Theorem idltrn 40151

Description: The identity function is a lattice translation. Remark below Lemma B in [Crawley] p. 112. (Contributed by NM, 18-May-2012.)

Hypotheses

Ref	Expression
idltrn.b	⊢ 𝐵 = (Base‘𝐾)
idltrn.h	⊢ 𝐻 = (LHyp‘𝐾)
idltrn.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)

Assertion

Ref	Expression
idltrn	⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( I ↾ 𝐵) ∈ 𝑇)

Proof of Theorem idltrn

Dummy variables 𝑞 𝑝 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		idltrn.b	. . 3 ⊢ 𝐵 = (Base‘𝐾)
2		idltrn.h	. . 3 ⊢ 𝐻 = (LHyp‘𝐾)
3		eqid 2730	. . 3 ⊢ ((LDil‘𝐾)‘𝑊) = ((LDil‘𝐾)‘𝑊)
4	1, 2, 3	idldil 40115	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( I ↾ 𝐵) ∈ ((LDil‘𝐾)‘𝑊))
5		simpll 766	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾))) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
6		simplrr 777	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾))) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → 𝑞 ∈ (Atoms‘𝐾))
7		simprr 772	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾))) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → ¬ 𝑞(le‘𝐾)𝑊)
8		eqid 2730	. . . . . . 7 ⊢ (le‘𝐾) = (le‘𝐾)
9		eqid 2730	. . . . . . 7 ⊢ (meet‘𝐾) = (meet‘𝐾)
10		eqid 2730	. . . . . . 7 ⊢ (0.‘𝐾) = (0.‘𝐾)
11		eqid 2730	. . . . . . 7 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
12	8, 9, 10, 11, 2	lhpmat 40031	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ ¬ 𝑞(le‘𝐾)𝑊)) → (𝑞(meet‘𝐾)𝑊) = (0.‘𝐾))
13	5, 6, 7, 12	syl12anc 836	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾))) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → (𝑞(meet‘𝐾)𝑊) = (0.‘𝐾))
14	1, 11	atbase 39289	. . . . . . . . 9 ⊢ (𝑞 ∈ (Atoms‘𝐾) → 𝑞 ∈ 𝐵)
15		fvresi 7150	. . . . . . . . 9 ⊢ (𝑞 ∈ 𝐵 → (( I ↾ 𝐵)‘𝑞) = 𝑞)
16	6, 14, 15	3syl 18	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾))) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → (( I ↾ 𝐵)‘𝑞) = 𝑞)
17	16	oveq2d 7406	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾))) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → (𝑞(join‘𝐾)(( I ↾ 𝐵)‘𝑞)) = (𝑞(join‘𝐾)𝑞))
18		simplll 774	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾))) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → 𝐾 ∈ HL)
19		eqid 2730	. . . . . . . . 9 ⊢ (join‘𝐾) = (join‘𝐾)
20	19, 11	hlatjidm 39369	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑞 ∈ (Atoms‘𝐾)) → (𝑞(join‘𝐾)𝑞) = 𝑞)
21	18, 6, 20	syl2anc 584	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾))) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → (𝑞(join‘𝐾)𝑞) = 𝑞)
22	17, 21	eqtrd 2765	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾))) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → (𝑞(join‘𝐾)(( I ↾ 𝐵)‘𝑞)) = 𝑞)
23	22	oveq1d 7405	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾))) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → ((𝑞(join‘𝐾)(( I ↾ 𝐵)‘𝑞))(meet‘𝐾)𝑊) = (𝑞(meet‘𝐾)𝑊))
24		simplrl 776	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾))) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → 𝑝 ∈ (Atoms‘𝐾))
25	1, 11	atbase 39289	. . . . . . . . . 10 ⊢ (𝑝 ∈ (Atoms‘𝐾) → 𝑝 ∈ 𝐵)
26		fvresi 7150	. . . . . . . . . 10 ⊢ (𝑝 ∈ 𝐵 → (( I ↾ 𝐵)‘𝑝) = 𝑝)
27	24, 25, 26	3syl 18	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾))) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → (( I ↾ 𝐵)‘𝑝) = 𝑝)
28	27	oveq2d 7406	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾))) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → (𝑝(join‘𝐾)(( I ↾ 𝐵)‘𝑝)) = (𝑝(join‘𝐾)𝑝))
29	19, 11	hlatjidm 39369	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾)) → (𝑝(join‘𝐾)𝑝) = 𝑝)
30	18, 24, 29	syl2anc 584	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾))) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → (𝑝(join‘𝐾)𝑝) = 𝑝)
31	28, 30	eqtrd 2765	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾))) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → (𝑝(join‘𝐾)(( I ↾ 𝐵)‘𝑝)) = 𝑝)
32	31	oveq1d 7405	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾))) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → ((𝑝(join‘𝐾)(( I ↾ 𝐵)‘𝑝))(meet‘𝐾)𝑊) = (𝑝(meet‘𝐾)𝑊))
33		simprl 770	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾))) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → ¬ 𝑝(le‘𝐾)𝑊)
34	8, 9, 10, 11, 2	lhpmat 40031	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ ¬ 𝑝(le‘𝐾)𝑊)) → (𝑝(meet‘𝐾)𝑊) = (0.‘𝐾))
35	5, 24, 33, 34	syl12anc 836	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾))) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → (𝑝(meet‘𝐾)𝑊) = (0.‘𝐾))
36	32, 35	eqtrd 2765	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾))) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → ((𝑝(join‘𝐾)(( I ↾ 𝐵)‘𝑝))(meet‘𝐾)𝑊) = (0.‘𝐾))
37	13, 23, 36	3eqtr4rd 2776	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾))) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → ((𝑝(join‘𝐾)(( I ↾ 𝐵)‘𝑝))(meet‘𝐾)𝑊) = ((𝑞(join‘𝐾)(( I ↾ 𝐵)‘𝑞))(meet‘𝐾)𝑊))
38	37	ex 412	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾))) → ((¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊) → ((𝑝(join‘𝐾)(( I ↾ 𝐵)‘𝑝))(meet‘𝐾)𝑊) = ((𝑞(join‘𝐾)(( I ↾ 𝐵)‘𝑞))(meet‘𝐾)𝑊)))
39	38	ralrimivva 3181	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ∀𝑝 ∈ (Atoms‘𝐾)∀𝑞 ∈ (Atoms‘𝐾)((¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊) → ((𝑝(join‘𝐾)(( I ↾ 𝐵)‘𝑝))(meet‘𝐾)𝑊) = ((𝑞(join‘𝐾)(( I ↾ 𝐵)‘𝑞))(meet‘𝐾)𝑊)))
40		idltrn.t	. . 3 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
41	8, 19, 9, 11, 2, 3, 40	isltrn 40120	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (( I ↾ 𝐵) ∈ 𝑇 ↔ (( I ↾ 𝐵) ∈ ((LDil‘𝐾)‘𝑊) ∧ ∀𝑝 ∈ (Atoms‘𝐾)∀𝑞 ∈ (Atoms‘𝐾)((¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊) → ((𝑝(join‘𝐾)(( I ↾ 𝐵)‘𝑝))(meet‘𝐾)𝑊) = ((𝑞(join‘𝐾)(( I ↾ 𝐵)‘𝑞))(meet‘𝐾)𝑊)))))
42	4, 39, 41	mpbir2and 713	1 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( I ↾ 𝐵) ∈ 𝑇)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3045   class class class wbr 5110   I cid 5535   ↾ cres 5643  ‘cfv 6514  (class class class)co 7390  Basecbs 17186  lecple 17234  joincjn 18279  meetcmee 18280  0.cp0 18389  Atomscatm 39263  HLchlt 39350  LHypclh 39985  LDilcldil 40101  LTrncltrn 40102

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-map 8804  df-proset 18262  df-poset 18281  df-plt 18296  df-lub 18312  df-glb 18313  df-join 18314  df-meet 18315  df-p0 18391  df-lat 18398  df-covers 39266  df-ats 39267  df-atl 39298  df-cvlat 39322  df-hlat 39351  df-lhyp 39989  df-laut 39990  df-ldil 40105  df-ltrn 40106

This theorem is referenced by:  trlid0  40177  tgrpgrplem  40750  tendoid  40774  tendo0cl  40791  cdlemkid2  40925  cdlemkid3N  40934  cdlemkid4  40935  cdlemkid5  40936  cdlemk35s-id  40939  dva0g  41028  dian0  41040  dia0  41053  dvhgrp  41108  dvh0g  41112  dvheveccl  41113  dvhopN  41117  dihmeetlem4preN  41307