Step |
Hyp |
Ref |
Expression |
1 |
|
ficardom |
⊢ ( 𝐴 ∈ Fin → ( card ‘ 𝐴 ) ∈ ω ) |
2 |
|
ficardom |
⊢ ( 𝐵 ∈ Fin → ( card ‘ 𝐵 ) ∈ ω ) |
3 |
|
nnadju |
⊢ ( ( ( card ‘ 𝐴 ) ∈ ω ∧ ( card ‘ 𝐵 ) ∈ ω ) → ( card ‘ ( ( card ‘ 𝐴 ) ⊔ ( card ‘ 𝐵 ) ) ) = ( ( card ‘ 𝐴 ) +o ( card ‘ 𝐵 ) ) ) |
4 |
|
df-dju |
⊢ ( ( card ‘ 𝐴 ) ⊔ ( card ‘ 𝐵 ) ) = ( ( { ∅ } × ( card ‘ 𝐴 ) ) ∪ ( { 1o } × ( card ‘ 𝐵 ) ) ) |
5 |
|
snfi |
⊢ { ∅ } ∈ Fin |
6 |
|
nnfi |
⊢ ( ( card ‘ 𝐴 ) ∈ ω → ( card ‘ 𝐴 ) ∈ Fin ) |
7 |
|
xpfi |
⊢ ( ( { ∅ } ∈ Fin ∧ ( card ‘ 𝐴 ) ∈ Fin ) → ( { ∅ } × ( card ‘ 𝐴 ) ) ∈ Fin ) |
8 |
5 6 7
|
sylancr |
⊢ ( ( card ‘ 𝐴 ) ∈ ω → ( { ∅ } × ( card ‘ 𝐴 ) ) ∈ Fin ) |
9 |
|
snfi |
⊢ { 1o } ∈ Fin |
10 |
|
nnfi |
⊢ ( ( card ‘ 𝐵 ) ∈ ω → ( card ‘ 𝐵 ) ∈ Fin ) |
11 |
|
xpfi |
⊢ ( ( { 1o } ∈ Fin ∧ ( card ‘ 𝐵 ) ∈ Fin ) → ( { 1o } × ( card ‘ 𝐵 ) ) ∈ Fin ) |
12 |
9 10 11
|
sylancr |
⊢ ( ( card ‘ 𝐵 ) ∈ ω → ( { 1o } × ( card ‘ 𝐵 ) ) ∈ Fin ) |
13 |
|
unfi |
⊢ ( ( ( { ∅ } × ( card ‘ 𝐴 ) ) ∈ Fin ∧ ( { 1o } × ( card ‘ 𝐵 ) ) ∈ Fin ) → ( ( { ∅ } × ( card ‘ 𝐴 ) ) ∪ ( { 1o } × ( card ‘ 𝐵 ) ) ) ∈ Fin ) |
14 |
8 12 13
|
syl2an |
⊢ ( ( ( card ‘ 𝐴 ) ∈ ω ∧ ( card ‘ 𝐵 ) ∈ ω ) → ( ( { ∅ } × ( card ‘ 𝐴 ) ) ∪ ( { 1o } × ( card ‘ 𝐵 ) ) ) ∈ Fin ) |
15 |
4 14
|
eqeltrid |
⊢ ( ( ( card ‘ 𝐴 ) ∈ ω ∧ ( card ‘ 𝐵 ) ∈ ω ) → ( ( card ‘ 𝐴 ) ⊔ ( card ‘ 𝐵 ) ) ∈ Fin ) |
16 |
|
ficardid |
⊢ ( ( ( card ‘ 𝐴 ) ⊔ ( card ‘ 𝐵 ) ) ∈ Fin → ( card ‘ ( ( card ‘ 𝐴 ) ⊔ ( card ‘ 𝐵 ) ) ) ≈ ( ( card ‘ 𝐴 ) ⊔ ( card ‘ 𝐵 ) ) ) |
17 |
15 16
|
syl |
⊢ ( ( ( card ‘ 𝐴 ) ∈ ω ∧ ( card ‘ 𝐵 ) ∈ ω ) → ( card ‘ ( ( card ‘ 𝐴 ) ⊔ ( card ‘ 𝐵 ) ) ) ≈ ( ( card ‘ 𝐴 ) ⊔ ( card ‘ 𝐵 ) ) ) |
18 |
3 17
|
eqbrtrrd |
⊢ ( ( ( card ‘ 𝐴 ) ∈ ω ∧ ( card ‘ 𝐵 ) ∈ ω ) → ( ( card ‘ 𝐴 ) +o ( card ‘ 𝐵 ) ) ≈ ( ( card ‘ 𝐴 ) ⊔ ( card ‘ 𝐵 ) ) ) |
19 |
1 2 18
|
syl2an |
⊢ ( ( 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ) → ( ( card ‘ 𝐴 ) +o ( card ‘ 𝐵 ) ) ≈ ( ( card ‘ 𝐴 ) ⊔ ( card ‘ 𝐵 ) ) ) |
20 |
|
ficardid |
⊢ ( 𝐴 ∈ Fin → ( card ‘ 𝐴 ) ≈ 𝐴 ) |
21 |
|
ficardid |
⊢ ( 𝐵 ∈ Fin → ( card ‘ 𝐵 ) ≈ 𝐵 ) |
22 |
|
djuen |
⊢ ( ( ( card ‘ 𝐴 ) ≈ 𝐴 ∧ ( card ‘ 𝐵 ) ≈ 𝐵 ) → ( ( card ‘ 𝐴 ) ⊔ ( card ‘ 𝐵 ) ) ≈ ( 𝐴 ⊔ 𝐵 ) ) |
23 |
20 21 22
|
syl2an |
⊢ ( ( 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ) → ( ( card ‘ 𝐴 ) ⊔ ( card ‘ 𝐵 ) ) ≈ ( 𝐴 ⊔ 𝐵 ) ) |
24 |
|
entr |
⊢ ( ( ( ( card ‘ 𝐴 ) +o ( card ‘ 𝐵 ) ) ≈ ( ( card ‘ 𝐴 ) ⊔ ( card ‘ 𝐵 ) ) ∧ ( ( card ‘ 𝐴 ) ⊔ ( card ‘ 𝐵 ) ) ≈ ( 𝐴 ⊔ 𝐵 ) ) → ( ( card ‘ 𝐴 ) +o ( card ‘ 𝐵 ) ) ≈ ( 𝐴 ⊔ 𝐵 ) ) |
25 |
19 23 24
|
syl2anc |
⊢ ( ( 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ) → ( ( card ‘ 𝐴 ) +o ( card ‘ 𝐵 ) ) ≈ ( 𝐴 ⊔ 𝐵 ) ) |
26 |
25
|
ensymd |
⊢ ( ( 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ) → ( 𝐴 ⊔ 𝐵 ) ≈ ( ( card ‘ 𝐴 ) +o ( card ‘ 𝐵 ) ) ) |