Theorem addcdi 6251

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)))

Proof of Theorem addcdi

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

Step	Hyp	Ref	Expression
1		ncaddccl 6145	. . 3 ⊢ ((B ∈ NC ∧ C ∈ NC ) → (B +c C) ∈ NC )
2	1	3adant1 973	. 2 ⊢ ((A ∈ NC ∧ B ∈ NC ∧ C ∈ NC ) → (B +c C) ∈ NC )
3		elncs 6120	. . 3 ⊢ ((B +c C) ∈ NC ↔ ∃x(B +c C) = Nc x)
4		vex 2863	. . . . . . 7 ⊢ x ∈ V
5	4	ncid 6124	. . . . . 6 ⊢ x ∈ Nc x
6		eleq2 2414	. . . . . 6 ⊢ ((B +c C) = Nc x → (x ∈ (B +c C) ↔ x ∈ Nc x))
7	5, 6	mpbiri 224	. . . . 5 ⊢ ((B +c C) = Nc x → x ∈ (B +c C))
8		eladdc 4399	. . . . . 6 ⊢ (x ∈ (B +c C) ↔ ∃y ∈ B ∃z ∈ C ((y ∩ z) = ∅ ∧ x = (y ∪ z)))
9		ncseqnc 6129	. . . . . . . . . 10 ⊢ (B ∈ NC → (B = Nc y ↔ y ∈ B))
10		ncseqnc 6129	. . . . . . . . . 10 ⊢ (C ∈ NC → (C = Nc z ↔ z ∈ C))
11	9, 10	bi2anan9 843	. . . . . . . . 9 ⊢ ((B ∈ NC ∧ C ∈ NC ) → ((B = Nc y ∧ C = Nc z) ↔ (y ∈ B ∧ z ∈ C)))
12	11	3adant1 973	. . . . . . . 8 ⊢ ((A ∈ NC ∧ B ∈ NC ∧ C ∈ NC ) → ((B = Nc y ∧ C = Nc z) ↔ (y ∈ B ∧ z ∈ C)))
13		elncs 6120	. . . . . . . . . . . 12 ⊢ (A ∈ NC ↔ ∃x A = Nc x)
14		vex 2863	. . . . . . . . . . . . . . . . 17 ⊢ y ∈ V
15		vex 2863	. . . . . . . . . . . . . . . . 17 ⊢ z ∈ V
16	14, 15	ncdisjun 6137	. . . . . . . . . . . . . . . 16 ⊢ ((y ∩ z) = ∅ → Nc (y ∪ z) = ( Nc y +c Nc z))
17	16	oveq2d 5539	. . . . . . . . . . . . . . 15 ⊢ ((y ∩ z) = ∅ → ( Nc x ·c Nc (y ∪ z)) = ( Nc x ·c ( Nc y +c Nc z)))
18		xpdisj2 5049	. . . . . . . . . . . . . . . . 17 ⊢ ((y ∩ z) = ∅ → ((x × y) ∩ (x × z)) = ∅)
19	4, 14	xpex 5116	. . . . . . . . . . . . . . . . . 18 ⊢ (x × y) ∈ V
20	4, 15	xpex 5116	. . . . . . . . . . . . . . . . . 18 ⊢ (x × z) ∈ V
21	19, 20	ncdisjun 6137	. . . . . . . . . . . . . . . . 17 ⊢ (((x × y) ∩ (x × z)) = ∅ → Nc ((x × y) ∪ (x × z)) = ( Nc (x × y) +c Nc (x × z)))
22	18, 21	syl 15	. . . . . . . . . . . . . . . 16 ⊢ ((y ∩ z) = ∅ → Nc ((x × y) ∪ (x × z)) = ( Nc (x × y) +c Nc (x × z)))
23	14, 15	unex 4107	. . . . . . . . . . . . . . . . . 18 ⊢ (y ∪ z) ∈ V
24	4, 23	mucnc 6132	. . . . . . . . . . . . . . . . 17 ⊢ ( Nc x ·c Nc (y ∪ z)) = Nc (x × (y ∪ z))
25		xpundi 4833	. . . . . . . . . . . . . . . . . 18 ⊢ (x × (y ∪ z)) = ((x × y) ∪ (x × z))
26	25	nceqi 6110	. . . . . . . . . . . . . . . . 17 ⊢ Nc (x × (y ∪ z)) = Nc ((x × y) ∪ (x × z))
27	24, 26	eqtri 2373	. . . . . . . . . . . . . . . 16 ⊢ ( Nc x ·c Nc (y ∪ z)) = Nc ((x × y) ∪ (x × z))
28	4, 14	mucnc 6132	. . . . . . . . . . . . . . . . 17 ⊢ ( Nc x ·c Nc y) = Nc (x × y)
29	4, 15	mucnc 6132	. . . . . . . . . . . . . . . . 17 ⊢ ( Nc x ·c Nc z) = Nc (x × z)
30	28, 29	addceq12i 4389	. . . . . . . . . . . . . . . 16 ⊢ (( Nc x ·c Nc y) +c ( Nc x ·c Nc z)) = ( Nc (x × y) +c Nc (x × z))
31	22, 27, 30	3eqtr4g 2410	. . . . . . . . . . . . . . 15 ⊢ ((y ∩ z) = ∅ → ( Nc x ·c Nc (y ∪ z)) = (( Nc x ·c Nc y) +c ( Nc x ·c Nc z)))
32	17, 31	eqtr3d 2387	. . . . . . . . . . . . . 14 ⊢ ((y ∩ z) = ∅ → ( Nc x ·c ( Nc y +c Nc z)) = (( Nc x ·c Nc y) +c ( Nc x ·c Nc z)))
33		oveq1 5531	. . . . . . . . . . . . . . 15 ⊢ (A = Nc x → (A ·c ( Nc y +c Nc z)) = ( Nc x ·c ( Nc y +c Nc z)))
34		oveq1 5531	. . . . . . . . . . . . . . . 16 ⊢ (A = Nc x → (A ·c Nc y) = ( Nc x ·c Nc y))
35		oveq1 5531	. . . . . . . . . . . . . . . 16 ⊢ (A = Nc x → (A ·c Nc z) = ( Nc x ·c Nc z))
36	34, 35	addceq12d 4392	. . . . . . . . . . . . . . 15 ⊢ (A = Nc x → ((A ·c Nc y) +c (A ·c Nc z)) = (( Nc x ·c Nc y) +c ( Nc x ·c Nc z)))
37	33, 36	eqeq12d 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))))
38	32, 37	syl5ibr 212	. . . . . . . . . . . . 13 ⊢ (A = Nc x → ((y ∩ z) = ∅ → (A ·c ( Nc y +c Nc z)) = ((A ·c Nc y) +c (A ·c Nc z))))
39	38	exlimiv 1634	. . . . . . . . . . . 12 ⊢ (∃x A = Nc x → ((y ∩ z) = ∅ → (A ·c ( Nc y +c Nc z)) = ((A ·c Nc y) +c (A ·c Nc z))))
40	13, 39	sylbi 187	. . . . . . . . . . 11 ⊢ (A ∈ NC → ((y ∩ z) = ∅ → (A ·c ( Nc y +c Nc z)) = ((A ·c Nc y) +c (A ·c Nc z))))
41	40	adantrd 454	. . . . . . . . . 10 ⊢ (A ∈ NC → (((y ∩ z) = ∅ ∧ x = (y ∪ z)) → (A ·c ( Nc y +c Nc z)) = ((A ·c Nc y) +c (A ·c Nc z))))
42		addceq12 4386	. . . . . . . . . . . . 13 ⊢ ((B = Nc y ∧ C = Nc z) → (B +c C) = ( Nc y +c Nc z))
43	42	oveq2d 5539	. . . . . . . . . . . 12 ⊢ ((B = Nc y ∧ C = Nc z) → (A ·c (B +c C)) = (A ·c ( Nc y +c Nc z)))
44		oveq2 5532	. . . . . . . . . . . . . 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 5532	. . . . . . . . . . . . . 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 4392	. . . . . . . . . . . 12 ⊢ ((B = Nc y ∧ C = Nc z) → ((A ·c B) +c (A ·c C)) = ((A ·c Nc y) +c (A ·c Nc z)))
49	43, 48	eqeq12d 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))))
50	49	imbi2d 307	. . . . . . . . . 10 ⊢ ((B = Nc y ∧ C = Nc z) → ((((y ∩ z) = ∅ ∧ x = (y ∪ z)) → (A ·c (B +c C)) = ((A ·c B) +c (A ·c C))) ↔ (((y ∩ z) = ∅ ∧ x = (y ∪ z)) → (A ·c ( Nc y +c Nc z)) = ((A ·c Nc y) +c (A ·c Nc z)))))
51	41, 50	syl5ibrcom 213	. . . . . . . . 9 ⊢ (A ∈ NC → ((B = Nc y ∧ C = Nc z) → (((y ∩ z) = ∅ ∧ x = (y ∪ z)) → (A ·c (B +c C)) = ((A ·c B) +c (A ·c C)))))
52	51	3ad2ant1 976	. . . . . . . 8 ⊢ ((A ∈ NC ∧ B ∈ NC ∧ C ∈ NC ) → ((B = Nc y ∧ C = Nc z) → (((y ∩ z) = ∅ ∧ x = (y ∪ z)) → (A ·c (B +c C)) = ((A ·c B) +c (A ·c C)))))
53	12, 52	sylbird 226	. . . . . . 7 ⊢ ((A ∈ NC ∧ B ∈ NC ∧ C ∈ NC ) → ((y ∈ B ∧ z ∈ C) → (((y ∩ z) = ∅ ∧ x = (y ∪ z)) → (A ·c (B +c C)) = ((A ·c B) +c (A ·c C)))))
54	53	rexlimdvv 2745	. . . . . 6 ⊢ ((A ∈ NC ∧ B ∈ NC ∧ C ∈ NC ) → (∃y ∈ B ∃z ∈ C ((y ∩ z) = ∅ ∧ x = (y ∪ z)) → (A ·c (B +c C)) = ((A ·c B) +c (A ·c C))))
55	8, 54	syl5bi 208	. . . . 5 ⊢ ((A ∈ NC ∧ B ∈ NC ∧ C ∈ NC ) → (x ∈ (B +c C) → (A ·c (B +c C)) = ((A ·c B) +c (A ·c C))))
56	7, 55	syl5 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))))
57	56	exlimdv 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))))
58	3, 57	syl5bi 208	. 2 ⊢ ((A ∈ NC ∧ B ∈ NC ∧ C ∈ NC ) → ((B +c C) ∈ NC → (A ·c (B +c C)) = ((A ·c B) +c (A ·c C))))
59	2, 58	mpd 14	1 ⊢ ((A ∈ NC ∧ B ∈ NC ∧ C ∈ NC ) → (A ·c (B +c C)) = ((A ·c B) +c (A ·c C)))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   ∧ w3a 934  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃wrex 2616   ∪ cun 3208   ∩ cin 3209  ∅c0 3551   +_c cplc 4376   × cxp 4771  (class class class)co 5526   NC cncs 6089   Nc cnc 6092   ·_c cmuc 6093

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 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

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 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-csb 3138  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-iun 3972  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-1st 4724  df-swap 4725  df-sset 4726  df-co 4727  df-ima 4728  df-si 4729  df-id 4768  df-xp 4785  df-cnv 4786  df-rn 4787  df-dm 4788  df-res 4789  df-fun 4790  df-fn 4791  df-f 4792  df-f1 4793  df-fo 4794  df-f1o 4795  df-fv 4796  df-2nd 4798  df-ov 5527  df-oprab 5529  df-mpt 5653  df-mpt2 5655  df-txp 5737  df-pprod 5739  df-ins2 5751  df-ins3 5753  df-image 5755  df-ins4 5757  df-si3 5759  df-funs 5761  df-fns 5763  df-cross 5765  df-trans 5900  df-sym 5909  df-er 5910  df-ec 5948  df-qs 5952  df-en 6030  df-ncs 6099  df-nc 6102  df-muc 6103

This theorem is referenced by:  addcdir  6252