Theorem lidlrsppropdg 14372

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 14332	. . . . . 6 ⊢ (𝐾 ∈ 𝑋 → (Base‘𝐾) = (Base‘(ringLMod‘𝐾)))
4	2, 3	syl 14	. . . . 5 ⊢ (𝜑 → (Base‘𝐾) = (Base‘(ringLMod‘𝐾)))
5	1, 4	eqtrd 2240	. . . 4 ⊢ (𝜑 → 𝐵 = (Base‘(ringLMod‘𝐾)))
6		lidlpropd.2	. . . . 5 ⊢ (𝜑 → 𝐵 = (Base‘𝐿))
7		lidlpropdg.l	. . . . . 6 ⊢ (𝜑 → 𝐿 ∈ 𝑌)
8		rlmbasg 14332	. . . . . 6 ⊢ (𝐿 ∈ 𝑌 → (Base‘𝐿) = (Base‘(ringLMod‘𝐿)))
9	7, 8	syl 14	. . . . 5 ⊢ (𝜑 → (Base‘𝐿) = (Base‘(ringLMod‘𝐿)))
10	6, 9	eqtrd 2240	. . . 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 14333	. . . . . . 7 ⊢ (𝐾 ∈ 𝑋 → (+g‘𝐾) = (+g‘(ringLMod‘𝐾)))
14	2, 13	syl 14	. . . . . 6 ⊢ (𝜑 → (+g‘𝐾) = (+g‘(ringLMod‘𝐾)))
15	14	oveqdr 5995	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘(ringLMod‘𝐾))𝑦))
16		rlmplusgg 14333	. . . . . . 7 ⊢ (𝐿 ∈ 𝑌 → (+g‘𝐿) = (+g‘(ringLMod‘𝐿)))
17	7, 16	syl 14	. . . . . 6 ⊢ (𝜑 → (+g‘𝐿) = (+g‘(ringLMod‘𝐿)))
18	17	oveqdr 5995	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊)) → (𝑥(+g‘𝐿)𝑦) = (𝑥(+g‘(ringLMod‘𝐿))𝑦))
19	12, 15, 18	3eqtr3d 2248	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊)) → (𝑥(+g‘(ringLMod‘𝐾))𝑦) = (𝑥(+g‘(ringLMod‘𝐿))𝑦))
20		rlmvscag 14338	. . . . . . 7 ⊢ (𝐾 ∈ 𝑋 → (.r‘𝐾) = ( ·𝑠 ‘(ringLMod‘𝐾)))
21	2, 20	syl 14	. . . . . 6 ⊢ (𝜑 → (.r‘𝐾) = ( ·𝑠 ‘(ringLMod‘𝐾)))
22	21	oveqdr 5995	. . . . 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 2285	. . . 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 14338	. . . . . . 7 ⊢ (𝐿 ∈ 𝑌 → (.r‘𝐿) = ( ·𝑠 ‘(ringLMod‘𝐿)))
27	7, 26	syl 14	. . . . . 6 ⊢ (𝜑 → (.r‘𝐿) = ( ·𝑠 ‘(ringLMod‘𝐿)))
28	27	oveqdr 5995	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝐿)𝑦) = (𝑥( ·𝑠 ‘(ringLMod‘𝐿))𝑦))
29	25, 22, 28	3eqtr3d 2248	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·𝑠 ‘(ringLMod‘𝐾))𝑦) = (𝑥( ·𝑠 ‘(ringLMod‘𝐿))𝑦))
30		rlmscabas 14337	. . . . . 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 2240	. . . 4 ⊢ (𝜑 → 𝐵 = (Base‘(Scalar‘(ringLMod‘𝐾))))
33		rlmscabas 14337	. . . . . 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 2240	. . . 4 ⊢ (𝜑 → 𝐵 = (Base‘(Scalar‘(ringLMod‘𝐿))))
36		rlmfn 14330	. . . . 5 ⊢ ringLMod Fn V
37	2	elexd 2790	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ V)
38		funfvex 5616	. . . . . 6 ⊢ ((Fun ringLMod ∧ 𝐾 ∈ dom ringLMod) → (ringLMod‘𝐾) ∈ V)
39	38	funfni 5395	. . . . 5 ⊢ ((ringLMod Fn V ∧ 𝐾 ∈ V) → (ringLMod‘𝐾) ∈ V)
40	36, 37, 39	sylancr 414	. . . 4 ⊢ (𝜑 → (ringLMod‘𝐾) ∈ V)
41	7	elexd 2790	. . . . 5 ⊢ (𝜑 → 𝐿 ∈ V)
42		funfvex 5616	. . . . . 6 ⊢ ((Fun ringLMod ∧ 𝐿 ∈ dom ringLMod) → (ringLMod‘𝐿) ∈ V)
43	42	funfni 5395	. . . . 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 14308	. . 3 ⊢ (𝜑 → (LSubSp‘(ringLMod‘𝐾)) = (LSubSp‘(ringLMod‘𝐿)))
46		lidlvalg 14348	. . . 4 ⊢ (𝐾 ∈ 𝑋 → (LIdeal‘𝐾) = (LSubSp‘(ringLMod‘𝐾)))
47	2, 46	syl 14	. . 3 ⊢ (𝜑 → (LIdeal‘𝐾) = (LSubSp‘(ringLMod‘𝐾)))
48		lidlvalg 14348	. . . 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 2250	. 2 ⊢ (𝜑 → (LIdeal‘𝐾) = (LIdeal‘𝐿))
51	5, 10, 11, 19, 24, 29, 32, 35, 40, 44	lsppropd 14309	. . 3 ⊢ (𝜑 → (LSpan‘(ringLMod‘𝐾)) = (LSpan‘(ringLMod‘𝐿)))
52		rspvalg 14349	. . . 4 ⊢ (𝐾 ∈ 𝑋 → (RSpan‘𝐾) = (LSpan‘(ringLMod‘𝐾)))
53	2, 52	syl 14	. . 3 ⊢ (𝜑 → (RSpan‘𝐾) = (LSpan‘(ringLMod‘𝐾)))
54		rspvalg 14349	. . . 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 2250	. 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 1373   ∈ wcel 2178  Vcvv 2776   ⊆ wss 3174   Fn wfn 5285  ‘cfv 5290  (class class class)co 5967  Basecbs 12947  +_gcplusg 13024  ._rcmulr 13025  Scalarcsca 13027   ·_𝑠 cvsca 13028  LSubSpclss 14229  LSpanclspn 14263  ringLModcrglmod 14311  LIdealclidl 14344  RSpancrsp 14345

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 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-addcom 8060  ax-addass 8062  ax-i2m1 8065  ax-0lt1 8066  ax-0id 8068  ax-rnegex 8069  ax-pre-ltirr 8072  ax-pre-lttrn 8074  ax-pre-ltadd 8076

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 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-ov 5970  df-oprab 5971  df-mpo 5972  df-pnf 8144  df-mnf 8145  df-ltxr 8147  df-inn 9072  df-2 9130  df-3 9131  df-4 9132  df-5 9133  df-6 9134  df-7 9135  df-8 9136  df-ndx 12950  df-slot 12951  df-base 12953  df-sets 12954  df-iress 12955  df-plusg 13037  df-mulr 13038  df-sca 13040  df-vsca 13041  df-ip 13042  df-lssm 14230  df-lsp 14264  df-sra 14312  df-rgmod 14313  df-lidl 14346  df-rsp 14347

This theorem is referenced by:  crngridl  14407