Step |
Hyp |
Ref |
Expression |
1 |
|
oveq1 |
⊢ ( 𝑥 = 𝐴 → ( 𝑥 ·o 𝐵 ) = ( 𝐴 ·o 𝐵 ) ) |
2 |
|
oveq2 |
⊢ ( 𝑥 = 𝐴 → ( 𝐵 ·o 𝑥 ) = ( 𝐵 ·o 𝐴 ) ) |
3 |
1 2
|
eqeq12d |
⊢ ( 𝑥 = 𝐴 → ( ( 𝑥 ·o 𝐵 ) = ( 𝐵 ·o 𝑥 ) ↔ ( 𝐴 ·o 𝐵 ) = ( 𝐵 ·o 𝐴 ) ) ) |
4 |
3
|
imbi2d |
⊢ ( 𝑥 = 𝐴 → ( ( 𝐵 ∈ ω → ( 𝑥 ·o 𝐵 ) = ( 𝐵 ·o 𝑥 ) ) ↔ ( 𝐵 ∈ ω → ( 𝐴 ·o 𝐵 ) = ( 𝐵 ·o 𝐴 ) ) ) ) |
5 |
|
oveq1 |
⊢ ( 𝑥 = ∅ → ( 𝑥 ·o 𝐵 ) = ( ∅ ·o 𝐵 ) ) |
6 |
|
oveq2 |
⊢ ( 𝑥 = ∅ → ( 𝐵 ·o 𝑥 ) = ( 𝐵 ·o ∅ ) ) |
7 |
5 6
|
eqeq12d |
⊢ ( 𝑥 = ∅ → ( ( 𝑥 ·o 𝐵 ) = ( 𝐵 ·o 𝑥 ) ↔ ( ∅ ·o 𝐵 ) = ( 𝐵 ·o ∅ ) ) ) |
8 |
|
oveq1 |
⊢ ( 𝑥 = 𝑦 → ( 𝑥 ·o 𝐵 ) = ( 𝑦 ·o 𝐵 ) ) |
9 |
|
oveq2 |
⊢ ( 𝑥 = 𝑦 → ( 𝐵 ·o 𝑥 ) = ( 𝐵 ·o 𝑦 ) ) |
10 |
8 9
|
eqeq12d |
⊢ ( 𝑥 = 𝑦 → ( ( 𝑥 ·o 𝐵 ) = ( 𝐵 ·o 𝑥 ) ↔ ( 𝑦 ·o 𝐵 ) = ( 𝐵 ·o 𝑦 ) ) ) |
11 |
|
oveq1 |
⊢ ( 𝑥 = suc 𝑦 → ( 𝑥 ·o 𝐵 ) = ( suc 𝑦 ·o 𝐵 ) ) |
12 |
|
oveq2 |
⊢ ( 𝑥 = suc 𝑦 → ( 𝐵 ·o 𝑥 ) = ( 𝐵 ·o suc 𝑦 ) ) |
13 |
11 12
|
eqeq12d |
⊢ ( 𝑥 = suc 𝑦 → ( ( 𝑥 ·o 𝐵 ) = ( 𝐵 ·o 𝑥 ) ↔ ( suc 𝑦 ·o 𝐵 ) = ( 𝐵 ·o suc 𝑦 ) ) ) |
14 |
|
nnm0r |
⊢ ( 𝐵 ∈ ω → ( ∅ ·o 𝐵 ) = ∅ ) |
15 |
|
nnm0 |
⊢ ( 𝐵 ∈ ω → ( 𝐵 ·o ∅ ) = ∅ ) |
16 |
14 15
|
eqtr4d |
⊢ ( 𝐵 ∈ ω → ( ∅ ·o 𝐵 ) = ( 𝐵 ·o ∅ ) ) |
17 |
|
oveq1 |
⊢ ( ( 𝑦 ·o 𝐵 ) = ( 𝐵 ·o 𝑦 ) → ( ( 𝑦 ·o 𝐵 ) +o 𝐵 ) = ( ( 𝐵 ·o 𝑦 ) +o 𝐵 ) ) |
18 |
|
nnmsucr |
⊢ ( ( 𝑦 ∈ ω ∧ 𝐵 ∈ ω ) → ( suc 𝑦 ·o 𝐵 ) = ( ( 𝑦 ·o 𝐵 ) +o 𝐵 ) ) |
19 |
|
nnmsuc |
⊢ ( ( 𝐵 ∈ ω ∧ 𝑦 ∈ ω ) → ( 𝐵 ·o suc 𝑦 ) = ( ( 𝐵 ·o 𝑦 ) +o 𝐵 ) ) |
20 |
19
|
ancoms |
⊢ ( ( 𝑦 ∈ ω ∧ 𝐵 ∈ ω ) → ( 𝐵 ·o suc 𝑦 ) = ( ( 𝐵 ·o 𝑦 ) +o 𝐵 ) ) |
21 |
18 20
|
eqeq12d |
⊢ ( ( 𝑦 ∈ ω ∧ 𝐵 ∈ ω ) → ( ( suc 𝑦 ·o 𝐵 ) = ( 𝐵 ·o suc 𝑦 ) ↔ ( ( 𝑦 ·o 𝐵 ) +o 𝐵 ) = ( ( 𝐵 ·o 𝑦 ) +o 𝐵 ) ) ) |
22 |
17 21
|
syl5ibr |
⊢ ( ( 𝑦 ∈ ω ∧ 𝐵 ∈ ω ) → ( ( 𝑦 ·o 𝐵 ) = ( 𝐵 ·o 𝑦 ) → ( suc 𝑦 ·o 𝐵 ) = ( 𝐵 ·o suc 𝑦 ) ) ) |
23 |
22
|
ex |
⊢ ( 𝑦 ∈ ω → ( 𝐵 ∈ ω → ( ( 𝑦 ·o 𝐵 ) = ( 𝐵 ·o 𝑦 ) → ( suc 𝑦 ·o 𝐵 ) = ( 𝐵 ·o suc 𝑦 ) ) ) ) |
24 |
7 10 13 16 23
|
finds2 |
⊢ ( 𝑥 ∈ ω → ( 𝐵 ∈ ω → ( 𝑥 ·o 𝐵 ) = ( 𝐵 ·o 𝑥 ) ) ) |
25 |
4 24
|
vtoclga |
⊢ ( 𝐴 ∈ ω → ( 𝐵 ∈ ω → ( 𝐴 ·o 𝐵 ) = ( 𝐵 ·o 𝐴 ) ) ) |
26 |
25
|
imp |
⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → ( 𝐴 ·o 𝐵 ) = ( 𝐵 ·o 𝐴 ) ) |