Step |
Hyp |
Ref |
Expression |
1 |
|
oveq2 |
⊢ ( 𝑥 = 1 → ( 1 · 𝑥 ) = ( 1 · 1 ) ) |
2 |
|
id |
⊢ ( 𝑥 = 1 → 𝑥 = 1 ) |
3 |
1 2
|
eqeq12d |
⊢ ( 𝑥 = 1 → ( ( 1 · 𝑥 ) = 𝑥 ↔ ( 1 · 1 ) = 1 ) ) |
4 |
|
oveq2 |
⊢ ( 𝑥 = 𝑦 → ( 1 · 𝑥 ) = ( 1 · 𝑦 ) ) |
5 |
|
id |
⊢ ( 𝑥 = 𝑦 → 𝑥 = 𝑦 ) |
6 |
4 5
|
eqeq12d |
⊢ ( 𝑥 = 𝑦 → ( ( 1 · 𝑥 ) = 𝑥 ↔ ( 1 · 𝑦 ) = 𝑦 ) ) |
7 |
|
oveq2 |
⊢ ( 𝑥 = ( 𝑦 + 1 ) → ( 1 · 𝑥 ) = ( 1 · ( 𝑦 + 1 ) ) ) |
8 |
|
id |
⊢ ( 𝑥 = ( 𝑦 + 1 ) → 𝑥 = ( 𝑦 + 1 ) ) |
9 |
7 8
|
eqeq12d |
⊢ ( 𝑥 = ( 𝑦 + 1 ) → ( ( 1 · 𝑥 ) = 𝑥 ↔ ( 1 · ( 𝑦 + 1 ) ) = ( 𝑦 + 1 ) ) ) |
10 |
|
oveq2 |
⊢ ( 𝑥 = 𝐴 → ( 1 · 𝑥 ) = ( 1 · 𝐴 ) ) |
11 |
|
id |
⊢ ( 𝑥 = 𝐴 → 𝑥 = 𝐴 ) |
12 |
10 11
|
eqeq12d |
⊢ ( 𝑥 = 𝐴 → ( ( 1 · 𝑥 ) = 𝑥 ↔ ( 1 · 𝐴 ) = 𝐴 ) ) |
13 |
|
1t1e1ALT |
⊢ ( 1 · 1 ) = 1 |
14 |
|
1cnd |
⊢ ( ( 𝑦 ∈ ℕ ∧ ( 1 · 𝑦 ) = 𝑦 ) → 1 ∈ ℂ ) |
15 |
|
simpl |
⊢ ( ( 𝑦 ∈ ℕ ∧ ( 1 · 𝑦 ) = 𝑦 ) → 𝑦 ∈ ℕ ) |
16 |
15
|
nncnd |
⊢ ( ( 𝑦 ∈ ℕ ∧ ( 1 · 𝑦 ) = 𝑦 ) → 𝑦 ∈ ℂ ) |
17 |
14 16 14
|
adddid |
⊢ ( ( 𝑦 ∈ ℕ ∧ ( 1 · 𝑦 ) = 𝑦 ) → ( 1 · ( 𝑦 + 1 ) ) = ( ( 1 · 𝑦 ) + ( 1 · 1 ) ) ) |
18 |
|
simpr |
⊢ ( ( 𝑦 ∈ ℕ ∧ ( 1 · 𝑦 ) = 𝑦 ) → ( 1 · 𝑦 ) = 𝑦 ) |
19 |
13
|
a1i |
⊢ ( ( 𝑦 ∈ ℕ ∧ ( 1 · 𝑦 ) = 𝑦 ) → ( 1 · 1 ) = 1 ) |
20 |
18 19
|
oveq12d |
⊢ ( ( 𝑦 ∈ ℕ ∧ ( 1 · 𝑦 ) = 𝑦 ) → ( ( 1 · 𝑦 ) + ( 1 · 1 ) ) = ( 𝑦 + 1 ) ) |
21 |
17 20
|
eqtrd |
⊢ ( ( 𝑦 ∈ ℕ ∧ ( 1 · 𝑦 ) = 𝑦 ) → ( 1 · ( 𝑦 + 1 ) ) = ( 𝑦 + 1 ) ) |
22 |
21
|
ex |
⊢ ( 𝑦 ∈ ℕ → ( ( 1 · 𝑦 ) = 𝑦 → ( 1 · ( 𝑦 + 1 ) ) = ( 𝑦 + 1 ) ) ) |
23 |
3 6 9 12 13 22
|
nnind |
⊢ ( 𝐴 ∈ ℕ → ( 1 · 𝐴 ) = 𝐴 ) |
24 |
|
nnre |
⊢ ( 𝐴 ∈ ℕ → 𝐴 ∈ ℝ ) |
25 |
|
ax-1rid |
⊢ ( 𝐴 ∈ ℝ → ( 𝐴 · 1 ) = 𝐴 ) |
26 |
24 25
|
syl |
⊢ ( 𝐴 ∈ ℕ → ( 𝐴 · 1 ) = 𝐴 ) |
27 |
23 26
|
eqtr4d |
⊢ ( 𝐴 ∈ ℕ → ( 1 · 𝐴 ) = ( 𝐴 · 1 ) ) |