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Theorem xpcdaen 8888
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
xpcdaen ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐴 × (𝐵 +𝑐 𝐶)) ≈ ((𝐴 × 𝐵) +𝑐 (𝐴 × 𝐶)))

Proof of Theorem xpcdaen
StepHypRef Expression
1 enrefg 7873 . . . . . 6 (𝐴𝑉𝐴𝐴)
213ad2ant1 1075 . . . . 5 ((𝐴𝑉𝐵𝑊𝐶𝑋) → 𝐴𝐴)
3 simp2 1055 . . . . . . 7 ((𝐴𝑉𝐵𝑊𝐶𝑋) → 𝐵𝑊)
4 0ex 4718 . . . . . . 7 ∅ ∈ V
5 xpsneng 7930 . . . . . . 7 ((𝐵𝑊 ∧ ∅ ∈ V) → (𝐵 × {∅}) ≈ 𝐵)
63, 4, 5sylancl 693 . . . . . 6 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐵 × {∅}) ≈ 𝐵)
76ensymd 7893 . . . . 5 ((𝐴𝑉𝐵𝑊𝐶𝑋) → 𝐵 ≈ (𝐵 × {∅}))
8 xpen 8008 . . . . 5 ((𝐴𝐴𝐵 ≈ (𝐵 × {∅})) → (𝐴 × 𝐵) ≈ (𝐴 × (𝐵 × {∅})))
92, 7, 8syl2anc 691 . . . 4 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐴 × 𝐵) ≈ (𝐴 × (𝐵 × {∅})))
10 simp3 1056 . . . . . . 7 ((𝐴𝑉𝐵𝑊𝐶𝑋) → 𝐶𝑋)
11 1on 7454 . . . . . . 7 1𝑜 ∈ On
12 xpsneng 7930 . . . . . . 7 ((𝐶𝑋 ∧ 1𝑜 ∈ On) → (𝐶 × {1𝑜}) ≈ 𝐶)
1310, 11, 12sylancl 693 . . . . . 6 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐶 × {1𝑜}) ≈ 𝐶)
1413ensymd 7893 . . . . 5 ((𝐴𝑉𝐵𝑊𝐶𝑋) → 𝐶 ≈ (𝐶 × {1𝑜}))
15 xpen 8008 . . . . 5 ((𝐴𝐴𝐶 ≈ (𝐶 × {1𝑜})) → (𝐴 × 𝐶) ≈ (𝐴 × (𝐶 × {1𝑜})))
162, 14, 15syl2anc 691 . . . 4 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐴 × 𝐶) ≈ (𝐴 × (𝐶 × {1𝑜})))
17 xp01disj 7463 . . . . . . 7 ((𝐵 × {∅}) ∩ (𝐶 × {1𝑜})) = ∅
1817xpeq2i 5060 . . . . . 6 (𝐴 × ((𝐵 × {∅}) ∩ (𝐶 × {1𝑜}))) = (𝐴 × ∅)
19 xpindi 5177 . . . . . 6 (𝐴 × ((𝐵 × {∅}) ∩ (𝐶 × {1𝑜}))) = ((𝐴 × (𝐵 × {∅})) ∩ (𝐴 × (𝐶 × {1𝑜})))
20 xp0 5471 . . . . . 6 (𝐴 × ∅) = ∅
2118, 19, 203eqtr3i 2640 . . . . 5 ((𝐴 × (𝐵 × {∅})) ∩ (𝐴 × (𝐶 × {1𝑜}))) = ∅
2221a1i 11 . . . 4 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ((𝐴 × (𝐵 × {∅})) ∩ (𝐴 × (𝐶 × {1𝑜}))) = ∅)
23 cdaenun 8879 . . . 4 (((𝐴 × 𝐵) ≈ (𝐴 × (𝐵 × {∅})) ∧ (𝐴 × 𝐶) ≈ (𝐴 × (𝐶 × {1𝑜})) ∧ ((𝐴 × (𝐵 × {∅})) ∩ (𝐴 × (𝐶 × {1𝑜}))) = ∅) → ((𝐴 × 𝐵) +𝑐 (𝐴 × 𝐶)) ≈ ((𝐴 × (𝐵 × {∅})) ∪ (𝐴 × (𝐶 × {1𝑜}))))
249, 16, 22, 23syl3anc 1318 . . 3 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ((𝐴 × 𝐵) +𝑐 (𝐴 × 𝐶)) ≈ ((𝐴 × (𝐵 × {∅})) ∪ (𝐴 × (𝐶 × {1𝑜}))))
25 cdaval 8875 . . . . . 6 ((𝐵𝑊𝐶𝑋) → (𝐵 +𝑐 𝐶) = ((𝐵 × {∅}) ∪ (𝐶 × {1𝑜})))
26253adant1 1072 . . . . 5 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐵 +𝑐 𝐶) = ((𝐵 × {∅}) ∪ (𝐶 × {1𝑜})))
2726xpeq2d 5063 . . . 4 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐴 × (𝐵 +𝑐 𝐶)) = (𝐴 × ((𝐵 × {∅}) ∪ (𝐶 × {1𝑜}))))
28 xpundi 5094 . . . 4 (𝐴 × ((𝐵 × {∅}) ∪ (𝐶 × {1𝑜}))) = ((𝐴 × (𝐵 × {∅})) ∪ (𝐴 × (𝐶 × {1𝑜})))
2927, 28syl6eq 2660 . . 3 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐴 × (𝐵 +𝑐 𝐶)) = ((𝐴 × (𝐵 × {∅})) ∪ (𝐴 × (𝐶 × {1𝑜}))))
3024, 29breqtrrd 4611 . 2 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ((𝐴 × 𝐵) +𝑐 (𝐴 × 𝐶)) ≈ (𝐴 × (𝐵 +𝑐 𝐶)))
3130ensymd 7893 1 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐴 × (𝐵 +𝑐 𝐶)) ≈ ((𝐴 × 𝐵) +𝑐 (𝐴 × 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1031   = wceq 1475  wcel 1977  Vcvv 3173  cun 3538  cin 3539  c0 3874  {csn 4125   class class class wbr 4583   × cxp 5036  Oncon0 5640  (class class class)co 6549  1𝑜c1o 7440  cen 7838   +𝑐 ccda 8872
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-ord 5643  df-on 5644  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-1o 7447  df-er 7629  df-en 7842  df-dom 7843  df-cda 8873
This theorem is referenced by: (None)
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