Theorem onfrALTlem5 38757

Description: Lemma for onfrALT 38764. (Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
onfrALTlem5	|- ( [. ( a i^i x ) / b ]. ( ( b C_ ( a i^i x ) /\ b =/= (/) ) -> E. y e. b ( b i^i y ) = (/) ) <-> ( ( ( a i^i x ) C_ ( a i^i x ) /\ ( a i^i x ) =/= (/) ) -> E. y e. ( a i^i x ) ( ( a i^i x ) i^i y ) = (/) ) )

Distinct variable groups:   a, b, y   x, b, y

Proof of Theorem onfrALTlem5

Step	Hyp	Ref	Expression
1		vex 3203	. . . 4 |- a e. _V
2	1	inex1 4799	. . 3 |- ( a i^i x ) e. _V
3		sbcimg 3477	. . 3 |- ( ( a i^i x ) e. _V -> ( [. ( a i^i x ) / b ]. ( ( b C_ ( a i^i x ) /\ b =/= (/) ) -> E. y e. b ( b i^i y ) = (/) ) <-> ( [. ( a i^i x ) / b ]. ( b C_ ( a i^i x ) /\ b =/= (/) ) -> [. ( a i^i x ) / b ]. E. y e. b ( b i^i y ) = (/) ) ) )
4	2, 3	ax-mp 5	. 2 |- ( [. ( a i^i x ) / b ]. ( ( b C_ ( a i^i x ) /\ b =/= (/) ) -> E. y e. b ( b i^i y ) = (/) ) <-> ( [. ( a i^i x ) / b ]. ( b C_ ( a i^i x ) /\ b =/= (/) ) -> [. ( a i^i x ) / b ]. E. y e. b ( b i^i y ) = (/) ) )
5		sbcan 3478	. . . 4 |- ( [. ( a i^i x ) / b ]. ( b C_ ( a i^i x ) /\ b =/= (/) ) <-> ( [. ( a i^i x ) / b ]. b C_ ( a i^i x ) /\ [. ( a i^i x ) / b ]. b =/= (/) ) )
6		sseq1 3626	. . . . . 6 |- ( b = ( a i^i x ) -> ( b C_ ( a i^i x ) <-> ( a i^i x ) C_ ( a i^i x ) ) )
7	2, 6	sbcie 3470	. . . . 5 |- ( [. ( a i^i x ) / b ]. b C_ ( a i^i x ) <-> ( a i^i x ) C_ ( a i^i x ) )
8		df-ne 2795	. . . . . . 7 |- ( b =/= (/) <-> -. b = (/) )
9	8	sbcbii 3491	. . . . . 6 |- ( [. ( a i^i x ) / b ]. b =/= (/) <-> [. ( a i^i x ) / b ]. -. b = (/) )
10		sbcng 3476	. . . . . . . 8 |- ( ( a i^i x ) e. _V -> ( [. ( a i^i x ) / b ]. -. b = (/) <-> -. [. ( a i^i x ) / b ]. b = (/) ) )
11	10	bicomd 213	. . . . . . 7 |- ( ( a i^i x ) e. _V -> ( -. [. ( a i^i x ) / b ]. b = (/) <-> [. ( a i^i x ) / b ]. -. b = (/) ) )
12	2, 11	ax-mp 5	. . . . . 6 |- ( -. [. ( a i^i x ) / b ]. b = (/) <-> [. ( a i^i x ) / b ]. -. b = (/) )
13		eqsbc3 3475	. . . . . . . 8 |- ( ( a i^i x ) e. _V -> ( [. ( a i^i x ) / b ]. b = (/) <-> ( a i^i x ) = (/) ) )
14	2, 13	ax-mp 5	. . . . . . 7 |- ( [. ( a i^i x ) / b ]. b = (/) <-> ( a i^i x ) = (/) )
15	14	necon3bbii 2841	. . . . . 6 |- ( -. [. ( a i^i x ) / b ]. b = (/) <-> ( a i^i x ) =/= (/) )
16	9, 12, 15	3bitr2i 288	. . . . 5 |- ( [. ( a i^i x ) / b ]. b =/= (/) <-> ( a i^i x ) =/= (/) )
17	7, 16	anbi12i 733	. . . 4 |- ( ( [. ( a i^i x ) / b ]. b C_ ( a i^i x ) /\ [. ( a i^i x ) / b ]. b =/= (/) ) <-> ( ( a i^i x ) C_ ( a i^i x ) /\ ( a i^i x ) =/= (/) ) )
18	5, 17	bitri 264	. . 3 |- ( [. ( a i^i x ) / b ]. ( b C_ ( a i^i x ) /\ b =/= (/) ) <-> ( ( a i^i x ) C_ ( a i^i x ) /\ ( a i^i x ) =/= (/) ) )
19		df-rex 2918	. . . . 5 |- ( E. y e. b ( b i^i y ) = (/) <-> E. y ( y e. b /\ ( b i^i y ) = (/) ) )
20	19	sbcbii 3491	. . . 4 |- ( [. ( a i^i x ) / b ]. E. y e. b ( b i^i y ) = (/) <-> [. ( a i^i x ) / b ]. E. y ( y e. b /\ ( b i^i y ) = (/) ) )
21		sbcan 3478	. . . . . . 7 |- ( [. ( a i^i x ) / b ]. ( y e. b /\ ( b i^i y ) = (/) ) <-> ( [. ( a i^i x ) / b ]. y e. b /\ [. ( a i^i x ) / b ]. ( b i^i y ) = (/) ) )
22		sbcel2gv 3496	. . . . . . . . 9 |- ( ( a i^i x ) e. _V -> ( [. ( a i^i x ) / b ]. y e. b <-> y e. ( a i^i x ) ) )
23	2, 22	ax-mp 5	. . . . . . . 8 |- ( [. ( a i^i x ) / b ]. y e. b <-> y e. ( a i^i x ) )
24		sbceqg 3984	. . . . . . . . . 10 |- ( ( a i^i x ) e. _V -> ( [. ( a i^i x ) / b ]. ( b i^i y ) = (/) <-> [_ ( a i^i x ) / b ]_ ( b i^i y ) = [_ ( a i^i x ) / b ]_ (/) ) )
25	2, 24	ax-mp 5	. . . . . . . . 9 |- ( [. ( a i^i x ) / b ]. ( b i^i y ) = (/) <-> [_ ( a i^i x ) / b ]_ ( b i^i y ) = [_ ( a i^i x ) / b ]_ (/) )
26		csbin 4010	. . . . . . . . . . 11 |- [_ ( a i^i x ) / b ]_ ( b i^i y ) = ( [_ ( a i^i x ) / b ]_ b i^i [_ ( a i^i x ) / b ]_ y )
27		csbvarg 4003	. . . . . . . . . . . . 13 |- ( ( a i^i x ) e. _V -> [_ ( a i^i x ) / b ]_ b = ( a i^i x ) )
28	2, 27	ax-mp 5	. . . . . . . . . . . 12 |- [_ ( a i^i x ) / b ]_ b = ( a i^i x )
29		csbconstg 3546	. . . . . . . . . . . . 13 |- ( ( a i^i x ) e. _V -> [_ ( a i^i x ) / b ]_ y = y )
30	2, 29	ax-mp 5	. . . . . . . . . . . 12 |- [_ ( a i^i x ) / b ]_ y = y
31	28, 30	ineq12i 3812	. . . . . . . . . . 11 |- ( [_ ( a i^i x ) / b ]_ b i^i [_ ( a i^i x ) / b ]_ y ) = ( ( a i^i x ) i^i y )
32	26, 31	eqtri 2644	. . . . . . . . . 10 |- [_ ( a i^i x ) / b ]_ ( b i^i y ) = ( ( a i^i x ) i^i y )
33		csb0 3982	. . . . . . . . . 10 |- [_ ( a i^i x ) / b ]_ (/) = (/)
34	32, 33	eqeq12i 2636	. . . . . . . . 9 |- ( [_ ( a i^i x ) / b ]_ ( b i^i y ) = [_ ( a i^i x ) / b ]_ (/) <-> ( ( a i^i x ) i^i y ) = (/) )
35	25, 34	bitri 264	. . . . . . . 8 |- ( [. ( a i^i x ) / b ]. ( b i^i y ) = (/) <-> ( ( a i^i x ) i^i y ) = (/) )
36	23, 35	anbi12i 733	. . . . . . 7 |- ( ( [. ( a i^i x ) / b ]. y e. b /\ [. ( a i^i x ) / b ]. ( b i^i y ) = (/) ) <-> ( y e. ( a i^i x ) /\ ( ( a i^i x ) i^i y ) = (/) ) )
37	21, 36	bitri 264	. . . . . 6 |- ( [. ( a i^i x ) / b ]. ( y e. b /\ ( b i^i y ) = (/) ) <-> ( y e. ( a i^i x ) /\ ( ( a i^i x ) i^i y ) = (/) ) )
38	37	exbii 1774	. . . . 5 |- ( E. y [. ( a i^i x ) / b ]. ( y e. b /\ ( b i^i y ) = (/) ) <-> E. y ( y e. ( a i^i x ) /\ ( ( a i^i x ) i^i y ) = (/) ) )
39		sbcex2 3486	. . . . 5 |- ( [. ( a i^i x ) / b ]. E. y ( y e. b /\ ( b i^i y ) = (/) ) <-> E. y [. ( a i^i x ) / b ]. ( y e. b /\ ( b i^i y ) = (/) ) )
40		df-rex 2918	. . . . 5 |- ( E. y e. ( a i^i x ) ( ( a i^i x ) i^i y ) = (/) <-> E. y ( y e. ( a i^i x ) /\ ( ( a i^i x ) i^i y ) = (/) ) )
41	38, 39, 40	3bitr4i 292	. . . 4 |- ( [. ( a i^i x ) / b ]. E. y ( y e. b /\ ( b i^i y ) = (/) ) <-> E. y e. ( a i^i x ) ( ( a i^i x ) i^i y ) = (/) )
42	20, 41	bitri 264	. . 3 |- ( [. ( a i^i x ) / b ]. E. y e. b ( b i^i y ) = (/) <-> E. y e. ( a i^i x ) ( ( a i^i x ) i^i y ) = (/) )
43	18, 42	imbi12i 340	. 2 |- ( ( [. ( a i^i x ) / b ]. ( b C_ ( a i^i x ) /\ b =/= (/) ) -> [. ( a i^i x ) / b ]. E. y e. b ( b i^i y ) = (/) ) <-> ( ( ( a i^i x ) C_ ( a i^i x ) /\ ( a i^i x ) =/= (/) ) -> E. y e. ( a i^i x ) ( ( a i^i x ) i^i y ) = (/) ) )
44	4, 43	bitri 264	1 |- ( [. ( a i^i x ) / b ]. ( ( b C_ ( a i^i x ) /\ b =/= (/) ) -> E. y e. b ( b i^i y ) = (/) ) <-> ( ( ( a i^i x ) C_ ( a i^i x ) /\ ( a i^i x ) =/= (/) ) -> E. y e. ( a i^i x ) ( ( a i^i x ) i^i y ) = (/) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   E.wrex 2913   _Vcvv 3200   [.wsbc 3435   [_csb 3533   i^i cin 3573   C_ wss 3574   (/)c0 3915

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

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-in 3581  df-ss 3588  df-nul 3916

This theorem is referenced by:  onfrALTlem3  38759  onfrALTlem3VD  39123