Step |
Hyp |
Ref |
Expression |
1 |
|
smfdiv.x |
⊢ Ⅎ 𝑥 𝜑 |
2 |
|
smfdiv.s |
⊢ ( 𝜑 → 𝑆 ∈ SAlg ) |
3 |
|
smfdiv.a |
⊢ ( 𝜑 → 𝐴 ∈ 𝑉 ) |
4 |
|
smfdiv.b |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ ℝ ) |
5 |
|
smfdiv.c |
⊢ ( 𝜑 → 𝐶 ∈ 𝑊 ) |
6 |
|
smfdiv.d |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐶 ) → 𝐷 ∈ ℝ ) |
7 |
|
smfdiv.m |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∈ ( SMblFn ‘ 𝑆 ) ) |
8 |
|
smfdiv.n |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐶 ↦ 𝐷 ) ∈ ( SMblFn ‘ 𝑆 ) ) |
9 |
|
smfdiv.e |
⊢ 𝐸 = { 𝑥 ∈ 𝐶 ∣ 𝐷 ≠ 0 } |
10 |
|
elinel1 |
⊢ ( 𝑥 ∈ ( 𝐴 ∩ 𝐸 ) → 𝑥 ∈ 𝐴 ) |
11 |
10
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝐴 ∩ 𝐸 ) ) → 𝑥 ∈ 𝐴 ) |
12 |
11 4
|
syldan |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝐴 ∩ 𝐸 ) ) → 𝐵 ∈ ℝ ) |
13 |
12
|
recnd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝐴 ∩ 𝐸 ) ) → 𝐵 ∈ ℂ ) |
14 |
|
ssrab2 |
⊢ { 𝑥 ∈ 𝐶 ∣ 𝐷 ≠ 0 } ⊆ 𝐶 |
15 |
9 14
|
eqsstri |
⊢ 𝐸 ⊆ 𝐶 |
16 |
|
elinel2 |
⊢ ( 𝑥 ∈ ( 𝐴 ∩ 𝐸 ) → 𝑥 ∈ 𝐸 ) |
17 |
15 16
|
sseldi |
⊢ ( 𝑥 ∈ ( 𝐴 ∩ 𝐸 ) → 𝑥 ∈ 𝐶 ) |
18 |
17
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝐴 ∩ 𝐸 ) ) → 𝑥 ∈ 𝐶 ) |
19 |
18 6
|
syldan |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝐴 ∩ 𝐸 ) ) → 𝐷 ∈ ℝ ) |
20 |
19
|
recnd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝐴 ∩ 𝐸 ) ) → 𝐷 ∈ ℂ ) |
21 |
9
|
eleq2i |
⊢ ( 𝑥 ∈ 𝐸 ↔ 𝑥 ∈ { 𝑥 ∈ 𝐶 ∣ 𝐷 ≠ 0 } ) |
22 |
21
|
biimpi |
⊢ ( 𝑥 ∈ 𝐸 → 𝑥 ∈ { 𝑥 ∈ 𝐶 ∣ 𝐷 ≠ 0 } ) |
23 |
|
rabidim2 |
⊢ ( 𝑥 ∈ { 𝑥 ∈ 𝐶 ∣ 𝐷 ≠ 0 } → 𝐷 ≠ 0 ) |
24 |
22 23
|
syl |
⊢ ( 𝑥 ∈ 𝐸 → 𝐷 ≠ 0 ) |
25 |
16 24
|
syl |
⊢ ( 𝑥 ∈ ( 𝐴 ∩ 𝐸 ) → 𝐷 ≠ 0 ) |
26 |
25
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝐴 ∩ 𝐸 ) ) → 𝐷 ≠ 0 ) |
27 |
13 20 26
|
divrecd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝐴 ∩ 𝐸 ) ) → ( 𝐵 / 𝐷 ) = ( 𝐵 · ( 1 / 𝐷 ) ) ) |
28 |
1 27
|
mpteq2da |
⊢ ( 𝜑 → ( 𝑥 ∈ ( 𝐴 ∩ 𝐸 ) ↦ ( 𝐵 / 𝐷 ) ) = ( 𝑥 ∈ ( 𝐴 ∩ 𝐸 ) ↦ ( 𝐵 · ( 1 / 𝐷 ) ) ) ) |
29 |
|
1red |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐸 ) → 1 ∈ ℝ ) |
30 |
15
|
sseli |
⊢ ( 𝑥 ∈ 𝐸 → 𝑥 ∈ 𝐶 ) |
31 |
30
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐸 ) → 𝑥 ∈ 𝐶 ) |
32 |
31 6
|
syldan |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐸 ) → 𝐷 ∈ ℝ ) |
33 |
24
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐸 ) → 𝐷 ≠ 0 ) |
34 |
29 32 33
|
redivcld |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐸 ) → ( 1 / 𝐷 ) ∈ ℝ ) |
35 |
1 2 5 6 8 9
|
smfrec |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐸 ↦ ( 1 / 𝐷 ) ) ∈ ( SMblFn ‘ 𝑆 ) ) |
36 |
1 2 3 4 34 7 35
|
smfmul |
⊢ ( 𝜑 → ( 𝑥 ∈ ( 𝐴 ∩ 𝐸 ) ↦ ( 𝐵 · ( 1 / 𝐷 ) ) ) ∈ ( SMblFn ‘ 𝑆 ) ) |
37 |
28 36
|
eqeltrd |
⊢ ( 𝜑 → ( 𝑥 ∈ ( 𝐴 ∩ 𝐸 ) ↦ ( 𝐵 / 𝐷 ) ) ∈ ( SMblFn ‘ 𝑆 ) ) |