Theorem xpdjuen 10230

Description: Cardinal multiplication distributes over cardinal addition. Theorem 6I(3) of [Enderton] p. 142. (Contributed by NM, 26-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)

Assertion

Ref	Expression
xpdjuen	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝐴 × (𝐵 ⊔ 𝐶)) ≈ ((𝐴 × 𝐵) ⊔ (𝐴 × 𝐶)))

Proof of Theorem xpdjuen

Step	Hyp	Ref	Expression
1		enrefg 9023	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ≈ 𝐴)
2	1	3ad2ant1 1130	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → 𝐴 ≈ 𝐴)
3		0ex 5315	. . . . . . 7 ⊢ ∅ ∈ V
4		simp2 1134	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → 𝐵 ∈ 𝑊)
5		xpsnen2g 9110	. . . . . . 7 ⊢ ((∅ ∈ V ∧ 𝐵 ∈ 𝑊) → ({∅} × 𝐵) ≈ 𝐵)
6	3, 4, 5	sylancr 585	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → ({∅} × 𝐵) ≈ 𝐵)
7	6	ensymd 9044	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → 𝐵 ≈ ({∅} × 𝐵))
8		xpen 9185	. . . . 5 ⊢ ((𝐴 ≈ 𝐴 ∧ 𝐵 ≈ ({∅} × 𝐵)) → (𝐴 × 𝐵) ≈ (𝐴 × ({∅} × 𝐵)))
9	2, 7, 8	syl2anc 582	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝐴 × 𝐵) ≈ (𝐴 × ({∅} × 𝐵)))
10		1on 8516	. . . . . . 7 ⊢ 1o ∈ On
11		simp3 1135	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → 𝐶 ∈ 𝑋)
12		xpsnen2g 9110	. . . . . . 7 ⊢ ((1o ∈ On ∧ 𝐶 ∈ 𝑋) → ({1o} × 𝐶) ≈ 𝐶)
13	10, 11, 12	sylancr 585	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → ({1o} × 𝐶) ≈ 𝐶)
14	13	ensymd 9044	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → 𝐶 ≈ ({1o} × 𝐶))
15		xpen 9185	. . . . 5 ⊢ ((𝐴 ≈ 𝐴 ∧ 𝐶 ≈ ({1o} × 𝐶)) → (𝐴 × 𝐶) ≈ (𝐴 × ({1o} × 𝐶)))
16	2, 14, 15	syl2anc 582	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝐴 × 𝐶) ≈ (𝐴 × ({1o} × 𝐶)))
17		xp01disjl 8530	. . . . . . 7 ⊢ (({∅} × 𝐵) ∩ ({1o} × 𝐶)) = ∅
18	17	xpeq2i 5713	. . . . . 6 ⊢ (𝐴 × (({∅} × 𝐵) ∩ ({1o} × 𝐶))) = (𝐴 × ∅)
19		xpindi 5844	. . . . . 6 ⊢ (𝐴 × (({∅} × 𝐵) ∩ ({1o} × 𝐶))) = ((𝐴 × ({∅} × 𝐵)) ∩ (𝐴 × ({1o} × 𝐶)))
20		xp0 6174	. . . . . 6 ⊢ (𝐴 × ∅) = ∅
21	18, 19, 20	3eqtr3i 2765	. . . . 5 ⊢ ((𝐴 × ({∅} × 𝐵)) ∩ (𝐴 × ({1o} × 𝐶))) = ∅
22	21	a1i 11	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → ((𝐴 × ({∅} × 𝐵)) ∩ (𝐴 × ({1o} × 𝐶))) = ∅)
23		djuenun 10221	. . . 4 ⊢ (((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 1368	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → ((𝐴 × 𝐵) ⊔ (𝐴 × 𝐶)) ≈ ((𝐴 × ({∅} × 𝐵)) ∪ (𝐴 × ({1o} × 𝐶))))
25		df-dju 9951	. . . . 5 ⊢ (𝐵 ⊔ 𝐶) = (({∅} × 𝐵) ∪ ({1o} × 𝐶))
26	25	xpeq2i 5713	. . . 4 ⊢ (𝐴 × (𝐵 ⊔ 𝐶)) = (𝐴 × (({∅} × 𝐵) ∪ ({1o} × 𝐶)))
27		xpundi 5754	. . . 4 ⊢ (𝐴 × (({∅} × 𝐵) ∪ ({1o} × 𝐶))) = ((𝐴 × ({∅} × 𝐵)) ∪ (𝐴 × ({1o} × 𝐶)))
28	26, 27	eqtri 2757	. . 3 ⊢ (𝐴 × (𝐵 ⊔ 𝐶)) = ((𝐴 × ({∅} × 𝐵)) ∪ (𝐴 × ({1o} × 𝐶)))
29	24, 28	breqtrrdi 5198	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → ((𝐴 × 𝐵) ⊔ (𝐴 × 𝐶)) ≈ (𝐴 × (𝐵 ⊔ 𝐶)))
30	29	ensymd 9044	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝐴 × (𝐵 ⊔ 𝐶)) ≈ ((𝐴 × 𝐵) ⊔ (𝐴 × 𝐶)))

Syntax hints:   → wi 4   ∧ w3a 1084   = wceq 1534   ∈ wcel 2100  Vcvv 3472   ∪ cun 3957   ∩ cin 3958  ∅c0 4335  {csn 4636   class class class wbr 5156   × cxp 5684  Oncon0 6380  1_oc1o 8497   ≈ cen 8979   ⊔ cdju 9948

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2102  ax-9 2110  ax-10 2133  ax-11 2150  ax-12 2170  ax-ext 2700  ax-sep 5307  ax-nul 5314  ax-pow 5373  ax-pr 5437  ax-un 7751

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2062  df-mo 2532  df-eu 2561  df-clab 2707  df-cleq 2721  df-clel 2806  df-nfc 2881  df-ne 2934  df-ral 3055  df-rex 3064  df-rab 3429  df-v 3474  df-sbc 3789  df-csb 3905  df-dif 3962  df-un 3964  df-in 3966  df-ss 3976  df-pss 3979  df-nul 4336  df-if 4537  df-pw 4612  df-sn 4637  df-pr 4639  df-op 4643  df-uni 4919  df-int 4960  df-br 5157  df-opab 5219  df-mpt 5240  df-tr 5274  df-id 5584  df-eprel 5590  df-po 5598  df-so 5599  df-fr 5641  df-we 5643  df-xp 5692  df-rel 5693  df-cnv 5694  df-co 5695  df-dm 5696  df-rn 5697  df-res 5698  df-ima 5699  df-ord 6383  df-on 6384  df-suc 6386  df-iota 6510  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-1st 8011  df-2nd 8012  df-1o 8504  df-er 8742  df-en 8983  df-dom 8984  df-dju 9951

This theorem is referenced by: (None)