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Theorem xpdjuen 9593
 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 8527 . . . . . 6 (𝐴𝑉𝐴𝐴)
213ad2ant1 1130 . . . . 5 ((𝐴𝑉𝐵𝑊𝐶𝑋) → 𝐴𝐴)
3 0ex 5176 . . . . . . 7 ∅ ∈ V
4 simp2 1134 . . . . . . 7 ((𝐴𝑉𝐵𝑊𝐶𝑋) → 𝐵𝑊)
5 xpsnen2g 8596 . . . . . . 7 ((∅ ∈ V ∧ 𝐵𝑊) → ({∅} × 𝐵) ≈ 𝐵)
63, 4, 5sylancr 590 . . . . . 6 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ({∅} × 𝐵) ≈ 𝐵)
76ensymd 8546 . . . . 5 ((𝐴𝑉𝐵𝑊𝐶𝑋) → 𝐵 ≈ ({∅} × 𝐵))
8 xpen 8667 . . . . 5 ((𝐴𝐴𝐵 ≈ ({∅} × 𝐵)) → (𝐴 × 𝐵) ≈ (𝐴 × ({∅} × 𝐵)))
92, 7, 8syl2anc 587 . . . 4 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐴 × 𝐵) ≈ (𝐴 × ({∅} × 𝐵)))
10 1on 8095 . . . . . . 7 1o ∈ On
11 simp3 1135 . . . . . . 7 ((𝐴𝑉𝐵𝑊𝐶𝑋) → 𝐶𝑋)
12 xpsnen2g 8596 . . . . . . 7 ((1o ∈ On ∧ 𝐶𝑋) → ({1o} × 𝐶) ≈ 𝐶)
1310, 11, 12sylancr 590 . . . . . 6 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ({1o} × 𝐶) ≈ 𝐶)
1413ensymd 8546 . . . . 5 ((𝐴𝑉𝐵𝑊𝐶𝑋) → 𝐶 ≈ ({1o} × 𝐶))
15 xpen 8667 . . . . 5 ((𝐴𝐴𝐶 ≈ ({1o} × 𝐶)) → (𝐴 × 𝐶) ≈ (𝐴 × ({1o} × 𝐶)))
162, 14, 15syl2anc 587 . . . 4 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐴 × 𝐶) ≈ (𝐴 × ({1o} × 𝐶)))
17 xp01disjl 8107 . . . . . . 7 (({∅} × 𝐵) ∩ ({1o} × 𝐶)) = ∅
1817xpeq2i 5547 . . . . . 6 (𝐴 × (({∅} × 𝐵) ∩ ({1o} × 𝐶))) = (𝐴 × ∅)
19 xpindi 5669 . . . . . 6 (𝐴 × (({∅} × 𝐵) ∩ ({1o} × 𝐶))) = ((𝐴 × ({∅} × 𝐵)) ∩ (𝐴 × ({1o} × 𝐶)))
20 xp0 5983 . . . . . 6 (𝐴 × ∅) = ∅
2118, 19, 203eqtr3i 2829 . . . . 5 ((𝐴 × ({∅} × 𝐵)) ∩ (𝐴 × ({1o} × 𝐶))) = ∅
2221a1i 11 . . . 4 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ((𝐴 × ({∅} × 𝐵)) ∩ (𝐴 × ({1o} × 𝐶))) = ∅)
23 djuenun 9584 . . . 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 1368 . . 3 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ((𝐴 × 𝐵) ⊔ (𝐴 × 𝐶)) ≈ ((𝐴 × ({∅} × 𝐵)) ∪ (𝐴 × ({1o} × 𝐶))))
25 df-dju 9317 . . . . 5 (𝐵𝐶) = (({∅} × 𝐵) ∪ ({1o} × 𝐶))
2625xpeq2i 5547 . . . 4 (𝐴 × (𝐵𝐶)) = (𝐴 × (({∅} × 𝐵) ∪ ({1o} × 𝐶)))
27 xpundi 5585 . . . 4 (𝐴 × (({∅} × 𝐵) ∪ ({1o} × 𝐶))) = ((𝐴 × ({∅} × 𝐵)) ∪ (𝐴 × ({1o} × 𝐶)))
2826, 27eqtri 2821 . . 3 (𝐴 × (𝐵𝐶)) = ((𝐴 × ({∅} × 𝐵)) ∪ (𝐴 × ({1o} × 𝐶)))
2924, 28breqtrrdi 5073 . 2 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ((𝐴 × 𝐵) ⊔ (𝐴 × 𝐶)) ≈ (𝐴 × (𝐵𝐶)))
3029ensymd 8546 1 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐴 × (𝐵𝐶)) ≈ ((𝐴 × 𝐵) ⊔ (𝐴 × 𝐶)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  Vcvv 3441   ∪ cun 3879   ∩ cin 3880  ∅c0 4243  {csn 4525   class class class wbr 5031   × cxp 5518  Oncon0 6160  1oc1o 8081   ≈ cen 8492   ⊔ cdju 9314 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7444 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-int 4840  df-br 5032  df-opab 5094  df-mpt 5112  df-tr 5138  df-id 5426  df-eprel 5431  df-po 5439  df-so 5440  df-fr 5479  df-we 5481  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-ord 6163  df-on 6164  df-suc 6166  df-iota 6284  df-fun 6327  df-fn 6328  df-f 6329  df-f1 6330  df-fo 6331  df-f1o 6332  df-fv 6333  df-1st 7674  df-2nd 7675  df-1o 8088  df-er 8275  df-en 8496  df-dom 8497  df-dju 9317 This theorem is referenced by: (None)
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