Theorem ltrnid 40833

Description: A lattice translation is the identity function iff all atoms not under the fiducial co-atom 𝑊 are equal to their values. (Contributed by NM, 24-May-2012.)

Hypotheses

Ref	Expression
ltrneq.b	⊢ 𝐵 = (Base‘𝐾)
ltrneq.l	⊢ ≤ = (le‘𝐾)
ltrneq.a	⊢ 𝐴 = (Atoms‘𝐾)
ltrneq.h	⊢ 𝐻 = (LHyp‘𝐾)
ltrneq.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)

Assertion

Ref	Expression
ltrnid	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 → (𝐹‘𝑝) = 𝑝) ↔ 𝐹 = ( I ↾ 𝐵)))

Distinct variable groups:   𝐴,𝑝   𝐵,𝑝   𝐹,𝑝   𝐻,𝑝   𝐾,𝑝   𝑇,𝑝   𝑊,𝑝

Allowed substitution hint:   ≤ (𝑝)

Proof of Theorem ltrnid

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp-4l 794	. . . . . . 7 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ ∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 → (𝐹‘𝑝) = 𝑝)) ∧ 𝑥 ∈ 𝐵) → 𝐾 ∈ HL)
2		ltrneq.h	. . . . . . . . 9 ⊢ 𝐻 = (LHyp‘𝐾)
3		eqid 2769	. . . . . . . . 9 ⊢ (LAut‘𝐾) = (LAut‘𝐾)
4		ltrneq.t	. . . . . . . . 9 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
5	2, 3, 4	ltrnlaut 40821	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → 𝐹 ∈ (LAut‘𝐾))
6	5	ad2antrr 738	. . . . . . 7 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ ∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 → (𝐹‘𝑝) = 𝑝)) ∧ 𝑥 ∈ 𝐵) → 𝐹 ∈ (LAut‘𝐾))
7		simpr 489	. . . . . . 7 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ ∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 → (𝐹‘𝑝) = 𝑝)) ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)
8		simplll 786	. . . . . . . . . . . . 13 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ 𝑝 ∈ 𝐴) ∧ 𝑝 ≤ 𝑊) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
9		simpllr 787	. . . . . . . . . . . . 13 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ 𝑝 ∈ 𝐴) ∧ 𝑝 ≤ 𝑊) → 𝐹 ∈ 𝑇)
10		ltrneq.b	. . . . . . . . . . . . . . 15 ⊢ 𝐵 = (Base‘𝐾)
11		ltrneq.a	. . . . . . . . . . . . . . 15 ⊢ 𝐴 = (Atoms‘𝐾)
12	10, 11	atbase 39987	. . . . . . . . . . . . . 14 ⊢ (𝑝 ∈ 𝐴 → 𝑝 ∈ 𝐵)
13	12	ad2antlr 739	. . . . . . . . . . . . 13 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ 𝑝 ∈ 𝐴) ∧ 𝑝 ≤ 𝑊) → 𝑝 ∈ 𝐵)
14		simpr 489	. . . . . . . . . . . . 13 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ 𝑝 ∈ 𝐴) ∧ 𝑝 ≤ 𝑊) → 𝑝 ≤ 𝑊)
15		ltrneq.l	. . . . . . . . . . . . . 14 ⊢ ≤ = (le‘𝐾)
16	10, 15, 2, 4	ltrnval1 40832	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑝 ∈ 𝐵 ∧ 𝑝 ≤ 𝑊)) → (𝐹‘𝑝) = 𝑝)
17	8, 9, 13, 14, 16	syl112anc 1399	. . . . . . . . . . . 12 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ 𝑝 ∈ 𝐴) ∧ 𝑝 ≤ 𝑊) → (𝐹‘𝑝) = 𝑝)
18	17	ex 417	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ 𝑝 ∈ 𝐴) → (𝑝 ≤ 𝑊 → (𝐹‘𝑝) = 𝑝))
19		pm2.61 194	. . . . . . . . . . 11 ⊢ ((𝑝 ≤ 𝑊 → (𝐹‘𝑝) = 𝑝) → ((¬ 𝑝 ≤ 𝑊 → (𝐹‘𝑝) = 𝑝) → (𝐹‘𝑝) = 𝑝))
20	18, 19	syl 18	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ 𝑝 ∈ 𝐴) → ((¬ 𝑝 ≤ 𝑊 → (𝐹‘𝑝) = 𝑝) → (𝐹‘𝑝) = 𝑝))
21	20	ralimdva 3183	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 → (𝐹‘𝑝) = 𝑝) → ∀𝑝 ∈ 𝐴 (𝐹‘𝑝) = 𝑝))
22	21	imp 411	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ ∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 → (𝐹‘𝑝) = 𝑝)) → ∀𝑝 ∈ 𝐴 (𝐹‘𝑝) = 𝑝)
23	22	adantr 485	. . . . . . 7 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ ∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 → (𝐹‘𝑝) = 𝑝)) ∧ 𝑥 ∈ 𝐵) → ∀𝑝 ∈ 𝐴 (𝐹‘𝑝) = 𝑝)
24	10, 11, 3	lauteq 40793	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝐹 ∈ (LAut‘𝐾) ∧ 𝑥 ∈ 𝐵) ∧ ∀𝑝 ∈ 𝐴 (𝐹‘𝑝) = 𝑝) → (𝐹‘𝑥) = 𝑥)
25	1, 6, 7, 23, 24	syl31anc 1398	. . . . . 6 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ ∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 → (𝐹‘𝑝) = 𝑝)) ∧ 𝑥 ∈ 𝐵) → (𝐹‘𝑥) = 𝑥)
26		fvresi 7172	. . . . . . 7 ⊢ (𝑥 ∈ 𝐵 → (( I ↾ 𝐵)‘𝑥) = 𝑥)
27	26	adantl 486	. . . . . 6 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ ∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 → (𝐹‘𝑝) = 𝑝)) ∧ 𝑥 ∈ 𝐵) → (( I ↾ 𝐵)‘𝑥) = 𝑥)
28	25, 27	eqtr4d 2807	. . . . 5 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ ∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 → (𝐹‘𝑝) = 𝑝)) ∧ 𝑥 ∈ 𝐵) → (𝐹‘𝑥) = (( I ↾ 𝐵)‘𝑥))
29	28	ralrimiva 3163	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ ∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 → (𝐹‘𝑝) = 𝑝)) → ∀𝑥 ∈ 𝐵 (𝐹‘𝑥) = (( I ↾ 𝐵)‘𝑥))
30	10, 2, 4	ltrn1o 40822	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → 𝐹:𝐵–1-1-onto→𝐵)
31	30	adantr 485	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ ∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 → (𝐹‘𝑝) = 𝑝)) → 𝐹:𝐵–1-1-onto→𝐵)
32		f1ofn 6822	. . . . . 6 ⊢ (𝐹:𝐵–1-1-onto→𝐵 → 𝐹 Fn 𝐵)
33	31, 32	syl 18	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ ∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 → (𝐹‘𝑝) = 𝑝)) → 𝐹 Fn 𝐵)
34		fnresi 6665	. . . . 5 ⊢ ( I ↾ 𝐵) Fn 𝐵
35		eqfnfv 7026	. . . . 5 ⊢ ((𝐹 Fn 𝐵 ∧ ( I ↾ 𝐵) Fn 𝐵) → (𝐹 = ( I ↾ 𝐵) ↔ ∀𝑥 ∈ 𝐵 (𝐹‘𝑥) = (( I ↾ 𝐵)‘𝑥)))
36	33, 34, 35	sylancl 597	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ ∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 → (𝐹‘𝑝) = 𝑝)) → (𝐹 = ( I ↾ 𝐵) ↔ ∀𝑥 ∈ 𝐵 (𝐹‘𝑥) = (( I ↾ 𝐵)‘𝑥)))
37	29, 36	mpbird 260	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ ∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 → (𝐹‘𝑝) = 𝑝)) → 𝐹 = ( I ↾ 𝐵))
38	37	ex 417	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 → (𝐹‘𝑝) = 𝑝) → 𝐹 = ( I ↾ 𝐵)))
39	12	adantl 486	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ 𝑝 ∈ 𝐴) → 𝑝 ∈ 𝐵)
40		fvresi 7172	. . . . . 6 ⊢ (𝑝 ∈ 𝐵 → (( I ↾ 𝐵)‘𝑝) = 𝑝)
41	39, 40	syl 18	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ 𝑝 ∈ 𝐴) → (( I ↾ 𝐵)‘𝑝) = 𝑝)
42		fveq1 6881	. . . . . 6 ⊢ (𝐹 = ( I ↾ 𝐵) → (𝐹‘𝑝) = (( I ↾ 𝐵)‘𝑝))
43	42	eqeq1d 2771	. . . . 5 ⊢ (𝐹 = ( I ↾ 𝐵) → ((𝐹‘𝑝) = 𝑝 ↔ (( I ↾ 𝐵)‘𝑝) = 𝑝))
44	41, 43	syl5ibrcom 250	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ 𝑝 ∈ 𝐴) → (𝐹 = ( I ↾ 𝐵) → (𝐹‘𝑝) = 𝑝))
45	44	a1dd 51	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ 𝑝 ∈ 𝐴) → (𝐹 = ( I ↾ 𝐵) → (¬ 𝑝 ≤ 𝑊 → (𝐹‘𝑝) = 𝑝)))
46	45	ralrimdva 3171	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (𝐹 = ( I ↾ 𝐵) → ∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 → (𝐹‘𝑝) = 𝑝)))
47	38, 46	impbid 215	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 → (𝐹‘𝑝) = 𝑝) ↔ 𝐹 = ( I ↾ 𝐵)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149  ∀wral 3085   class class class wbr 5113   I cid 5556   ↾ cres 5664   Fn wfn 6532  –1-1-onto→wf1o 6536  ‘cfv 6537  Basecbs 17269  lecple 17317  Atomscatm 39961  HLchlt 40048  LHypclh 40682  LAutclaut 40683  LTrncltrn 40799

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7368  df-ov 7414  df-oprab 7415  df-mpo 7416  df-map 8826  df-proset 18350  df-poset 18369  df-plt 18384  df-lub 18400  df-glb 18401  df-join 18402  df-meet 18403  df-p0 18479  df-lat 18488  df-clat 18555  df-oposet 39874  df-ol 39876  df-oml 39877  df-covers 39964  df-ats 39965  df-atl 39996  df-cvlat 40020  df-hlat 40049  df-laut 40687  df-ldil 40802  df-ltrn 40803

This theorem is referenced by:  ltrnnid  40834