Theorem ltrnid 40129

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 782	. . . . . . 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 2729	. . . . . . . . 9 ⊢ (LAut‘𝐾) = (LAut‘𝐾)
4		ltrneq.t	. . . . . . . . 9 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
5	2, 3, 4	ltrnlaut 40117	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → 𝐹 ∈ (LAut‘𝐾))
6	5	ad2antrr 726	. . . . . . 7 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ ∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 → (𝐹‘𝑝) = 𝑝)) ∧ 𝑥 ∈ 𝐵) → 𝐹 ∈ (LAut‘𝐾))
7		simpr 484	. . . . . . 7 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ ∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 → (𝐹‘𝑝) = 𝑝)) ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)
8		simplll 774	. . . . . . . . . . . . 13 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ 𝑝 ∈ 𝐴) ∧ 𝑝 ≤ 𝑊) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
9		simpllr 775	. . . . . . . . . . . . 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 39282	. . . . . . . . . . . . . 14 ⊢ (𝑝 ∈ 𝐴 → 𝑝 ∈ 𝐵)
13	12	ad2antlr 727	. . . . . . . . . . . . 13 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ 𝑝 ∈ 𝐴) ∧ 𝑝 ≤ 𝑊) → 𝑝 ∈ 𝐵)
14		simpr 484	. . . . . . . . . . . . 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 40128	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑝 ∈ 𝐵 ∧ 𝑝 ≤ 𝑊)) → (𝐹‘𝑝) = 𝑝)
17	8, 9, 13, 14, 16	syl112anc 1376	. . . . . . . . . . . 12 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ 𝑝 ∈ 𝐴) ∧ 𝑝 ≤ 𝑊) → (𝐹‘𝑝) = 𝑝)
18	17	ex 412	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ 𝑝 ∈ 𝐴) → (𝑝 ≤ 𝑊 → (𝐹‘𝑝) = 𝑝))
19		pm2.61 192	. . . . . . . . . . 11 ⊢ ((𝑝 ≤ 𝑊 → (𝐹‘𝑝) = 𝑝) → ((¬ 𝑝 ≤ 𝑊 → (𝐹‘𝑝) = 𝑝) → (𝐹‘𝑝) = 𝑝))
20	18, 19	syl 17	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ 𝑝 ∈ 𝐴) → ((¬ 𝑝 ≤ 𝑊 → (𝐹‘𝑝) = 𝑝) → (𝐹‘𝑝) = 𝑝))
21	20	ralimdva 3145	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 → (𝐹‘𝑝) = 𝑝) → ∀𝑝 ∈ 𝐴 (𝐹‘𝑝) = 𝑝))
22	21	imp 406	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ ∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 → (𝐹‘𝑝) = 𝑝)) → ∀𝑝 ∈ 𝐴 (𝐹‘𝑝) = 𝑝)
23	22	adantr 480	. . . . . . 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 40089	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝐹 ∈ (LAut‘𝐾) ∧ 𝑥 ∈ 𝐵) ∧ ∀𝑝 ∈ 𝐴 (𝐹‘𝑝) = 𝑝) → (𝐹‘𝑥) = 𝑥)
25	1, 6, 7, 23, 24	syl31anc 1375	. . . . . 6 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ ∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 → (𝐹‘𝑝) = 𝑝)) ∧ 𝑥 ∈ 𝐵) → (𝐹‘𝑥) = 𝑥)
26		fvresi 7147	. . . . . . 7 ⊢ (𝑥 ∈ 𝐵 → (( I ↾ 𝐵)‘𝑥) = 𝑥)
27	26	adantl 481	. . . . . 6 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ ∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 → (𝐹‘𝑝) = 𝑝)) ∧ 𝑥 ∈ 𝐵) → (( I ↾ 𝐵)‘𝑥) = 𝑥)
28	25, 27	eqtr4d 2767	. . . . 5 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ ∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 → (𝐹‘𝑝) = 𝑝)) ∧ 𝑥 ∈ 𝐵) → (𝐹‘𝑥) = (( I ↾ 𝐵)‘𝑥))
29	28	ralrimiva 3125	. . . 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 40118	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → 𝐹:𝐵–1-1-onto→𝐵)
31	30	adantr 480	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ ∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 → (𝐹‘𝑝) = 𝑝)) → 𝐹:𝐵–1-1-onto→𝐵)
32		f1ofn 6801	. . . . . 6 ⊢ (𝐹:𝐵–1-1-onto→𝐵 → 𝐹 Fn 𝐵)
33	31, 32	syl 17	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ ∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 → (𝐹‘𝑝) = 𝑝)) → 𝐹 Fn 𝐵)
34		fnresi 6647	. . . . 5 ⊢ ( I ↾ 𝐵) Fn 𝐵
35		eqfnfv 7003	. . . . 5 ⊢ ((𝐹 Fn 𝐵 ∧ ( I ↾ 𝐵) Fn 𝐵) → (𝐹 = ( I ↾ 𝐵) ↔ ∀𝑥 ∈ 𝐵 (𝐹‘𝑥) = (( I ↾ 𝐵)‘𝑥)))
36	33, 34, 35	sylancl 586	. . . 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 257	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ ∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 → (𝐹‘𝑝) = 𝑝)) → 𝐹 = ( I ↾ 𝐵))
38	37	ex 412	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 → (𝐹‘𝑝) = 𝑝) → 𝐹 = ( I ↾ 𝐵)))
39	12	adantl 481	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ 𝑝 ∈ 𝐴) → 𝑝 ∈ 𝐵)
40		fvresi 7147	. . . . . 6 ⊢ (𝑝 ∈ 𝐵 → (( I ↾ 𝐵)‘𝑝) = 𝑝)
41	39, 40	syl 17	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ 𝑝 ∈ 𝐴) → (( I ↾ 𝐵)‘𝑝) = 𝑝)
42		fveq1 6857	. . . . . 6 ⊢ (𝐹 = ( I ↾ 𝐵) → (𝐹‘𝑝) = (( I ↾ 𝐵)‘𝑝))
43	42	eqeq1d 2731	. . . . 5 ⊢ (𝐹 = ( I ↾ 𝐵) → ((𝐹‘𝑝) = 𝑝 ↔ (( I ↾ 𝐵)‘𝑝) = 𝑝))
44	41, 43	syl5ibrcom 247	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ 𝑝 ∈ 𝐴) → (𝐹 = ( I ↾ 𝐵) → (𝐹‘𝑝) = 𝑝))
45	44	a1dd 50	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ 𝑝 ∈ 𝐴) → (𝐹 = ( I ↾ 𝐵) → (¬ 𝑝 ≤ 𝑊 → (𝐹‘𝑝) = 𝑝)))
46	45	ralrimdva 3133	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (𝐹 = ( I ↾ 𝐵) → ∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 → (𝐹‘𝑝) = 𝑝)))
47	38, 46	impbid 212	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 → (𝐹‘𝑝) = 𝑝) ↔ 𝐹 = ( I ↾ 𝐵)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044   class class class wbr 5107   I cid 5532   ↾ cres 5640   Fn wfn 6506  –1-1-onto→wf1o 6510  ‘cfv 6511  Basecbs 17179  lecple 17227  Atomscatm 39256  HLchlt 39343  LHypclh 39978  LAutclaut 39979  LTrncltrn 40095

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 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-map 8801  df-proset 18255  df-poset 18274  df-plt 18289  df-lub 18305  df-glb 18306  df-join 18307  df-meet 18308  df-p0 18384  df-lat 18391  df-clat 18458  df-oposet 39169  df-ol 39171  df-oml 39172  df-covers 39259  df-ats 39260  df-atl 39291  df-cvlat 39315  df-hlat 39344  df-laut 39983  df-ldil 40098  df-ltrn 40099

This theorem is referenced by:  ltrnnid  40130