Theorem lidlrsppropdg 14774

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	⊢ (𝜑 → 𝐵 = (Base‘𝐾))
lidlpropd.2	⊢ (𝜑 → 𝐵 = (Base‘𝐿))
lidlpropd.3	⊢ (𝜑 → 𝐵 ⊆ 𝑊)
lidlpropd.4	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦))
lidlpropd.5	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝐾)𝑦) ∈ 𝑊)
lidlpropd.6	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝐾)𝑦) = (𝑥(.r‘𝐿)𝑦))
lidlpropdg.k	⊢ (𝜑 → 𝐾 ∈ 𝑋)
lidlpropdg.l	⊢ (𝜑 → 𝐿 ∈ 𝑌)

Assertion

Ref	Expression
lidlrsppropdg	⊢ (𝜑 → ((LIdeal‘𝐾) = (LIdeal‘𝐿) ∧ (RSpan‘𝐾) = (RSpan‘𝐿)))

Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐾,𝑦   𝑥,𝐿,𝑦   𝜑,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 ⊢ (𝜑 → 𝐵 = (Base‘𝐾))
2		lidlpropdg.k	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ 𝑋)
3		rlmbasg 14734	. . . . . 6 ⊢ (𝐾 ∈ 𝑋 → (Base‘𝐾) = (Base‘(ringLMod‘𝐾)))
4	2, 3	syl 14	. . . . 5 ⊢ (𝜑 → (Base‘𝐾) = (Base‘(ringLMod‘𝐾)))
5	1, 4	eqtrd 2267	. . . 4 ⊢ (𝜑 → 𝐵 = (Base‘(ringLMod‘𝐾)))
6		lidlpropd.2	. . . . 5 ⊢ (𝜑 → 𝐵 = (Base‘𝐿))
7		lidlpropdg.l	. . . . . 6 ⊢ (𝜑 → 𝐿 ∈ 𝑌)
8		rlmbasg 14734	. . . . . 6 ⊢ (𝐿 ∈ 𝑌 → (Base‘𝐿) = (Base‘(ringLMod‘𝐿)))
9	7, 8	syl 14	. . . . 5 ⊢ (𝜑 → (Base‘𝐿) = (Base‘(ringLMod‘𝐿)))
10	6, 9	eqtrd 2267	. . . 4 ⊢ (𝜑 → 𝐵 = (Base‘(ringLMod‘𝐿)))
11		lidlpropd.3	. . . 4 ⊢ (𝜑 → 𝐵 ⊆ 𝑊)
12		lidlpropd.4	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦))
13		rlmplusgg 14735	. . . . . . 7 ⊢ (𝐾 ∈ 𝑋 → (+g‘𝐾) = (+g‘(ringLMod‘𝐾)))
14	2, 13	syl 14	. . . . . 6 ⊢ (𝜑 → (+g‘𝐾) = (+g‘(ringLMod‘𝐾)))
15	14	oveqdr 6087	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘(ringLMod‘𝐾))𝑦))
16		rlmplusgg 14735	. . . . . . 7 ⊢ (𝐿 ∈ 𝑌 → (+g‘𝐿) = (+g‘(ringLMod‘𝐿)))
17	7, 16	syl 14	. . . . . 6 ⊢ (𝜑 → (+g‘𝐿) = (+g‘(ringLMod‘𝐿)))
18	17	oveqdr 6087	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊)) → (𝑥(+g‘𝐿)𝑦) = (𝑥(+g‘(ringLMod‘𝐿))𝑦))
19	12, 15, 18	3eqtr3d 2275	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊)) → (𝑥(+g‘(ringLMod‘𝐾))𝑦) = (𝑥(+g‘(ringLMod‘𝐿))𝑦))
20		rlmvscag 14740	. . . . . . 7 ⊢ (𝐾 ∈ 𝑋 → (.r‘𝐾) = ( ·𝑠 ‘(ringLMod‘𝐾)))
21	2, 20	syl 14	. . . . . 6 ⊢ (𝜑 → (.r‘𝐾) = ( ·𝑠 ‘(ringLMod‘𝐾)))
22	21	oveqdr 6087	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝐾)𝑦) = (𝑥( ·𝑠 ‘(ringLMod‘𝐾))𝑦))
23		lidlpropd.5	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝐾)𝑦) ∈ 𝑊)
24	22, 23	eqeltrrd 2312	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·𝑠 ‘(ringLMod‘𝐾))𝑦) ∈ 𝑊)
25		lidlpropd.6	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝐾)𝑦) = (𝑥(.r‘𝐿)𝑦))
26		rlmvscag 14740	. . . . . . 7 ⊢ (𝐿 ∈ 𝑌 → (.r‘𝐿) = ( ·𝑠 ‘(ringLMod‘𝐿)))
27	7, 26	syl 14	. . . . . 6 ⊢ (𝜑 → (.r‘𝐿) = ( ·𝑠 ‘(ringLMod‘𝐿)))
28	27	oveqdr 6087	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝐿)𝑦) = (𝑥( ·𝑠 ‘(ringLMod‘𝐿))𝑦))
29	25, 22, 28	3eqtr3d 2275	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·𝑠 ‘(ringLMod‘𝐾))𝑦) = (𝑥( ·𝑠 ‘(ringLMod‘𝐿))𝑦))
30		rlmscabas 14739	. . . . . 6 ⊢ (𝐾 ∈ 𝑋 → (Base‘𝐾) = (Base‘(Scalar‘(ringLMod‘𝐾))))
31	2, 30	syl 14	. . . . 5 ⊢ (𝜑 → (Base‘𝐾) = (Base‘(Scalar‘(ringLMod‘𝐾))))
32	1, 31	eqtrd 2267	. . . 4 ⊢ (𝜑 → 𝐵 = (Base‘(Scalar‘(ringLMod‘𝐾))))
33		rlmscabas 14739	. . . . . 6 ⊢ (𝐿 ∈ 𝑌 → (Base‘𝐿) = (Base‘(Scalar‘(ringLMod‘𝐿))))
34	7, 33	syl 14	. . . . 5 ⊢ (𝜑 → (Base‘𝐿) = (Base‘(Scalar‘(ringLMod‘𝐿))))
35	6, 34	eqtrd 2267	. . . 4 ⊢ (𝜑 → 𝐵 = (Base‘(Scalar‘(ringLMod‘𝐿))))
36		rlmfn 14732	. . . . 5 ⊢ ringLMod Fn V
37	2	elexd 2829	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ V)
38		funfvex 5693	. . . . . 6 ⊢ ((Fun ringLMod ∧ 𝐾 ∈ dom ringLMod) → (ringLMod‘𝐾) ∈ V)
39	38	funfni 5464	. . . . 5 ⊢ ((ringLMod Fn V ∧ 𝐾 ∈ V) → (ringLMod‘𝐾) ∈ V)
40	36, 37, 39	sylancr 414	. . . 4 ⊢ (𝜑 → (ringLMod‘𝐾) ∈ V)
41	7	elexd 2829	. . . . 5 ⊢ (𝜑 → 𝐿 ∈ V)
42		funfvex 5693	. . . . . 6 ⊢ ((Fun ringLMod ∧ 𝐿 ∈ dom ringLMod) → (ringLMod‘𝐿) ∈ V)
43	42	funfni 5464	. . . . 5 ⊢ ((ringLMod Fn V ∧ 𝐿 ∈ V) → (ringLMod‘𝐿) ∈ V)
44	36, 41, 43	sylancr 414	. . . 4 ⊢ (𝜑 → (ringLMod‘𝐿) ∈ V)
45	5, 10, 11, 19, 24, 29, 32, 35, 40, 44	lsspropdg 14710	. . 3 ⊢ (𝜑 → (LSubSp‘(ringLMod‘𝐾)) = (LSubSp‘(ringLMod‘𝐿)))
46		lidlvalg 14750	. . . 4 ⊢ (𝐾 ∈ 𝑋 → (LIdeal‘𝐾) = (LSubSp‘(ringLMod‘𝐾)))
47	2, 46	syl 14	. . 3 ⊢ (𝜑 → (LIdeal‘𝐾) = (LSubSp‘(ringLMod‘𝐾)))
48		lidlvalg 14750	. . . 4 ⊢ (𝐿 ∈ 𝑌 → (LIdeal‘𝐿) = (LSubSp‘(ringLMod‘𝐿)))
49	7, 48	syl 14	. . 3 ⊢ (𝜑 → (LIdeal‘𝐿) = (LSubSp‘(ringLMod‘𝐿)))
50	45, 47, 49	3eqtr4d 2277	. 2 ⊢ (𝜑 → (LIdeal‘𝐾) = (LIdeal‘𝐿))
51	5, 10, 11, 19, 24, 29, 32, 35, 40, 44	lsppropd 14711	. . 3 ⊢ (𝜑 → (LSpan‘(ringLMod‘𝐾)) = (LSpan‘(ringLMod‘𝐿)))
52		rspvalg 14751	. . . 4 ⊢ (𝐾 ∈ 𝑋 → (RSpan‘𝐾) = (LSpan‘(ringLMod‘𝐾)))
53	2, 52	syl 14	. . 3 ⊢ (𝜑 → (RSpan‘𝐾) = (LSpan‘(ringLMod‘𝐾)))
54		rspvalg 14751	. . . 4 ⊢ (𝐿 ∈ 𝑌 → (RSpan‘𝐿) = (LSpan‘(ringLMod‘𝐿)))
55	7, 54	syl 14	. . 3 ⊢ (𝜑 → (RSpan‘𝐿) = (LSpan‘(ringLMod‘𝐿)))
56	51, 53, 55	3eqtr4d 2277	. 2 ⊢ (𝜑 → (RSpan‘𝐾) = (RSpan‘𝐿))
57	50, 56	jca 306	1 ⊢ (𝜑 → ((LIdeal‘𝐾) = (LIdeal‘𝐿) ∧ (RSpan‘𝐾) = (RSpan‘𝐿)))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1398   ∈ wcel 2205  Vcvv 2815   ⊆ wss 3214   Fn wfn 5353  ‘cfv 5358  (class class class)co 6059  Basecbs 13301  +_gcplusg 13379  ._rcmulr 13380  Scalarcsca 13382   ·_𝑠 cvsca 13383  LSubSpclss 14631  LSpanclspn 14665  ringLModcrglmod 14713  LIdealclidl 14746  RSpancrsp 14747

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4231  ax-sep 4234  ax-pow 4293  ax-pr 4328  ax-un 4560  ax-setind 4665  ax-cnex 8235  ax-resscn 8236  ax-1cn 8237  ax-1re 8238  ax-icn 8239  ax-addcl 8240  ax-addrcl 8241  ax-mulcl 8242  ax-addcom 8244  ax-addass 8246  ax-i2m1 8249  ax-0lt1 8250  ax-0id 8252  ax-rnegex 8253  ax-pre-ltirr 8256  ax-pre-lttrn 8258  ax-pre-ltadd 8260

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3677  df-sn 3701  df-pr 3702  df-op 3704  df-uni 3921  df-int 3956  df-iun 3999  df-br 4116  df-opab 4178  df-mpt 4179  df-id 4420  df-xp 4761  df-rel 4762  df-cnv 4763  df-co 4764  df-dm 4765  df-rn 4766  df-res 4767  df-ima 4768  df-iota 5318  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 6062  df-oprab 6063  df-mpo 6064  df-pnf 8327  df-mnf 8328  df-ltxr 8330  df-inn 9259  df-2 9317  df-3 9318  df-4 9319  df-5 9320  df-6 9321  df-7 9322  df-8 9323  df-ndx 13304  df-slot 13305  df-base 13307  df-sets 13308  df-iress 13309  df-plusg 13392  df-mulr 13393  df-sca 13395  df-vsca 13396  df-ip 13397  df-lssm 14632  df-lsp 14666  df-sra 14714  df-rgmod 14715  df-lidl 14748  df-rsp 14749

This theorem is referenced by:  crngridl  14809