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Theorem mapcdaen 8889
 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 8875 . . . . 5 ((𝐵𝑊𝐶𝑋) → (𝐵 +𝑐 𝐶) = ((𝐵 × {∅}) ∪ (𝐶 × {1𝑜})))
213adant1 1072 . . . 4 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐵 +𝑐 𝐶) = ((𝐵 × {∅}) ∪ (𝐶 × {1𝑜})))
32oveq2d 6565 . . 3 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐴𝑚 (𝐵 +𝑐 𝐶)) = (𝐴𝑚 ((𝐵 × {∅}) ∪ (𝐶 × {1𝑜}))))
4 simp2 1055 . . . . 5 ((𝐴𝑉𝐵𝑊𝐶𝑋) → 𝐵𝑊)
5 snex 4835 . . . . 5 {∅} ∈ V
6 xpexg 6858 . . . . 5 ((𝐵𝑊 ∧ {∅} ∈ V) → (𝐵 × {∅}) ∈ V)
74, 5, 6sylancl 693 . . . 4 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐵 × {∅}) ∈ V)
8 simp3 1056 . . . . 5 ((𝐴𝑉𝐵𝑊𝐶𝑋) → 𝐶𝑋)
9 snex 4835 . . . . 5 {1𝑜} ∈ V
10 xpexg 6858 . . . . 5 ((𝐶𝑋 ∧ {1𝑜} ∈ V) → (𝐶 × {1𝑜}) ∈ V)
118, 9, 10sylancl 693 . . . 4 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐶 × {1𝑜}) ∈ V)
12 simp1 1054 . . . 4 ((𝐴𝑉𝐵𝑊𝐶𝑋) → 𝐴𝑉)
13 xp01disj 7463 . . . . 5 ((𝐵 × {∅}) ∩ (𝐶 × {1𝑜})) = ∅
1413a1i 11 . . . 4 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ((𝐵 × {∅}) ∩ (𝐶 × {1𝑜})) = ∅)
15 mapunen 8014 . . . 4 ((((𝐵 × {∅}) ∈ V ∧ (𝐶 × {1𝑜}) ∈ V ∧ 𝐴𝑉) ∧ ((𝐵 × {∅}) ∩ (𝐶 × {1𝑜})) = ∅) → (𝐴𝑚 ((𝐵 × {∅}) ∪ (𝐶 × {1𝑜}))) ≈ ((𝐴𝑚 (𝐵 × {∅})) × (𝐴𝑚 (𝐶 × {1𝑜}))))
167, 11, 12, 14, 15syl31anc 1321 . . 3 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐴𝑚 ((𝐵 × {∅}) ∪ (𝐶 × {1𝑜}))) ≈ ((𝐴𝑚 (𝐵 × {∅})) × (𝐴𝑚 (𝐶 × {1𝑜}))))
173, 16eqbrtrd 4605 . 2 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐴𝑚 (𝐵 +𝑐 𝐶)) ≈ ((𝐴𝑚 (𝐵 × {∅})) × (𝐴𝑚 (𝐶 × {1𝑜}))))
18 enrefg 7873 . . . . 5 (𝐴𝑉𝐴𝐴)
1912, 18syl 17 . . . 4 ((𝐴𝑉𝐵𝑊𝐶𝑋) → 𝐴𝐴)
20 0ex 4718 . . . . 5 ∅ ∈ V
21 xpsneng 7930 . . . . 5 ((𝐵𝑊 ∧ ∅ ∈ V) → (𝐵 × {∅}) ≈ 𝐵)
224, 20, 21sylancl 693 . . . 4 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐵 × {∅}) ≈ 𝐵)
23 mapen 8009 . . . 4 ((𝐴𝐴 ∧ (𝐵 × {∅}) ≈ 𝐵) → (𝐴𝑚 (𝐵 × {∅})) ≈ (𝐴𝑚 𝐵))
2419, 22, 23syl2anc 691 . . 3 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐴𝑚 (𝐵 × {∅})) ≈ (𝐴𝑚 𝐵))
25 1on 7454 . . . . 5 1𝑜 ∈ On
26 xpsneng 7930 . . . . 5 ((𝐶𝑋 ∧ 1𝑜 ∈ On) → (𝐶 × {1𝑜}) ≈ 𝐶)
278, 25, 26sylancl 693 . . . 4 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐶 × {1𝑜}) ≈ 𝐶)
28 mapen 8009 . . . 4 ((𝐴𝐴 ∧ (𝐶 × {1𝑜}) ≈ 𝐶) → (𝐴𝑚 (𝐶 × {1𝑜})) ≈ (𝐴𝑚 𝐶))
2919, 27, 28syl2anc 691 . . 3 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐴𝑚 (𝐶 × {1𝑜})) ≈ (𝐴𝑚 𝐶))
30 xpen 8008 . . 3 (((𝐴𝑚 (𝐵 × {∅})) ≈ (𝐴𝑚 𝐵) ∧ (𝐴𝑚 (𝐶 × {1𝑜})) ≈ (𝐴𝑚 𝐶)) → ((𝐴𝑚 (𝐵 × {∅})) × (𝐴𝑚 (𝐶 × {1𝑜}))) ≈ ((𝐴𝑚 𝐵) × (𝐴𝑚 𝐶)))
3124, 29, 30syl2anc 691 . 2 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ((𝐴𝑚 (𝐵 × {∅})) × (𝐴𝑚 (𝐶 × {1𝑜}))) ≈ ((𝐴𝑚 𝐵) × (𝐴𝑚 𝐶)))
32 entr 7894 . 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 691 1 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐴𝑚 (𝐵 +𝑐 𝐶)) ≈ ((𝐴𝑚 𝐵) × (𝐴𝑚 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   ↑𝑚 cmap 7744   ≈ 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-iun 4457  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-map 7746  df-en 7842  df-dom 7843  df-cda 8873 This theorem is referenced by:  pwcdaen  8890
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