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Theorem xpdjuen 10218
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 9023 . . . . . 6 (𝐴𝑉𝐴𝐴)
213ad2ant1 1132 . . . . 5 ((𝐴𝑉𝐵𝑊𝐶𝑋) → 𝐴𝐴)
3 0ex 5313 . . . . . . 7 ∅ ∈ V
4 simp2 1136 . . . . . . 7 ((𝐴𝑉𝐵𝑊𝐶𝑋) → 𝐵𝑊)
5 xpsnen2g 9104 . . . . . . 7 ((∅ ∈ V ∧ 𝐵𝑊) → ({∅} × 𝐵) ≈ 𝐵)
63, 4, 5sylancr 587 . . . . . 6 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ({∅} × 𝐵) ≈ 𝐵)
76ensymd 9044 . . . . 5 ((𝐴𝑉𝐵𝑊𝐶𝑋) → 𝐵 ≈ ({∅} × 𝐵))
8 xpen 9179 . . . . 5 ((𝐴𝐴𝐵 ≈ ({∅} × 𝐵)) → (𝐴 × 𝐵) ≈ (𝐴 × ({∅} × 𝐵)))
92, 7, 8syl2anc 584 . . . 4 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐴 × 𝐵) ≈ (𝐴 × ({∅} × 𝐵)))
10 1on 8517 . . . . . . 7 1o ∈ On
11 simp3 1137 . . . . . . 7 ((𝐴𝑉𝐵𝑊𝐶𝑋) → 𝐶𝑋)
12 xpsnen2g 9104 . . . . . . 7 ((1o ∈ On ∧ 𝐶𝑋) → ({1o} × 𝐶) ≈ 𝐶)
1310, 11, 12sylancr 587 . . . . . 6 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ({1o} × 𝐶) ≈ 𝐶)
1413ensymd 9044 . . . . 5 ((𝐴𝑉𝐵𝑊𝐶𝑋) → 𝐶 ≈ ({1o} × 𝐶))
15 xpen 9179 . . . . 5 ((𝐴𝐴𝐶 ≈ ({1o} × 𝐶)) → (𝐴 × 𝐶) ≈ (𝐴 × ({1o} × 𝐶)))
162, 14, 15syl2anc 584 . . . 4 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐴 × 𝐶) ≈ (𝐴 × ({1o} × 𝐶)))
17 xp01disjl 8529 . . . . . . 7 (({∅} × 𝐵) ∩ ({1o} × 𝐶)) = ∅
1817xpeq2i 5716 . . . . . 6 (𝐴 × (({∅} × 𝐵) ∩ ({1o} × 𝐶))) = (𝐴 × ∅)
19 xpindi 5847 . . . . . 6 (𝐴 × (({∅} × 𝐵) ∩ ({1o} × 𝐶))) = ((𝐴 × ({∅} × 𝐵)) ∩ (𝐴 × ({1o} × 𝐶)))
20 xp0 6180 . . . . . 6 (𝐴 × ∅) = ∅
2118, 19, 203eqtr3i 2771 . . . . 5 ((𝐴 × ({∅} × 𝐵)) ∩ (𝐴 × ({1o} × 𝐶))) = ∅
2221a1i 11 . . . 4 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ((𝐴 × ({∅} × 𝐵)) ∩ (𝐴 × ({1o} × 𝐶))) = ∅)
23 djuenun 10209 . . . 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 1370 . . 3 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ((𝐴 × 𝐵) ⊔ (𝐴 × 𝐶)) ≈ ((𝐴 × ({∅} × 𝐵)) ∪ (𝐴 × ({1o} × 𝐶))))
25 df-dju 9939 . . . . 5 (𝐵𝐶) = (({∅} × 𝐵) ∪ ({1o} × 𝐶))
2625xpeq2i 5716 . . . 4 (𝐴 × (𝐵𝐶)) = (𝐴 × (({∅} × 𝐵) ∪ ({1o} × 𝐶)))
27 xpundi 5757 . . . 4 (𝐴 × (({∅} × 𝐵) ∪ ({1o} × 𝐶))) = ((𝐴 × ({∅} × 𝐵)) ∪ (𝐴 × ({1o} × 𝐶)))
2826, 27eqtri 2763 . . 3 (𝐴 × (𝐵𝐶)) = ((𝐴 × ({∅} × 𝐵)) ∪ (𝐴 × ({1o} × 𝐶)))
2924, 28breqtrrdi 5190 . 2 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ((𝐴 × 𝐵) ⊔ (𝐴 × 𝐶)) ≈ (𝐴 × (𝐵𝐶)))
3029ensymd 9044 1 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐴 × (𝐵𝐶)) ≈ ((𝐴 × 𝐵) ⊔ (𝐴 × 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1086   = wceq 1537  wcel 2106  Vcvv 3478  cun 3961  cin 3962  c0 4339  {csn 4631   class class class wbr 5148   × cxp 5687  Oncon0 6386  1oc1o 8498  cen 8981  cdju 9936
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-1st 8013  df-2nd 8014  df-1o 8505  df-er 8744  df-en 8985  df-dom 8986  df-dju 9939
This theorem is referenced by: (None)
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