Theorem lidlrsppropdg 14467

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 14427	. . . . . 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 2262	. . . 4 |- ( ph -> B = ( Base ` (ringLMod ` K ) ) )
6		lidlpropd.2	. . . . 5 |- ( ph -> B = ( Base ` L ) )
7		lidlpropdg.l	. . . . . 6 |- ( ph -> L e. Y )
8		rlmbasg 14427	. . . . . 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 2262	. . . 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 14428	. . . . . . 7 |- ( K e. X -> ( +g ` K ) = ( +g ` (ringLMod ` K ) ) )
14	2, 13	syl 14	. . . . . 6 |- ( ph -> ( +g ` K ) = ( +g ` (ringLMod ` K ) ) )
15	14	oveqdr 6035	. . . . 5 |- ( ( ph /\ ( x e. W /\ y e. W ) ) -> ( x ( +g ` K ) y ) = ( x ( +g ` (ringLMod ` K ) ) y ) )
16		rlmplusgg 14428	. . . . . . 7 |- ( L e. Y -> ( +g ` L ) = ( +g ` (ringLMod ` L ) ) )
17	7, 16	syl 14	. . . . . 6 |- ( ph -> ( +g ` L ) = ( +g ` (ringLMod ` L ) ) )
18	17	oveqdr 6035	. . . . 5 |- ( ( ph /\ ( x e. W /\ y e. W ) ) -> ( x ( +g ` L ) y ) = ( x ( +g ` (ringLMod ` L ) ) y ) )
19	12, 15, 18	3eqtr3d 2270	. . . 4 |- ( ( ph /\ ( x e. W /\ y e. W ) ) -> ( x ( +g ` (ringLMod ` K ) ) y ) = ( x ( +g ` (ringLMod ` L ) ) y ) )
20		rlmvscag 14433	. . . . . . 7 |- ( K e. X -> ( .r ` K ) = ( .s ` (ringLMod ` K ) ) )
21	2, 20	syl 14	. . . . . 6 |- ( ph -> ( .r ` K ) = ( .s ` (ringLMod ` K ) ) )
22	21	oveqdr 6035	. . . . 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 2307	. . . 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 14433	. . . . . . 7 |- ( L e. Y -> ( .r ` L ) = ( .s ` (ringLMod ` L ) ) )
27	7, 26	syl 14	. . . . . 6 |- ( ph -> ( .r ` L ) = ( .s ` (ringLMod ` L ) ) )
28	27	oveqdr 6035	. . . . 5 |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( .r ` L ) y ) = ( x ( .s ` (ringLMod ` L ) ) y ) )
29	25, 22, 28	3eqtr3d 2270	. . . 4 |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( .s ` (ringLMod ` K ) ) y ) = ( x ( .s ` (ringLMod ` L ) ) y ) )
30		rlmscabas 14432	. . . . . 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 2262	. . . 4 |- ( ph -> B = ( Base ` (Scalar ` (ringLMod ` K ) ) ) )
33		rlmscabas 14432	. . . . . 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 2262	. . . 4 |- ( ph -> B = ( Base ` (Scalar ` (ringLMod ` L ) ) ) )
36		rlmfn 14425	. . . . 5 |- ringLMod Fn _V
37	2	elexd 2813	. . . . 5 |- ( ph -> K e. _V )
38		funfvex 5646	. . . . . 6 |- ( ( Fun ringLMod /\ K e. dom ringLMod ) -> (ringLMod ` K ) e. _V )
39	38	funfni 5423	. . . . 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 2813	. . . . 5 |- ( ph -> L e. _V )
42		funfvex 5646	. . . . . 6 |- ( ( Fun ringLMod /\ L e. dom ringLMod ) -> (ringLMod ` L ) e. _V )
43	42	funfni 5423	. . . . 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 14403	. . 3 |- ( ph -> ( LSubSp ` (ringLMod ` K ) ) = ( LSubSp ` (ringLMod ` L ) ) )
46		lidlvalg 14443	. . . 4 |- ( K e. X -> (LIdeal ` K ) = ( LSubSp ` (ringLMod ` K ) ) )
47	2, 46	syl 14	. . 3 |- ( ph -> (LIdeal ` K ) = ( LSubSp ` (ringLMod ` K ) ) )
48		lidlvalg 14443	. . . 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 2272	. 2 |- ( ph -> (LIdeal ` K ) = (LIdeal ` L ) )
51	5, 10, 11, 19, 24, 29, 32, 35, 40, 44	lsppropd 14404	. . 3 |- ( ph -> ( LSpan ` (ringLMod ` K ) ) = ( LSpan ` (ringLMod ` L ) ) )
52		rspvalg 14444	. . . 4 |- ( K e. X -> (RSpan ` K ) = ( LSpan ` (ringLMod ` K ) ) )
53	2, 52	syl 14	. . 3 |- ( ph -> (RSpan ` K ) = ( LSpan ` (ringLMod ` K ) ) )
54		rspvalg 14444	. . . 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 2272	. 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 1395   e. wcel 2200   _Vcvv 2799   C_ wss 3197   Fn wfn 5313   ` cfv 5318  (class class class)co 6007   Basecbs 13040   +g cplusg 13118   .rcmulr 13119  Scalarcsca 13121   .scvsca 13122   LSubSpclss 14324   LSpanclspn 14358  ringLModcrglmod 14406  LIdealclidl 14439  RSpancrsp 14440

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8098  ax-resscn 8099  ax-1cn 8100  ax-1re 8101  ax-icn 8102  ax-addcl 8103  ax-addrcl 8104  ax-mulcl 8105  ax-addcom 8107  ax-addass 8109  ax-i2m1 8112  ax-0lt1 8113  ax-0id 8115  ax-rnegex 8116  ax-pre-ltirr 8119  ax-pre-lttrn 8121  ax-pre-ltadd 8123

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-ov 6010  df-oprab 6011  df-mpo 6012  df-pnf 8191  df-mnf 8192  df-ltxr 8194  df-inn 9119  df-2 9177  df-3 9178  df-4 9179  df-5 9180  df-6 9181  df-7 9182  df-8 9183  df-ndx 13043  df-slot 13044  df-base 13046  df-sets 13047  df-iress 13048  df-plusg 13131  df-mulr 13132  df-sca 13134  df-vsca 13135  df-ip 13136  df-lssm 14325  df-lsp 14359  df-sra 14407  df-rgmod 14408  df-lidl 14441  df-rsp 14442

This theorem is referenced by:  crngridl  14502