Theorem axaddf 8183

Description: Addition is an operation on the complex numbers. This theorem can be used as an alternate axiom for complex numbers in place of the less specific axaddcl 8179. This construction-dependent theorem should not be referenced directly; instead, use ax-addf 8249. (Contributed by NM, 8-Feb-2005.) (New usage is discouraged.)

Assertion

Ref	Expression
axaddf	|- + : ( CC X. CC ) --> CC

Proof of Theorem axaddf

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

Step	Hyp	Ref	Expression
1		moeq 2992	. . . . . . . . 9 |- E* z z = <. ( w +R u ) , ( v +R f ) >.
2	1	mosubop 4816	. . . . . . . 8 |- E* z E. u E. f ( y = <. u , f >. /\ z = <. ( w +R u ) , ( v +R f ) >. )
3	2	mosubop 4816	. . . . . . 7 |- E* z E. w E. v ( x = <. w , v >. /\ E. u E. f ( y = <. u , f >. /\ z = <. ( w +R u ) , ( v +R f ) >. ) )
4		anass 401	. . . . . . . . . . 11 |- ( ( ( x = <. w , v >. /\ y = <. u , f >. ) /\ z = <. ( w +R u ) , ( v +R f ) >. ) <-> ( x = <. w , v >. /\ ( y = <. u , f >. /\ z = <. ( w +R u ) , ( v +R f ) >. ) ) )
5	4	2exbii 1655	. . . . . . . . . 10 |- ( E. u E. f ( ( x = <. w , v >. /\ y = <. u , f >. ) /\ z = <. ( w +R u ) , ( v +R f ) >. ) <-> E. u E. f ( x = <. w , v >. /\ ( y = <. u , f >. /\ z = <. ( w +R u ) , ( v +R f ) >. ) ) )
6		19.42vv 1961	. . . . . . . . . 10 |- ( E. u E. f ( x = <. w , v >. /\ ( y = <. u , f >. /\ z = <. ( w +R u ) , ( v +R f ) >. ) ) <-> ( x = <. w , v >. /\ E. u E. f ( y = <. u , f >. /\ z = <. ( w +R u ) , ( v +R f ) >. ) ) )
7	5, 6	bitri 184	. . . . . . . . 9 |- ( E. u E. f ( ( x = <. w , v >. /\ y = <. u , f >. ) /\ z = <. ( w +R u ) , ( v +R f ) >. ) <-> ( x = <. w , v >. /\ E. u E. f ( y = <. u , f >. /\ z = <. ( w +R u ) , ( v +R f ) >. ) ) )
8	7	2exbii 1655	. . . . . . . 8 |- ( E. w E. v E. u E. f ( ( x = <. w , v >. /\ y = <. u , f >. ) /\ z = <. ( w +R u ) , ( v +R f ) >. ) <-> E. w E. v ( x = <. w , v >. /\ E. u E. f ( y = <. u , f >. /\ z = <. ( w +R u ) , ( v +R f ) >. ) ) )
9	8	mobii 2117	. . . . . . 7 |- ( E* z E. w E. v E. u E. f ( ( x = <. w , v >. /\ y = <. u , f >. ) /\ z = <. ( w +R u ) , ( v +R f ) >. ) <-> E* z E. w E. v ( x = <. w , v >. /\ E. u E. f ( y = <. u , f >. /\ z = <. ( w +R u ) , ( v +R f ) >. ) ) )
10	3, 9	mpbir 146	. . . . . 6 |- E* z E. w E. v E. u E. f ( ( x = <. w , v >. /\ y = <. u , f >. ) /\ z = <. ( w +R u ) , ( v +R f ) >. )
11	10	moani 2151	. . . . 5 |- E* z ( ( x e. CC /\ y e. CC ) /\ E. w E. v E. u E. f ( ( x = <. w , v >. /\ y = <. u , f >. ) /\ z = <. ( w +R u ) , ( v +R f ) >. ) )
12	11	funoprab 6153	. . . 4 |- Fun { <. <. x , y >. , z >. | ( ( x e. CC /\ y e. CC ) /\ E. w E. v E. u E. f ( ( x = <. w , v >. /\ y = <. u , f >. ) /\ z = <. ( w +R u ) , ( v +R f ) >. ) ) }
13		df-add 8138	. . . . 5 |- + = { <. <. x , y >. , z >. | ( ( x e. CC /\ y e. CC ) /\ E. w E. v E. u E. f ( ( x = <. w , v >. /\ y = <. u , f >. ) /\ z = <. ( w +R u ) , ( v +R f ) >. ) ) }
14	13	funeqi 5373	. . . 4 |- ( Fun + <-> Fun { <. <. x , y >. , z >. | ( ( x e. CC /\ y e. CC ) /\ E. w E. v E. u E. f ( ( x = <. w , v >. /\ y = <. u , f >. ) /\ z = <. ( w +R u ) , ( v +R f ) >. ) ) } )
15	12, 14	mpbir 146	. . 3 |- Fun +
16	13	dmeqi 4957	. . . . 5 |- dom + = dom { <. <. x , y >. , z >. | ( ( x e. CC /\ y e. CC ) /\ E. w E. v E. u E. f ( ( x = <. w , v >. /\ y = <. u , f >. ) /\ z = <. ( w +R u ) , ( v +R f ) >. ) ) }
17		dmoprabss 6135	. . . . 5 |- dom { <. <. x , y >. , z >. | ( ( x e. CC /\ y e. CC ) /\ E. w E. v E. u E. f ( ( x = <. w , v >. /\ y = <. u , f >. ) /\ z = <. ( w +R u ) , ( v +R f ) >. ) ) } C_ ( CC X. CC )
18	16, 17	eqsstri 3270	. . . 4 |- dom + C_ ( CC X. CC )
19		cnm 8147	. . . . . . 7 |- ( a e. CC -> E. b b e. a )
20	19	adantl 277	. . . . . 6 |- ( ( T. /\ a e. CC ) -> E. b b e. a )
21		axaddcl 8179	. . . . . . 7 |- ( ( x e. CC /\ y e. CC ) -> ( x + y ) e. CC )
22	21	adantl 277	. . . . . 6 |- ( ( T. /\ ( x e. CC /\ y e. CC ) ) -> ( x + y ) e. CC )
23		funrel 5369	. . . . . . 7 |- ( Fun + -> Rel + )
24	15, 23	mp1i 10	. . . . . 6 |- ( T. -> Rel + )
25	20, 22, 24	oprssdmm 6365	. . . . 5 |- ( T. -> ( CC X. CC ) C_ dom + )
26	25	mptru 1407	. . . 4 |- ( CC X. CC ) C_ dom +
27	18, 26	eqssi 3254	. . 3 |- dom + = ( CC X. CC )
28		df-fn 5355	. . 3 |- ( + Fn ( CC X. CC ) <-> ( Fun + /\ dom + = ( CC X. CC ) ) )
29	15, 27, 28	mpbir2an 951	. 2 |- + Fn ( CC X. CC )
30	21	rgen2a 2596	. 2 |- A. x e. CC A. y e. CC ( x + y ) e. CC
31		ffnov 6157	. 2 |- ( + : ( CC X. CC ) --> CC <-> ( + Fn ( CC X. CC ) /\ A. x e. CC A. y e. CC ( x + y ) e. CC ) )
32	29, 30, 31	mpbir2an 951	1 |- + : ( CC X. CC ) --> CC

Syntax hints:   /\ wa 104   = wceq 1398   T. wtru 1399   E.wex 1541   E*wmo 2081   e. wcel 2203   A.wral 2520   C_ wss 3211   <.cop 3692   X. cxp 4747   dom cdm 4749   Rel wrel 4754   Fun wfun 5346   Fn wfn 5347   -->wf 5348  (class class class)co 6050   {coprab 6051   +R cplr 7616   CCcc 8125   + caddc 8130

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-eprel 4410  df-id 4414  df-po 4417  df-iso 4418  df-iord 4487  df-on 4489  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-recs 6536  df-irdg 6601  df-1o 6647  df-2o 6648  df-oadd 6651  df-omul 6652  df-er 6767  df-ec 6769  df-qs 6773  df-ni 7619  df-pli 7620  df-mi 7621  df-lti 7622  df-plpq 7659  df-mpq 7660  df-enq 7662  df-nqqs 7663  df-plqqs 7664  df-mqqs 7665  df-1nqqs 7666  df-rq 7667  df-ltnqqs 7668  df-enq0 7739  df-nq0 7740  df-0nq0 7741  df-plq0 7742  df-mq0 7743  df-inp 7781  df-iplp 7783  df-enr 8041  df-nr 8042  df-plr 8043  df-c 8133  df-add 8138

This theorem is referenced by: (None)