Metamath Proof Explorer


Theorem mapdjuen

Description: Sum of exponents law for cardinal arithmetic. Theorem 6I(4) of Enderton p. 142. (Contributed by NM, 27-Sep-2004) (Revised by Mario Carneiro, 29-Apr-2015)

Ref Expression
Assertion mapdjuen ( ( 𝐴𝑉𝐵𝑊𝐶𝑋 ) → ( 𝐴m ( 𝐵𝐶 ) ) ≈ ( ( 𝐴m 𝐵 ) × ( 𝐴m 𝐶 ) ) )

Proof

Step Hyp Ref Expression
1 df-dju ( 𝐵𝐶 ) = ( ( { ∅ } × 𝐵 ) ∪ ( { 1o } × 𝐶 ) )
2 1 oveq2i ( 𝐴m ( 𝐵𝐶 ) ) = ( 𝐴m ( ( { ∅ } × 𝐵 ) ∪ ( { 1o } × 𝐶 ) ) )
3 snex { ∅ } ∈ V
4 simp2 ( ( 𝐴𝑉𝐵𝑊𝐶𝑋 ) → 𝐵𝑊 )
5 xpexg ( ( { ∅ } ∈ V ∧ 𝐵𝑊 ) → ( { ∅ } × 𝐵 ) ∈ V )
6 3 4 5 sylancr ( ( 𝐴𝑉𝐵𝑊𝐶𝑋 ) → ( { ∅ } × 𝐵 ) ∈ V )
7 snex { 1o } ∈ V
8 simp3 ( ( 𝐴𝑉𝐵𝑊𝐶𝑋 ) → 𝐶𝑋 )
9 xpexg ( ( { 1o } ∈ V ∧ 𝐶𝑋 ) → ( { 1o } × 𝐶 ) ∈ V )
10 7 8 9 sylancr ( ( 𝐴𝑉𝐵𝑊𝐶𝑋 ) → ( { 1o } × 𝐶 ) ∈ V )
11 simp1 ( ( 𝐴𝑉𝐵𝑊𝐶𝑋 ) → 𝐴𝑉 )
12 xp01disjl ( ( { ∅ } × 𝐵 ) ∩ ( { 1o } × 𝐶 ) ) = ∅
13 12 a1i ( ( 𝐴𝑉𝐵𝑊𝐶𝑋 ) → ( ( { ∅ } × 𝐵 ) ∩ ( { 1o } × 𝐶 ) ) = ∅ )
14 mapunen ( ( ( ( { ∅ } × 𝐵 ) ∈ V ∧ ( { 1o } × 𝐶 ) ∈ V ∧ 𝐴𝑉 ) ∧ ( ( { ∅ } × 𝐵 ) ∩ ( { 1o } × 𝐶 ) ) = ∅ ) → ( 𝐴m ( ( { ∅ } × 𝐵 ) ∪ ( { 1o } × 𝐶 ) ) ) ≈ ( ( 𝐴m ( { ∅ } × 𝐵 ) ) × ( 𝐴m ( { 1o } × 𝐶 ) ) ) )
15 6 10 11 13 14 syl31anc ( ( 𝐴𝑉𝐵𝑊𝐶𝑋 ) → ( 𝐴m ( ( { ∅ } × 𝐵 ) ∪ ( { 1o } × 𝐶 ) ) ) ≈ ( ( 𝐴m ( { ∅ } × 𝐵 ) ) × ( 𝐴m ( { 1o } × 𝐶 ) ) ) )
16 2 15 eqbrtrid ( ( 𝐴𝑉𝐵𝑊𝐶𝑋 ) → ( 𝐴m ( 𝐵𝐶 ) ) ≈ ( ( 𝐴m ( { ∅ } × 𝐵 ) ) × ( 𝐴m ( { 1o } × 𝐶 ) ) ) )
17 enrefg ( 𝐴𝑉𝐴𝐴 )
18 11 17 syl ( ( 𝐴𝑉𝐵𝑊𝐶𝑋 ) → 𝐴𝐴 )
19 0ex ∅ ∈ V
20 xpsnen2g ( ( ∅ ∈ V ∧ 𝐵𝑊 ) → ( { ∅ } × 𝐵 ) ≈ 𝐵 )
21 19 4 20 sylancr ( ( 𝐴𝑉𝐵𝑊𝐶𝑋 ) → ( { ∅ } × 𝐵 ) ≈ 𝐵 )
22 mapen ( ( 𝐴𝐴 ∧ ( { ∅ } × 𝐵 ) ≈ 𝐵 ) → ( 𝐴m ( { ∅ } × 𝐵 ) ) ≈ ( 𝐴m 𝐵 ) )
23 18 21 22 syl2anc ( ( 𝐴𝑉𝐵𝑊𝐶𝑋 ) → ( 𝐴m ( { ∅ } × 𝐵 ) ) ≈ ( 𝐴m 𝐵 ) )
24 1on 1o ∈ On
25 xpsnen2g ( ( 1o ∈ On ∧ 𝐶𝑋 ) → ( { 1o } × 𝐶 ) ≈ 𝐶 )
26 24 8 25 sylancr ( ( 𝐴𝑉𝐵𝑊𝐶𝑋 ) → ( { 1o } × 𝐶 ) ≈ 𝐶 )
27 mapen ( ( 𝐴𝐴 ∧ ( { 1o } × 𝐶 ) ≈ 𝐶 ) → ( 𝐴m ( { 1o } × 𝐶 ) ) ≈ ( 𝐴m 𝐶 ) )
28 18 26 27 syl2anc ( ( 𝐴𝑉𝐵𝑊𝐶𝑋 ) → ( 𝐴m ( { 1o } × 𝐶 ) ) ≈ ( 𝐴m 𝐶 ) )
29 xpen ( ( ( 𝐴m ( { ∅ } × 𝐵 ) ) ≈ ( 𝐴m 𝐵 ) ∧ ( 𝐴m ( { 1o } × 𝐶 ) ) ≈ ( 𝐴m 𝐶 ) ) → ( ( 𝐴m ( { ∅ } × 𝐵 ) ) × ( 𝐴m ( { 1o } × 𝐶 ) ) ) ≈ ( ( 𝐴m 𝐵 ) × ( 𝐴m 𝐶 ) ) )
30 23 28 29 syl2anc ( ( 𝐴𝑉𝐵𝑊𝐶𝑋 ) → ( ( 𝐴m ( { ∅ } × 𝐵 ) ) × ( 𝐴m ( { 1o } × 𝐶 ) ) ) ≈ ( ( 𝐴m 𝐵 ) × ( 𝐴m 𝐶 ) ) )
31 entr ( ( ( 𝐴m ( 𝐵𝐶 ) ) ≈ ( ( 𝐴m ( { ∅ } × 𝐵 ) ) × ( 𝐴m ( { 1o } × 𝐶 ) ) ) ∧ ( ( 𝐴m ( { ∅ } × 𝐵 ) ) × ( 𝐴m ( { 1o } × 𝐶 ) ) ) ≈ ( ( 𝐴m 𝐵 ) × ( 𝐴m 𝐶 ) ) ) → ( 𝐴m ( 𝐵𝐶 ) ) ≈ ( ( 𝐴m 𝐵 ) × ( 𝐴m 𝐶 ) ) )
32 16 30 31 syl2anc ( ( 𝐴𝑉𝐵𝑊𝐶𝑋 ) → ( 𝐴m ( 𝐵𝐶 ) ) ≈ ( ( 𝐴m 𝐵 ) × ( 𝐴m 𝐶 ) ) )