Theorem lidlrsppropdg 14533

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 14493	. . . . . 6 ⊢ (𝐾 ∈ 𝑋 → (Base‘𝐾) = (Base‘(ringLMod‘𝐾)))
4	2, 3	syl 14	. . . . 5 ⊢ (𝜑 → (Base‘𝐾) = (Base‘(ringLMod‘𝐾)))
5	1, 4	eqtrd 2263	. . . 4 ⊢ (𝜑 → 𝐵 = (Base‘(ringLMod‘𝐾)))
6		lidlpropd.2	. . . . 5 ⊢ (𝜑 → 𝐵 = (Base‘𝐿))
7		lidlpropdg.l	. . . . . 6 ⊢ (𝜑 → 𝐿 ∈ 𝑌)
8		rlmbasg 14493	. . . . . 6 ⊢ (𝐿 ∈ 𝑌 → (Base‘𝐿) = (Base‘(ringLMod‘𝐿)))
9	7, 8	syl 14	. . . . 5 ⊢ (𝜑 → (Base‘𝐿) = (Base‘(ringLMod‘𝐿)))
10	6, 9	eqtrd 2263	. . . 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 14494	. . . . . . 7 ⊢ (𝐾 ∈ 𝑋 → (+g‘𝐾) = (+g‘(ringLMod‘𝐾)))
14	2, 13	syl 14	. . . . . 6 ⊢ (𝜑 → (+g‘𝐾) = (+g‘(ringLMod‘𝐾)))
15	14	oveqdr 6051	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘(ringLMod‘𝐾))𝑦))
16		rlmplusgg 14494	. . . . . . 7 ⊢ (𝐿 ∈ 𝑌 → (+g‘𝐿) = (+g‘(ringLMod‘𝐿)))
17	7, 16	syl 14	. . . . . 6 ⊢ (𝜑 → (+g‘𝐿) = (+g‘(ringLMod‘𝐿)))
18	17	oveqdr 6051	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊)) → (𝑥(+g‘𝐿)𝑦) = (𝑥(+g‘(ringLMod‘𝐿))𝑦))
19	12, 15, 18	3eqtr3d 2271	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊)) → (𝑥(+g‘(ringLMod‘𝐾))𝑦) = (𝑥(+g‘(ringLMod‘𝐿))𝑦))
20		rlmvscag 14499	. . . . . . 7 ⊢ (𝐾 ∈ 𝑋 → (.r‘𝐾) = ( ·𝑠 ‘(ringLMod‘𝐾)))
21	2, 20	syl 14	. . . . . 6 ⊢ (𝜑 → (.r‘𝐾) = ( ·𝑠 ‘(ringLMod‘𝐾)))
22	21	oveqdr 6051	. . . . 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 2308	. . . 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 14499	. . . . . . 7 ⊢ (𝐿 ∈ 𝑌 → (.r‘𝐿) = ( ·𝑠 ‘(ringLMod‘𝐿)))
27	7, 26	syl 14	. . . . . 6 ⊢ (𝜑 → (.r‘𝐿) = ( ·𝑠 ‘(ringLMod‘𝐿)))
28	27	oveqdr 6051	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝐿)𝑦) = (𝑥( ·𝑠 ‘(ringLMod‘𝐿))𝑦))
29	25, 22, 28	3eqtr3d 2271	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·𝑠 ‘(ringLMod‘𝐾))𝑦) = (𝑥( ·𝑠 ‘(ringLMod‘𝐿))𝑦))
30		rlmscabas 14498	. . . . . 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 2263	. . . 4 ⊢ (𝜑 → 𝐵 = (Base‘(Scalar‘(ringLMod‘𝐾))))
33		rlmscabas 14498	. . . . . 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 2263	. . . 4 ⊢ (𝜑 → 𝐵 = (Base‘(Scalar‘(ringLMod‘𝐿))))
36		rlmfn 14491	. . . . 5 ⊢ ringLMod Fn V
37	2	elexd 2815	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ V)
38		funfvex 5659	. . . . . 6 ⊢ ((Fun ringLMod ∧ 𝐾 ∈ dom ringLMod) → (ringLMod‘𝐾) ∈ V)
39	38	funfni 5434	. . . . 5 ⊢ ((ringLMod Fn V ∧ 𝐾 ∈ V) → (ringLMod‘𝐾) ∈ V)
40	36, 37, 39	sylancr 414	. . . 4 ⊢ (𝜑 → (ringLMod‘𝐾) ∈ V)
41	7	elexd 2815	. . . . 5 ⊢ (𝜑 → 𝐿 ∈ V)
42		funfvex 5659	. . . . . 6 ⊢ ((Fun ringLMod ∧ 𝐿 ∈ dom ringLMod) → (ringLMod‘𝐿) ∈ V)
43	42	funfni 5434	. . . . 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 14469	. . 3 ⊢ (𝜑 → (LSubSp‘(ringLMod‘𝐾)) = (LSubSp‘(ringLMod‘𝐿)))
46		lidlvalg 14509	. . . 4 ⊢ (𝐾 ∈ 𝑋 → (LIdeal‘𝐾) = (LSubSp‘(ringLMod‘𝐾)))
47	2, 46	syl 14	. . 3 ⊢ (𝜑 → (LIdeal‘𝐾) = (LSubSp‘(ringLMod‘𝐾)))
48		lidlvalg 14509	. . . 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 2273	. 2 ⊢ (𝜑 → (LIdeal‘𝐾) = (LIdeal‘𝐿))
51	5, 10, 11, 19, 24, 29, 32, 35, 40, 44	lsppropd 14470	. . 3 ⊢ (𝜑 → (LSpan‘(ringLMod‘𝐾)) = (LSpan‘(ringLMod‘𝐿)))
52		rspvalg 14510	. . . 4 ⊢ (𝐾 ∈ 𝑋 → (RSpan‘𝐾) = (LSpan‘(ringLMod‘𝐾)))
53	2, 52	syl 14	. . 3 ⊢ (𝜑 → (RSpan‘𝐾) = (LSpan‘(ringLMod‘𝐾)))
54		rspvalg 14510	. . . 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 2273	. 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 1397   ∈ wcel 2201  Vcvv 2801   ⊆ wss 3199   Fn wfn 5323  ‘cfv 5328  (class class class)co 6023  Basecbs 13105  +_gcplusg 13183  ._rcmulr 13184  Scalarcsca 13186   ·_𝑠 cvsca 13187  LSubSpclss 14390  LSpanclspn 14424  ringLModcrglmod 14472  LIdealclidl 14505  RSpancrsp 14506

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2203  ax-14 2204  ax-ext 2212  ax-coll 4205  ax-sep 4208  ax-pow 4266  ax-pr 4301  ax-un 4532  ax-setind 4637  ax-cnex 8128  ax-resscn 8129  ax-1cn 8130  ax-1re 8131  ax-icn 8132  ax-addcl 8133  ax-addrcl 8134  ax-mulcl 8135  ax-addcom 8137  ax-addass 8139  ax-i2m1 8142  ax-0lt1 8143  ax-0id 8145  ax-rnegex 8146  ax-pre-ltirr 8149  ax-pre-lttrn 8151  ax-pre-ltadd 8153

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ne 2402  df-nel 2497  df-ral 2514  df-rex 2515  df-reu 2516  df-rab 2518  df-v 2803  df-sbc 3031  df-csb 3127  df-dif 3201  df-un 3203  df-in 3205  df-ss 3212  df-nul 3494  df-pw 3655  df-sn 3676  df-pr 3677  df-op 3679  df-uni 3895  df-int 3930  df-iun 3973  df-br 4090  df-opab 4152  df-mpt 4153  df-id 4392  df-xp 4733  df-rel 4734  df-cnv 4735  df-co 4736  df-dm 4737  df-rn 4738  df-res 4739  df-ima 4740  df-iota 5288  df-fun 5330  df-fn 5331  df-f 5332  df-f1 5333  df-fo 5334  df-f1o 5335  df-fv 5336  df-ov 6026  df-oprab 6027  df-mpo 6028  df-pnf 8221  df-mnf 8222  df-ltxr 8224  df-inn 9149  df-2 9207  df-3 9208  df-4 9209  df-5 9210  df-6 9211  df-7 9212  df-8 9213  df-ndx 13108  df-slot 13109  df-base 13111  df-sets 13112  df-iress 13113  df-plusg 13196  df-mulr 13197  df-sca 13199  df-vsca 13200  df-ip 13201  df-lssm 14391  df-lsp 14425  df-sra 14473  df-rgmod 14474  df-lidl 14507  df-rsp 14508

This theorem is referenced by:  crngridl  14568