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Theorem nncaddccl 4419
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.
StepHypRef Expression
1 addceq1 4383 . . . . 5 (a = A → (a +c B) = (A +c B))
21eleq1d 2419 . . . 4 (a = A → ((a +c B) Nn ↔ (A +c B) Nn ))
32imbi2d 307 . . 3 (a = A → ((B Nn → (a +c B) Nn ) ↔ (B Nn → (A +c B) Nn )))
4 unab 3521 . . . . . . 7 ({b ¬ a Nn } ∪ {b (a +c b) Nn }) = {b a Nn (a +c b) Nn )}
5 vex 2862 . . . . . . . . . . . . 13 b V
6 vex 2862 . . . . . . . . . . . . 13 x V
7 opkelimagekg 4271 . . . . . . . . . . . . 13 ((b V x V) → (⟪b, x Imagek(( Ins3k ∼ (( Ins3k SkIns2k Sk ) “k 111c) (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k SkIns3k SIk SIk Sk )) “k 11111c)) “k 11a) ↔ x = ((( Ins3k ∼ (( Ins3k SkIns2k Sk ) “k 111c) (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k SkIns3k SIk SIk Sk )) “k 11111c)) “k 11a) “k b)))
85, 6, 7mp2an 653 . . . . . . . . . . . 12 (⟪b, x Imagek(( Ins3k ∼ (( Ins3k SkIns2k Sk ) “k 111c) (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k SkIns3k SIk SIk Sk )) “k 11111c)) “k 11a) ↔ x = ((( Ins3k ∼ (( Ins3k SkIns2k Sk ) “k 111c) (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k SkIns3k SIk SIk Sk )) “k 11111c)) “k 11a) “k b))
96, 5opkelcnvk 4250 . . . . . . . . . . . 12 (⟪x, b kImagek(( Ins3k ∼ (( Ins3k SkIns2k Sk ) “k 111c) (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k SkIns3k SIk SIk Sk )) “k 11111c)) “k 11a) ↔ ⟪b, x Imagek(( Ins3k ∼ (( Ins3k SkIns2k Sk ) “k 111c) (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k SkIns3k SIk SIk Sk )) “k 11111c)) “k 11a))
10 addccom 4406 . . . . . . . . . . . . . 14 (a +c b) = (b +c a)
11 dfaddc2 4381 . . . . . . . . . . . . . 14 (b +c a) = ((( Ins3k ∼ (( Ins3k SkIns2k Sk ) “k 111c) (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k SkIns3k SIk SIk Sk )) “k 11111c)) “k 11a) “k b)
1210, 11eqtri 2373 . . . . . . . . . . . . 13 (a +c b) = ((( Ins3k ∼ (( Ins3k SkIns2k Sk ) “k 111c) (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k SkIns3k SIk SIk Sk )) “k 11111c)) “k 11a) “k b)
1312eqeq2i 2363 . . . . . . . . . . . 12 (x = (a +c b) ↔ x = ((( Ins3k ∼ (( Ins3k SkIns2k Sk ) “k 111c) (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k SkIns3k SIk SIk Sk )) “k 11111c)) “k 11a) “k b))
148, 9, 133bitr4i 268 . . . . . . . . . . 11 (⟪x, b kImagek(( Ins3k ∼ (( Ins3k SkIns2k Sk ) “k 111c) (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k SkIns3k SIk SIk Sk )) “k 11111c)) “k 11a) ↔ x = (a +c b))
1514rexbii 2639 . . . . . . . . . 10 (x Nnx, b kImagek(( Ins3k ∼ (( Ins3k SkIns2k Sk ) “k 111c) (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k SkIns3k SIk SIk Sk )) “k 11111c)) “k 11a) ↔ x Nn x = (a +c b))
165elimak 4259 . . . . . . . . . 10 (b (kImagek(( Ins3k ∼ (( Ins3k SkIns2k Sk ) “k 111c) (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k SkIns3k SIk SIk Sk )) “k 11111c)) “k 11a) “k Nn ) ↔ x Nnx, b kImagek(( Ins3k ∼ (( Ins3k SkIns2k Sk ) “k 111c) (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k SkIns3k SIk SIk Sk )) “k 11111c)) “k 11a))
17 risset 2661 . . . . . . . . . 10 ((a +c b) Nnx Nn x = (a +c b))
1815, 16, 173bitr4i 268 . . . . . . . . 9 (b (kImagek(( Ins3k ∼ (( Ins3k SkIns2k Sk ) “k 111c) (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k SkIns3k SIk SIk Sk )) “k 11111c)) “k 11a) “k Nn ) ↔ (a +c b) Nn )
1918abbi2i 2464 . . . . . . . 8 (kImagek(( Ins3k ∼ (( Ins3k SkIns2k Sk ) “k 111c) (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k SkIns3k SIk SIk Sk )) “k 11111c)) “k 11a) “k Nn ) = {b (a +c b) Nn }
2019uneq2i 3415 . . . . . . 7 ({b ¬ a Nn } ∪ (kImagek(( Ins3k ∼ (( Ins3k SkIns2k Sk ) “k 111c) (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k SkIns3k SIk SIk Sk )) “k 11111c)) “k 11a) “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 ))
2221abbii 2465 . . . . . . 7 {b (a Nn → (a +c b) Nn )} = {b a Nn (a +c b) Nn )}
234, 20, 223eqtr4i 2383 . . . . . 6 ({b ¬ a Nn } ∪ (kImagek(( Ins3k ∼ (( Ins3k SkIns2k Sk ) “k 111c) (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k SkIns3k SIk SIk Sk )) “k 11111c)) “k 11a) “k Nn )) = {b (a Nn → (a +c b) Nn )}
24 abexv 4324 . . . . . . 7 {b ¬ a Nn } V
25 addcexlem 4382 . . . . . . . . . . 11 ( Ins3k ∼ (( Ins3k SkIns2k Sk ) “k 111c) (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k SkIns3k SIk SIk Sk )) “k 11111c)) V
26 vex 2862 . . . . . . . . . . . . 13 a V
2726pw1ex 4303 . . . . . . . . . . . 12 1a V
2827pw1ex 4303 . . . . . . . . . . 11 11a V
2925, 28imakex 4300 . . . . . . . . . 10 (( Ins3k ∼ (( Ins3k SkIns2k Sk ) “k 111c) (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k SkIns3k SIk SIk Sk )) “k 11111c)) “k 11a) V
3029imagekex 4312 . . . . . . . . 9 Imagek(( Ins3k ∼ (( Ins3k SkIns2k Sk ) “k 111c) (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k SkIns3k SIk SIk Sk )) “k 11111c)) “k 11a) V
3130cnvkex 4287 . . . . . . . 8 kImagek(( Ins3k ∼ (( Ins3k SkIns2k Sk ) “k 111c) (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k SkIns3k SIk SIk Sk )) “k 11111c)) “k 11a) V
32 nncex 4396 . . . . . . . 8 Nn V
3331, 32imakex 4300 . . . . . . 7 (kImagek(( Ins3k ∼ (( Ins3k SkIns2k Sk ) “k 111c) (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k SkIns3k SIk SIk Sk )) “k 11111c)) “k 11a) “k Nn ) V
3424, 33unex 4106 . . . . . 6 ({b ¬ a Nn } ∪ (kImagek(( Ins3k ∼ (( Ins3k SkIns2k Sk ) “k 111c) (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k SkIns3k SIk SIk Sk )) “k 11111c)) “k 11a) “k Nn )) V
3523, 34eqeltrri 2424 . . . . 5 {b (a Nn → (a +c b) Nn )} V
36 addceq2 4384 . . . . . . 7 (b = 0c → (a +c b) = (a +c 0c))
3736eleq1d 2419 . . . . . 6 (b = 0c → ((a +c b) Nn ↔ (a +c 0c) Nn ))
3837imbi2d 307 . . . . 5 (b = 0c → ((a Nn → (a +c b) Nn ) ↔ (a Nn → (a +c 0c) Nn )))
39 addceq2 4384 . . . . . . 7 (b = c → (a +c b) = (a +c c))
4039eleq1d 2419 . . . . . 6 (b = c → ((a +c b) Nn ↔ (a +c c) Nn ))
4140imbi2d 307 . . . . 5 (b = c → ((a Nn → (a +c b) Nn ) ↔ (a Nn → (a +c c) Nn )))
42 addceq2 4384 . . . . . . 7 (b = (c +c 1c) → (a +c b) = (a +c (c +c 1c)))
4342eleq1d 2419 . . . . . 6 (b = (c +c 1c) → ((a +c b) Nn ↔ (a +c (c +c 1c)) Nn ))
4443imbi2d 307 . . . . 5 (b = (c +c 1c) → ((a Nn → (a +c b) Nn ) ↔ (a Nn → (a +c (c +c 1c)) Nn )))
45 addceq2 4384 . . . . . . 7 (b = B → (a +c b) = (a +c B))
4645eleq1d 2419 . . . . . 6 (b = B → ((a +c b) Nn ↔ (a +c B) Nn ))
4746imbi2d 307 . . . . 5 (b = B → ((a Nn → (a +c b) Nn ) ↔ (a Nn → (a +c B) Nn )))
48 addcid1 4405 . . . . . 6 (a +c 0c) = a
49 id 19 . . . . . 6 (a Nna Nn )
5048, 49syl5eqel 2437 . . . . 5 (a Nn → (a +c 0c) Nn )
51 addcass 4415 . . . . . . . 8 ((a +c c) +c 1c) = (a +c (c +c 1c))
52 peano2 4403 . . . . . . . 8 ((a +c c) Nn → ((a +c c) +c 1c) Nn )
5351, 52syl5eqelr 2438 . . . . . . 7 ((a +c c) Nn → (a +c (c +c 1c)) Nn )
5453imim2i 13 . . . . . 6 ((a Nn → (a +c c) Nn ) → (a Nn → (a +c (c +c 1c)) Nn ))
5554a1i 10 . . . . 5 (c Nn → ((a Nn → (a +c c) Nn ) → (a Nn → (a +c (c +c 1c)) Nn )))
5635, 38, 41, 44, 47, 50, 55finds 4411 . . . 4 (B Nn → (a Nn → (a +c B) Nn ))
5756com12 27 . . 3 (a Nn → (B Nn → (a +c B) Nn ))
583, 57vtoclga 2920 . 2 (A Nn → (B Nn → (A +c B) Nn ))
5958imp 418 1 ((A Nn B Nn ) → (A +c B) Nn )
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 176   wo 357   wa 358   = wceq 1642   wcel 1710  {cab 2339  wrex 2615  Vcvv 2859  ccompl 3205   cdif 3206  cun 3207  cin 3208  csymdif 3209  copk 4057  1cc1c 4134  1cpw1 4135  kccnvk 4175   Ins2k cins2k 4176   Ins3k cins3k 4177  k cimak 4179   SIk csik 4181  Imagekcimagek 4182   Sk cssetk 4183   Nn cnnc 4373  0cc0c 4374   +c cplc 4375
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-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-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  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-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-0c 4377  df-addc 4378  df-nnc 4379
This theorem is referenced by:  preaddccan2  4455  leltfintr  4458  ltfintr  4459  ncfindi  4475  tfindi  4496  tfinltfinlem1  4500  evennn  4506  oddnn  4507  evenodddisj  4516  eventfin  4517  oddtfin  4518  nnpweq  4523  sfindbl  4530  sfinltfin  4535  addccan2  4559  nnc3n3p1  6278  nnc3n3p2  6279  nnc3p1n3p2  6280  nchoicelem1  6289  nchoicelem2  6290
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