Theorem axaddass 9977

Description: Addition of complex numbers is associative. This theorem transfers the associative laws for the real and imaginary signed real components of complex number pairs, to complex number addition itself. Axiom 9 of 22 for real and complex numbers, derived from ZF set theory. This construction-dependent theorem should not be referenced directly, nor should the proven axiom ax-addass 10001 be used later. Instead, use addass 10023. (Contributed by NM, 2-Sep-1995.) (New usage is discouraged.)

Assertion

Ref	Expression
axaddass	|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + B ) + C ) = ( A + ( B + C ) ) )

Proof of Theorem axaddass

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

Step	Hyp	Ref	Expression
1		dfcnqs 9963	. 2 |- CC = ( ( R. X. R. ) /. `' _E )
2		addcnsrec 9964	. 2 |- ( ( ( x e. R. /\ y e. R. ) /\ ( z e. R. /\ w e. R. ) ) -> ( [ <. x , y >. ] `' _E + [ <. z , w >. ] `' _E ) = [ <. ( x +R z ) , ( y +R w ) >. ] `' _E )
3		addcnsrec 9964	. 2 |- ( ( ( z e. R. /\ w e. R. ) /\ ( v e. R. /\ u e. R. ) ) -> ( [ <. z , w >. ] `' _E + [ <. v , u >. ] `' _E ) = [ <. ( z +R v ) , ( w +R u ) >. ] `' _E )
4		addcnsrec 9964	. 2 |- ( ( ( ( x +R z ) e. R. /\ ( y +R w ) e. R. ) /\ ( v e. R. /\ u e. R. ) ) -> ( [ <. ( x +R z ) , ( y +R w ) >. ] `' _E + [ <. v , u >. ] `' _E ) = [ <. ( ( x +R z ) +R v ) , ( ( y +R w ) +R u ) >. ] `' _E )
5		addcnsrec 9964	. 2 |- ( ( ( x e. R. /\ y e. R. ) /\ ( ( z +R v ) e. R. /\ ( w +R u ) e. R. ) ) -> ( [ <. x , y >. ] `' _E + [ <. ( z +R v ) , ( w +R u ) >. ] `' _E ) = [ <. ( x +R ( z +R v ) ) , ( y +R ( w +R u ) ) >. ] `' _E )
6		addclsr 9904	. . . 4 |- ( ( x e. R. /\ z e. R. ) -> ( x +R z ) e. R. )
7		addclsr 9904	. . . 4 |- ( ( y e. R. /\ w e. R. ) -> ( y +R w ) e. R. )
8	6, 7	anim12i 590	. . 3 |- ( ( ( x e. R. /\ z e. R. ) /\ ( y e. R. /\ w e. R. ) ) -> ( ( x +R z ) e. R. /\ ( y +R w ) e. R. ) )
9	8	an4s 869	. 2 |- ( ( ( x e. R. /\ y e. R. ) /\ ( z e. R. /\ w e. R. ) ) -> ( ( x +R z ) e. R. /\ ( y +R w ) e. R. ) )
10		addclsr 9904	. . . 4 |- ( ( z e. R. /\ v e. R. ) -> ( z +R v ) e. R. )
11		addclsr 9904	. . . 4 |- ( ( w e. R. /\ u e. R. ) -> ( w +R u ) e. R. )
12	10, 11	anim12i 590	. . 3 |- ( ( ( z e. R. /\ v e. R. ) /\ ( w e. R. /\ u e. R. ) ) -> ( ( z +R v ) e. R. /\ ( w +R u ) e. R. ) )
13	12	an4s 869	. 2 |- ( ( ( z e. R. /\ w e. R. ) /\ ( v e. R. /\ u e. R. ) ) -> ( ( z +R v ) e. R. /\ ( w +R u ) e. R. ) )
14		addasssr 9909	. 2 |- ( ( x +R z ) +R v ) = ( x +R ( z +R v ) )
15		addasssr 9909	. 2 |- ( ( y +R w ) +R u ) = ( y +R ( w +R u ) )
16	1, 2, 3, 4, 5, 9, 13, 14, 15	ecovass 7855	1 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + B ) + C ) = ( A + ( B + C ) ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   _E cep 5028   `'ccnv 5113  (class class class)co 6650   R.cnr 9687   +R cplr 9691   CCcc 9934   + caddc 9939

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-omul 7565  df-er 7742  df-ec 7744  df-qs 7748  df-ni 9694  df-pli 9695  df-mi 9696  df-lti 9697  df-plpq 9730  df-mpq 9731  df-ltpq 9732  df-enq 9733  df-nq 9734  df-erq 9735  df-plq 9736  df-mq 9737  df-1nq 9738  df-rq 9739  df-ltnq 9740  df-np 9803  df-plp 9805  df-ltp 9807  df-enr 9877  df-nr 9878  df-plr 9879  df-c 9942  df-add 9947

This theorem is referenced by: (None)