Theorem ltrnid 35421

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 806	. . . . . . 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 2622	. . . . . . . . 9 ⊢ (LAut‘𝐾) = (LAut‘𝐾)
4		ltrneq.t	. . . . . . . . 9 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
5	2, 3, 4	ltrnlaut 35409	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → 𝐹 ∈ (LAut‘𝐾))
6	5	ad2antrr 762	. . . . . . 7 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ ∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 → (𝐹‘𝑝) = 𝑝)) ∧ 𝑥 ∈ 𝐵) → 𝐹 ∈ (LAut‘𝐾))
7		simpr 477	. . . . . . 7 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ ∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 → (𝐹‘𝑝) = 𝑝)) ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)
8		simplll 798	. . . . . . . . . . . . 13 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ 𝑝 ∈ 𝐴) ∧ 𝑝 ≤ 𝑊) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
9		simpllr 799	. . . . . . . . . . . . 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 34576	. . . . . . . . . . . . . 14 ⊢ (𝑝 ∈ 𝐴 → 𝑝 ∈ 𝐵)
13	12	ad2antlr 763	. . . . . . . . . . . . 13 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ 𝑝 ∈ 𝐴) ∧ 𝑝 ≤ 𝑊) → 𝑝 ∈ 𝐵)
14		simpr 477	. . . . . . . . . . . . 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 35420	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑝 ∈ 𝐵 ∧ 𝑝 ≤ 𝑊)) → (𝐹‘𝑝) = 𝑝)
17	8, 9, 13, 14, 16	syl112anc 1330	. . . . . . . . . . . 12 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ 𝑝 ∈ 𝐴) ∧ 𝑝 ≤ 𝑊) → (𝐹‘𝑝) = 𝑝)
18	17	ex 450	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ 𝑝 ∈ 𝐴) → (𝑝 ≤ 𝑊 → (𝐹‘𝑝) = 𝑝))
19		pm2.61 183	. . . . . . . . . . 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 2962	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 → (𝐹‘𝑝) = 𝑝) → ∀𝑝 ∈ 𝐴 (𝐹‘𝑝) = 𝑝))
22	21	imp 445	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ ∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 → (𝐹‘𝑝) = 𝑝)) → ∀𝑝 ∈ 𝐴 (𝐹‘𝑝) = 𝑝)
23	22	adantr 481	. . . . . . 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 35381	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝐹 ∈ (LAut‘𝐾) ∧ 𝑥 ∈ 𝐵) ∧ ∀𝑝 ∈ 𝐴 (𝐹‘𝑝) = 𝑝) → (𝐹‘𝑥) = 𝑥)
25	1, 6, 7, 23, 24	syl31anc 1329	. . . . . 6 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ ∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 → (𝐹‘𝑝) = 𝑝)) ∧ 𝑥 ∈ 𝐵) → (𝐹‘𝑥) = 𝑥)
26		fvresi 6439	. . . . . . 7 ⊢ (𝑥 ∈ 𝐵 → (( I ↾ 𝐵)‘𝑥) = 𝑥)
27	26	adantl 482	. . . . . 6 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ ∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 → (𝐹‘𝑝) = 𝑝)) ∧ 𝑥 ∈ 𝐵) → (( I ↾ 𝐵)‘𝑥) = 𝑥)
28	25, 27	eqtr4d 2659	. . . . 5 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ ∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 → (𝐹‘𝑝) = 𝑝)) ∧ 𝑥 ∈ 𝐵) → (𝐹‘𝑥) = (( I ↾ 𝐵)‘𝑥))
29	28	ralrimiva 2966	. . . 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 35410	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → 𝐹:𝐵–1-1-onto→𝐵)
31	30	adantr 481	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ ∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 → (𝐹‘𝑝) = 𝑝)) → 𝐹:𝐵–1-1-onto→𝐵)
32		f1ofn 6138	. . . . . 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 6008	. . . . 5 ⊢ ( I ↾ 𝐵) Fn 𝐵
35		eqfnfv 6311	. . . . 5 ⊢ ((𝐹 Fn 𝐵 ∧ ( I ↾ 𝐵) Fn 𝐵) → (𝐹 = ( I ↾ 𝐵) ↔ ∀𝑥 ∈ 𝐵 (𝐹‘𝑥) = (( I ↾ 𝐵)‘𝑥)))
36	33, 34, 35	sylancl 694	. . . 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 247	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ ∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 → (𝐹‘𝑝) = 𝑝)) → 𝐹 = ( I ↾ 𝐵))
38	37	ex 450	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 → (𝐹‘𝑝) = 𝑝) → 𝐹 = ( I ↾ 𝐵)))
39	12	adantl 482	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ 𝑝 ∈ 𝐴) → 𝑝 ∈ 𝐵)
40		fvresi 6439	. . . . . 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 6190	. . . . . 6 ⊢ (𝐹 = ( I ↾ 𝐵) → (𝐹‘𝑝) = (( I ↾ 𝐵)‘𝑝))
43	42	eqeq1d 2624	. . . . 5 ⊢ (𝐹 = ( I ↾ 𝐵) → ((𝐹‘𝑝) = 𝑝 ↔ (( I ↾ 𝐵)‘𝑝) = 𝑝))
44	41, 43	syl5ibrcom 237	. . . 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 2969	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (𝐹 = ( I ↾ 𝐵) → ∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 → (𝐹‘𝑝) = 𝑝)))
47	38, 46	impbid 202	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 → (𝐹‘𝑝) = 𝑝) ↔ 𝐹 = ( I ↾ 𝐵)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912   class class class wbr 4653   I cid 5023   ↾ cres 5116   Fn wfn 5883  –1-1-onto→wf1o 5887  ‘cfv 5888  Basecbs 15857  lecple 15948  Atomscatm 34550  HLchlt 34637  LHypclh 35270  LAutclaut 35271  LTrncltrn 35387

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859  df-preset 16928  df-poset 16946  df-plt 16958  df-lub 16974  df-glb 16975  df-join 16976  df-meet 16977  df-p0 17039  df-lat 17046  df-clat 17108  df-oposet 34463  df-ol 34465  df-oml 34466  df-covers 34553  df-ats 34554  df-atl 34585  df-cvlat 34609  df-hlat 34638  df-laut 35275  df-ldil 35390  df-ltrn 35391

This theorem is referenced by:  ltrnnid  35422