Step |
Hyp |
Ref |
Expression |
1 |
|
enrefg |
⊢ ( 𝐴 ∈ 𝑉 → 𝐴 ≈ 𝐴 ) |
2 |
1
|
3ad2ant1 |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋 ) → 𝐴 ≈ 𝐴 ) |
3 |
|
0ex |
⊢ ∅ ∈ V |
4 |
|
simp2 |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋 ) → 𝐵 ∈ 𝑊 ) |
5 |
|
xpsnen2g |
⊢ ( ( ∅ ∈ V ∧ 𝐵 ∈ 𝑊 ) → ( { ∅ } × 𝐵 ) ≈ 𝐵 ) |
6 |
3 4 5
|
sylancr |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋 ) → ( { ∅ } × 𝐵 ) ≈ 𝐵 ) |
7 |
6
|
ensymd |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋 ) → 𝐵 ≈ ( { ∅ } × 𝐵 ) ) |
8 |
|
xpen |
⊢ ( ( 𝐴 ≈ 𝐴 ∧ 𝐵 ≈ ( { ∅ } × 𝐵 ) ) → ( 𝐴 × 𝐵 ) ≈ ( 𝐴 × ( { ∅ } × 𝐵 ) ) ) |
9 |
2 7 8
|
syl2anc |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋 ) → ( 𝐴 × 𝐵 ) ≈ ( 𝐴 × ( { ∅ } × 𝐵 ) ) ) |
10 |
|
1on |
⊢ 1o ∈ On |
11 |
|
simp3 |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋 ) → 𝐶 ∈ 𝑋 ) |
12 |
|
xpsnen2g |
⊢ ( ( 1o ∈ On ∧ 𝐶 ∈ 𝑋 ) → ( { 1o } × 𝐶 ) ≈ 𝐶 ) |
13 |
10 11 12
|
sylancr |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋 ) → ( { 1o } × 𝐶 ) ≈ 𝐶 ) |
14 |
13
|
ensymd |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋 ) → 𝐶 ≈ ( { 1o } × 𝐶 ) ) |
15 |
|
xpen |
⊢ ( ( 𝐴 ≈ 𝐴 ∧ 𝐶 ≈ ( { 1o } × 𝐶 ) ) → ( 𝐴 × 𝐶 ) ≈ ( 𝐴 × ( { 1o } × 𝐶 ) ) ) |
16 |
2 14 15
|
syl2anc |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋 ) → ( 𝐴 × 𝐶 ) ≈ ( 𝐴 × ( { 1o } × 𝐶 ) ) ) |
17 |
|
xp01disjl |
⊢ ( ( { ∅ } × 𝐵 ) ∩ ( { 1o } × 𝐶 ) ) = ∅ |
18 |
17
|
xpeq2i |
⊢ ( 𝐴 × ( ( { ∅ } × 𝐵 ) ∩ ( { 1o } × 𝐶 ) ) ) = ( 𝐴 × ∅ ) |
19 |
|
xpindi |
⊢ ( 𝐴 × ( ( { ∅ } × 𝐵 ) ∩ ( { 1o } × 𝐶 ) ) ) = ( ( 𝐴 × ( { ∅ } × 𝐵 ) ) ∩ ( 𝐴 × ( { 1o } × 𝐶 ) ) ) |
20 |
|
xp0 |
⊢ ( 𝐴 × ∅ ) = ∅ |
21 |
18 19 20
|
3eqtr3i |
⊢ ( ( 𝐴 × ( { ∅ } × 𝐵 ) ) ∩ ( 𝐴 × ( { 1o } × 𝐶 ) ) ) = ∅ |
22 |
21
|
a1i |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋 ) → ( ( 𝐴 × ( { ∅ } × 𝐵 ) ) ∩ ( 𝐴 × ( { 1o } × 𝐶 ) ) ) = ∅ ) |
23 |
|
djuenun |
⊢ ( ( ( 𝐴 × 𝐵 ) ≈ ( 𝐴 × ( { ∅ } × 𝐵 ) ) ∧ ( 𝐴 × 𝐶 ) ≈ ( 𝐴 × ( { 1o } × 𝐶 ) ) ∧ ( ( 𝐴 × ( { ∅ } × 𝐵 ) ) ∩ ( 𝐴 × ( { 1o } × 𝐶 ) ) ) = ∅ ) → ( ( 𝐴 × 𝐵 ) ⊔ ( 𝐴 × 𝐶 ) ) ≈ ( ( 𝐴 × ( { ∅ } × 𝐵 ) ) ∪ ( 𝐴 × ( { 1o } × 𝐶 ) ) ) ) |
24 |
9 16 22 23
|
syl3anc |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋 ) → ( ( 𝐴 × 𝐵 ) ⊔ ( 𝐴 × 𝐶 ) ) ≈ ( ( 𝐴 × ( { ∅ } × 𝐵 ) ) ∪ ( 𝐴 × ( { 1o } × 𝐶 ) ) ) ) |
25 |
|
df-dju |
⊢ ( 𝐵 ⊔ 𝐶 ) = ( ( { ∅ } × 𝐵 ) ∪ ( { 1o } × 𝐶 ) ) |
26 |
25
|
xpeq2i |
⊢ ( 𝐴 × ( 𝐵 ⊔ 𝐶 ) ) = ( 𝐴 × ( ( { ∅ } × 𝐵 ) ∪ ( { 1o } × 𝐶 ) ) ) |
27 |
|
xpundi |
⊢ ( 𝐴 × ( ( { ∅ } × 𝐵 ) ∪ ( { 1o } × 𝐶 ) ) ) = ( ( 𝐴 × ( { ∅ } × 𝐵 ) ) ∪ ( 𝐴 × ( { 1o } × 𝐶 ) ) ) |
28 |
26 27
|
eqtri |
⊢ ( 𝐴 × ( 𝐵 ⊔ 𝐶 ) ) = ( ( 𝐴 × ( { ∅ } × 𝐵 ) ) ∪ ( 𝐴 × ( { 1o } × 𝐶 ) ) ) |
29 |
24 28
|
breqtrrdi |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋 ) → ( ( 𝐴 × 𝐵 ) ⊔ ( 𝐴 × 𝐶 ) ) ≈ ( 𝐴 × ( 𝐵 ⊔ 𝐶 ) ) ) |
30 |
29
|
ensymd |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋 ) → ( 𝐴 × ( 𝐵 ⊔ 𝐶 ) ) ≈ ( ( 𝐴 × 𝐵 ) ⊔ ( 𝐴 × 𝐶 ) ) ) |