Theorem divvalap 8604

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 997	. 2 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> A e. CC )
2		simp2 998	. . 3 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> B e. CC )
3		0cn 7924	. . . . . 6 |- 0 e. CC
4		apne 8554	. . . . . 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 1200	. . 3 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> B =/= 0 )
8		eldifsn 3716	. . 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 8599	. . 3 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> E! x e. CC ( B x. x ) = A )
11		riotacl 5835	. . 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 2185	. . . 4 |- ( z = A -> ( ( y x. x ) = z <-> ( y x. x ) = A ) )
14	13	riotabidv 5823	. . 3 |- ( z = A -> ( iota_ x e. CC ( y x. x ) = z ) = ( iota_ x e. CC ( y x. x ) = A ) )
15		oveq1 5872	. . . . 5 |- ( y = B -> ( y x. x ) = ( B x. x ) )
16	15	eqeq1d 2184	. . . 4 |- ( y = B -> ( ( y x. x ) = A <-> ( B x. x ) = A ) )
17	16	riotabidv 5823	. . 3 |- ( y = B -> ( iota_ x e. CC ( y x. x ) = A ) = ( iota_ x e. CC ( B x. x ) = A ) )
18		df-div 8603	. . 3 |- / = ( z e. CC , y e. ( CC \ { 0 } ) |-> ( iota_ x e. CC ( y x. x ) = z ) )
19	14, 17, 18	ovmpog 5999	. 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 1238	1 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> ( A / B ) = ( iota_ x e. CC ( B x. x ) = A ) )

Syntax hints:   -> wi 4   /\ w3a 978   = wceq 1353   e. wcel 2146   =/= wne 2345   E!wreu 2455   \ cdif 3124   {csn 3589   class class class wbr 3998   iota_crio 5820  (class class class)co 5865   CCcc 7784   0cc0 7786   x. cmul 7791   # cap 8512   / cdiv 8602

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 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-cnex 7877  ax-resscn 7878  ax-1cn 7879  ax-1re 7880  ax-icn 7881  ax-addcl 7882  ax-addrcl 7883  ax-mulcl 7884  ax-mulrcl 7885  ax-addcom 7886  ax-mulcom 7887  ax-addass 7888  ax-mulass 7889  ax-distr 7890  ax-i2m1 7891  ax-0lt1 7892  ax-1rid 7893  ax-0id 7894  ax-rnegex 7895  ax-precex 7896  ax-cnre 7897  ax-pre-ltirr 7898  ax-pre-ltwlin 7899  ax-pre-lttrn 7900  ax-pre-apti 7901  ax-pre-ltadd 7902  ax-pre-mulgt0 7903  ax-pre-mulext 7904

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-nel 2441  df-ral 2458  df-rex 2459  df-reu 2460  df-rmo 2461  df-rab 2462  df-v 2737  df-sbc 2961  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-br 3999  df-opab 4060  df-id 4287  df-po 4290  df-iso 4291  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-iota 5170  df-fun 5210  df-fv 5216  df-riota 5821  df-ov 5868  df-oprab 5869  df-mpo 5870  df-pnf 7968  df-mnf 7969  df-xr 7970  df-ltxr 7971  df-le 7972  df-sub 8104  df-neg 8105  df-reap 8506  df-ap 8513  df-div 8603

This theorem is referenced by:  divmulap  8605  divclap  8608