Theorem divvalap 8458

Description: Value of division: the (unique) element x such that ( B x. x ) = A. This is meaningful only when B is apart from zero. (Contributed by Jim Kingdon, 21-Feb-2020.)

Assertion

Ref	Expression
divvalap	|- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> ( A / B ) = ( iota_ x e. CC ( B x. x ) = A ) )

Distinct variable groups:   x, A   x, B

Proof of Theorem divvalap

Dummy variables y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp1 982	. 2 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> A e. CC )
2		simp2 983	. . 3 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> B e. CC )
3		0cn 7782	. . . . . 6 |- 0 e. CC
4		apne 8409	. . . . . 6 |- ( ( B e. CC /\ 0 e. CC ) -> ( B # 0 -> B =/= 0 ) )
5	3, 4	mpan2 422	. . . . 5 |- ( B e. CC -> ( B # 0 -> B =/= 0 ) )
6	5	adantl 275	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( B # 0 -> B =/= 0 ) )
7	6	3impia 1179	. . 3 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> B =/= 0 )
8		eldifsn 3658	. . 3 |- ( B e. ( CC \ { 0 } ) <-> ( B e. CC /\ B =/= 0 ) )
9	2, 7, 8	sylanbrc 414	. 2 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> B e. ( CC \ { 0 } ) )
10		receuap 8454	. . 3 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> E! x e. CC ( B x. x ) = A )
11		riotacl 5752	. . 3 |- ( E! x e. CC ( B x. x ) = A -> ( iota_ x e. CC ( B x. x ) = A ) e. CC )
12	10, 11	syl 14	. 2 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> ( iota_ x e. CC ( B x. x ) = A ) e. CC )
13		eqeq2 2150	. . . 4 |- ( z = A -> ( ( y x. x ) = z <-> ( y x. x ) = A ) )
14	13	riotabidv 5740	. . 3 |- ( z = A -> ( iota_ x e. CC ( y x. x ) = z ) = ( iota_ x e. CC ( y x. x ) = A ) )
15		oveq1 5789	. . . . 5 |- ( y = B -> ( y x. x ) = ( B x. x ) )
16	15	eqeq1d 2149	. . . 4 |- ( y = B -> ( ( y x. x ) = A <-> ( B x. x ) = A ) )
17	16	riotabidv 5740	. . 3 |- ( y = B -> ( iota_ x e. CC ( y x. x ) = A ) = ( iota_ x e. CC ( B x. x ) = A ) )
18		df-div 8457	. . 3 |- / = ( z e. CC , y e. ( CC \ { 0 } ) |-> ( iota_ x e. CC ( y x. x ) = z ) )
19	14, 17, 18	ovmpog 5913	. 2 |- ( ( A e. CC /\ B e. ( CC \ { 0 } ) /\ ( iota_ x e. CC ( B x. x ) = A ) e. CC ) -> ( A / B ) = ( iota_ x e. CC ( B x. x ) = A ) )
20	1, 9, 12, 19	syl3anc 1217	1 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> ( A / B ) = ( iota_ x e. CC ( B x. x ) = A ) )

Syntax hints:   -> wi 4   /\ w3a 963   = wceq 1332   e. wcel 1481   =/= wne 2309   E!wreu 2419   \ cdif 3073   {csn 3532   class class class wbr 3937   iota_crio 5737  (class class class)co 5782   CCcc 7642   0cc0 7644   x. cmul 7649   # cap 8367   / cdiv 8456

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-cnex 7735  ax-resscn 7736  ax-1cn 7737  ax-1re 7738  ax-icn 7739  ax-addcl 7740  ax-addrcl 7741  ax-mulcl 7742  ax-mulrcl 7743  ax-addcom 7744  ax-mulcom 7745  ax-addass 7746  ax-mulass 7747  ax-distr 7748  ax-i2m1 7749  ax-0lt1 7750  ax-1rid 7751  ax-0id 7752  ax-rnegex 7753  ax-precex 7754  ax-cnre 7755  ax-pre-ltirr 7756  ax-pre-ltwlin 7757  ax-pre-lttrn 7758  ax-pre-apti 7759  ax-pre-ltadd 7760  ax-pre-mulgt0 7761  ax-pre-mulext 7762

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rmo 2425  df-rab 2426  df-v 2691  df-sbc 2914  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-opab 3998  df-id 4223  df-po 4226  df-iso 4227  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-iota 5096  df-fun 5133  df-fv 5139  df-riota 5738  df-ov 5785  df-oprab 5786  df-mpo 5787  df-pnf 7826  df-mnf 7827  df-xr 7828  df-ltxr 7829  df-le 7830  df-sub 7959  df-neg 7960  df-reap 8361  df-ap 8368  df-div 8457

This theorem is referenced by:  divmulap  8459  divclap  8462