Theorem xdivval 29627

Description: Value of division: the (unique) element x such that ( B x. x ) = A. This is meaningful only when B is nonzero. (Contributed by Thierry Arnoux, 17-Dec-2016.)

Assertion

Ref	Expression
xdivval	|- ( ( A e. RR* /\ B e. RR /\ B =/= 0 ) -> ( A /𝑒 B ) = ( iota_ x e. RR* ( B xe x ) = A ) )

Distinct variable groups:   x, A   x, B

Proof of Theorem xdivval

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

Step	Hyp	Ref	Expression
1		eldifsn 4317	. . 3 |- ( B e. ( RR \ { 0 } ) <-> ( B e. RR /\ B =/= 0 ) )
2		simpl 473	. . . . . 6 |- ( ( y = A /\ x e. RR* ) -> y = A )
3	2	eqeq2d 2632	. . . . 5 |- ( ( y = A /\ x e. RR* ) -> ( ( z xe x ) = y <-> ( z xe x ) = A ) )
4	3	riotabidva 6627	. . . 4 |- ( y = A -> ( iota_ x e. RR* ( z xe x ) = y ) = ( iota_ x e. RR* ( z xe x ) = A ) )
5		simpl 473	. . . . . . 7 |- ( ( z = B /\ x e. RR* ) -> z = B )
6	5	oveq1d 6665	. . . . . 6 |- ( ( z = B /\ x e. RR* ) -> ( z xe x ) = ( B xe x ) )
7	6	eqeq1d 2624	. . . . 5 |- ( ( z = B /\ x e. RR* ) -> ( ( z xe x ) = A <-> ( B xe x ) = A ) )
8	7	riotabidva 6627	. . . 4 |- ( z = B -> ( iota_ x e. RR* ( z xe x ) = A ) = ( iota_ x e. RR* ( B xe x ) = A ) )
9		df-xdiv 29626	. . . 4 |- /𝑒 = ( y e. RR* , z e. ( RR \ { 0 } ) |-> ( iota_ x e. RR* ( z xe x ) = y ) )
10		riotaex 6615	. . . 4 |- ( iota_ x e. RR* ( B xe x ) = A ) e. _V
11	4, 8, 9, 10	ovmpt2 6796	. . 3 |- ( ( A e. RR* /\ B e. ( RR \ { 0 } ) ) -> ( A /𝑒 B ) = ( iota_ x e. RR* ( B xe x ) = A ) )
12	1, 11	sylan2br 493	. 2 |- ( ( A e. RR* /\ ( B e. RR /\ B =/= 0 ) ) -> ( A /𝑒 B ) = ( iota_ x e. RR* ( B xe x ) = A ) )
13	12	3impb 1260	1 |- ( ( A e. RR* /\ B e. RR /\ B =/= 0 ) -> ( A /𝑒 B ) = ( iota_ x e. RR* ( B xe x ) = A ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   \ cdif 3571   {csn 4177   iota_crio 6610  (class class class)co 6650   RRcr 9935   0cc0 9936   RR*cxr 10073   xecxmu 11945   /_𝑒 cxdiv 29625

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-xdiv 29626

This theorem is referenced by:  xdivcld  29631  xdivmul  29633  rexdiv  29634  xdivpnfrp  29641