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 Description: Distributivity law for cardinal addition and multiplication. Theorem XI.2.31 of [Rosser] p. 379. (Contributed by Scott Fenton, 31-Jul-2019.)
Assertion
Ref Expression
addcdi ((A NC B NC C NC ) → (A ·c (B +c C)) = ((A ·c B) +c (A ·c C)))

Dummy variables x y z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ncaddccl 6144 . . 3 ((B NC C NC ) → (B +c C) NC )
213adant1 973 . 2 ((A NC B NC C NC ) → (B +c C) NC )
3 elncs 6119 . . 3 ((B +c C) NCx(B +c C) = Nc x)
4 vex 2862 . . . . . . 7 x V
54ncid 6123 . . . . . 6 x Nc x
6 eleq2 2414 . . . . . 6 ((B +c C) = Nc x → (x (B +c C) ↔ x Nc x))
75, 6mpbiri 224 . . . . 5 ((B +c C) = Nc xx (B +c C))
8 eladdc 4398 . . . . . 6 (x (B +c C) ↔ y B z C ((yz) = x = (yz)))
9 ncseqnc 6128 . . . . . . . . . 10 (B NC → (B = Nc yy B))
10 ncseqnc 6128 . . . . . . . . . 10 (C NC → (C = Nc zz C))
119, 10bi2anan9 843 . . . . . . . . 9 ((B NC C NC ) → ((B = Nc y C = Nc z) ↔ (y B z C)))
12113adant1 973 . . . . . . . 8 ((A NC B NC C NC ) → ((B = Nc y C = Nc z) ↔ (y B z C)))
13 elncs 6119 . . . . . . . . . . . 12 (A NCx A = Nc x)
14 vex 2862 . . . . . . . . . . . . . . . . 17 y V
15 vex 2862 . . . . . . . . . . . . . . . . 17 z V
1614, 15ncdisjun 6136 . . . . . . . . . . . . . . . 16 ((yz) = Nc (yz) = ( Nc y +c Nc z))
1716oveq2d 5538 . . . . . . . . . . . . . . 15 ((yz) = → ( Nc x ·c Nc (yz)) = ( Nc x ·c ( Nc y +c Nc z)))
18 xpdisj2 5048 . . . . . . . . . . . . . . . . 17 ((yz) = → ((x × y) ∩ (x × z)) = )
194, 14xpex 5115 . . . . . . . . . . . . . . . . . 18 (x × y) V
204, 15xpex 5115 . . . . . . . . . . . . . . . . . 18 (x × z) V
2119, 20ncdisjun 6136 . . . . . . . . . . . . . . . . 17 (((x × y) ∩ (x × z)) = Nc ((x × y) ∪ (x × z)) = ( Nc (x × y) +c Nc (x × z)))
2218, 21syl 15 . . . . . . . . . . . . . . . 16 ((yz) = Nc ((x × y) ∪ (x × z)) = ( Nc (x × y) +c Nc (x × z)))
2314, 15unex 4106 . . . . . . . . . . . . . . . . . 18 (yz) V
244, 23mucnc 6131 . . . . . . . . . . . . . . . . 17 ( Nc x ·c Nc (yz)) = Nc (x × (yz))
25 xpundi 4832 . . . . . . . . . . . . . . . . . 18 (x × (yz)) = ((x × y) ∪ (x × z))
2625nceqi 6109 . . . . . . . . . . . . . . . . 17 Nc (x × (yz)) = Nc ((x × y) ∪ (x × z))
2724, 26eqtri 2373 . . . . . . . . . . . . . . . 16 ( Nc x ·c Nc (yz)) = Nc ((x × y) ∪ (x × z))
284, 14mucnc 6131 . . . . . . . . . . . . . . . . 17 ( Nc x ·c Nc y) = Nc (x × y)
294, 15mucnc 6131 . . . . . . . . . . . . . . . . 17 ( Nc x ·c Nc z) = Nc (x × z)
3028, 29addceq12i 4388 . . . . . . . . . . . . . . . 16 (( Nc x ·c Nc y) +c ( Nc x ·c Nc z)) = ( Nc (x × y) +c Nc (x × z))
3122, 27, 303eqtr4g 2410 . . . . . . . . . . . . . . 15 ((yz) = → ( Nc x ·c Nc (yz)) = (( Nc x ·c Nc y) +c ( Nc x ·c Nc z)))
3217, 31eqtr3d 2387 . . . . . . . . . . . . . 14 ((yz) = → ( Nc x ·c ( Nc y +c Nc z)) = (( Nc x ·c Nc y) +c ( Nc x ·c Nc z)))
33 oveq1 5530 . . . . . . . . . . . . . . 15 (A = Nc x → (A ·c ( Nc y +c Nc z)) = ( Nc x ·c ( Nc y +c Nc z)))
34 oveq1 5530 . . . . . . . . . . . . . . . 16 (A = Nc x → (A ·c Nc y) = ( Nc x ·c Nc y))
35 oveq1 5530 . . . . . . . . . . . . . . . 16 (A = Nc x → (A ·c Nc z) = ( Nc x ·c Nc z))
3634, 35addceq12d 4391 . . . . . . . . . . . . . . 15 (A = Nc x → ((A ·c Nc y) +c (A ·c Nc z)) = (( Nc x ·c Nc y) +c ( Nc x ·c Nc z)))
3733, 36eqeq12d 2367 . . . . . . . . . . . . . 14 (A = Nc x → ((A ·c ( Nc y +c Nc z)) = ((A ·c Nc y) +c (A ·c Nc z)) ↔ ( Nc x ·c ( Nc y +c Nc z)) = (( Nc x ·c Nc y) +c ( Nc x ·c Nc z))))
3832, 37syl5ibr 212 . . . . . . . . . . . . 13 (A = Nc x → ((yz) = → (A ·c ( Nc y +c Nc z)) = ((A ·c Nc y) +c (A ·c Nc z))))
3938exlimiv 1634 . . . . . . . . . . . 12 (x A = Nc x → ((yz) = → (A ·c ( Nc y +c Nc z)) = ((A ·c Nc y) +c (A ·c Nc z))))
4013, 39sylbi 187 . . . . . . . . . . 11 (A NC → ((yz) = → (A ·c ( Nc y +c Nc z)) = ((A ·c Nc y) +c (A ·c Nc z))))
4140adantrd 454 . . . . . . . . . 10 (A NC → (((yz) = x = (yz)) → (A ·c ( Nc y +c Nc z)) = ((A ·c Nc y) +c (A ·c Nc z))))
42 addceq12 4385 . . . . . . . . . . . . 13 ((B = Nc y C = Nc z) → (B +c C) = ( Nc y +c Nc z))
4342oveq2d 5538 . . . . . . . . . . . 12 ((B = Nc y C = Nc z) → (A ·c (B +c C)) = (A ·c ( Nc y +c Nc z)))
44 oveq2 5531 . . . . . . . . . . . . . 14 (B = Nc y → (A ·c B) = (A ·c Nc y))
4544adantr 451 . . . . . . . . . . . . 13 ((B = Nc y C = Nc z) → (A ·c B) = (A ·c Nc y))
46 oveq2 5531 . . . . . . . . . . . . . 14 (C = Nc z → (A ·c C) = (A ·c Nc z))
4746adantl 452 . . . . . . . . . . . . 13 ((B = Nc y C = Nc z) → (A ·c C) = (A ·c Nc z))
4845, 47addceq12d 4391 . . . . . . . . . . . 12 ((B = Nc y C = Nc z) → ((A ·c B) +c (A ·c C)) = ((A ·c Nc y) +c (A ·c Nc z)))
4943, 48eqeq12d 2367 . . . . . . . . . . 11 ((B = Nc y C = Nc z) → ((A ·c (B +c C)) = ((A ·c B) +c (A ·c C)) ↔ (A ·c ( Nc y +c Nc z)) = ((A ·c Nc y) +c (A ·c Nc z))))
5049imbi2d 307 . . . . . . . . . 10 ((B = Nc y C = Nc z) → ((((yz) = x = (yz)) → (A ·c (B +c C)) = ((A ·c B) +c (A ·c C))) ↔ (((yz) = x = (yz)) → (A ·c ( Nc y +c Nc z)) = ((A ·c Nc y) +c (A ·c Nc z)))))
5141, 50syl5ibrcom 213 . . . . . . . . 9 (A NC → ((B = Nc y C = Nc z) → (((yz) = x = (yz)) → (A ·c (B +c C)) = ((A ·c B) +c (A ·c C)))))
52513ad2ant1 976 . . . . . . . 8 ((A NC B NC C NC ) → ((B = Nc y C = Nc z) → (((yz) = x = (yz)) → (A ·c (B +c C)) = ((A ·c B) +c (A ·c C)))))
5312, 52sylbird 226 . . . . . . 7 ((A NC B NC C NC ) → ((y B z C) → (((yz) = x = (yz)) → (A ·c (B +c C)) = ((A ·c B) +c (A ·c C)))))
5453rexlimdvv 2744 . . . . . 6 ((A NC B NC C NC ) → (y B z C ((yz) = x = (yz)) → (A ·c (B +c C)) = ((A ·c B) +c (A ·c C))))
558, 54syl5bi 208 . . . . 5 ((A NC B NC C NC ) → (x (B +c C) → (A ·c (B +c C)) = ((A ·c B) +c (A ·c C))))
567, 55syl5 28 . . . 4 ((A NC B NC C NC ) → ((B +c C) = Nc x → (A ·c (B +c C)) = ((A ·c B) +c (A ·c C))))
5756exlimdv 1636 . . 3 ((A NC B NC C NC ) → (x(B +c C) = Nc x → (A ·c (B +c C)) = ((A ·c B) +c (A ·c C))))
583, 57syl5bi 208 . 2 ((A NC B NC C NC ) → ((B +c C) NC → (A ·c (B +c C)) = ((A ·c B) +c (A ·c C))))
592, 58mpd 14 1 ((A NC B NC C NC ) → (A ·c (B +c C)) = ((A ·c B) +c (A ·c C)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   ∧ w3a 934  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃wrex 2615   ∪ cun 3207   ∩ cin 3208  ∅c0 3550   +c cplc 4375   × cxp 4770  (class class class)co 5525   NC cncs 6088   Nc cnc 6091   ·c cmuc 6092 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  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-3or 935  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-csb 3137  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-iun 3971  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-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-1st 4723  df-swap 4724  df-sset 4725  df-co 4726  df-ima 4727  df-si 4728  df-id 4767  df-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787  df-res 4788  df-fun 4789  df-fn 4790  df-f 4791  df-f1 4792  df-fo 4793  df-f1o 4794  df-fv 4795  df-2nd 4797  df-ov 5526  df-oprab 5528  df-mpt 5652  df-mpt2 5654  df-txp 5736  df-pprod 5738  df-ins2 5750  df-ins3 5752  df-image 5754  df-ins4 5756  df-si3 5758  df-funs 5760  df-fns 5762  df-cross 5764  df-trans 5899  df-sym 5908  df-er 5909  df-ec 5947  df-qs 5951  df-en 6029  df-ncs 6098  df-nc 6101  df-muc 6102 This theorem is referenced by:  addcdir  6251
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