Theorem divvalap 8701

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 8018	. . . . . 6 |- 0 e. CC
4		apne 8650	. . . . . 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 3749	. . 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 8696	. . 3 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> E! x e. CC ( B x. x ) = A )
11		riotacl 5892	. . 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 2206	. . . 4 |- ( z = A -> ( ( y x. x ) = z <-> ( y x. x ) = A ) )
14	13	riotabidv 5879	. . 3 |- ( z = A -> ( iota_ x e. CC ( y x. x ) = z ) = ( iota_ x e. CC ( y x. x ) = A ) )
15		oveq1 5929	. . . . 5 |- ( y = B -> ( y x. x ) = ( B x. x ) )
16	15	eqeq1d 2205	. . . 4 |- ( y = B -> ( ( y x. x ) = A <-> ( B x. x ) = A ) )
17	16	riotabidv 5879	. . 3 |- ( y = B -> ( iota_ x e. CC ( y x. x ) = A ) = ( iota_ x e. CC ( B x. x ) = A ) )
18		df-div 8700	. . 3 |- / = ( z e. CC , y e. ( CC \ { 0 } ) |-> ( iota_ x e. CC ( y x. x ) = z ) )
19	14, 17, 18	ovmpog 6057	. 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 2167   =/= wne 2367   E!wreu 2477   \ cdif 3154   {csn 3622   class class class wbr 4033   iota_crio 5876  (class class class)co 5922   CCcc 7877   0cc0 7879   x. cmul 7884   # cap 8608   / cdiv 8699

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-mulrcl 7978  ax-addcom 7979  ax-mulcom 7980  ax-addass 7981  ax-mulass 7982  ax-distr 7983  ax-i2m1 7984  ax-0lt1 7985  ax-1rid 7986  ax-0id 7987  ax-rnegex 7988  ax-precex 7989  ax-cnre 7990  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-lttrn 7993  ax-pre-apti 7994  ax-pre-ltadd 7995  ax-pre-mulgt0 7996  ax-pre-mulext 7997

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-opab 4095  df-id 4328  df-po 4331  df-iso 4332  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-iota 5219  df-fun 5260  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-sub 8199  df-neg 8200  df-reap 8602  df-ap 8609  df-div 8700

This theorem is referenced by:  divmulap  8702  divclap  8705