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Theorem ncfindi 4475
 Description: Distribution law for finite cardinality. (Contributed by SF, 30-Jan-2015.)
Assertion
Ref Expression
ncfindi (((V Fin A V) B W (AB) = ) → Ncfin (AB) = ( Ncfin A +c Ncfin B))

Proof of Theorem ncfindi
StepHypRef Expression
1 simp1l 979 . . . 4 (((V Fin A V) B W (AB) = ) → V Fin )
2 simp1r 980 . . . . 5 (((V Fin A V) B W (AB) = ) → A V)
3 simp2 956 . . . . 5 (((V Fin A V) B W (AB) = ) → B W)
4 unexg 4101 . . . . 5 ((A V B W) → (AB) V)
52, 3, 4syl2anc 642 . . . 4 (((V Fin A V) B W (AB) = ) → (AB) V)
6 ncfinprop 4474 . . . 4 ((V Fin (AB) V) → ( Ncfin (AB) Nn (AB) Ncfin (AB)))
71, 5, 6syl2anc 642 . . 3 (((V Fin A V) B W (AB) = ) → ( Ncfin (AB) Nn (AB) Ncfin (AB)))
87simpld 445 . 2 (((V Fin A V) B W (AB) = ) → Ncfin (AB) Nn )
9 ncfinprop 4474 . . . . 5 ((V Fin A V) → ( Ncfin A Nn A Ncfin A))
101, 2, 9syl2anc 642 . . . 4 (((V Fin A V) B W (AB) = ) → ( Ncfin A Nn A Ncfin A))
1110simpld 445 . . 3 (((V Fin A V) B W (AB) = ) → Ncfin A Nn )
12 ncfinprop 4474 . . . . 5 ((V Fin B W) → ( Ncfin B Nn B Ncfin B))
131, 3, 12syl2anc 642 . . . 4 (((V Fin A V) B W (AB) = ) → ( Ncfin B Nn B Ncfin B))
1413simpld 445 . . 3 (((V Fin A V) B W (AB) = ) → Ncfin B Nn )
15 nncaddccl 4419 . . 3 (( Ncfin A Nn Ncfin B Nn ) → ( Ncfin A +c Ncfin B) Nn )
1611, 14, 15syl2anc 642 . 2 (((V Fin A V) B W (AB) = ) → ( Ncfin A +c Ncfin B) Nn )
177simprd 449 . 2 (((V Fin A V) B W (AB) = ) → (AB) Ncfin (AB))
1810simprd 449 . . 3 (((V Fin A V) B W (AB) = ) → A Ncfin A)
1913simprd 449 . . 3 (((V Fin A V) B W (AB) = ) → B Ncfin B)
20 simp3 957 . . 3 (((V Fin A V) B W (AB) = ) → (AB) = )
21 eladdci 4399 . . 3 ((A Ncfin A B Ncfin B (AB) = ) → (AB) ( Ncfin A +c Ncfin B))
2218, 19, 20, 21syl3anc 1182 . 2 (((V Fin A V) B W (AB) = ) → (AB) ( Ncfin A +c Ncfin B))
23 nnceleq 4430 . 2 ((( Ncfin (AB) Nn ( Ncfin A +c Ncfin B) Nn ) ((AB) Ncfin (AB) (AB) ( Ncfin A +c Ncfin B))) → Ncfin (AB) = ( Ncfin A +c Ncfin B))
248, 16, 17, 22, 23syl22anc 1183 1 (((V Fin A V) B W (AB) = ) → Ncfin (AB) = ( Ncfin A +c Ncfin B))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358   ∧ w3a 934   = wceq 1642   ∈ wcel 1710  Vcvv 2859   ∪ cun 3207   ∩ cin 3208  ∅c0 3550   Nn cnnc 4373   +c cplc 4375   Fin cfin 4376   Ncfin cncfin 4434 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-ncfin 4442 This theorem is referenced by:  vfintle  4546  vfin1cltv  4547  vfinncsp  4554
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