Theorem onfrALTlem5 33457

Description: Lemma for onfrALT 33464. (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 3112	. . . 4 |- a e. _V
2	1	inex1 4597	. . 3 |- ( a i^i x ) e. _V
3		sbcimg 3369	. . 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 3370	. . . 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 3520	. . . . . 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 3362	. . . . 5 |- ( [. ( a i^i x ) / b ]. b C_ ( a i^i x ) <-> ( a i^i x ) C_ ( a i^i x ) )
8		df-ne 2654	. . . . . . 7 |- ( b =/= (/) <-> -. b = (/) )
9	8	sbcbii 3387	. . . . . 6 |- ( [. ( a i^i x ) / b ]. b =/= (/) <-> [. ( a i^i x ) / b ]. -. b = (/) )
10		sbcng 3368	. . . . . . . 8 |- ( ( a i^i x ) e. _V -> ( [. ( a i^i x ) / b ]. -. b = (/) <-> -. [. ( a i^i x ) / b ]. b = (/) ) )
11	10	bicomd 201	. . . . . . 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 3367	. . . . . . . 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 2718	. . . . . 6 |- ( -. [. ( a i^i x ) / b ]. b = (/) <-> ( a i^i x ) =/= (/) )
16	9, 12, 15	3bitr2i 273	. . . . 5 |- ( [. ( a i^i x ) / b ]. b =/= (/) <-> ( a i^i x ) =/= (/) )
17	7, 16	anbi12i 697	. . . 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 249	. . 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 2813	. . . . 5 |- ( E. y e. b ( b i^i y ) = (/) <-> E. y ( y e. b /\ ( b i^i y ) = (/) ) )
20	19	sbcbii 3387	. . . 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 3370	. . . . . . 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 3393	. . . . . . . . 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 3834	. . . . . . . . . 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 3863	. . . . . . . . . . 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 3855	. . . . . . . . . . . . 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 3443	. . . . . . . . . . . . 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 3694	. . . . . . . . . . 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 2486	. . . . . . . . . 10 |- [_ ( a i^i x ) / b ]_ ( b i^i y ) = ( ( a i^i x ) i^i y )
33		csb0 3831	. . . . . . . . . 10 |- [_ ( a i^i x ) / b ]_ (/) = (/)
34	32, 33	eqeq12i 2477	. . . . . . . . 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 249	. . . . . . . 8 |- ( [. ( a i^i x ) / b ]. ( b i^i y ) = (/) <-> ( ( a i^i x ) i^i y ) = (/) )
36	23, 35	anbi12i 697	. . . . . . 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 249	. . . . . 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 1668	. . . . 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 3381	. . . . 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 2813	. . . . 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 277	. . . 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 249	. . 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 326	. 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 249	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 184   /\ wa 369   = wceq 1395   E.wex 1613   e. wcel 1819   =/= wne 2652   E.wrex 2808   _Vcvv 3109   [.wsbc 3327   [_csb 3430   i^i cin 3470   C_ wss 3471   (/)c0 3793

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-in 3478  df-ss 3485  df-nul 3794

This theorem is referenced by:  onfrALTlem3  33459  onfrALTlem3VD  33830