 ∈ 𝑋 ) |
26 |
1 3
|
nvsass |
⊢ ( ( 𝑈 ∈ NrmCVec ∧ ( 𝑗 ∈ ℂ ∧ ( 1 / 𝑘 ) ∈ ℂ ∧ 𝐴 ∈ 𝑋 ) ) → ( ( 𝑗 · ( 1 / 𝑘 ) ) 𝑆 𝐴 ) = ( 𝑗 𝑆 ( ( 1 / 𝑘 ) 𝑆 𝐴 ) ) ) |
27 |
11 26
|
mpan |
⊢ ( ( 𝑗 ∈ ℂ ∧ ( 1 / 𝑘 ) ∈ ℂ ∧ 𝐴 ∈ 𝑋 ) → ( ( 𝑗 · ( 1 / 𝑘 ) ) 𝑆 𝐴 ) = ( 𝑗 𝑆 ( ( 1 / 𝑘 ) 𝑆 𝐴 ) ) ) |
28 |
8 10 25 27
|
syl3an |
⊢ ( ( 𝑗 ∈ ℤ ∧ 𝑘 ∈ ℕ ∧ 𝐴 ∈ 𝑋 ) → ( ( 𝑗 · ( 1 / 𝑘 ) ) 𝑆 𝐴 ) = ( 𝑗 𝑆 ( ( 1 / 𝑘 ) 𝑆 𝐴 ) ) ) |
29 |
24 28
|
eqtrd |
⊢ ( ( 𝑗 ∈ ℤ ∧ 𝑘 ∈ ℕ ∧ 𝐴 ∈ 𝑋 ) → ( ( 𝑗 / 𝑘 ) 𝑆 𝐴 ) = ( 𝑗 𝑆 ( ( 1 / 𝑘 ) 𝑆 𝐴 ) ) ) |
30 |
29
|
oveq1d |
⊢ ( ( 𝑗 ∈ ℤ ∧ 𝑘 ∈ ℕ ∧ 𝐴 ∈ 𝑋 ) → ( ( ( 𝑗 / 𝑘 ) 𝑆 𝐴 ) 𝑃 𝐵 ) = ( ( 𝑗 𝑆 ( ( 1 / 𝑘 ) 𝑆 𝐴 ) ) 𝑃 𝐵 ) ) |
31 |
1 3
|
nvscl |
⊢ ( ( 𝑈 ∈ NrmCVec ∧ ( 1 / 𝑘 ) ∈ ℂ ∧ 𝐴 ∈ 𝑋 ) → ( ( 1 / 𝑘 ) 𝑆 𝐴 ) ∈ 𝑋 ) |
32 |
11 31
|
mp3an1 |
⊢ ( ( ( 1 / 𝑘 ) ∈ ℂ ∧ 𝐴 ∈ 𝑋 ) → ( ( 1 / 𝑘 ) 𝑆 𝐴 ) ∈ 𝑋 ) |
33 |
10 32
|
sylan |
⊢ ( ( 𝑘 ∈ ℕ ∧ 𝐴 ∈ 𝑋 ) → ( ( 1 / 𝑘 ) 𝑆 𝐴 ) ∈ 𝑋 ) |
34 |
1 2 3 4 5 6
|
ipasslem3 |
⊢ ( ( 𝑗 ∈ ℤ ∧ ( ( 1 / 𝑘 ) 𝑆 𝐴 ) ∈ 𝑋 ) → ( ( 𝑗 𝑆 ( ( 1 / 𝑘 ) 𝑆 𝐴 ) ) 𝑃 𝐵 ) = ( 𝑗 · ( ( ( 1 / 𝑘 ) 𝑆 𝐴 ) 𝑃 𝐵 ) ) ) |
35 |
33 34
|
sylan2 |
⊢ ( ( 𝑗 ∈ ℤ ∧ ( 𝑘 ∈ ℕ ∧ 𝐴 ∈ 𝑋 ) ) → ( ( 𝑗 𝑆 ( ( 1 / 𝑘 ) 𝑆 𝐴 ) ) 𝑃 𝐵 ) = ( 𝑗 · ( ( ( 1 / 𝑘 ) 𝑆 𝐴 ) 𝑃 𝐵 ) ) ) |
36 |
35
|
3impb |
⊢ ( ( 𝑗 ∈ ℤ ∧ 𝑘 ∈ ℕ ∧ 𝐴 ∈ 𝑋 ) → ( ( 𝑗 𝑆 ( ( 1 / 𝑘 ) 𝑆 𝐴 ) ) 𝑃 𝐵 ) = ( 𝑗 · ( ( ( 1 / 𝑘 ) 𝑆 𝐴 ) 𝑃 𝐵 ) ) ) |
37 |
1 2 3 4 5 6
|
ipasslem4 |
⊢ ( ( 𝑘 ∈ ℕ ∧ 𝐴 ∈ 𝑋 ) → ( ( ( 1 / 𝑘 ) 𝑆 𝐴 ) 𝑃 𝐵 ) = ( ( 1 / 𝑘 ) · ( 𝐴 𝑃 𝐵 ) ) ) |
38 |
37
|
3adant1 |
⊢ ( ( 𝑗 ∈ ℤ ∧ 𝑘 ∈ ℕ ∧ 𝐴 ∈ 𝑋 ) → ( ( ( 1 / 𝑘 ) 𝑆 𝐴 ) 𝑃 𝐵 ) = ( ( 1 / 𝑘 ) · ( 𝐴 𝑃 𝐵 ) ) ) |
39 |
38
|
oveq2d |
⊢ ( ( 𝑗 ∈ ℤ ∧ 𝑘 ∈ ℕ ∧ 𝐴 ∈ 𝑋 ) → ( 𝑗 · ( ( ( 1 / 𝑘 ) 𝑆 𝐴 ) 𝑃 𝐵 ) ) = ( 𝑗 · ( ( 1 / 𝑘 ) · ( 𝐴 𝑃 𝐵 ) ) ) ) |
40 |
30 36 39
|
3eqtrd |
⊢ ( ( 𝑗 ∈ ℤ ∧ 𝑘 ∈ ℕ ∧ 𝐴 ∈ 𝑋 ) → ( ( ( 𝑗 / 𝑘 ) 𝑆 𝐴 ) 𝑃 𝐵 ) = ( 𝑗 · ( ( 1 / 𝑘 ) · ( 𝐴 𝑃 𝐵 ) ) ) ) |
41 |
15 23 40
|
3eqtr4rd |
⊢ ( ( 𝑗 ∈ ℤ ∧ 𝑘 ∈ ℕ ∧ 𝐴 ∈ 𝑋 ) → ( ( ( 𝑗 / 𝑘 ) 𝑆 𝐴 ) 𝑃 𝐵 ) = ( ( 𝑗 / 𝑘 ) · ( 𝐴 𝑃 𝐵 ) ) ) |
42 |
|
oveq1 |
⊢ ( 𝐶 = ( 𝑗 / 𝑘 ) → ( 𝐶 𝑆 𝐴 ) = ( ( 𝑗 / 𝑘 ) 𝑆 𝐴 ) ) |
43 |
42
|
oveq1d |
⊢ ( 𝐶 = ( 𝑗 / 𝑘 ) → ( ( 𝐶 𝑆 𝐴 ) 𝑃 𝐵 ) = ( ( ( 𝑗 / 𝑘 ) 𝑆 𝐴 ) 𝑃 𝐵 ) ) |
44 |
|
oveq1 |
⊢ ( 𝐶 = ( 𝑗 / 𝑘 ) → ( 𝐶 · ( 𝐴 𝑃 𝐵 ) ) = ( ( 𝑗 / 𝑘 ) · ( 𝐴 𝑃 𝐵 ) ) ) |
45 |
43 44
|
eqeq12d |
⊢ ( 𝐶 = ( 𝑗 / 𝑘 ) → ( ( ( 𝐶 𝑆 𝐴 ) 𝑃 𝐵 ) = ( 𝐶 · ( 𝐴 𝑃 𝐵 ) ) ↔ ( ( ( 𝑗 / 𝑘 ) 𝑆 𝐴 ) 𝑃 𝐵 ) = ( ( 𝑗 / 𝑘 ) · ( 𝐴 𝑃 𝐵 ) ) ) ) |
46 |
41 45
|
syl5ibrcom |
⊢ ( ( 𝑗 ∈ ℤ ∧ 𝑘 ∈ ℕ ∧ 𝐴 ∈ 𝑋 ) → ( 𝐶 = ( 𝑗 / 𝑘 ) → ( ( 𝐶 𝑆 𝐴 ) 𝑃 𝐵 ) = ( 𝐶 · ( 𝐴 𝑃 𝐵 ) ) ) ) |
47 |
46
|
3expia |
⊢ ( ( 𝑗 ∈ ℤ ∧ 𝑘 ∈ ℕ ) → ( 𝐴 ∈ 𝑋 → ( 𝐶 = ( 𝑗 / 𝑘 ) → ( ( 𝐶 𝑆 𝐴 ) 𝑃 𝐵 ) = ( 𝐶 · ( 𝐴 𝑃 𝐵 ) ) ) ) ) |
48 |
47
|
com23 |
⊢ ( ( 𝑗 ∈ ℤ ∧ 𝑘 ∈ ℕ ) → ( 𝐶 = ( 𝑗 / 𝑘 ) → ( 𝐴 ∈ 𝑋 → ( ( 𝐶 𝑆 𝐴 ) 𝑃 𝐵 ) = ( 𝐶 · ( 𝐴 𝑃 𝐵 ) ) ) ) ) |
49 |
48
|
rexlimivv |
⊢ ( ∃ 𝑗 ∈ ℤ ∃ 𝑘 ∈ ℕ 𝐶 = ( 𝑗 / 𝑘 ) → ( 𝐴 ∈ 𝑋 → ( ( 𝐶 𝑆 𝐴 ) 𝑃 𝐵 ) = ( 𝐶 · ( 𝐴 𝑃 𝐵 ) ) ) ) |
50 |
7 49
|
sylbi |
⊢ ( 𝐶 ∈ ℚ → ( 𝐴 ∈ 𝑋 → ( ( 𝐶 𝑆 𝐴 ) 𝑃 𝐵 ) = ( 𝐶 · ( 𝐴 𝑃 𝐵 ) ) ) ) |
51 |
50
|
imp |
⊢ ( ( 𝐶 ∈ ℚ ∧ 𝐴 ∈ 𝑋 ) → ( ( 𝐶 𝑆 𝐴 ) 𝑃 𝐵 ) = ( 𝐶 · ( 𝐴 𝑃 𝐵 ) ) ) |