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Theorem mapcdaen 8950
Description: Sum of exponents law for cardinal arithmetic. Theorem 6I(4) of [Enderton] p. 142. (Contributed by NM, 27-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
Assertion
Ref Expression
mapcdaen ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐴𝑚 (𝐵 +𝑐 𝐶)) ≈ ((𝐴𝑚 𝐵) × (𝐴𝑚 𝐶)))

Proof of Theorem mapcdaen
StepHypRef Expression
1 cdaval 8936 . . . . 5 ((𝐵𝑊𝐶𝑋) → (𝐵 +𝑐 𝐶) = ((𝐵 × {∅}) ∪ (𝐶 × {1𝑜})))
213adant1 1077 . . . 4 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐵 +𝑐 𝐶) = ((𝐵 × {∅}) ∪ (𝐶 × {1𝑜})))
32oveq2d 6620 . . 3 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐴𝑚 (𝐵 +𝑐 𝐶)) = (𝐴𝑚 ((𝐵 × {∅}) ∪ (𝐶 × {1𝑜}))))
4 simp2 1060 . . . . 5 ((𝐴𝑉𝐵𝑊𝐶𝑋) → 𝐵𝑊)
5 snex 4869 . . . . 5 {∅} ∈ V
6 xpexg 6913 . . . . 5 ((𝐵𝑊 ∧ {∅} ∈ V) → (𝐵 × {∅}) ∈ V)
74, 5, 6sylancl 693 . . . 4 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐵 × {∅}) ∈ V)
8 simp3 1061 . . . . 5 ((𝐴𝑉𝐵𝑊𝐶𝑋) → 𝐶𝑋)
9 snex 4869 . . . . 5 {1𝑜} ∈ V
10 xpexg 6913 . . . . 5 ((𝐶𝑋 ∧ {1𝑜} ∈ V) → (𝐶 × {1𝑜}) ∈ V)
118, 9, 10sylancl 693 . . . 4 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐶 × {1𝑜}) ∈ V)
12 simp1 1059 . . . 4 ((𝐴𝑉𝐵𝑊𝐶𝑋) → 𝐴𝑉)
13 xp01disj 7521 . . . . 5 ((𝐵 × {∅}) ∩ (𝐶 × {1𝑜})) = ∅
1413a1i 11 . . . 4 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ((𝐵 × {∅}) ∩ (𝐶 × {1𝑜})) = ∅)
15 mapunen 8073 . . . 4 ((((𝐵 × {∅}) ∈ V ∧ (𝐶 × {1𝑜}) ∈ V ∧ 𝐴𝑉) ∧ ((𝐵 × {∅}) ∩ (𝐶 × {1𝑜})) = ∅) → (𝐴𝑚 ((𝐵 × {∅}) ∪ (𝐶 × {1𝑜}))) ≈ ((𝐴𝑚 (𝐵 × {∅})) × (𝐴𝑚 (𝐶 × {1𝑜}))))
167, 11, 12, 14, 15syl31anc 1326 . . 3 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐴𝑚 ((𝐵 × {∅}) ∪ (𝐶 × {1𝑜}))) ≈ ((𝐴𝑚 (𝐵 × {∅})) × (𝐴𝑚 (𝐶 × {1𝑜}))))
173, 16eqbrtrd 4635 . 2 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐴𝑚 (𝐵 +𝑐 𝐶)) ≈ ((𝐴𝑚 (𝐵 × {∅})) × (𝐴𝑚 (𝐶 × {1𝑜}))))
18 enrefg 7931 . . . . 5 (𝐴𝑉𝐴𝐴)
1912, 18syl 17 . . . 4 ((𝐴𝑉𝐵𝑊𝐶𝑋) → 𝐴𝐴)
20 0ex 4750 . . . . 5 ∅ ∈ V
21 xpsneng 7989 . . . . 5 ((𝐵𝑊 ∧ ∅ ∈ V) → (𝐵 × {∅}) ≈ 𝐵)
224, 20, 21sylancl 693 . . . 4 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐵 × {∅}) ≈ 𝐵)
23 mapen 8068 . . . 4 ((𝐴𝐴 ∧ (𝐵 × {∅}) ≈ 𝐵) → (𝐴𝑚 (𝐵 × {∅})) ≈ (𝐴𝑚 𝐵))
2419, 22, 23syl2anc 692 . . 3 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐴𝑚 (𝐵 × {∅})) ≈ (𝐴𝑚 𝐵))
25 1on 7512 . . . . 5 1𝑜 ∈ On
26 xpsneng 7989 . . . . 5 ((𝐶𝑋 ∧ 1𝑜 ∈ On) → (𝐶 × {1𝑜}) ≈ 𝐶)
278, 25, 26sylancl 693 . . . 4 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐶 × {1𝑜}) ≈ 𝐶)
28 mapen 8068 . . . 4 ((𝐴𝐴 ∧ (𝐶 × {1𝑜}) ≈ 𝐶) → (𝐴𝑚 (𝐶 × {1𝑜})) ≈ (𝐴𝑚 𝐶))
2919, 27, 28syl2anc 692 . . 3 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐴𝑚 (𝐶 × {1𝑜})) ≈ (𝐴𝑚 𝐶))
30 xpen 8067 . . 3 (((𝐴𝑚 (𝐵 × {∅})) ≈ (𝐴𝑚 𝐵) ∧ (𝐴𝑚 (𝐶 × {1𝑜})) ≈ (𝐴𝑚 𝐶)) → ((𝐴𝑚 (𝐵 × {∅})) × (𝐴𝑚 (𝐶 × {1𝑜}))) ≈ ((𝐴𝑚 𝐵) × (𝐴𝑚 𝐶)))
3124, 29, 30syl2anc 692 . 2 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ((𝐴𝑚 (𝐵 × {∅})) × (𝐴𝑚 (𝐶 × {1𝑜}))) ≈ ((𝐴𝑚 𝐵) × (𝐴𝑚 𝐶)))
32 entr 7952 . 2 (((𝐴𝑚 (𝐵 +𝑐 𝐶)) ≈ ((𝐴𝑚 (𝐵 × {∅})) × (𝐴𝑚 (𝐶 × {1𝑜}))) ∧ ((𝐴𝑚 (𝐵 × {∅})) × (𝐴𝑚 (𝐶 × {1𝑜}))) ≈ ((𝐴𝑚 𝐵) × (𝐴𝑚 𝐶))) → (𝐴𝑚 (𝐵 +𝑐 𝐶)) ≈ ((𝐴𝑚 𝐵) × (𝐴𝑚 𝐶)))
3317, 31, 32syl2anc 692 1 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐴𝑚 (𝐵 +𝑐 𝐶)) ≈ ((𝐴𝑚 𝐵) × (𝐴𝑚 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1036   = wceq 1480  wcel 1987  Vcvv 3186  cun 3553  cin 3554  c0 3891  {csn 4148   class class class wbr 4613   × cxp 5072  Oncon0 5682  (class class class)co 6604  1𝑜c1o 7498  𝑚 cmap 7802  cen 7896   +𝑐 ccda 8933
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-ord 5685  df-on 5686  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-1st 7113  df-2nd 7114  df-1o 7505  df-er 7687  df-map 7804  df-en 7900  df-dom 7901  df-cda 8934
This theorem is referenced by:  pwcdaen  8951
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