MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  xpcdaen Structured version   Visualization version   GIF version

Theorem xpcdaen 8865
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 7850 . . . . . 6 (𝐴𝑉𝐴𝐴)
213ad2ant1 1074 . . . . 5 ((𝐴𝑉𝐵𝑊𝐶𝑋) → 𝐴𝐴)
3 simp2 1054 . . . . . . 7 ((𝐴𝑉𝐵𝑊𝐶𝑋) → 𝐵𝑊)
4 0ex 4713 . . . . . . 7 ∅ ∈ V
5 xpsneng 7907 . . . . . . 7 ((𝐵𝑊 ∧ ∅ ∈ V) → (𝐵 × {∅}) ≈ 𝐵)
63, 4, 5sylancl 692 . . . . . 6 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐵 × {∅}) ≈ 𝐵)
76ensymd 7870 . . . . 5 ((𝐴𝑉𝐵𝑊𝐶𝑋) → 𝐵 ≈ (𝐵 × {∅}))
8 xpen 7985 . . . . 5 ((𝐴𝐴𝐵 ≈ (𝐵 × {∅})) → (𝐴 × 𝐵) ≈ (𝐴 × (𝐵 × {∅})))
92, 7, 8syl2anc 690 . . . 4 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐴 × 𝐵) ≈ (𝐴 × (𝐵 × {∅})))
10 simp3 1055 . . . . . . 7 ((𝐴𝑉𝐵𝑊𝐶𝑋) → 𝐶𝑋)
11 1on 7431 . . . . . . 7 1𝑜 ∈ On
12 xpsneng 7907 . . . . . . 7 ((𝐶𝑋 ∧ 1𝑜 ∈ On) → (𝐶 × {1𝑜}) ≈ 𝐶)
1310, 11, 12sylancl 692 . . . . . 6 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐶 × {1𝑜}) ≈ 𝐶)
1413ensymd 7870 . . . . 5 ((𝐴𝑉𝐵𝑊𝐶𝑋) → 𝐶 ≈ (𝐶 × {1𝑜}))
15 xpen 7985 . . . . 5 ((𝐴𝐴𝐶 ≈ (𝐶 × {1𝑜})) → (𝐴 × 𝐶) ≈ (𝐴 × (𝐶 × {1𝑜})))
162, 14, 15syl2anc 690 . . . 4 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐴 × 𝐶) ≈ (𝐴 × (𝐶 × {1𝑜})))
17 xp01disj 7440 . . . . . . 7 ((𝐵 × {∅}) ∩ (𝐶 × {1𝑜})) = ∅
1817xpeq2i 5050 . . . . . 6 (𝐴 × ((𝐵 × {∅}) ∩ (𝐶 × {1𝑜}))) = (𝐴 × ∅)
19 xpindi 5165 . . . . . 6 (𝐴 × ((𝐵 × {∅}) ∩ (𝐶 × {1𝑜}))) = ((𝐴 × (𝐵 × {∅})) ∩ (𝐴 × (𝐶 × {1𝑜})))
20 xp0 5457 . . . . . 6 (𝐴 × ∅) = ∅
2118, 19, 203eqtr3i 2639 . . . . 5 ((𝐴 × (𝐵 × {∅})) ∩ (𝐴 × (𝐶 × {1𝑜}))) = ∅
2221a1i 11 . . . 4 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ((𝐴 × (𝐵 × {∅})) ∩ (𝐴 × (𝐶 × {1𝑜}))) = ∅)
23 cdaenun 8856 . . . 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 1317 . . 3 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ((𝐴 × 𝐵) +𝑐 (𝐴 × 𝐶)) ≈ ((𝐴 × (𝐵 × {∅})) ∪ (𝐴 × (𝐶 × {1𝑜}))))
25 cdaval 8852 . . . . . 6 ((𝐵𝑊𝐶𝑋) → (𝐵 +𝑐 𝐶) = ((𝐵 × {∅}) ∪ (𝐶 × {1𝑜})))
26253adant1 1071 . . . . 5 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐵 +𝑐 𝐶) = ((𝐵 × {∅}) ∪ (𝐶 × {1𝑜})))
2726xpeq2d 5053 . . . 4 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐴 × (𝐵 +𝑐 𝐶)) = (𝐴 × ((𝐵 × {∅}) ∪ (𝐶 × {1𝑜}))))
28 xpundi 5084 . . . 4 (𝐴 × ((𝐵 × {∅}) ∪ (𝐶 × {1𝑜}))) = ((𝐴 × (𝐵 × {∅})) ∪ (𝐴 × (𝐶 × {1𝑜})))
2927, 28syl6eq 2659 . . 3 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐴 × (𝐵 +𝑐 𝐶)) = ((𝐴 × (𝐵 × {∅})) ∪ (𝐴 × (𝐶 × {1𝑜}))))
3024, 29breqtrrd 4605 . 2 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ((𝐴 × 𝐵) +𝑐 (𝐴 × 𝐶)) ≈ (𝐴 × (𝐵 +𝑐 𝐶)))
3130ensymd 7870 1 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐴 × (𝐵 +𝑐 𝐶)) ≈ ((𝐴 × 𝐵) +𝑐 (𝐴 × 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1030   = wceq 1474  wcel 1976  Vcvv 3172  cun 3537  cin 3538  c0 3873  {csn 4124   class class class wbr 4577   × cxp 5026  Oncon0 5626  (class class class)co 6527  1𝑜c1o 7417  cen 7815   +𝑐 ccda 8849
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-ord 5629  df-on 5630  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-1st 7036  df-2nd 7037  df-1o 7424  df-er 7606  df-en 7819  df-dom 7820  df-cda 8850
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator