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Theorem mapcdaen 9044
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 9030 . . . . 5 ((𝐵𝑊𝐶𝑋) → (𝐵 +𝑐 𝐶) = ((𝐵 × {∅}) ∪ (𝐶 × {1𝑜})))
213adant1 1099 . . . 4 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐵 +𝑐 𝐶) = ((𝐵 × {∅}) ∪ (𝐶 × {1𝑜})))
32oveq2d 6706 . . 3 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐴𝑚 (𝐵 +𝑐 𝐶)) = (𝐴𝑚 ((𝐵 × {∅}) ∪ (𝐶 × {1𝑜}))))
4 simp2 1082 . . . . 5 ((𝐴𝑉𝐵𝑊𝐶𝑋) → 𝐵𝑊)
5 snex 4938 . . . . 5 {∅} ∈ V
6 xpexg 7002 . . . . 5 ((𝐵𝑊 ∧ {∅} ∈ V) → (𝐵 × {∅}) ∈ V)
74, 5, 6sylancl 695 . . . 4 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐵 × {∅}) ∈ V)
8 simp3 1083 . . . . 5 ((𝐴𝑉𝐵𝑊𝐶𝑋) → 𝐶𝑋)
9 snex 4938 . . . . 5 {1𝑜} ∈ V
10 xpexg 7002 . . . . 5 ((𝐶𝑋 ∧ {1𝑜} ∈ V) → (𝐶 × {1𝑜}) ∈ V)
118, 9, 10sylancl 695 . . . 4 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐶 × {1𝑜}) ∈ V)
12 simp1 1081 . . . 4 ((𝐴𝑉𝐵𝑊𝐶𝑋) → 𝐴𝑉)
13 xp01disj 7621 . . . . 5 ((𝐵 × {∅}) ∩ (𝐶 × {1𝑜})) = ∅
1413a1i 11 . . . 4 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ((𝐵 × {∅}) ∩ (𝐶 × {1𝑜})) = ∅)
15 mapunen 8170 . . . 4 ((((𝐵 × {∅}) ∈ V ∧ (𝐶 × {1𝑜}) ∈ V ∧ 𝐴𝑉) ∧ ((𝐵 × {∅}) ∩ (𝐶 × {1𝑜})) = ∅) → (𝐴𝑚 ((𝐵 × {∅}) ∪ (𝐶 × {1𝑜}))) ≈ ((𝐴𝑚 (𝐵 × {∅})) × (𝐴𝑚 (𝐶 × {1𝑜}))))
167, 11, 12, 14, 15syl31anc 1369 . . 3 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐴𝑚 ((𝐵 × {∅}) ∪ (𝐶 × {1𝑜}))) ≈ ((𝐴𝑚 (𝐵 × {∅})) × (𝐴𝑚 (𝐶 × {1𝑜}))))
173, 16eqbrtrd 4707 . 2 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐴𝑚 (𝐵 +𝑐 𝐶)) ≈ ((𝐴𝑚 (𝐵 × {∅})) × (𝐴𝑚 (𝐶 × {1𝑜}))))
18 enrefg 8029 . . . . 5 (𝐴𝑉𝐴𝐴)
1912, 18syl 17 . . . 4 ((𝐴𝑉𝐵𝑊𝐶𝑋) → 𝐴𝐴)
20 0ex 4823 . . . . 5 ∅ ∈ V
21 xpsneng 8086 . . . . 5 ((𝐵𝑊 ∧ ∅ ∈ V) → (𝐵 × {∅}) ≈ 𝐵)
224, 20, 21sylancl 695 . . . 4 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐵 × {∅}) ≈ 𝐵)
23 mapen 8165 . . . 4 ((𝐴𝐴 ∧ (𝐵 × {∅}) ≈ 𝐵) → (𝐴𝑚 (𝐵 × {∅})) ≈ (𝐴𝑚 𝐵))
2419, 22, 23syl2anc 694 . . 3 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐴𝑚 (𝐵 × {∅})) ≈ (𝐴𝑚 𝐵))
25 1on 7612 . . . . 5 1𝑜 ∈ On
26 xpsneng 8086 . . . . 5 ((𝐶𝑋 ∧ 1𝑜 ∈ On) → (𝐶 × {1𝑜}) ≈ 𝐶)
278, 25, 26sylancl 695 . . . 4 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐶 × {1𝑜}) ≈ 𝐶)
28 mapen 8165 . . . 4 ((𝐴𝐴 ∧ (𝐶 × {1𝑜}) ≈ 𝐶) → (𝐴𝑚 (𝐶 × {1𝑜})) ≈ (𝐴𝑚 𝐶))
2919, 27, 28syl2anc 694 . . 3 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐴𝑚 (𝐶 × {1𝑜})) ≈ (𝐴𝑚 𝐶))
30 xpen 8164 . . 3 (((𝐴𝑚 (𝐵 × {∅})) ≈ (𝐴𝑚 𝐵) ∧ (𝐴𝑚 (𝐶 × {1𝑜})) ≈ (𝐴𝑚 𝐶)) → ((𝐴𝑚 (𝐵 × {∅})) × (𝐴𝑚 (𝐶 × {1𝑜}))) ≈ ((𝐴𝑚 𝐵) × (𝐴𝑚 𝐶)))
3124, 29, 30syl2anc 694 . 2 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ((𝐴𝑚 (𝐵 × {∅})) × (𝐴𝑚 (𝐶 × {1𝑜}))) ≈ ((𝐴𝑚 𝐵) × (𝐴𝑚 𝐶)))
32 entr 8049 . 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 694 1 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐴𝑚 (𝐵 +𝑐 𝐶)) ≈ ((𝐴𝑚 𝐵) × (𝐴𝑚 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1054   = wceq 1523  wcel 2030  Vcvv 3231  cun 3605  cin 3606  c0 3948  {csn 4210   class class class wbr 4685   × cxp 5141  Oncon0 5761  (class class class)co 6690  1𝑜c1o 7598  𝑚 cmap 7899  cen 7994   +𝑐 ccda 9027
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-ord 5764  df-on 5765  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211  df-1o 7605  df-er 7787  df-map 7901  df-en 7998  df-dom 7999  df-cda 9028
This theorem is referenced by:  pwcdaen  9045
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