Theorem igenval 16209

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

Hypotheses

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

Assertion

Ref	Expression
igenval	|- ((R e. Ring /\ S C_ X) -> (R IdlGen S) = |^|{j e. (Idl` R) | S C_ j})

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

Proof of Theorem igenval

Step	Hyp	Ref	Expression
1		simpl 346	. . 3 |- ((R e. Ring /\ S C_ X) -> R e. Ring)
2		igenval.2	. . . . . 6 |- X = ran G
3		igenval.1	. . . . . . . 8 |- G = (1st` R)
4		fvex 4689	. . . . . . . 8 |- (1st` R) e. _V
5	3, 4	eqeltri 1967	. . . . . . 7 |- G e. _V
6	5	rnex 4209	. . . . . 6 |- ran G e. _V
7	2, 6	eqeltri 1967	. . . . 5 |- X e. _V
8	7	ssex 3455	. . . 4 |- (S C_ X -> S e. _V)
9	8	adantl 424	. . 3 |- ((R e. Ring /\ S C_ X) -> S e. _V)
10		sseq2 2639	. . . . . . 7 |- (j = X -> (S C_ j <-> S C_ X))
11	10	rcla4ev 2381	. . . . . 6 |- ((X e. (Idl` R) /\ S C_ X) -> E.j e. (Idl` R)S C_ j)
12	3, 2	rngidl 16172	. . . . . 6 |- (R e. Ring -> X e. (Idl` R))
13	11, 12	sylan 497	. . . . 5 |- ((R e. Ring /\ S C_ X) -> E.j e. (Idl` R)S C_ j)
14		rabn0 2893	. . . . 5 |- ({j e. (Idl` R) | S C_ j} =/= (/) <-> E.j e. (Idl` R)S C_ j)
15	13, 14	sylibr 217	. . . 4 |- ((R e. Ring /\ S C_ X) -> {j e. (Idl` R) | S C_ j} =/= (/))
16		intex 3465	. . . 4 |- ({j e. (Idl` R) | S C_ j} =/= (/) <-> |^|{j e. (Idl` R) | S C_ j} e. _V)
17	15, 16	sylib 215	. . 3 |- ((R e. Ring /\ S C_ X) -> |^|{j e. (Idl` R) | S C_ j} e. _V)
18	1, 9, 17	3jca 1050	. 2 |- ((R e. Ring /\ S C_ X) -> (R e. Ring /\ S e. _V /\ |^|{j e. (Idl` R) | S C_ j} e. _V))
19		eleq1 1957	. . . . . . 7 |- (r = R -> (r e. Ring <-> R e. Ring))
20		fveq2 4681	. . . . . . . . . . . 12 |- (r = R -> (1st` r) = (1st` R))
21	20, 3	syl6eqr 1946	. . . . . . . . . . 11 |- (r = R -> (1st` r) = G)
22	21	rneqd 4188	. . . . . . . . . 10 |- (r = R -> ran (1st` r) = ran G)
23	22, 2	syl6eqr 1946	. . . . . . . . 9 |- (r = R -> ran (1st` r) = X)
24		pweq 3036	. . . . . . . . 9 |- (ran (1st` r) = X -> ~Pran (1st` r) = ~PX)
25	23, 24	syl 12	. . . . . . . 8 |- (r = R -> ~Pran (1st` r) = ~PX)
26	25	eleq2d 1964	. . . . . . 7 |- (r = R -> (s e. ~Pran (1st` r) <-> s e. ~PX))
27	19, 26	anbi12d 690	. . . . . 6 |- (r = R -> ((r e. Ring /\ s e. ~Pran (1st` r)) <-> (R e. Ring /\ s e. ~PX)))
28		fveq2 4681	. . . . . . . . 9 |- (r = R -> (Idl` r) = (Idl` R))
29		rabeq 2289	. . . . . . . . 9 |- ((Idl` r) = (Idl` R) -> {j e. (Idl` r) | s C_ j} = {j e. (Idl` R) | s C_ j})
30	28, 29	syl 12	. . . . . . . 8 |- (r = R -> {j e. (Idl` r) | s C_ j} = {j e. (Idl` R) | s C_ j})
31	30	inteqd 3219	. . . . . . 7 |- (r = R -> |^|{j e. (Idl` r) | s C_ j} = |^|{j e. (Idl` R) | s C_ j})
32	31	eqeq2d 1895	. . . . . 6 |- (r = R -> (i = |^|{j e. (Idl` r) | s C_ j} <-> i = |^|{j e. (Idl` R) | s C_ j}))
33	27, 32	anbi12d 690	. . . . 5 |- (r = R -> (((r e. Ring /\ s e. ~Pran (1st` r)) /\ i = |^|{j e. (Idl` r) | s C_ j}) <-> ((R e. Ring /\ s e. ~PX) /\ i = |^|{j e. (Idl` R) | s C_ j})))
34		eleq1 1957	. . . . . . 7 |- (s = S -> (s e. ~PX <-> S e. ~PX))
35	34	anbi2d 678	. . . . . 6 |- (s = S -> ((R e. Ring /\ s e. ~PX) <-> (R e. Ring /\ S e. ~PX)))
36		sseq1 2637	. . . . . . . . 9 |- (s = S -> (s C_ j <-> S C_ j))
37	36	rabbidv 2287	. . . . . . . 8 |- (s = S -> {j e. (Idl` R) | s C_ j} = {j e. (Idl` R) | S C_ j})
38	37	inteqd 3219	. . . . . . 7 |- (s = S -> |^|{j e. (Idl` R) | s C_ j} = |^|{j e. (Idl` R) | S C_ j})
39	38	eqeq2d 1895	. . . . . 6 |- (s = S -> (i = |^|{j e. (Idl` R) | s C_ j} <-> i = |^|{j e. (Idl` R) | S C_ j}))
40	35, 39	anbi12d 690	. . . . 5 |- (s = S -> (((R e. Ring /\ s e. ~PX) /\ i = |^|{j e. (Idl` R) | s C_ j}) <-> ((R e. Ring /\ S e. ~PX) /\ i = |^|{j e. (Idl` R) | S C_ j})))
41		simpl 346	. . . . . 6 |- (((R e. Ring /\ S e. ~PX) /\ i = |^|{j e. (Idl` R) | S C_ j}) -> (R e. Ring /\ S e. ~PX))
42		pm3.21 306	. . . . . 6 |- (i = |^|{j e. (Idl` R) | S C_ j} -> ((R e. Ring /\ S e. ~PX) -> ((R e. Ring /\ S e. ~PX) /\ i = |^|{j e. (Idl` R) | S C_ j})))
43	41, 42	impbid2 576	. . . . 5 |- (i = |^|{j e. (Idl` R) | S C_ j} -> (((R e. Ring /\ S e. ~PX) /\ i = |^|{j e. (Idl` R) | S C_ j}) <-> (R e. Ring /\ S e. ~PX)))
44		moeq 2431	. . . . . 6 |- E*i i = |^|{j e. (Idl` r) | s C_ j}
45	44	moani 1820	. . . . 5 |- E*i((r e. Ring /\ s e. ~Pran (1st` r)) /\ i = |^|{j e. (Idl` r) | s C_ j})
46		df-igen 16208	. . . . 5 |- IdlGen = {<.<.r, s>., i>. | ((r e. Ring /\ s e. ~Pran (1st` r)) /\ i = |^|{j e. (Idl` r) | s C_ j})}
47	33, 40, 43, 45, 46	oprabvaligg 10154	. . . 4 |- ((R e. Ring /\ S e. _V /\ |^|{j e. (Idl` R) | S C_ j} e. _V) -> ((R e. Ring /\ S e. ~PX) -> (R IdlGen S) = |^|{j e. (Idl` R) | S C_ j}))
48	47	com12 14	. . 3 |- ((R e. Ring /\ S e. ~PX) -> ((R e. Ring /\ S e. _V /\ |^|{j e. (Idl` R) | S C_ j} e. _V) -> (R IdlGen S) = |^|{j e. (Idl` R) | S C_ j}))
49	7	elpw2 3464	. . 3 |- (S e. ~PX <-> S C_ X)
50	48, 49	sylan2br 502	. 2 |- ((R e. Ring /\ S C_ X) -> ((R e. Ring /\ S e. _V /\ |^|{j e. (Idl` R) | S C_ j} e. _V) -> (R IdlGen S) = |^|{j e. (Idl` R) | S C_ j}))
51	18, 50	mpd 29	1 |- ((R e. Ring /\ S C_ X) -> (R IdlGen S) = |^|{j e. (Idl` R) | S C_ j})

Syntax hints:   -> wi 3   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300   =/= wne 2017  E.wrex 2106  {crab 2108  _Vcvv 2292   C_ wss 2593  (/)c0 2875  ~Pcpw 3032  |^|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:  igenss 16210  igenidl 16211  igenmin 16212  igenidl2 16213  igenval2 16214

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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