Theorem divvalap 8662

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 999	. 2 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> A e. CC )
2		simp2 1000	. . 3 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> B e. CC )
3		0cn 7980	. . . . . 6 |- 0 e. CC
4		apne 8611	. . . . . 6 |- ( ( B e. CC /\ 0 e. CC ) -> ( B # 0 -> B =/= 0 ) )
5	3, 4	mpan2 425	. . . . 5 |- ( B e. CC -> ( B # 0 -> B =/= 0 ) )
6	5	adantl 277	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( B # 0 -> B =/= 0 ) )
7	6	3impia 1202	. . 3 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> B =/= 0 )
8		eldifsn 3734	. . 3 |- ( B e. ( CC \ { 0 } ) <-> ( B e. CC /\ B =/= 0 ) )
9	2, 7, 8	sylanbrc 417	. 2 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> B e. ( CC \ { 0 } ) )
10		receuap 8657	. . 3 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> E! x e. CC ( B x. x ) = A )
11		riotacl 5867	. . 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 2199	. . . 4 |- ( z = A -> ( ( y x. x ) = z <-> ( y x. x ) = A ) )
14	13	riotabidv 5854	. . 3 |- ( z = A -> ( iota_ x e. CC ( y x. x ) = z ) = ( iota_ x e. CC ( y x. x ) = A ) )
15		oveq1 5904	. . . . 5 |- ( y = B -> ( y x. x ) = ( B x. x ) )
16	15	eqeq1d 2198	. . . 4 |- ( y = B -> ( ( y x. x ) = A <-> ( B x. x ) = A ) )
17	16	riotabidv 5854	. . 3 |- ( y = B -> ( iota_ x e. CC ( y x. x ) = A ) = ( iota_ x e. CC ( B x. x ) = A ) )
18		df-div 8661	. . 3 |- / = ( z e. CC , y e. ( CC \ { 0 } ) |-> ( iota_ x e. CC ( y x. x ) = z ) )
19	14, 17, 18	ovmpog 6032	. 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 1249	1 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> ( A / B ) = ( iota_ x e. CC ( B x. x ) = A ) )

Syntax hints:   -> wi 4   /\ w3a 980   = wceq 1364   e. wcel 2160   =/= wne 2360   E!wreu 2470   \ cdif 3141   {csn 3607   class class class wbr 4018   iota_crio 5851  (class class class)co 5897   CCcc 7840   0cc0 7842   x. cmul 7847   # cap 8569   / cdiv 8660

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-cnex 7933  ax-resscn 7934  ax-1cn 7935  ax-1re 7936  ax-icn 7937  ax-addcl 7938  ax-addrcl 7939  ax-mulcl 7940  ax-mulrcl 7941  ax-addcom 7942  ax-mulcom 7943  ax-addass 7944  ax-mulass 7945  ax-distr 7946  ax-i2m1 7947  ax-0lt1 7948  ax-1rid 7949  ax-0id 7950  ax-rnegex 7951  ax-precex 7952  ax-cnre 7953  ax-pre-ltirr 7954  ax-pre-ltwlin 7955  ax-pre-lttrn 7956  ax-pre-apti 7957  ax-pre-ltadd 7958  ax-pre-mulgt0 7959  ax-pre-mulext 7960

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-id 4311  df-po 4314  df-iso 4315  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-iota 5196  df-fun 5237  df-fv 5243  df-riota 5852  df-ov 5900  df-oprab 5901  df-mpo 5902  df-pnf 8025  df-mnf 8026  df-xr 8027  df-ltxr 8028  df-le 8029  df-sub 8161  df-neg 8162  df-reap 8563  df-ap 8570  df-div 8661

This theorem is referenced by:  divmulap  8663  divclap  8666