Theorem addcdi 6250

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 e. NC /\ B e. NC /\ C e. NC ) -> (A ·c (B +o C)) = ((A ·c B) +o (A ·c C)))

Proof of Theorem addcdi

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ncaddccl 6144	. . 3 |- ((B e. NC /\ C e. NC ) -> (B +o C) e. NC )
2	1	3adant1 973	. 2 |- ((A e. NC /\ B e. NC /\ C e. NC ) -> (B +o C) e. NC )
3		elncs 6119	. . 3 |- ((B +o C) e. NC <-> E.x(B +o C) = Nc x)
4		vex 2862	. . . . . . 7 |- x e. _V
5	4	ncid 6123	. . . . . 6 |- x e. Nc x
6		eleq2 2414	. . . . . 6 |- ((B +o C) = Nc x -> (x e. (B +o C) <-> x e. Nc x))
7	5, 6	mpbiri 224	. . . . 5 |- ((B +o C) = Nc x -> x e. (B +o C))
8		eladdc 4398	. . . . . 6 |- (x e. (B +o C) <-> E.y e. B E.z e. C ((y i^i z) = (/) /\ x = (y u. z)))
9		ncseqnc 6128	. . . . . . . . . 10 |- (B e. NC -> (B = Nc y <-> y e. B))
10		ncseqnc 6128	. . . . . . . . . 10 |- (C e. NC -> (C = Nc z <-> z e. C))
11	9, 10	bi2anan9 843	. . . . . . . . 9 |- ((B e. NC /\ C e. NC ) -> ((B = Nc y /\ C = Nc z) <-> (y e. B /\ z e. C)))
12	11	3adant1 973	. . . . . . . 8 |- ((A e. NC /\ B e. NC /\ C e. NC ) -> ((B = Nc y /\ C = Nc z) <-> (y e. B /\ z e. C)))
13		elncs 6119	. . . . . . . . . . . 12 |- (A e. NC <-> E.x A = Nc x)
14		vex 2862	. . . . . . . . . . . . . . . . 17 |- y e. _V
15		vex 2862	. . . . . . . . . . . . . . . . 17 |- z e. _V
16	14, 15	ncdisjun 6136	. . . . . . . . . . . . . . . 16 |- ((y i^i z) = (/) -> Nc (y u. z) = ( Nc y +o Nc z))
17	16	oveq2d 5538	. . . . . . . . . . . . . . 15 |- ((y i^i z) = (/) -> ( Nc x ·c Nc (y u. z)) = ( Nc x ·c ( Nc y +o Nc z)))
18		xpdisj2 5048	. . . . . . . . . . . . . . . . 17 |- ((y i^i z) = (/) -> ((x X. y) i^i (x X. z)) = (/))
19	4, 14	xpex 5115	. . . . . . . . . . . . . . . . . 18 |- (x X. y) e. _V
20	4, 15	xpex 5115	. . . . . . . . . . . . . . . . . 18 |- (x X. z) e. _V
21	19, 20	ncdisjun 6136	. . . . . . . . . . . . . . . . 17 |- (((x X. y) i^i (x X. z)) = (/) -> Nc ((x X. y) u. (x X. z)) = ( Nc (x X. y) +o Nc (x X. z)))
22	18, 21	syl 15	. . . . . . . . . . . . . . . 16 |- ((y i^i z) = (/) -> Nc ((x X. y) u. (x X. z)) = ( Nc (x X. y) +o Nc (x X. z)))
23	14, 15	unex 4106	. . . . . . . . . . . . . . . . . 18 |- (y u. z) e. _V
24	4, 23	mucnc 6131	. . . . . . . . . . . . . . . . 17 |- ( Nc x ·c Nc (y u. z)) = Nc (x X. (y u. z))
25		xpundi 4832	. . . . . . . . . . . . . . . . . 18 |- (x X. (y u. z)) = ((x X. y) u. (x X. z))
26	25	nceqi 6109	. . . . . . . . . . . . . . . . 17 |- Nc (x X. (y u. z)) = Nc ((x X. y) u. (x X. z))
27	24, 26	eqtri 2373	. . . . . . . . . . . . . . . 16 |- ( Nc x ·c Nc (y u. z)) = Nc ((x X. y) u. (x X. z))
28	4, 14	mucnc 6131	. . . . . . . . . . . . . . . . 17 |- ( Nc x ·c Nc y) = Nc (x X. y)
29	4, 15	mucnc 6131	. . . . . . . . . . . . . . . . 17 |- ( Nc x ·c Nc z) = Nc (x X. z)
30	28, 29	addceq12i 4388	. . . . . . . . . . . . . . . 16 |- (( Nc x ·c Nc y) +o ( Nc x ·c Nc z)) = ( Nc (x X. y) +o Nc (x X. z))
31	22, 27, 30	3eqtr4g 2410	. . . . . . . . . . . . . . 15 |- ((y i^i z) = (/) -> ( Nc x ·c Nc (y u. z)) = (( Nc x ·c Nc y) +o ( Nc x ·c Nc z)))
32	17, 31	eqtr3d 2387	. . . . . . . . . . . . . 14 |- ((y i^i z) = (/) -> ( Nc x ·c ( Nc y +o Nc z)) = (( Nc x ·c Nc y) +o ( Nc x ·c Nc z)))
33		oveq1 5530	. . . . . . . . . . . . . . 15 |- (A = Nc x -> (A ·c ( Nc y +o Nc z)) = ( Nc x ·c ( Nc y +o 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))
36	34, 35	addceq12d 4391	. . . . . . . . . . . . . . 15 |- (A = Nc x -> ((A ·c Nc y) +o (A ·c Nc z)) = (( Nc x ·c Nc y) +o ( Nc x ·c Nc z)))
37	33, 36	eqeq12d 2367	. . . . . . . . . . . . . 14 |- (A = Nc x -> ((A ·c ( Nc y +o Nc z)) = ((A ·c Nc y) +o (A ·c Nc z)) <-> ( Nc x ·c ( Nc y +o Nc z)) = (( Nc x ·c Nc y) +o ( Nc x ·c Nc z))))
38	32, 37	syl5ibr 212	. . . . . . . . . . . . 13 |- (A = Nc x -> ((y i^i z) = (/) -> (A ·c ( Nc y +o Nc z)) = ((A ·c Nc y) +o (A ·c Nc z))))
39	38	exlimiv 1634	. . . . . . . . . . . 12 |- (E.x A = Nc x -> ((y i^i z) = (/) -> (A ·c ( Nc y +o Nc z)) = ((A ·c Nc y) +o (A ·c Nc z))))
40	13, 39	sylbi 187	. . . . . . . . . . 11 |- (A e. NC -> ((y i^i z) = (/) -> (A ·c ( Nc y +o Nc z)) = ((A ·c Nc y) +o (A ·c Nc z))))
41	40	adantrd 454	. . . . . . . . . 10 |- (A e. NC -> (((y i^i z) = (/) /\ x = (y u. z)) -> (A ·c ( Nc y +o Nc z)) = ((A ·c Nc y) +o (A ·c Nc z))))
42		addceq12 4385	. . . . . . . . . . . . 13 |- ((B = Nc y /\ C = Nc z) -> (B +o C) = ( Nc y +o Nc z))
43	42	oveq2d 5538	. . . . . . . . . . . 12 |- ((B = Nc y /\ C = Nc z) -> (A ·c (B +o C)) = (A ·c ( Nc y +o Nc z)))
44		oveq2 5531	. . . . . . . . . . . . . 14 |- (B = Nc y -> (A ·c B) = (A ·c Nc y))
45	44	adantr 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))
47	46	adantl 452	. . . . . . . . . . . . 13 |- ((B = Nc y /\ C = Nc z) -> (A ·c C) = (A ·c Nc z))
48	45, 47	addceq12d 4391	. . . . . . . . . . . 12 |- ((B = Nc y /\ C = Nc z) -> ((A ·c B) +o (A ·c C)) = ((A ·c Nc y) +o (A ·c Nc z)))
49	43, 48	eqeq12d 2367	. . . . . . . . . . 11 |- ((B = Nc y /\ C = Nc z) -> ((A ·c (B +o C)) = ((A ·c B) +o (A ·c C)) <-> (A ·c ( Nc y +o Nc z)) = ((A ·c Nc y) +o (A ·c Nc z))))
50	49	imbi2d 307	. . . . . . . . . 10 |- ((B = Nc y /\ C = Nc z) -> ((((y i^i z) = (/) /\ x = (y u. z)) -> (A ·c (B +o C)) = ((A ·c B) +o (A ·c C))) <-> (((y i^i z) = (/) /\ x = (y u. z)) -> (A ·c ( Nc y +o Nc z)) = ((A ·c Nc y) +o (A ·c Nc z)))))
51	41, 50	syl5ibrcom 213	. . . . . . . . 9 |- (A e. NC -> ((B = Nc y /\ C = Nc z) -> (((y i^i z) = (/) /\ x = (y u. z)) -> (A ·c (B +o C)) = ((A ·c B) +o (A ·c C)))))
52	51	3ad2ant1 976	. . . . . . . 8 |- ((A e. NC /\ B e. NC /\ C e. NC ) -> ((B = Nc y /\ C = Nc z) -> (((y i^i z) = (/) /\ x = (y u. z)) -> (A ·c (B +o C)) = ((A ·c B) +o (A ·c C)))))
53	12, 52	sylbird 226	. . . . . . 7 |- ((A e. NC /\ B e. NC /\ C e. NC ) -> ((y e. B /\ z e. C) -> (((y i^i z) = (/) /\ x = (y u. z)) -> (A ·c (B +o C)) = ((A ·c B) +o (A ·c C)))))
54	53	rexlimdvv 2744	. . . . . 6 |- ((A e. NC /\ B e. NC /\ C e. NC ) -> (E.y e. B E.z e. C ((y i^i z) = (/) /\ x = (y u. z)) -> (A ·c (B +o C)) = ((A ·c B) +o (A ·c C))))
55	8, 54	syl5bi 208	. . . . 5 |- ((A e. NC /\ B e. NC /\ C e. NC ) -> (x e. (B +o C) -> (A ·c (B +o C)) = ((A ·c B) +o (A ·c C))))
56	7, 55	syl5 28	. . . 4 |- ((A e. NC /\ B e. NC /\ C e. NC ) -> ((B +o C) = Nc x -> (A ·c (B +o C)) = ((A ·c B) +o (A ·c C))))
57	56	exlimdv 1636	. . 3 |- ((A e. NC /\ B e. NC /\ C e. NC ) -> (E.x(B +o C) = Nc x -> (A ·c (B +o C)) = ((A ·c B) +o (A ·c C))))
58	3, 57	syl5bi 208	. 2 |- ((A e. NC /\ B e. NC /\ C e. NC ) -> ((B +o C) e. NC -> (A ·c (B +o C)) = ((A ·c B) +o (A ·c C))))
59	2, 58	mpd 14	1 |- ((A e. NC /\ B e. NC /\ C e. NC ) -> (A ·c (B +o C)) = ((A ·c B) +o (A ·c C)))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358   /\ w3a 934  E.wex 1541   = wceq 1642   e. wcel 1710  E.wrex 2615   u. cun 3207   i^i cin 3208  (/)c0 3550   +o cplc 4375   X. cxp 4770  (class class class)co 5525   NC cncs 6088   Nc cnc 6091   ·_c cmuc 6092

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-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