Theorem addceq1 4383

Description: Equality law for cardinal addition. (Contributed by SF, 15-Jan-2015.)

Assertion

Ref	Expression
addceq1	⊢ (A = B → (A +c C) = (B +c C))

Proof of Theorem addceq1

Step	Hyp	Ref	Expression
1		imakeq2 4225	. 2 ⊢ (A = 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℘1C) “k 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℘1C) “k B))
2		dfaddc2 4381	. 2 ⊢ (A +c C) = ((( 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℘1C) “k A)
3		dfaddc2 4381	. 2 ⊢ (B +c C) = ((( 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℘1C) “k B)
4	1, 2, 3	3eqtr4g 2410	1 ⊢ (A = B → (A +c C) = (B +c C))

Syntax hints:   → wi 4   = wceq 1642   ∼ ccompl 3205   ∖ cdif 3206   ∪ cun 3207   ∩ cin 3208   ⊕ csymdif 3209  1_cc1c 4134  ℘₁cpw1 4135   Ins2_k cins2k 4176   Ins3_k cins3k 4177   “_k cimak 4179   SI_k csik 4181   S_k cssetk 4183   +_c cplc 4375

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 4078  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-rex 2620  df-v 2861  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-pw 3724  df-sn 3741  df-pr 3742  df-opk 4058  df-1c 4136  df-pw1 4137  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-sik 4192  df-ssetk 4193  df-addc 4378

This theorem is referenced by:  addceq12  4385  addceq1i  4386  addceq1d  4389  peano2  4403  nnc0suc  4412  nncaddccl  4419  nnsucelr  4428  nndisjeq  4429  opklefing  4448  opkltfing  4449  addcnnul  4453  preaddccan2  4455  leltfintr  4458  ltfintr  4459  ltfinex  4464  sucevenodd  4510  sucoddeven  4511  evenodddisjlem1  4515  oddtfin  4518  suc11nnc  4558  phi11lem1  4595  phialllem1  4616  braddcfn  5826  dfnnc3  5885  peano4nc  6150  dflec2  6210  lectr  6211  leaddc1  6214  addceq0  6219  csucex  6259  brcsuc  6260  nnltp1c  6262  addccan2nc  6265  nncdiv3lem1  6275  nncdiv3lem2  6276  nncdiv3  6277  nnc3n3p1  6278  nchoicelem9  6297  nchoicelem16  6304  nchoicelem17  6305  fnfreclem2  6318  fnfreclem3  6319  frecsuc  6322