Theorem ltrnid 40244

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 2731	. . . . . . . . 9 ⊢ (LAut‘𝐾) = (LAut‘𝐾)
4		ltrneq.t	. . . . . . . . 9 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
5	2, 3, 4	ltrnlaut 40232	. . . . . . . 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 39398	. . . . . . . . . . . . . 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 40243	. . . . . . . . . . . . 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 3144	. . . . . . . . 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 40204	. . . . . . 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 7107	. . . . . . 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 2769	. . . . 5 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ ∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 → (𝐹‘𝑝) = 𝑝)) ∧ 𝑥 ∈ 𝐵) → (𝐹‘𝑥) = (( I ↾ 𝐵)‘𝑥))
29	28	ralrimiva 3124	. . . 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 40233	. . . . . . 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 6764	. . . . . 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 6610	. . . . 5 ⊢ ( I ↾ 𝐵) Fn 𝐵
35		eqfnfv 6964	. . . . 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 7107	. . . . . 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 6821	. . . . . 6 ⊢ (𝐹 = ( I ↾ 𝐵) → (𝐹‘𝑝) = (( I ↾ 𝐵)‘𝑝))
43	42	eqeq1d 2733	. . . . 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 3132	. 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 1541   ∈ wcel 2111  ∀wral 3047   class class class wbr 5089   I cid 5508   ↾ cres 5616   Fn wfn 6476  –1-1-onto→wf1o 6480  ‘cfv 6481  Basecbs 17120  lecple 17168  Atomscatm 39372  HLchlt 39459  LHypclh 40093  LAutclaut 40094  LTrncltrn 40210

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-map 8752  df-proset 18200  df-poset 18219  df-plt 18234  df-lub 18250  df-glb 18251  df-join 18252  df-meet 18253  df-p0 18329  df-lat 18338  df-clat 18405  df-oposet 39285  df-ol 39287  df-oml 39288  df-covers 39375  df-ats 39376  df-atl 39407  df-cvlat 39431  df-hlat 39460  df-laut 40098  df-ldil 40213  df-ltrn 40214

This theorem is referenced by:  ltrnnid  40245