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Theorem xpdjuen 10220
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 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐴 × (𝐵𝐶)) ≈ ((𝐴 × 𝐵) ⊔ (𝐴 × 𝐶)))

Proof of Theorem xpdjuen
StepHypRef Expression
1 enrefg 9024 . . . . . 6 (𝐴𝑉𝐴𝐴)
213ad2ant1 1134 . . . . 5 ((𝐴𝑉𝐵𝑊𝐶𝑋) → 𝐴𝐴)
3 0ex 5307 . . . . . . 7 ∅ ∈ V
4 simp2 1138 . . . . . . 7 ((𝐴𝑉𝐵𝑊𝐶𝑋) → 𝐵𝑊)
5 xpsnen2g 9105 . . . . . . 7 ((∅ ∈ V ∧ 𝐵𝑊) → ({∅} × 𝐵) ≈ 𝐵)
63, 4, 5sylancr 587 . . . . . 6 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ({∅} × 𝐵) ≈ 𝐵)
76ensymd 9045 . . . . 5 ((𝐴𝑉𝐵𝑊𝐶𝑋) → 𝐵 ≈ ({∅} × 𝐵))
8 xpen 9180 . . . . 5 ((𝐴𝐴𝐵 ≈ ({∅} × 𝐵)) → (𝐴 × 𝐵) ≈ (𝐴 × ({∅} × 𝐵)))
92, 7, 8syl2anc 584 . . . 4 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐴 × 𝐵) ≈ (𝐴 × ({∅} × 𝐵)))
10 1on 8518 . . . . . . 7 1o ∈ On
11 simp3 1139 . . . . . . 7 ((𝐴𝑉𝐵𝑊𝐶𝑋) → 𝐶𝑋)
12 xpsnen2g 9105 . . . . . . 7 ((1o ∈ On ∧ 𝐶𝑋) → ({1o} × 𝐶) ≈ 𝐶)
1310, 11, 12sylancr 587 . . . . . 6 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ({1o} × 𝐶) ≈ 𝐶)
1413ensymd 9045 . . . . 5 ((𝐴𝑉𝐵𝑊𝐶𝑋) → 𝐶 ≈ ({1o} × 𝐶))
15 xpen 9180 . . . . 5 ((𝐴𝐴𝐶 ≈ ({1o} × 𝐶)) → (𝐴 × 𝐶) ≈ (𝐴 × ({1o} × 𝐶)))
162, 14, 15syl2anc 584 . . . 4 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐴 × 𝐶) ≈ (𝐴 × ({1o} × 𝐶)))
17 xp01disjl 8530 . . . . . . 7 (({∅} × 𝐵) ∩ ({1o} × 𝐶)) = ∅
1817xpeq2i 5712 . . . . . 6 (𝐴 × (({∅} × 𝐵) ∩ ({1o} × 𝐶))) = (𝐴 × ∅)
19 xpindi 5844 . . . . . 6 (𝐴 × (({∅} × 𝐵) ∩ ({1o} × 𝐶))) = ((𝐴 × ({∅} × 𝐵)) ∩ (𝐴 × ({1o} × 𝐶)))
20 xp0 6178 . . . . . 6 (𝐴 × ∅) = ∅
2118, 19, 203eqtr3i 2773 . . . . 5 ((𝐴 × ({∅} × 𝐵)) ∩ (𝐴 × ({1o} × 𝐶))) = ∅
2221a1i 11 . . . 4 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ((𝐴 × ({∅} × 𝐵)) ∩ (𝐴 × ({1o} × 𝐶))) = ∅)
23 djuenun 10211 . . . 4 (((𝐴 × 𝐵) ≈ (𝐴 × ({∅} × 𝐵)) ∧ (𝐴 × 𝐶) ≈ (𝐴 × ({1o} × 𝐶)) ∧ ((𝐴 × ({∅} × 𝐵)) ∩ (𝐴 × ({1o} × 𝐶))) = ∅) → ((𝐴 × 𝐵) ⊔ (𝐴 × 𝐶)) ≈ ((𝐴 × ({∅} × 𝐵)) ∪ (𝐴 × ({1o} × 𝐶))))
249, 16, 22, 23syl3anc 1373 . . 3 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ((𝐴 × 𝐵) ⊔ (𝐴 × 𝐶)) ≈ ((𝐴 × ({∅} × 𝐵)) ∪ (𝐴 × ({1o} × 𝐶))))
25 df-dju 9941 . . . . 5 (𝐵𝐶) = (({∅} × 𝐵) ∪ ({1o} × 𝐶))
2625xpeq2i 5712 . . . 4 (𝐴 × (𝐵𝐶)) = (𝐴 × (({∅} × 𝐵) ∪ ({1o} × 𝐶)))
27 xpundi 5754 . . . 4 (𝐴 × (({∅} × 𝐵) ∪ ({1o} × 𝐶))) = ((𝐴 × ({∅} × 𝐵)) ∪ (𝐴 × ({1o} × 𝐶)))
2826, 27eqtri 2765 . . 3 (𝐴 × (𝐵𝐶)) = ((𝐴 × ({∅} × 𝐵)) ∪ (𝐴 × ({1o} × 𝐶)))
2924, 28breqtrrdi 5185 . 2 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ((𝐴 × 𝐵) ⊔ (𝐴 × 𝐶)) ≈ (𝐴 × (𝐵𝐶)))
3029ensymd 9045 1 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐴 × (𝐵𝐶)) ≈ ((𝐴 × 𝐵) ⊔ (𝐴 × 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1087   = wceq 1540  wcel 2108  Vcvv 3480  cun 3949  cin 3950  c0 4333  {csn 4626   class class class wbr 5143   × cxp 5683  Oncon0 6384  1oc1o 8499  cen 8982  cdju 9938
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ord 6387  df-on 6388  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-1st 8014  df-2nd 8015  df-1o 8506  df-er 8745  df-en 8986  df-dom 8987  df-dju 9941
This theorem is referenced by: (None)
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