Theorem igenval2 16214

Description: The ideal generated by a subset of a ring.

Hypotheses

Ref	Expression
igenval2.1	|- G = (1st` R)
igenval2.2	|- X = ran G

Assertion

Ref	Expression
igenval2	|- ((R e. Ring /\ S C_ X) -> ((R IdlGen S) = I <-> (I e. (Idl` R) /\ S C_ I /\ A.j e. (Idl` R)(S C_ j -> I C_ j))))

Distinct variable groups:   R,j   S,j   j,I

Proof of Theorem igenval2

Step	Hyp	Ref	Expression
1		eleq1 1957	. . . 4 |- ((R IdlGen S) = I -> ((R IdlGen S) e. (Idl` R) <-> I e. (Idl` R)))
2		sseq2 2639	. . . 4 |- ((R IdlGen S) = I -> (S C_ (R IdlGen S) <-> S C_ I))
3		sseq1 2637	. . . . . 6 |- ((R IdlGen S) = I -> ((R IdlGen S) C_ j <-> I C_ j))
4	3	imbi2d 674	. . . . 5 |- ((R IdlGen S) = I -> ((S C_ j -> (R IdlGen S) C_ j) <-> (S C_ j -> I C_ j)))
5	4	ralbidv 2123	. . . 4 |- ((R IdlGen S) = I -> (A.j e. (Idl` R)(S C_ j -> (R IdlGen S) C_ j) <-> A.j e. (Idl` R)(S C_ j -> I C_ j)))
6	1, 2, 5	3anbi123d 1168	. . 3 |- ((R IdlGen S) = I -> (((R IdlGen S) e. (Idl` R) /\ S C_ (R IdlGen S) /\ A.j e. (Idl` R)(S C_ j -> (R IdlGen S) C_ j)) <-> (I e. (Idl` R) /\ S C_ I /\ A.j e. (Idl` R)(S C_ j -> I C_ j))))
7		igenval2.1	. . . . 5 |- G = (1st` R)
8		igenval2.2	. . . . 5 |- X = ran G
9	7, 8	igenidl 16211	. . . 4 |- ((R e. Ring /\ S C_ X) -> (R IdlGen S) e. (Idl` R))
10	7, 8	igenss 16210	. . . 4 |- ((R e. Ring /\ S C_ X) -> S C_ (R IdlGen S))
11		igenmin 16212	. . . . . . 7 |- ((R e. Ring /\ j e. (Idl` R) /\ S C_ j) -> (R IdlGen S) C_ j)
12	11	3expia 1069	. . . . . 6 |- ((R e. Ring /\ j e. (Idl` R)) -> (S C_ j -> (R IdlGen S) C_ j))
13	12	r19.21aiva 2176	. . . . 5 |- (R e. Ring -> A.j e. (Idl` R)(S C_ j -> (R IdlGen S) C_ j))
14	13	adantr 425	. . . 4 |- ((R e. Ring /\ S C_ X) -> A.j e. (Idl` R)(S C_ j -> (R IdlGen S) C_ j))
15	9, 10, 14	3jca 1050	. . 3 |- ((R e. Ring /\ S C_ X) -> ((R IdlGen S) e. (Idl` R) /\ S C_ (R IdlGen S) /\ A.j e. (Idl` R)(S C_ j -> (R IdlGen S) C_ j)))
16	6, 15	syl5cbi 226	. 2 |- ((R e. Ring /\ S C_ X) -> ((R IdlGen S) = I -> (I e. (Idl` R) /\ S C_ I /\ A.j e. (Idl` R)(S C_ j -> I C_ j))))
17		igenmin 16212	. . . . . 6 |- ((R e. Ring /\ I e. (Idl` R) /\ S C_ I) -> (R IdlGen S) C_ I)
18	17	3adant3r3 1079	. . . . 5 |- ((R e. Ring /\ (I e. (Idl` R) /\ S C_ I /\ A.j e. (Idl` R)(S C_ j -> I C_ j))) -> (R IdlGen S) C_ I)
19	18	adantlr 429	. . . 4 |- (((R e. Ring /\ S C_ X) /\ (I e. (Idl` R) /\ S C_ I /\ A.j e. (Idl` R)(S C_ j -> I C_ j))) -> (R IdlGen S) C_ I)
20		sseq2 2639	. . . . . . . . . 10 |- (i = j -> (S C_ i <-> S C_ j))
21	20	ralrab 2418	. . . . . . . . 9 |- (A.j e. {i e. (Idl` R) | S C_ i}I C_ j <-> A.j e. (Idl` R)(S C_ j -> I C_ j))
22	21	biimpri 169	. . . . . . . 8 |- (A.j e. (Idl` R)(S C_ j -> I C_ j) -> A.j e. {i e. (Idl` R) | S C_ i}I C_ j)
23		ssint 3232	. . . . . . . 8 |- (I C_ |^|{i e. (Idl` R) | S C_ i} <-> A.j e. {i e. (Idl` R) | S C_ i}I C_ j)
24	22, 23	sylibr 217	. . . . . . 7 |- (A.j e. (Idl` R)(S C_ j -> I C_ j) -> I C_ |^|{i e. (Idl` R) | S C_ i})
25	24	3ad2ant3 899	. . . . . 6 |- ((I e. (Idl` R) /\ S C_ I /\ A.j e. (Idl` R)(S C_ j -> I C_ j)) -> I C_ |^|{i e. (Idl` R) | S C_ i})
26	25	adantl 424	. . . . 5 |- (((R e. Ring /\ S C_ X) /\ (I e. (Idl` R) /\ S C_ I /\ A.j e. (Idl` R)(S C_ j -> I C_ j))) -> I C_ |^|{i e. (Idl` R) | S C_ i})
27	7, 8	igenval 16209	. . . . . 6 |- ((R e. Ring /\ S C_ X) -> (R IdlGen S) = |^|{i e. (Idl` R) | S C_ i})
28	27	adantr 425	. . . . 5 |- (((R e. Ring /\ S C_ X) /\ (I e. (Idl` R) /\ S C_ I /\ A.j e. (Idl` R)(S C_ j -> I C_ j))) -> (R IdlGen S) = |^|{i e. (Idl` R) | S C_ i})
29	26, 28	sseqtr4d 2654	. . . 4 |- (((R e. Ring /\ S C_ X) /\ (I e. (Idl` R) /\ S C_ I /\ A.j e. (Idl` R)(S C_ j -> I C_ j))) -> I C_ (R IdlGen S))
30	19, 29	eqssd 2633	. . 3 |- (((R e. Ring /\ S C_ X) /\ (I e. (Idl` R) /\ S C_ I /\ A.j e. (Idl` R)(S C_ j -> I C_ j))) -> (R IdlGen S) = I)
31	30	ex 402	. 2 |- ((R e. Ring /\ S C_ X) -> ((I e. (Idl` R) /\ S C_ I /\ A.j e. (Idl` R)(S C_ j -> I C_ j)) -> (R IdlGen S) = I))
32	16, 31	impbid 574	1 |- ((R e. Ring /\ S C_ X) -> ((R IdlGen S) = I <-> (I e. (Idl` R) /\ S C_ I /\ A.j e. (Idl` R)(S C_ j -> I C_ j))))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  A.wral 2105  {crab 2108   C_ wss 2593  |^|cint 3214  ran crn 3987  ` cfv 3998  (class class class)co 4884  1stc1st 5018  Ringcring 9463  Idlcidl 16155   IdlGen cigen 16207

This theorem is referenced by:  prnc 16215

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-int 3215  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-1st 5020  df-2nd 5021  df-grp 9316  df-gid 9317  df-abl 9408  df-ring 9464  df-idl 16158  df-igen 16208

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