Theorem nncaddccl 4420

Description: The finite cardinals are closed under addition. Theorem X.1.14 of [Rosser] p. 278. (Contributed by SF, 17-Jan-2015.)

Assertion

Ref	Expression
nncaddccl	⊢ ((A ∈ Nn ∧ B ∈ Nn ) → (A +c B) ∈ Nn )

Proof of Theorem nncaddccl

Dummy variables a b c x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		addceq1 4384	. . . . 5 ⊢ (a = A → (a +c B) = (A +c B))
2	1	eleq1d 2419	. . . 4 ⊢ (a = A → ((a +c B) ∈ Nn ↔ (A +c B) ∈ Nn ))
3	2	imbi2d 307	. . 3 ⊢ (a = A → ((B ∈ Nn → (a +c B) ∈ Nn ) ↔ (B ∈ Nn → (A +c B) ∈ Nn )))
4		unab 3522	. . . . . . 7 ⊢ ({b ∣ ¬ a ∈ Nn } ∪ {b ∣ (a +c b) ∈ Nn }) = {b ∣ (¬ a ∈ Nn ∨ (a +c b) ∈ Nn )}
5		vex 2863	. . . . . . . . . . . . 13 ⊢ b ∈ V
6		vex 2863	. . . . . . . . . . . . 13 ⊢ x ∈ V
7		opkelimagekg 4272	. . . . . . . . . . . . 13 ⊢ ((b ∈ V ∧ x ∈ V) → (⟪b, x⟫ ∈ Imagek(( Ins3k ∼ (( Ins3k Sk ∩ Ins2k Sk ) “k ℘1℘11c) ∖ (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k Sk ∪ Ins3k SIk SIk Sk )) “k ℘1℘1℘1℘11c)) “k ℘1℘1a) ↔ x = ((( Ins3k ∼ (( Ins3k Sk ∩ Ins2k Sk ) “k ℘1℘11c) ∖ (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k Sk ∪ Ins3k SIk SIk Sk )) “k ℘1℘1℘1℘11c)) “k ℘1℘1a) “k b)))
8	5, 6, 7	mp2an 653	. . . . . . . . . . . 12 ⊢ (⟪b, x⟫ ∈ Imagek(( Ins3k ∼ (( Ins3k Sk ∩ Ins2k Sk ) “k ℘1℘11c) ∖ (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k Sk ∪ Ins3k SIk SIk Sk )) “k ℘1℘1℘1℘11c)) “k ℘1℘1a) ↔ x = ((( Ins3k ∼ (( Ins3k Sk ∩ Ins2k Sk ) “k ℘1℘11c) ∖ (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k Sk ∪ Ins3k SIk SIk Sk )) “k ℘1℘1℘1℘11c)) “k ℘1℘1a) “k b))
9	6, 5	opkelcnvk 4251	. . . . . . . . . . . 12 ⊢ (⟪x, b⟫ ∈ ◡kImagek(( Ins3k ∼ (( Ins3k Sk ∩ Ins2k Sk ) “k ℘1℘11c) ∖ (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k Sk ∪ Ins3k SIk SIk Sk )) “k ℘1℘1℘1℘11c)) “k ℘1℘1a) ↔ ⟪b, x⟫ ∈ Imagek(( Ins3k ∼ (( Ins3k Sk ∩ Ins2k Sk ) “k ℘1℘11c) ∖ (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k Sk ∪ Ins3k SIk SIk Sk )) “k ℘1℘1℘1℘11c)) “k ℘1℘1a))
10		addccom 4407	. . . . . . . . . . . . . 14 ⊢ (a +c b) = (b +c a)
11		dfaddc2 4382	. . . . . . . . . . . . . 14 ⊢ (b +c a) = ((( Ins3k ∼ (( Ins3k Sk ∩ Ins2k Sk ) “k ℘1℘11c) ∖ (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k Sk ∪ Ins3k SIk SIk Sk )) “k ℘1℘1℘1℘11c)) “k ℘1℘1a) “k b)
12	10, 11	eqtri 2373	. . . . . . . . . . . . 13 ⊢ (a +c b) = ((( Ins3k ∼ (( Ins3k Sk ∩ Ins2k Sk ) “k ℘1℘11c) ∖ (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k Sk ∪ Ins3k SIk SIk Sk )) “k ℘1℘1℘1℘11c)) “k ℘1℘1a) “k b)
13	12	eqeq2i 2363	. . . . . . . . . . . 12 ⊢ (x = (a +c b) ↔ x = ((( Ins3k ∼ (( Ins3k Sk ∩ Ins2k Sk ) “k ℘1℘11c) ∖ (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k Sk ∪ Ins3k SIk SIk Sk )) “k ℘1℘1℘1℘11c)) “k ℘1℘1a) “k b))
14	8, 9, 13	3bitr4i 268	. . . . . . . . . . 11 ⊢ (⟪x, b⟫ ∈ ◡kImagek(( Ins3k ∼ (( Ins3k Sk ∩ Ins2k Sk ) “k ℘1℘11c) ∖ (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k Sk ∪ Ins3k SIk SIk Sk )) “k ℘1℘1℘1℘11c)) “k ℘1℘1a) ↔ x = (a +c b))
15	14	rexbii 2640	. . . . . . . . . 10 ⊢ (∃x ∈ Nn ⟪x, b⟫ ∈ ◡kImagek(( Ins3k ∼ (( Ins3k Sk ∩ Ins2k Sk ) “k ℘1℘11c) ∖ (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k Sk ∪ Ins3k SIk SIk Sk )) “k ℘1℘1℘1℘11c)) “k ℘1℘1a) ↔ ∃x ∈ Nn x = (a +c b))
16	5	elimak 4260	. . . . . . . . . 10 ⊢ (b ∈ (◡kImagek(( Ins3k ∼ (( Ins3k Sk ∩ Ins2k Sk ) “k ℘1℘11c) ∖ (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k Sk ∪ Ins3k SIk SIk Sk )) “k ℘1℘1℘1℘11c)) “k ℘1℘1a) “k Nn ) ↔ ∃x ∈ Nn ⟪x, b⟫ ∈ ◡kImagek(( Ins3k ∼ (( Ins3k Sk ∩ Ins2k Sk ) “k ℘1℘11c) ∖ (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k Sk ∪ Ins3k SIk SIk Sk )) “k ℘1℘1℘1℘11c)) “k ℘1℘1a))
17		risset 2662	. . . . . . . . . 10 ⊢ ((a +c b) ∈ Nn ↔ ∃x ∈ Nn x = (a +c b))
18	15, 16, 17	3bitr4i 268	. . . . . . . . 9 ⊢ (b ∈ (◡kImagek(( Ins3k ∼ (( Ins3k Sk ∩ Ins2k Sk ) “k ℘1℘11c) ∖ (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k Sk ∪ Ins3k SIk SIk Sk )) “k ℘1℘1℘1℘11c)) “k ℘1℘1a) “k Nn ) ↔ (a +c b) ∈ Nn )
19	18	abbi2i 2465	. . . . . . . 8 ⊢ (◡kImagek(( Ins3k ∼ (( Ins3k Sk ∩ Ins2k Sk ) “k ℘1℘11c) ∖ (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k Sk ∪ Ins3k SIk SIk Sk )) “k ℘1℘1℘1℘11c)) “k ℘1℘1a) “k Nn ) = {b ∣ (a +c b) ∈ Nn }
20	19	uneq2i 3416	. . . . . . 7 ⊢ ({b ∣ ¬ a ∈ Nn } ∪ (◡kImagek(( Ins3k ∼ (( Ins3k Sk ∩ Ins2k Sk ) “k ℘1℘11c) ∖ (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k Sk ∪ Ins3k SIk SIk Sk )) “k ℘1℘1℘1℘11c)) “k ℘1℘1a) “k Nn )) = ({b ∣ ¬ a ∈ Nn } ∪ {b ∣ (a +c b) ∈ Nn })
21		imor 401	. . . . . . . 8 ⊢ ((a ∈ Nn → (a +c b) ∈ Nn ) ↔ (¬ a ∈ Nn ∨ (a +c b) ∈ Nn ))
22	21	abbii 2466	. . . . . . 7 ⊢ {b ∣ (a ∈ Nn → (a +c b) ∈ Nn )} = {b ∣ (¬ a ∈ Nn ∨ (a +c b) ∈ Nn )}
23	4, 20, 22	3eqtr4i 2383	. . . . . 6 ⊢ ({b ∣ ¬ a ∈ Nn } ∪ (◡kImagek(( Ins3k ∼ (( Ins3k Sk ∩ Ins2k Sk ) “k ℘1℘11c) ∖ (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k Sk ∪ Ins3k SIk SIk Sk )) “k ℘1℘1℘1℘11c)) “k ℘1℘1a) “k Nn )) = {b ∣ (a ∈ Nn → (a +c b) ∈ Nn )}
24		abexv 4325	. . . . . . 7 ⊢ {b ∣ ¬ a ∈ Nn } ∈ V
25		addcexlem 4383	. . . . . . . . . . 11 ⊢ ( Ins3k ∼ (( Ins3k Sk ∩ Ins2k Sk ) “k ℘1℘11c) ∖ (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k Sk ∪ Ins3k SIk SIk Sk )) “k ℘1℘1℘1℘11c)) ∈ V
26		vex 2863	. . . . . . . . . . . . 13 ⊢ a ∈ V
27	26	pw1ex 4304	. . . . . . . . . . . 12 ⊢ ℘1a ∈ V
28	27	pw1ex 4304	. . . . . . . . . . 11 ⊢ ℘1℘1a ∈ V
29	25, 28	imakex 4301	. . . . . . . . . 10 ⊢ (( Ins3k ∼ (( Ins3k Sk ∩ Ins2k Sk ) “k ℘1℘11c) ∖ (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k Sk ∪ Ins3k SIk SIk Sk )) “k ℘1℘1℘1℘11c)) “k ℘1℘1a) ∈ V
30	29	imagekex 4313	. . . . . . . . 9 ⊢ Imagek(( Ins3k ∼ (( Ins3k Sk ∩ Ins2k Sk ) “k ℘1℘11c) ∖ (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k Sk ∪ Ins3k SIk SIk Sk )) “k ℘1℘1℘1℘11c)) “k ℘1℘1a) ∈ V
31	30	cnvkex 4288	. . . . . . . 8 ⊢ ◡kImagek(( Ins3k ∼ (( Ins3k Sk ∩ Ins2k Sk ) “k ℘1℘11c) ∖ (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k Sk ∪ Ins3k SIk SIk Sk )) “k ℘1℘1℘1℘11c)) “k ℘1℘1a) ∈ V
32		nncex 4397	. . . . . . . 8 ⊢ Nn ∈ V
33	31, 32	imakex 4301	. . . . . . 7 ⊢ (◡kImagek(( Ins3k ∼ (( Ins3k Sk ∩ Ins2k Sk ) “k ℘1℘11c) ∖ (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k Sk ∪ Ins3k SIk SIk Sk )) “k ℘1℘1℘1℘11c)) “k ℘1℘1a) “k Nn ) ∈ V
34	24, 33	unex 4107	. . . . . 6 ⊢ ({b ∣ ¬ a ∈ Nn } ∪ (◡kImagek(( Ins3k ∼ (( Ins3k Sk ∩ Ins2k Sk ) “k ℘1℘11c) ∖ (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k Sk ∪ Ins3k SIk SIk Sk )) “k ℘1℘1℘1℘11c)) “k ℘1℘1a) “k Nn )) ∈ V
35	23, 34	eqeltrri 2424	. . . . 5 ⊢ {b ∣ (a ∈ Nn → (a +c b) ∈ Nn )} ∈ V
36		addceq2 4385	. . . . . . 7 ⊢ (b = 0c → (a +c b) = (a +c 0c))
37	36	eleq1d 2419	. . . . . 6 ⊢ (b = 0c → ((a +c b) ∈ Nn ↔ (a +c 0c) ∈ Nn ))
38	37	imbi2d 307	. . . . 5 ⊢ (b = 0c → ((a ∈ Nn → (a +c b) ∈ Nn ) ↔ (a ∈ Nn → (a +c 0c) ∈ Nn )))
39		addceq2 4385	. . . . . . 7 ⊢ (b = c → (a +c b) = (a +c c))
40	39	eleq1d 2419	. . . . . 6 ⊢ (b = c → ((a +c b) ∈ Nn ↔ (a +c c) ∈ Nn ))
41	40	imbi2d 307	. . . . 5 ⊢ (b = c → ((a ∈ Nn → (a +c b) ∈ Nn ) ↔ (a ∈ Nn → (a +c c) ∈ Nn )))
42		addceq2 4385	. . . . . . 7 ⊢ (b = (c +c 1c) → (a +c b) = (a +c (c +c 1c)))
43	42	eleq1d 2419	. . . . . 6 ⊢ (b = (c +c 1c) → ((a +c b) ∈ Nn ↔ (a +c (c +c 1c)) ∈ Nn ))
44	43	imbi2d 307	. . . . 5 ⊢ (b = (c +c 1c) → ((a ∈ Nn → (a +c b) ∈ Nn ) ↔ (a ∈ Nn → (a +c (c +c 1c)) ∈ Nn )))
45		addceq2 4385	. . . . . . 7 ⊢ (b = B → (a +c b) = (a +c B))
46	45	eleq1d 2419	. . . . . 6 ⊢ (b = B → ((a +c b) ∈ Nn ↔ (a +c B) ∈ Nn ))
47	46	imbi2d 307	. . . . 5 ⊢ (b = B → ((a ∈ Nn → (a +c b) ∈ Nn ) ↔ (a ∈ Nn → (a +c B) ∈ Nn )))
48		addcid1 4406	. . . . . 6 ⊢ (a +c 0c) = a
49		id 19	. . . . . 6 ⊢ (a ∈ Nn → a ∈ Nn )
50	48, 49	syl5eqel 2437	. . . . 5 ⊢ (a ∈ Nn → (a +c 0c) ∈ Nn )
51		addcass 4416	. . . . . . . 8 ⊢ ((a +c c) +c 1c) = (a +c (c +c 1c))
52		peano2 4404	. . . . . . . 8 ⊢ ((a +c c) ∈ Nn → ((a +c c) +c 1c) ∈ Nn )
53	51, 52	syl5eqelr 2438	. . . . . . 7 ⊢ ((a +c c) ∈ Nn → (a +c (c +c 1c)) ∈ Nn )
54	53	imim2i 13	. . . . . 6 ⊢ ((a ∈ Nn → (a +c c) ∈ Nn ) → (a ∈ Nn → (a +c (c +c 1c)) ∈ Nn ))
55	54	a1i 10	. . . . 5 ⊢ (c ∈ Nn → ((a ∈ Nn → (a +c c) ∈ Nn ) → (a ∈ Nn → (a +c (c +c 1c)) ∈ Nn )))
56	35, 38, 41, 44, 47, 50, 55	finds 4412	. . . 4 ⊢ (B ∈ Nn → (a ∈ Nn → (a +c B) ∈ Nn ))
57	56	com12 27	. . 3 ⊢ (a ∈ Nn → (B ∈ Nn → (a +c B) ∈ Nn ))
58	3, 57	vtoclga 2921	. 2 ⊢ (A ∈ Nn → (B ∈ Nn → (A +c B) ∈ Nn ))
59	58	imp 418	1 ⊢ ((A ∈ Nn ∧ B ∈ Nn ) → (A +c B) ∈ Nn )

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∨ wo 357   ∧ wa 358   = wceq 1642   ∈ wcel 1710  {cab 2339  ∃wrex 2616  Vcvv 2860   ∼ ccompl 3206   ∖ cdif 3207   ∪ cun 3208   ∩ cin 3209   ⊕ csymdif 3210  ⟪copk 4058  1_cc1c 4135  ℘₁cpw1 4136  ^◡_kccnvk 4176   Ins2_k cins2k 4177   Ins3_k cins3k 4178   “_k cimak 4180   SI_k csik 4182  Image_kcimagek 4183   S_k cssetk 4184   Nn cnnc 4374  0_cc0c 4375   +_c cplc 4376

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-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-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  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-0c 4378  df-addc 4379  df-nnc 4380

This theorem is referenced by:  preaddccan2  4456  leltfintr  4459  ltfintr  4460  ncfindi  4476  tfindi  4497  tfinltfinlem1  4501  evennn  4507  oddnn  4508  evenodddisj  4517  eventfin  4518  oddtfin  4519  nnpweq  4524  sfindbl  4531  sfinltfin  4536  addccan2  4560  nnc3n3p1  6279  nnc3n3p2  6280  nnc3p1n3p2  6281  nchoicelem1  6290  nchoicelem2  6291