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Theorem xpcdaen 9284
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 8218 . . . . . 6 (𝐴𝑉𝐴𝐴)
213ad2ant1 1156 . . . . 5 ((𝐴𝑉𝐵𝑊𝐶𝑋) → 𝐴𝐴)
3 simp2 1160 . . . . . . 7 ((𝐴𝑉𝐵𝑊𝐶𝑋) → 𝐵𝑊)
4 0ex 4978 . . . . . . 7 ∅ ∈ V
5 xpsneng 8278 . . . . . . 7 ((𝐵𝑊 ∧ ∅ ∈ V) → (𝐵 × {∅}) ≈ 𝐵)
63, 4, 5sylancl 576 . . . . . 6 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐵 × {∅}) ≈ 𝐵)
76ensymd 8237 . . . . 5 ((𝐴𝑉𝐵𝑊𝐶𝑋) → 𝐵 ≈ (𝐵 × {∅}))
8 xpen 8356 . . . . 5 ((𝐴𝐴𝐵 ≈ (𝐵 × {∅})) → (𝐴 × 𝐵) ≈ (𝐴 × (𝐵 × {∅})))
92, 7, 8syl2anc 575 . . . 4 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐴 × 𝐵) ≈ (𝐴 × (𝐵 × {∅})))
10 simp3 1161 . . . . . . 7 ((𝐴𝑉𝐵𝑊𝐶𝑋) → 𝐶𝑋)
11 1on 7797 . . . . . . 7 1𝑜 ∈ On
12 xpsneng 8278 . . . . . . 7 ((𝐶𝑋 ∧ 1𝑜 ∈ On) → (𝐶 × {1𝑜}) ≈ 𝐶)
1310, 11, 12sylancl 576 . . . . . 6 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐶 × {1𝑜}) ≈ 𝐶)
1413ensymd 8237 . . . . 5 ((𝐴𝑉𝐵𝑊𝐶𝑋) → 𝐶 ≈ (𝐶 × {1𝑜}))
15 xpen 8356 . . . . 5 ((𝐴𝐴𝐶 ≈ (𝐶 × {1𝑜})) → (𝐴 × 𝐶) ≈ (𝐴 × (𝐶 × {1𝑜})))
162, 14, 15syl2anc 575 . . . 4 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐴 × 𝐶) ≈ (𝐴 × (𝐶 × {1𝑜})))
17 xp01disj 7807 . . . . . . 7 ((𝐵 × {∅}) ∩ (𝐶 × {1𝑜})) = ∅
1817xpeq2i 5331 . . . . . 6 (𝐴 × ((𝐵 × {∅}) ∩ (𝐶 × {1𝑜}))) = (𝐴 × ∅)
19 xpindi 5451 . . . . . 6 (𝐴 × ((𝐵 × {∅}) ∩ (𝐶 × {1𝑜}))) = ((𝐴 × (𝐵 × {∅})) ∩ (𝐴 × (𝐶 × {1𝑜})))
20 xp0 5757 . . . . . 6 (𝐴 × ∅) = ∅
2118, 19, 203eqtr3i 2832 . . . . 5 ((𝐴 × (𝐵 × {∅})) ∩ (𝐴 × (𝐶 × {1𝑜}))) = ∅
2221a1i 11 . . . 4 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ((𝐴 × (𝐵 × {∅})) ∩ (𝐴 × (𝐶 × {1𝑜}))) = ∅)
23 cdaenun 9275 . . . 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 1483 . . 3 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ((𝐴 × 𝐵) +𝑐 (𝐴 × 𝐶)) ≈ ((𝐴 × (𝐵 × {∅})) ∪ (𝐴 × (𝐶 × {1𝑜}))))
25 cdaval 9271 . . . . . 6 ((𝐵𝑊𝐶𝑋) → (𝐵 +𝑐 𝐶) = ((𝐵 × {∅}) ∪ (𝐶 × {1𝑜})))
26253adant1 1153 . . . . 5 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐵 +𝑐 𝐶) = ((𝐵 × {∅}) ∪ (𝐶 × {1𝑜})))
2726xpeq2d 5334 . . . 4 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐴 × (𝐵 +𝑐 𝐶)) = (𝐴 × ((𝐵 × {∅}) ∪ (𝐶 × {1𝑜}))))
28 xpundi 5365 . . . 4 (𝐴 × ((𝐵 × {∅}) ∪ (𝐶 × {1𝑜}))) = ((𝐴 × (𝐵 × {∅})) ∪ (𝐴 × (𝐶 × {1𝑜})))
2927, 28syl6eq 2852 . . 3 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐴 × (𝐵 +𝑐 𝐶)) = ((𝐴 × (𝐵 × {∅})) ∪ (𝐴 × (𝐶 × {1𝑜}))))
3024, 29breqtrrd 4865 . 2 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ((𝐴 × 𝐵) +𝑐 (𝐴 × 𝐶)) ≈ (𝐴 × (𝐵 +𝑐 𝐶)))
3130ensymd 8237 1 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐴 × (𝐵 +𝑐 𝐶)) ≈ ((𝐴 × 𝐵) +𝑐 (𝐴 × 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1100   = wceq 1637  wcel 2155  Vcvv 3387  cun 3761  cin 3762  c0 4110  {csn 4364   class class class wbr 4837   × cxp 5303  Oncon0 5930  (class class class)co 6868  1𝑜c1o 7783  cen 8183   +𝑐 ccda 9268
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2067  ax-7 2103  ax-8 2157  ax-9 2164  ax-10 2184  ax-11 2200  ax-12 2213  ax-13 2419  ax-ext 2781  ax-sep 4968  ax-nul 4977  ax-pow 5029  ax-pr 5090  ax-un 7173
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2060  df-eu 2633  df-mo 2634  df-clab 2789  df-cleq 2795  df-clel 2798  df-nfc 2933  df-ne 2975  df-ral 3097  df-rex 3098  df-rab 3101  df-v 3389  df-sbc 3628  df-csb 3723  df-dif 3766  df-un 3768  df-in 3770  df-ss 3777  df-pss 3779  df-nul 4111  df-if 4274  df-pw 4347  df-sn 4365  df-pr 4367  df-tp 4369  df-op 4371  df-uni 4624  df-int 4663  df-br 4838  df-opab 4900  df-mpt 4917  df-tr 4940  df-id 5213  df-eprel 5218  df-po 5226  df-so 5227  df-fr 5264  df-we 5266  df-xp 5311  df-rel 5312  df-cnv 5313  df-co 5314  df-dm 5315  df-rn 5316  df-res 5317  df-ima 5318  df-ord 5933  df-on 5934  df-suc 5936  df-iota 6058  df-fun 6097  df-fn 6098  df-f 6099  df-f1 6100  df-fo 6101  df-f1o 6102  df-fv 6103  df-ov 6871  df-oprab 6872  df-mpt2 6873  df-1st 7392  df-2nd 7393  df-1o 7790  df-er 7973  df-en 8187  df-dom 8188  df-cda 9269
This theorem is referenced by: (None)
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