Theorem ltrnid 41193

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 795	. . . . . . 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 2760	. . . . . . . . 9 ⊢ (LAut‘𝐾) = (LAut‘𝐾)
4		ltrneq.t	. . . . . . . . 9 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
5	2, 3, 4	ltrnlaut 41181	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → 𝐹 ∈ (LAut‘𝐾))
6	5	ad2antrr 739	. . . . . . 7 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ ∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 → (𝐹‘𝑝) = 𝑝)) ∧ 𝑥 ∈ 𝐵) → 𝐹 ∈ (LAut‘𝐾))
7		simpr 490	. . . . . . 7 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ ∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 → (𝐹‘𝑝) = 𝑝)) ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)
8		simplll 787	. . . . . . . . . . . . 13 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ 𝑝 ∈ 𝐴) ∧ 𝑝 ≤ 𝑊) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
9		simpllr 788	. . . . . . . . . . . . 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 40347	. . . . . . . . . . . . . 14 ⊢ (𝑝 ∈ 𝐴 → 𝑝 ∈ 𝐵)
13	12	ad2antlr 740	. . . . . . . . . . . . 13 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ 𝑝 ∈ 𝐴) ∧ 𝑝 ≤ 𝑊) → 𝑝 ∈ 𝐵)
14		simpr 490	. . . . . . . . . . . . 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 41192	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑝 ∈ 𝐵 ∧ 𝑝 ≤ 𝑊)) → (𝐹‘𝑝) = 𝑝)
17	8, 9, 13, 14, 16	syl112anc 1401	. . . . . . . . . . . 12 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ 𝑝 ∈ 𝐴) ∧ 𝑝 ≤ 𝑊) → (𝐹‘𝑝) = 𝑝)
18	17	ex 418	. . . . . . . . . . 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 3174	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 → (𝐹‘𝑝) = 𝑝) → ∀𝑝 ∈ 𝐴 (𝐹‘𝑝) = 𝑝))
22	21	imp 412	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ ∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 → (𝐹‘𝑝) = 𝑝)) → ∀𝑝 ∈ 𝐴 (𝐹‘𝑝) = 𝑝)
23	22	adantr 486	. . . . . . 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 41153	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝐹 ∈ (LAut‘𝐾) ∧ 𝑥 ∈ 𝐵) ∧ ∀𝑝 ∈ 𝐴 (𝐹‘𝑝) = 𝑝) → (𝐹‘𝑥) = 𝑥)
25	1, 6, 7, 23, 24	syl31anc 1400	. . . . . 6 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ ∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 → (𝐹‘𝑝) = 𝑝)) ∧ 𝑥 ∈ 𝐵) → (𝐹‘𝑥) = 𝑥)
26		fvresi 7178	. . . . . . 7 ⊢ (𝑥 ∈ 𝐵 → (( I ↾ 𝐵)‘𝑥) = 𝑥)
27	26	adantl 487	. . . . . 6 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ ∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 → (𝐹‘𝑝) = 𝑝)) ∧ 𝑥 ∈ 𝐵) → (( I ↾ 𝐵)‘𝑥) = 𝑥)
28	25, 27	eqtr4d 2798	. . . . 5 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ ∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 → (𝐹‘𝑝) = 𝑝)) ∧ 𝑥 ∈ 𝐵) → (𝐹‘𝑥) = (( I ↾ 𝐵)‘𝑥))
29	28	ralrimiva 3154	. . . 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 41182	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → 𝐹:𝐵–1-1-onto→𝐵)
31	30	adantr 486	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ ∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 → (𝐹‘𝑝) = 𝑝)) → 𝐹:𝐵–1-1-onto→𝐵)
32		f1ofn 6825	. . . . . 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 6668	. . . . 5 ⊢ ( I ↾ 𝐵) Fn 𝐵
35		eqfnfv 7029	. . . . 5 ⊢ ((𝐹 Fn 𝐵 ∧ ( I ↾ 𝐵) Fn 𝐵) → (𝐹 = ( I ↾ 𝐵) ↔ ∀𝑥 ∈ 𝐵 (𝐹‘𝑥) = (( I ↾ 𝐵)‘𝑥)))
36	33, 34, 35	sylancl 598	. . . 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 418	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 → (𝐹‘𝑝) = 𝑝) → 𝐹 = ( I ↾ 𝐵)))
39	12	adantl 487	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ 𝑝 ∈ 𝐴) → 𝑝 ∈ 𝐵)
40		fvresi 7178	. . . . . 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 6884	. . . . . 6 ⊢ (𝐹 = ( I ↾ 𝐵) → (𝐹‘𝑝) = (( I ↾ 𝐵)‘𝑝))
43	42	eqeq1d 2762	. . . . 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 3162	. 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)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∀wral 3076   class class class wbr 5102   I cid 5544   ↾ cres 5652   Fn wfn 6533  –1-1-onto→wf1o 6537  ‘cfv 6538  Basecbs 17387  lecple 17435  Atomscatm 40321  HLchlt 40408  LHypclh 41042  LAutclaut 41043  LTrncltrn 41159

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-map 8849  df-proset 18468  df-poset 18487  df-plt 18502  df-lub 18518  df-glb 18519  df-join 18520  df-meet 18521  df-p0 18597  df-lat 18606  df-clat 18673  df-oposet 40234  df-ol 40236  df-oml 40237  df-covers 40324  df-ats 40325  df-atl 40356  df-cvlat 40380  df-hlat 40409  df-laut 41047  df-ldil 41162  df-ltrn 41163

This theorem is used by:  ltrnnid  41194