Theorem idltrn 40786

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 2765	. . 3 ⊢ ((LDil‘𝐾)‘𝑊) = ((LDil‘𝐾)‘𝑊)
4	1, 2, 3	idldil 40750	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( I ↾ 𝐵) ∈ ((LDil‘𝐾)‘𝑊))
5		simpll 778	. . . . . 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 789	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾))) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → 𝑞 ∈ (Atoms‘𝐾))
7		simprr 784	. . . . . 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 2765	. . . . . . 7 ⊢ (le‘𝐾) = (le‘𝐾)
9		eqid 2765	. . . . . . 7 ⊢ (meet‘𝐾) = (meet‘𝐾)
10		eqid 2765	. . . . . . 7 ⊢ (0.‘𝐾) = (0.‘𝐾)
11		eqid 2765	. . . . . . 7 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
12	8, 9, 10, 11, 2	lhpmat 40666	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ ¬ 𝑞(le‘𝐾)𝑊)) → (𝑞(meet‘𝐾)𝑊) = (0.‘𝐾))
13	5, 6, 7, 12	syl12anc 849	. . . . 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 39925	. . . . . . . . 9 ⊢ (𝑞 ∈ (Atoms‘𝐾) → 𝑞 ∈ 𝐵)
15		fvresi 7161	. . . . . . . . 9 ⊢ (𝑞 ∈ 𝐵 → (( I ↾ 𝐵)‘𝑞) = 𝑞)
16	6, 14, 15	3syl 19	. . . . . . . 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 7416	. . . . . . 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 786	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾))) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → 𝐾 ∈ HL)
19		eqid 2765	. . . . . . . . 9 ⊢ (join‘𝐾) = (join‘𝐾)
20	19, 11	hlatjidm 40005	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑞 ∈ (Atoms‘𝐾)) → (𝑞(join‘𝐾)𝑞) = 𝑞)
21	18, 6, 20	syl2anc 595	. . . . . . 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 2800	. . . . . 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 7415	. . . . 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 788	. . . . . . . . . 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 39925	. . . . . . . . . 10 ⊢ (𝑝 ∈ (Atoms‘𝐾) → 𝑝 ∈ 𝐵)
26		fvresi 7161	. . . . . . . . . 10 ⊢ (𝑝 ∈ 𝐵 → (( I ↾ 𝐵)‘𝑝) = 𝑝)
27	24, 25, 26	3syl 19	. . . . . . . . 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 7416	. . . . . . . 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 40005	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾)) → (𝑝(join‘𝐾)𝑝) = 𝑝)
30	18, 24, 29	syl2anc 595	. . . . . . . 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 2800	. . . . . . 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 7415	. . . . . 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 782	. . . . . . 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 40666	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ ¬ 𝑝(le‘𝐾)𝑊)) → (𝑝(meet‘𝐾)𝑊) = (0.‘𝐾))
35	5, 24, 33, 34	syl12anc 849	. . . . . 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 2800	. . . . 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 2811	. . . 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 417	. . 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 3208	. 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 40755	. 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 725	1 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( I ↾ 𝐵) ∈ 𝑇)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   = wceq 1563   ∈ wcel 2145  ∀wral 3079   class class class wbr 5105   I cid 5546   ↾ cres 5654  ‘cfv 6525  (class class class)co 7400  Basecbs 17259  lecple 17307  joincjn 18357  meetcmee 18358  0.cp0 18467  Atomscatm 39899  HLchlt 39986  LHypclh 40620  LDilcldil 40736  LTrncltrn 40737

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rmo 3370  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-riota 7357  df-ov 7403  df-oprab 7404  df-mpo 7405  df-map 8814  df-proset 18340  df-poset 18359  df-plt 18374  df-lub 18390  df-glb 18391  df-join 18392  df-meet 18393  df-p0 18469  df-lat 18478  df-covers 39902  df-ats 39903  df-atl 39934  df-cvlat 39958  df-hlat 39987  df-lhyp 40624  df-laut 40625  df-ldil 40740  df-ltrn 40741

This theorem is referenced by:  trlid0  40812  tgrpgrplem  41385  tendoid  41409  tendo0cl  41426  cdlemkid2  41560  cdlemkid3N  41569  cdlemkid4  41570  cdlemkid5  41571  cdlemk35s-id  41574  dva0g  41663  dian0  41675  dia0  41688  dvhgrp  41743  dvh0g  41747  dvheveccl  41748  dvhopN  41752  dihmeetlem4preN  41942