Theorem lidlrsppropdg 14061

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 14021	. . . . . 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 2229	. . . 4 |- ( ph -> B = ( Base ` (ringLMod ` K ) ) )
6		lidlpropd.2	. . . . 5 |- ( ph -> B = ( Base ` L ) )
7		lidlpropdg.l	. . . . . 6 |- ( ph -> L e. Y )
8		rlmbasg 14021	. . . . . 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 2229	. . . 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 14022	. . . . . . 7 |- ( K e. X -> ( +g ` K ) = ( +g ` (ringLMod ` K ) ) )
14	2, 13	syl 14	. . . . . 6 |- ( ph -> ( +g ` K ) = ( +g ` (ringLMod ` K ) ) )
15	14	oveqdr 5951	. . . . 5 |- ( ( ph /\ ( x e. W /\ y e. W ) ) -> ( x ( +g ` K ) y ) = ( x ( +g ` (ringLMod ` K ) ) y ) )
16		rlmplusgg 14022	. . . . . . 7 |- ( L e. Y -> ( +g ` L ) = ( +g ` (ringLMod ` L ) ) )
17	7, 16	syl 14	. . . . . 6 |- ( ph -> ( +g ` L ) = ( +g ` (ringLMod ` L ) ) )
18	17	oveqdr 5951	. . . . 5 |- ( ( ph /\ ( x e. W /\ y e. W ) ) -> ( x ( +g ` L ) y ) = ( x ( +g ` (ringLMod ` L ) ) y ) )
19	12, 15, 18	3eqtr3d 2237	. . . 4 |- ( ( ph /\ ( x e. W /\ y e. W ) ) -> ( x ( +g ` (ringLMod ` K ) ) y ) = ( x ( +g ` (ringLMod ` L ) ) y ) )
20		rlmvscag 14027	. . . . . . 7 |- ( K e. X -> ( .r ` K ) = ( .s ` (ringLMod ` K ) ) )
21	2, 20	syl 14	. . . . . 6 |- ( ph -> ( .r ` K ) = ( .s ` (ringLMod ` K ) ) )
22	21	oveqdr 5951	. . . . 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 2274	. . . 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 14027	. . . . . . 7 |- ( L e. Y -> ( .r ` L ) = ( .s ` (ringLMod ` L ) ) )
27	7, 26	syl 14	. . . . . 6 |- ( ph -> ( .r ` L ) = ( .s ` (ringLMod ` L ) ) )
28	27	oveqdr 5951	. . . . 5 |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( .r ` L ) y ) = ( x ( .s ` (ringLMod ` L ) ) y ) )
29	25, 22, 28	3eqtr3d 2237	. . . 4 |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( .s ` (ringLMod ` K ) ) y ) = ( x ( .s ` (ringLMod ` L ) ) y ) )
30		rlmscabas 14026	. . . . . 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 2229	. . . 4 |- ( ph -> B = ( Base ` (Scalar ` (ringLMod ` K ) ) ) )
33		rlmscabas 14026	. . . . . 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 2229	. . . 4 |- ( ph -> B = ( Base ` (Scalar ` (ringLMod ` L ) ) ) )
36		rlmfn 14019	. . . . 5 |- ringLMod Fn _V
37	2	elexd 2776	. . . . 5 |- ( ph -> K e. _V )
38		funfvex 5576	. . . . . 6 |- ( ( Fun ringLMod /\ K e. dom ringLMod ) -> (ringLMod ` K ) e. _V )
39	38	funfni 5359	. . . . 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 2776	. . . . 5 |- ( ph -> L e. _V )
42		funfvex 5576	. . . . . 6 |- ( ( Fun ringLMod /\ L e. dom ringLMod ) -> (ringLMod ` L ) e. _V )
43	42	funfni 5359	. . . . 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 13997	. . 3 |- ( ph -> ( LSubSp ` (ringLMod ` K ) ) = ( LSubSp ` (ringLMod ` L ) ) )
46		lidlvalg 14037	. . . 4 |- ( K e. X -> (LIdeal ` K ) = ( LSubSp ` (ringLMod ` K ) ) )
47	2, 46	syl 14	. . 3 |- ( ph -> (LIdeal ` K ) = ( LSubSp ` (ringLMod ` K ) ) )
48		lidlvalg 14037	. . . 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 2239	. 2 |- ( ph -> (LIdeal ` K ) = (LIdeal ` L ) )
51	5, 10, 11, 19, 24, 29, 32, 35, 40, 44	lsppropd 13998	. . 3 |- ( ph -> ( LSpan ` (ringLMod ` K ) ) = ( LSpan ` (ringLMod ` L ) ) )
52		rspvalg 14038	. . . 4 |- ( K e. X -> (RSpan ` K ) = ( LSpan ` (ringLMod ` K ) ) )
53	2, 52	syl 14	. . 3 |- ( ph -> (RSpan ` K ) = ( LSpan ` (ringLMod ` K ) ) )
54		rspvalg 14038	. . . 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 2239	. 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 2167   _Vcvv 2763   C_ wss 3157   Fn wfn 5254   ` cfv 5259  (class class class)co 5923   Basecbs 12688   +g cplusg 12765   .rcmulr 12766  Scalarcsca 12768   .scvsca 12769   LSubSpclss 13918   LSpanclspn 13952  ringLModcrglmod 14000  LIdealclidl 14033  RSpancrsp 14034

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7972  ax-resscn 7973  ax-1cn 7974  ax-1re 7975  ax-icn 7976  ax-addcl 7977  ax-addrcl 7978  ax-mulcl 7979  ax-addcom 7981  ax-addass 7983  ax-i2m1 7986  ax-0lt1 7987  ax-0id 7989  ax-rnegex 7990  ax-pre-ltirr 7993  ax-pre-lttrn 7995  ax-pre-ltadd 7997

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-ov 5926  df-oprab 5927  df-mpo 5928  df-pnf 8065  df-mnf 8066  df-ltxr 8068  df-inn 8993  df-2 9051  df-3 9052  df-4 9053  df-5 9054  df-6 9055  df-7 9056  df-8 9057  df-ndx 12691  df-slot 12692  df-base 12694  df-sets 12695  df-iress 12696  df-plusg 12778  df-mulr 12779  df-sca 12781  df-vsca 12782  df-ip 12783  df-lssm 13919  df-lsp 13953  df-sra 14001  df-rgmod 14002  df-lidl 14035  df-rsp 14036

This theorem is referenced by:  crngridl  14096