Theorem lidlrsppropdg 13975

Description: The left ideals and ring span of a ring depend only on the ring components. Here W is expected to be either B (when closure is available) or _V (when strong equality is available). (Contributed by Mario Carneiro, 14-Jun-2015.)

Hypotheses

Ref	Expression
lidlpropd.1	|- ( ph -> B = ( Base ` K ) )
lidlpropd.2	|- ( ph -> B = ( Base ` L ) )
lidlpropd.3	|- ( ph -> B C_ W )
lidlpropd.4	|- ( ( ph /\ ( x e. W /\ y e. W ) ) -> ( x ( +g ` K ) y ) = ( x ( +g ` L ) y ) )
lidlpropd.5	|- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( .r ` K ) y ) e. W )
lidlpropd.6	|- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( .r ` K ) y ) = ( x ( .r ` L ) y ) )
lidlpropdg.k	|- ( ph -> K e. X )
lidlpropdg.l	|- ( ph -> L e. Y )

Assertion

Ref	Expression
lidlrsppropdg	|- ( ph -> ( (LIdeal ` K ) = (LIdeal ` L ) /\ (RSpan ` K ) = (RSpan ` L ) ) )

Distinct variable groups:   x, y, B   x, K, y   x, L, y   ph, x, y   x, W, y

Allowed substitution hints:   X( x, y)   Y( x, y)

Proof of Theorem lidlrsppropdg

Step	Hyp	Ref	Expression
1		lidlpropd.1	. . . . 5 |- ( ph -> B = ( Base ` K ) )
2		lidlpropdg.k	. . . . . 6 |- ( ph -> K e. X )
3		rlmbasg 13935	. . . . . 6 |- ( K e. X -> ( Base ` K ) = ( Base ` (ringLMod ` K ) ) )
4	2, 3	syl 14	. . . . 5 |- ( ph -> ( Base ` K ) = ( Base ` (ringLMod ` K ) ) )
5	1, 4	eqtrd 2226	. . . 4 |- ( ph -> B = ( Base ` (ringLMod ` K ) ) )
6		lidlpropd.2	. . . . 5 |- ( ph -> B = ( Base ` L ) )
7		lidlpropdg.l	. . . . . 6 |- ( ph -> L e. Y )
8		rlmbasg 13935	. . . . . 6 |- ( L e. Y -> ( Base ` L ) = ( Base ` (ringLMod ` L ) ) )
9	7, 8	syl 14	. . . . 5 |- ( ph -> ( Base ` L ) = ( Base ` (ringLMod ` L ) ) )
10	6, 9	eqtrd 2226	. . . 4 |- ( ph -> B = ( Base ` (ringLMod ` L ) ) )
11		lidlpropd.3	. . . 4 |- ( ph -> B C_ W )
12		lidlpropd.4	. . . . 5 |- ( ( ph /\ ( x e. W /\ y e. W ) ) -> ( x ( +g ` K ) y ) = ( x ( +g ` L ) y ) )
13		rlmplusgg 13936	. . . . . . 7 |- ( K e. X -> ( +g ` K ) = ( +g ` (ringLMod ` K ) ) )
14	2, 13	syl 14	. . . . . 6 |- ( ph -> ( +g ` K ) = ( +g ` (ringLMod ` K ) ) )
15	14	oveqdr 5938	. . . . 5 |- ( ( ph /\ ( x e. W /\ y e. W ) ) -> ( x ( +g ` K ) y ) = ( x ( +g ` (ringLMod ` K ) ) y ) )
16		rlmplusgg 13936	. . . . . . 7 |- ( L e. Y -> ( +g ` L ) = ( +g ` (ringLMod ` L ) ) )
17	7, 16	syl 14	. . . . . 6 |- ( ph -> ( +g ` L ) = ( +g ` (ringLMod ` L ) ) )
18	17	oveqdr 5938	. . . . 5 |- ( ( ph /\ ( x e. W /\ y e. W ) ) -> ( x ( +g ` L ) y ) = ( x ( +g ` (ringLMod ` L ) ) y ) )
19	12, 15, 18	3eqtr3d 2234	. . . 4 |- ( ( ph /\ ( x e. W /\ y e. W ) ) -> ( x ( +g ` (ringLMod ` K ) ) y ) = ( x ( +g ` (ringLMod ` L ) ) y ) )
20		rlmvscag 13941	. . . . . . 7 |- ( K e. X -> ( .r ` K ) = ( .s ` (ringLMod ` K ) ) )
21	2, 20	syl 14	. . . . . 6 |- ( ph -> ( .r ` K ) = ( .s ` (ringLMod ` K ) ) )
22	21	oveqdr 5938	. . . . 5 |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( .r ` K ) y ) = ( x ( .s ` (ringLMod ` K ) ) y ) )
23		lidlpropd.5	. . . . 5 |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( .r ` K ) y ) e. W )
24	22, 23	eqeltrrd 2271	. . . 4 |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( .s ` (ringLMod ` K ) ) y ) e. W )
25		lidlpropd.6	. . . . 5 |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( .r ` K ) y ) = ( x ( .r ` L ) y ) )
26		rlmvscag 13941	. . . . . . 7 |- ( L e. Y -> ( .r ` L ) = ( .s ` (ringLMod ` L ) ) )
27	7, 26	syl 14	. . . . . 6 |- ( ph -> ( .r ` L ) = ( .s ` (ringLMod ` L ) ) )
28	27	oveqdr 5938	. . . . 5 |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( .r ` L ) y ) = ( x ( .s ` (ringLMod ` L ) ) y ) )
29	25, 22, 28	3eqtr3d 2234	. . . 4 |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( .s ` (ringLMod ` K ) ) y ) = ( x ( .s ` (ringLMod ` L ) ) y ) )
30		rlmscabas 13940	. . . . . 6 |- ( K e. X -> ( Base ` K ) = ( Base ` (Scalar ` (ringLMod ` K ) ) ) )
31	2, 30	syl 14	. . . . 5 |- ( ph -> ( Base ` K ) = ( Base ` (Scalar ` (ringLMod ` K ) ) ) )
32	1, 31	eqtrd 2226	. . . 4 |- ( ph -> B = ( Base ` (Scalar ` (ringLMod ` K ) ) ) )
33		rlmscabas 13940	. . . . . 6 |- ( L e. Y -> ( Base ` L ) = ( Base ` (Scalar ` (ringLMod ` L ) ) ) )
34	7, 33	syl 14	. . . . 5 |- ( ph -> ( Base ` L ) = ( Base ` (Scalar ` (ringLMod ` L ) ) ) )
35	6, 34	eqtrd 2226	. . . 4 |- ( ph -> B = ( Base ` (Scalar ` (ringLMod ` L ) ) ) )
36		rlmfn 13933	. . . . 5 |- ringLMod Fn _V
37	2	elexd 2773	. . . . 5 |- ( ph -> K e. _V )
38		funfvex 5563	. . . . . 6 |- ( ( Fun ringLMod /\ K e. dom ringLMod ) -> (ringLMod ` K ) e. _V )
39	38	funfni 5346	. . . . 5 |- ( (ringLMod Fn _V /\ K e. _V ) -> (ringLMod ` K ) e. _V )
40	36, 37, 39	sylancr 414	. . . 4 |- ( ph -> (ringLMod ` K ) e. _V )
41	7	elexd 2773	. . . . 5 |- ( ph -> L e. _V )
42		funfvex 5563	. . . . . 6 |- ( ( Fun ringLMod /\ L e. dom ringLMod ) -> (ringLMod ` L ) e. _V )
43	42	funfni 5346	. . . . 5 |- ( (ringLMod Fn _V /\ L e. _V ) -> (ringLMod ` L ) e. _V )
44	36, 41, 43	sylancr 414	. . . 4 |- ( ph -> (ringLMod ` L ) e. _V )
45	5, 10, 11, 19, 24, 29, 32, 35, 40, 44	lsspropdg 13911	. . 3 |- ( ph -> ( LSubSp ` (ringLMod ` K ) ) = ( LSubSp ` (ringLMod ` L ) ) )
46		lidlvalg 13951	. . . 4 |- ( K e. X -> (LIdeal ` K ) = ( LSubSp ` (ringLMod ` K ) ) )
47	2, 46	syl 14	. . 3 |- ( ph -> (LIdeal ` K ) = ( LSubSp ` (ringLMod ` K ) ) )
48		lidlvalg 13951	. . . 4 |- ( L e. Y -> (LIdeal ` L ) = ( LSubSp ` (ringLMod ` L ) ) )
49	7, 48	syl 14	. . 3 |- ( ph -> (LIdeal ` L ) = ( LSubSp ` (ringLMod ` L ) ) )
50	45, 47, 49	3eqtr4d 2236	. 2 |- ( ph -> (LIdeal ` K ) = (LIdeal ` L ) )
51	5, 10, 11, 19, 24, 29, 32, 35, 40, 44	lsppropd 13912	. . 3 |- ( ph -> ( LSpan ` (ringLMod ` K ) ) = ( LSpan ` (ringLMod ` L ) ) )
52		rspvalg 13952	. . . 4 |- ( K e. X -> (RSpan ` K ) = ( LSpan ` (ringLMod ` K ) ) )
53	2, 52	syl 14	. . 3 |- ( ph -> (RSpan ` K ) = ( LSpan ` (ringLMod ` K ) ) )
54		rspvalg 13952	. . . 4 |- ( L e. Y -> (RSpan ` L ) = ( LSpan ` (ringLMod ` L ) ) )
55	7, 54	syl 14	. . 3 |- ( ph -> (RSpan ` L ) = ( LSpan ` (ringLMod ` L ) ) )
56	51, 53, 55	3eqtr4d 2236	. 2 |- ( ph -> (RSpan ` K ) = (RSpan ` L ) )
57	50, 56	jca 306	1 |- ( ph -> ( (LIdeal ` K ) = (LIdeal ` L ) /\ (RSpan ` K ) = (RSpan ` L ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2164   _Vcvv 2760   C_ wss 3153   Fn wfn 5241   ` cfv 5246  (class class class)co 5910   Basecbs 12608   +g cplusg 12685   .rcmulr 12686  Scalarcsca 12688   .scvsca 12689   LSubSpclss 13832   LSpanclspn 13866  ringLModcrglmod 13914  LIdealclidl 13947  RSpancrsp 13948

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4462  ax-setind 4565  ax-cnex 7953  ax-resscn 7954  ax-1cn 7955  ax-1re 7956  ax-icn 7957  ax-addcl 7958  ax-addrcl 7959  ax-mulcl 7960  ax-addcom 7962  ax-addass 7964  ax-i2m1 7967  ax-0lt1 7968  ax-0id 7970  ax-rnegex 7971  ax-pre-ltirr 7974  ax-pre-lttrn 7976  ax-pre-ltadd 7978

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4322  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-iota 5207  df-fun 5248  df-fn 5249  df-f 5250  df-f1 5251  df-fo 5252  df-f1o 5253  df-fv 5254  df-ov 5913  df-oprab 5914  df-mpo 5915  df-pnf 8046  df-mnf 8047  df-ltxr 8049  df-inn 8973  df-2 9031  df-3 9032  df-4 9033  df-5 9034  df-6 9035  df-7 9036  df-8 9037  df-ndx 12611  df-slot 12612  df-base 12614  df-sets 12615  df-iress 12616  df-plusg 12698  df-mulr 12699  df-sca 12701  df-vsca 12702  df-ip 12703  df-lssm 13833  df-lsp 13867  df-sra 13915  df-rgmod 13916  df-lidl 13949  df-rsp 13950

This theorem is referenced by:  crngridl  14010