Theorem lidlrsppropdg 14307

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 14267	. . . . . 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 2239	. . . 4 |- ( ph -> B = ( Base ` (ringLMod ` K ) ) )
6		lidlpropd.2	. . . . 5 |- ( ph -> B = ( Base ` L ) )
7		lidlpropdg.l	. . . . . 6 |- ( ph -> L e. Y )
8		rlmbasg 14267	. . . . . 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 2239	. . . 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 14268	. . . . . . 7 |- ( K e. X -> ( +g ` K ) = ( +g ` (ringLMod ` K ) ) )
14	2, 13	syl 14	. . . . . 6 |- ( ph -> ( +g ` K ) = ( +g ` (ringLMod ` K ) ) )
15	14	oveqdr 5982	. . . . 5 |- ( ( ph /\ ( x e. W /\ y e. W ) ) -> ( x ( +g ` K ) y ) = ( x ( +g ` (ringLMod ` K ) ) y ) )
16		rlmplusgg 14268	. . . . . . 7 |- ( L e. Y -> ( +g ` L ) = ( +g ` (ringLMod ` L ) ) )
17	7, 16	syl 14	. . . . . 6 |- ( ph -> ( +g ` L ) = ( +g ` (ringLMod ` L ) ) )
18	17	oveqdr 5982	. . . . 5 |- ( ( ph /\ ( x e. W /\ y e. W ) ) -> ( x ( +g ` L ) y ) = ( x ( +g ` (ringLMod ` L ) ) y ) )
19	12, 15, 18	3eqtr3d 2247	. . . 4 |- ( ( ph /\ ( x e. W /\ y e. W ) ) -> ( x ( +g ` (ringLMod ` K ) ) y ) = ( x ( +g ` (ringLMod ` L ) ) y ) )
20		rlmvscag 14273	. . . . . . 7 |- ( K e. X -> ( .r ` K ) = ( .s ` (ringLMod ` K ) ) )
21	2, 20	syl 14	. . . . . 6 |- ( ph -> ( .r ` K ) = ( .s ` (ringLMod ` K ) ) )
22	21	oveqdr 5982	. . . . 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 2284	. . . 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 14273	. . . . . . 7 |- ( L e. Y -> ( .r ` L ) = ( .s ` (ringLMod ` L ) ) )
27	7, 26	syl 14	. . . . . 6 |- ( ph -> ( .r ` L ) = ( .s ` (ringLMod ` L ) ) )
28	27	oveqdr 5982	. . . . 5 |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( .r ` L ) y ) = ( x ( .s ` (ringLMod ` L ) ) y ) )
29	25, 22, 28	3eqtr3d 2247	. . . 4 |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( .s ` (ringLMod ` K ) ) y ) = ( x ( .s ` (ringLMod ` L ) ) y ) )
30		rlmscabas 14272	. . . . . 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 2239	. . . 4 |- ( ph -> B = ( Base ` (Scalar ` (ringLMod ` K ) ) ) )
33		rlmscabas 14272	. . . . . 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 2239	. . . 4 |- ( ph -> B = ( Base ` (Scalar ` (ringLMod ` L ) ) ) )
36		rlmfn 14265	. . . . 5 |- ringLMod Fn _V
37	2	elexd 2787	. . . . 5 |- ( ph -> K e. _V )
38		funfvex 5603	. . . . . 6 |- ( ( Fun ringLMod /\ K e. dom ringLMod ) -> (ringLMod ` K ) e. _V )
39	38	funfni 5382	. . . . 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 2787	. . . . 5 |- ( ph -> L e. _V )
42		funfvex 5603	. . . . . 6 |- ( ( Fun ringLMod /\ L e. dom ringLMod ) -> (ringLMod ` L ) e. _V )
43	42	funfni 5382	. . . . 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 14243	. . 3 |- ( ph -> ( LSubSp ` (ringLMod ` K ) ) = ( LSubSp ` (ringLMod ` L ) ) )
46		lidlvalg 14283	. . . 4 |- ( K e. X -> (LIdeal ` K ) = ( LSubSp ` (ringLMod ` K ) ) )
47	2, 46	syl 14	. . 3 |- ( ph -> (LIdeal ` K ) = ( LSubSp ` (ringLMod ` K ) ) )
48		lidlvalg 14283	. . . 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 2249	. 2 |- ( ph -> (LIdeal ` K ) = (LIdeal ` L ) )
51	5, 10, 11, 19, 24, 29, 32, 35, 40, 44	lsppropd 14244	. . 3 |- ( ph -> ( LSpan ` (ringLMod ` K ) ) = ( LSpan ` (ringLMod ` L ) ) )
52		rspvalg 14284	. . . 4 |- ( K e. X -> (RSpan ` K ) = ( LSpan ` (ringLMod ` K ) ) )
53	2, 52	syl 14	. . 3 |- ( ph -> (RSpan ` K ) = ( LSpan ` (ringLMod ` K ) ) )
54		rspvalg 14284	. . . 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 2249	. 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 1373   e. wcel 2177   _Vcvv 2773   C_ wss 3168   Fn wfn 5272   ` cfv 5277  (class class class)co 5954   Basecbs 12882   +g cplusg 12959   .rcmulr 12960  Scalarcsca 12962   .scvsca 12963   LSubSpclss 14164   LSpanclspn 14198  ringLModcrglmod 14246  LIdealclidl 14279  RSpancrsp 14280

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4164  ax-sep 4167  ax-pow 4223  ax-pr 4258  ax-un 4485  ax-setind 4590  ax-cnex 8029  ax-resscn 8030  ax-1cn 8031  ax-1re 8032  ax-icn 8033  ax-addcl 8034  ax-addrcl 8035  ax-mulcl 8036  ax-addcom 8038  ax-addass 8040  ax-i2m1 8043  ax-0lt1 8044  ax-0id 8046  ax-rnegex 8047  ax-pre-ltirr 8050  ax-pre-lttrn 8052  ax-pre-ltadd 8054

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3001  df-csb 3096  df-dif 3170  df-un 3172  df-in 3174  df-ss 3181  df-nul 3463  df-pw 3620  df-sn 3641  df-pr 3642  df-op 3644  df-uni 3854  df-int 3889  df-iun 3932  df-br 4049  df-opab 4111  df-mpt 4112  df-id 4345  df-xp 4686  df-rel 4687  df-cnv 4688  df-co 4689  df-dm 4690  df-rn 4691  df-res 4692  df-ima 4693  df-iota 5238  df-fun 5279  df-fn 5280  df-f 5281  df-f1 5282  df-fo 5283  df-f1o 5284  df-fv 5285  df-ov 5957  df-oprab 5958  df-mpo 5959  df-pnf 8122  df-mnf 8123  df-ltxr 8125  df-inn 9050  df-2 9108  df-3 9109  df-4 9110  df-5 9111  df-6 9112  df-7 9113  df-8 9114  df-ndx 12885  df-slot 12886  df-base 12888  df-sets 12889  df-iress 12890  df-plusg 12972  df-mulr 12973  df-sca 12975  df-vsca 12976  df-ip 12977  df-lssm 14165  df-lsp 14199  df-sra 14247  df-rgmod 14248  df-lidl 14281  df-rsp 14282

This theorem is referenced by:  crngridl  14342