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Theorem xpcdaen 9189
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 8145 . . . . . 6 (𝐴𝑉𝐴𝐴)
213ad2ant1 1127 . . . . 5 ((𝐴𝑉𝐵𝑊𝐶𝑋) → 𝐴𝐴)
3 simp2 1131 . . . . . . 7 ((𝐴𝑉𝐵𝑊𝐶𝑋) → 𝐵𝑊)
4 0ex 4934 . . . . . . 7 ∅ ∈ V
5 xpsneng 8202 . . . . . . 7 ((𝐵𝑊 ∧ ∅ ∈ V) → (𝐵 × {∅}) ≈ 𝐵)
63, 4, 5sylancl 697 . . . . . 6 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐵 × {∅}) ≈ 𝐵)
76ensymd 8164 . . . . 5 ((𝐴𝑉𝐵𝑊𝐶𝑋) → 𝐵 ≈ (𝐵 × {∅}))
8 xpen 8280 . . . . 5 ((𝐴𝐴𝐵 ≈ (𝐵 × {∅})) → (𝐴 × 𝐵) ≈ (𝐴 × (𝐵 × {∅})))
92, 7, 8syl2anc 696 . . . 4 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐴 × 𝐵) ≈ (𝐴 × (𝐵 × {∅})))
10 simp3 1132 . . . . . . 7 ((𝐴𝑉𝐵𝑊𝐶𝑋) → 𝐶𝑋)
11 1on 7728 . . . . . . 7 1𝑜 ∈ On
12 xpsneng 8202 . . . . . . 7 ((𝐶𝑋 ∧ 1𝑜 ∈ On) → (𝐶 × {1𝑜}) ≈ 𝐶)
1310, 11, 12sylancl 697 . . . . . 6 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐶 × {1𝑜}) ≈ 𝐶)
1413ensymd 8164 . . . . 5 ((𝐴𝑉𝐵𝑊𝐶𝑋) → 𝐶 ≈ (𝐶 × {1𝑜}))
15 xpen 8280 . . . . 5 ((𝐴𝐴𝐶 ≈ (𝐶 × {1𝑜})) → (𝐴 × 𝐶) ≈ (𝐴 × (𝐶 × {1𝑜})))
162, 14, 15syl2anc 696 . . . 4 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐴 × 𝐶) ≈ (𝐴 × (𝐶 × {1𝑜})))
17 xp01disj 7737 . . . . . . 7 ((𝐵 × {∅}) ∩ (𝐶 × {1𝑜})) = ∅
1817xpeq2i 5285 . . . . . 6 (𝐴 × ((𝐵 × {∅}) ∩ (𝐶 × {1𝑜}))) = (𝐴 × ∅)
19 xpindi 5403 . . . . . 6 (𝐴 × ((𝐵 × {∅}) ∩ (𝐶 × {1𝑜}))) = ((𝐴 × (𝐵 × {∅})) ∩ (𝐴 × (𝐶 × {1𝑜})))
20 xp0 5702 . . . . . 6 (𝐴 × ∅) = ∅
2118, 19, 203eqtr3i 2782 . . . . 5 ((𝐴 × (𝐵 × {∅})) ∩ (𝐴 × (𝐶 × {1𝑜}))) = ∅
2221a1i 11 . . . 4 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ((𝐴 × (𝐵 × {∅})) ∩ (𝐴 × (𝐶 × {1𝑜}))) = ∅)
23 cdaenun 9180 . . . 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 1473 . . 3 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ((𝐴 × 𝐵) +𝑐 (𝐴 × 𝐶)) ≈ ((𝐴 × (𝐵 × {∅})) ∪ (𝐴 × (𝐶 × {1𝑜}))))
25 cdaval 9176 . . . . . 6 ((𝐵𝑊𝐶𝑋) → (𝐵 +𝑐 𝐶) = ((𝐵 × {∅}) ∪ (𝐶 × {1𝑜})))
26253adant1 1124 . . . . 5 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐵 +𝑐 𝐶) = ((𝐵 × {∅}) ∪ (𝐶 × {1𝑜})))
2726xpeq2d 5288 . . . 4 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐴 × (𝐵 +𝑐 𝐶)) = (𝐴 × ((𝐵 × {∅}) ∪ (𝐶 × {1𝑜}))))
28 xpundi 5320 . . . 4 (𝐴 × ((𝐵 × {∅}) ∪ (𝐶 × {1𝑜}))) = ((𝐴 × (𝐵 × {∅})) ∪ (𝐴 × (𝐶 × {1𝑜})))
2927, 28syl6eq 2802 . . 3 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐴 × (𝐵 +𝑐 𝐶)) = ((𝐴 × (𝐵 × {∅})) ∪ (𝐴 × (𝐶 × {1𝑜}))))
3024, 29breqtrrd 4824 . 2 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ((𝐴 × 𝐵) +𝑐 (𝐴 × 𝐶)) ≈ (𝐴 × (𝐵 +𝑐 𝐶)))
3130ensymd 8164 1 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐴 × (𝐵 +𝑐 𝐶)) ≈ ((𝐴 × 𝐵) +𝑐 (𝐴 × 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1072   = wceq 1624  wcel 2131  Vcvv 3332  cun 3705  cin 3706  c0 4050  {csn 4313   class class class wbr 4796   × cxp 5256  Oncon0 5876  (class class class)co 6805  1𝑜c1o 7714  cen 8110   +𝑐 ccda 9173
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-ral 3047  df-rex 3048  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-pss 3723  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-tp 4318  df-op 4320  df-uni 4581  df-int 4620  df-br 4797  df-opab 4857  df-mpt 4874  df-tr 4897  df-id 5166  df-eprel 5171  df-po 5179  df-so 5180  df-fr 5217  df-we 5219  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-ord 5879  df-on 5880  df-suc 5882  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-f1 6046  df-fo 6047  df-f1o 6048  df-fv 6049  df-ov 6808  df-oprab 6809  df-mpt2 6810  df-1st 7325  df-2nd 7326  df-1o 7721  df-er 7903  df-en 8114  df-dom 8115  df-cda 9174
This theorem is referenced by: (None)
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