Theorem lidlrsppropd 19230

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 ) )

Assertion

Ref	Expression
lidlrsppropd	|- ( 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

Proof of Theorem lidlrsppropd

Step	Hyp	Ref	Expression
1		lidlpropd.1	. . . . 5 |- ( ph -> B = ( Base ` K ) )
2		rlmbas 19195	. . . . 5 |- ( Base ` K ) = ( Base ` (ringLMod ` K ) )
3	1, 2	syl6eq 2672	. . . 4 |- ( ph -> B = ( Base ` (ringLMod ` K ) ) )
4		lidlpropd.2	. . . . 5 |- ( ph -> B = ( Base ` L ) )
5		rlmbas 19195	. . . . 5 |- ( Base ` L ) = ( Base ` (ringLMod ` L ) )
6	4, 5	syl6eq 2672	. . . 4 |- ( ph -> B = ( Base ` (ringLMod ` L ) ) )
7		lidlpropd.3	. . . 4 |- ( ph -> B C_ W )
8		lidlpropd.4	. . . . 5 |- ( ( ph /\ ( x e. W /\ y e. W ) ) -> ( x ( +g ` K ) y ) = ( x ( +g ` L ) y ) )
9		rlmplusg 19196	. . . . . 6 |- ( +g ` K ) = ( +g ` (ringLMod ` K ) )
10	9	oveqi 6663	. . . . 5 |- ( x ( +g ` K ) y ) = ( x ( +g ` (ringLMod ` K ) ) y )
11		rlmplusg 19196	. . . . . 6 |- ( +g ` L ) = ( +g ` (ringLMod ` L ) )
12	11	oveqi 6663	. . . . 5 |- ( x ( +g ` L ) y ) = ( x ( +g ` (ringLMod ` L ) ) y )
13	8, 10, 12	3eqtr3g 2679	. . . 4 |- ( ( ph /\ ( x e. W /\ y e. W ) ) -> ( x ( +g ` (ringLMod ` K ) ) y ) = ( x ( +g ` (ringLMod ` L ) ) y ) )
14		rlmvsca 19202	. . . . . 6 |- ( .r ` K ) = ( .s ` (ringLMod ` K ) )
15	14	oveqi 6663	. . . . 5 |- ( x ( .r ` K ) y ) = ( x ( .s ` (ringLMod ` K ) ) y )
16		lidlpropd.5	. . . . 5 |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( .r ` K ) y ) e. W )
17	15, 16	syl5eqelr 2706	. . . 4 |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( .s ` (ringLMod ` K ) ) y ) e. W )
18		lidlpropd.6	. . . . 5 |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( .r ` K ) y ) = ( x ( .r ` L ) y ) )
19		rlmvsca 19202	. . . . . 6 |- ( .r ` L ) = ( .s ` (ringLMod ` L ) )
20	19	oveqi 6663	. . . . 5 |- ( x ( .r ` L ) y ) = ( x ( .s ` (ringLMod ` L ) ) y )
21	18, 15, 20	3eqtr3g 2679	. . . 4 |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( .s ` (ringLMod ` K ) ) y ) = ( x ( .s ` (ringLMod ` L ) ) y ) )
22		baseid 15919	. . . . . . 7 |- Base = Slot ( Base ` ndx )
23		eqid 2622	. . . . . . 7 |- ( Base ` K ) = ( Base ` K )
24	22, 23	strfvi 15913	. . . . . 6 |- ( Base ` K ) = ( Base ` ( _I ` K ) )
25		rlmsca2 19201	. . . . . . 7 |- ( _I ` K ) = (Scalar ` (ringLMod ` K ) )
26	25	fveq2i 6194	. . . . . 6 |- ( Base ` ( _I ` K ) ) = ( Base ` (Scalar ` (ringLMod ` K ) ) )
27	24, 26	eqtri 2644	. . . . 5 |- ( Base ` K ) = ( Base ` (Scalar ` (ringLMod ` K ) ) )
28	1, 27	syl6eq 2672	. . . 4 |- ( ph -> B = ( Base ` (Scalar ` (ringLMod ` K ) ) ) )
29		eqid 2622	. . . . . . 7 |- ( Base ` L ) = ( Base ` L )
30	22, 29	strfvi 15913	. . . . . 6 |- ( Base ` L ) = ( Base ` ( _I ` L ) )
31		rlmsca2 19201	. . . . . . 7 |- ( _I ` L ) = (Scalar ` (ringLMod ` L ) )
32	31	fveq2i 6194	. . . . . 6 |- ( Base ` ( _I ` L ) ) = ( Base ` (Scalar ` (ringLMod ` L ) ) )
33	30, 32	eqtri 2644	. . . . 5 |- ( Base ` L ) = ( Base ` (Scalar ` (ringLMod ` L ) ) )
34	4, 33	syl6eq 2672	. . . 4 |- ( ph -> B = ( Base ` (Scalar ` (ringLMod ` L ) ) ) )
35	3, 6, 7, 13, 17, 21, 28, 34	lsspropd 19017	. . 3 |- ( ph -> ( LSubSp ` (ringLMod ` K ) ) = ( LSubSp ` (ringLMod ` L ) ) )
36		lidlval 19192	. . 3 |- (LIdeal ` K ) = ( LSubSp ` (ringLMod ` K ) )
37		lidlval 19192	. . 3 |- (LIdeal ` L ) = ( LSubSp ` (ringLMod ` L ) )
38	35, 36, 37	3eqtr4g 2681	. 2 |- ( ph -> (LIdeal ` K ) = (LIdeal ` L ) )
39		fvexd 6203	. . . 4 |- ( ph -> (ringLMod ` K ) e. _V )
40		fvexd 6203	. . . 4 |- ( ph -> (ringLMod ` L ) e. _V )
41	3, 6, 7, 13, 17, 21, 28, 34, 39, 40	lsppropd 19018	. . 3 |- ( ph -> ( LSpan ` (ringLMod ` K ) ) = ( LSpan ` (ringLMod ` L ) ) )
42		rspval 19193	. . 3 |- (RSpan ` K ) = ( LSpan ` (ringLMod ` K ) )
43		rspval 19193	. . 3 |- (RSpan ` L ) = ( LSpan ` (ringLMod ` L ) )
44	41, 42, 43	3eqtr4g 2681	. 2 |- ( ph -> (RSpan ` K ) = (RSpan ` L ) )
45	38, 44	jca 554	1 |- ( ph -> ( (LIdeal ` K ) = (LIdeal ` L ) /\ (RSpan ` K ) = (RSpan ` L ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   C_ wss 3574   _I cid 5023   ` cfv 5888  (class class class)co 6650   ndxcnx 15854   Basecbs 15857   +g cplusg 15941   .rcmulr 15942  Scalarcsca 15944   .scvsca 15945   LSubSpclss 18932   LSpanclspn 18971  ringLModcrglmod 19169  LIdealclidl 19170  RSpancrsp 19171

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-sca 15957  df-vsca 15958  df-ip 15959  df-lss 18933  df-lsp 18972  df-sra 19172  df-rgmod 19173  df-lidl 19174  df-rsp 19175

This theorem is referenced by:  crngridl  19238