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Theorem ltrnid 39001
Description: A lattice translation is the identity function iff all atoms not under the fiducial co-atom π‘Š are equal to their values. (Contributed by NM, 24-May-2012.)
Hypotheses
Ref Expression
ltrneq.b 𝐡 = (Baseβ€˜πΎ)
ltrneq.l ≀ = (leβ€˜πΎ)
ltrneq.a 𝐴 = (Atomsβ€˜πΎ)
ltrneq.h 𝐻 = (LHypβ€˜πΎ)
ltrneq.t 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
Assertion
Ref Expression
ltrnid (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) β†’ (βˆ€π‘ ∈ 𝐴 (Β¬ 𝑝 ≀ π‘Š β†’ (πΉβ€˜π‘) = 𝑝) ↔ 𝐹 = ( I β†Ύ 𝐡)))
Distinct variable groups:   𝐴,𝑝   𝐡,𝑝   𝐹,𝑝   𝐻,𝑝   𝐾,𝑝   𝑇,𝑝   π‘Š,𝑝
Allowed substitution hint:   ≀ (𝑝)

Proof of Theorem ltrnid
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 simp-4l 781 . . . . . . 7 (((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ βˆ€π‘ ∈ 𝐴 (Β¬ 𝑝 ≀ π‘Š β†’ (πΉβ€˜π‘) = 𝑝)) ∧ π‘₯ ∈ 𝐡) β†’ 𝐾 ∈ HL)
2 ltrneq.h . . . . . . . . 9 𝐻 = (LHypβ€˜πΎ)
3 eqid 2732 . . . . . . . . 9 (LAutβ€˜πΎ) = (LAutβ€˜πΎ)
4 ltrneq.t . . . . . . . . 9 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
52, 3, 4ltrnlaut 38989 . . . . . . . 8 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) β†’ 𝐹 ∈ (LAutβ€˜πΎ))
65ad2antrr 724 . . . . . . 7 (((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ βˆ€π‘ ∈ 𝐴 (Β¬ 𝑝 ≀ π‘Š β†’ (πΉβ€˜π‘) = 𝑝)) ∧ π‘₯ ∈ 𝐡) β†’ 𝐹 ∈ (LAutβ€˜πΎ))
7 simpr 485 . . . . . . 7 (((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ βˆ€π‘ ∈ 𝐴 (Β¬ 𝑝 ≀ π‘Š β†’ (πΉβ€˜π‘) = 𝑝)) ∧ π‘₯ ∈ 𝐡) β†’ π‘₯ ∈ 𝐡)
8 simplll 773 . . . . . . . . . . . . 13 (((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ 𝑝 ∈ 𝐴) ∧ 𝑝 ≀ π‘Š) β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))
9 simpllr 774 . . . . . . . . . . . . 13 (((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ 𝑝 ∈ 𝐴) ∧ 𝑝 ≀ π‘Š) β†’ 𝐹 ∈ 𝑇)
10 ltrneq.b . . . . . . . . . . . . . . 15 𝐡 = (Baseβ€˜πΎ)
11 ltrneq.a . . . . . . . . . . . . . . 15 𝐴 = (Atomsβ€˜πΎ)
1210, 11atbase 38154 . . . . . . . . . . . . . 14 (𝑝 ∈ 𝐴 β†’ 𝑝 ∈ 𝐡)
1312ad2antlr 725 . . . . . . . . . . . . 13 (((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ 𝑝 ∈ 𝐴) ∧ 𝑝 ≀ π‘Š) β†’ 𝑝 ∈ 𝐡)
14 simpr 485 . . . . . . . . . . . . 13 (((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ 𝑝 ∈ 𝐴) ∧ 𝑝 ≀ π‘Š) β†’ 𝑝 ≀ π‘Š)
15 ltrneq.l . . . . . . . . . . . . . 14 ≀ = (leβ€˜πΎ)
1610, 15, 2, 4ltrnval1 39000 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑝 ∈ 𝐡 ∧ 𝑝 ≀ π‘Š)) β†’ (πΉβ€˜π‘) = 𝑝)
178, 9, 13, 14, 16syl112anc 1374 . . . . . . . . . . . 12 (((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ 𝑝 ∈ 𝐴) ∧ 𝑝 ≀ π‘Š) β†’ (πΉβ€˜π‘) = 𝑝)
1817ex 413 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ 𝑝 ∈ 𝐴) β†’ (𝑝 ≀ π‘Š β†’ (πΉβ€˜π‘) = 𝑝))
19 pm2.61 191 . . . . . . . . . . 11 ((𝑝 ≀ π‘Š β†’ (πΉβ€˜π‘) = 𝑝) β†’ ((Β¬ 𝑝 ≀ π‘Š β†’ (πΉβ€˜π‘) = 𝑝) β†’ (πΉβ€˜π‘) = 𝑝))
2018, 19syl 17 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ 𝑝 ∈ 𝐴) β†’ ((Β¬ 𝑝 ≀ π‘Š β†’ (πΉβ€˜π‘) = 𝑝) β†’ (πΉβ€˜π‘) = 𝑝))
2120ralimdva 3167 . . . . . . . . 9 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) β†’ (βˆ€π‘ ∈ 𝐴 (Β¬ 𝑝 ≀ π‘Š β†’ (πΉβ€˜π‘) = 𝑝) β†’ βˆ€π‘ ∈ 𝐴 (πΉβ€˜π‘) = 𝑝))
2221imp 407 . . . . . . . 8 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ βˆ€π‘ ∈ 𝐴 (Β¬ 𝑝 ≀ π‘Š β†’ (πΉβ€˜π‘) = 𝑝)) β†’ βˆ€π‘ ∈ 𝐴 (πΉβ€˜π‘) = 𝑝)
2322adantr 481 . . . . . . 7 (((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ βˆ€π‘ ∈ 𝐴 (Β¬ 𝑝 ≀ π‘Š β†’ (πΉβ€˜π‘) = 𝑝)) ∧ π‘₯ ∈ 𝐡) β†’ βˆ€π‘ ∈ 𝐴 (πΉβ€˜π‘) = 𝑝)
2410, 11, 3lauteq 38961 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝐹 ∈ (LAutβ€˜πΎ) ∧ π‘₯ ∈ 𝐡) ∧ βˆ€π‘ ∈ 𝐴 (πΉβ€˜π‘) = 𝑝) β†’ (πΉβ€˜π‘₯) = π‘₯)
251, 6, 7, 23, 24syl31anc 1373 . . . . . 6 (((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ βˆ€π‘ ∈ 𝐴 (Β¬ 𝑝 ≀ π‘Š β†’ (πΉβ€˜π‘) = 𝑝)) ∧ π‘₯ ∈ 𝐡) β†’ (πΉβ€˜π‘₯) = π‘₯)
26 fvresi 7170 . . . . . . 7 (π‘₯ ∈ 𝐡 β†’ (( I β†Ύ 𝐡)β€˜π‘₯) = π‘₯)
2726adantl 482 . . . . . 6 (((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ βˆ€π‘ ∈ 𝐴 (Β¬ 𝑝 ≀ π‘Š β†’ (πΉβ€˜π‘) = 𝑝)) ∧ π‘₯ ∈ 𝐡) β†’ (( I β†Ύ 𝐡)β€˜π‘₯) = π‘₯)
2825, 27eqtr4d 2775 . . . . 5 (((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ βˆ€π‘ ∈ 𝐴 (Β¬ 𝑝 ≀ π‘Š β†’ (πΉβ€˜π‘) = 𝑝)) ∧ π‘₯ ∈ 𝐡) β†’ (πΉβ€˜π‘₯) = (( I β†Ύ 𝐡)β€˜π‘₯))
2928ralrimiva 3146 . . . 4 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ βˆ€π‘ ∈ 𝐴 (Β¬ 𝑝 ≀ π‘Š β†’ (πΉβ€˜π‘) = 𝑝)) β†’ βˆ€π‘₯ ∈ 𝐡 (πΉβ€˜π‘₯) = (( I β†Ύ 𝐡)β€˜π‘₯))
3010, 2, 4ltrn1o 38990 . . . . . . 7 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) β†’ 𝐹:𝐡–1-1-onto→𝐡)
3130adantr 481 . . . . . 6 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ βˆ€π‘ ∈ 𝐴 (Β¬ 𝑝 ≀ π‘Š β†’ (πΉβ€˜π‘) = 𝑝)) β†’ 𝐹:𝐡–1-1-onto→𝐡)
32 f1ofn 6834 . . . . . 6 (𝐹:𝐡–1-1-onto→𝐡 β†’ 𝐹 Fn 𝐡)
3331, 32syl 17 . . . . 5 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ βˆ€π‘ ∈ 𝐴 (Β¬ 𝑝 ≀ π‘Š β†’ (πΉβ€˜π‘) = 𝑝)) β†’ 𝐹 Fn 𝐡)
34 fnresi 6679 . . . . 5 ( I β†Ύ 𝐡) Fn 𝐡
35 eqfnfv 7032 . . . . 5 ((𝐹 Fn 𝐡 ∧ ( I β†Ύ 𝐡) Fn 𝐡) β†’ (𝐹 = ( I β†Ύ 𝐡) ↔ βˆ€π‘₯ ∈ 𝐡 (πΉβ€˜π‘₯) = (( I β†Ύ 𝐡)β€˜π‘₯)))
3633, 34, 35sylancl 586 . . . 4 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ βˆ€π‘ ∈ 𝐴 (Β¬ 𝑝 ≀ π‘Š β†’ (πΉβ€˜π‘) = 𝑝)) β†’ (𝐹 = ( I β†Ύ 𝐡) ↔ βˆ€π‘₯ ∈ 𝐡 (πΉβ€˜π‘₯) = (( I β†Ύ 𝐡)β€˜π‘₯)))
3729, 36mpbird 256 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ βˆ€π‘ ∈ 𝐴 (Β¬ 𝑝 ≀ π‘Š β†’ (πΉβ€˜π‘) = 𝑝)) β†’ 𝐹 = ( I β†Ύ 𝐡))
3837ex 413 . 2 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) β†’ (βˆ€π‘ ∈ 𝐴 (Β¬ 𝑝 ≀ π‘Š β†’ (πΉβ€˜π‘) = 𝑝) β†’ 𝐹 = ( I β†Ύ 𝐡)))
3912adantl 482 . . . . . 6 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ 𝑝 ∈ 𝐴) β†’ 𝑝 ∈ 𝐡)
40 fvresi 7170 . . . . . 6 (𝑝 ∈ 𝐡 β†’ (( I β†Ύ 𝐡)β€˜π‘) = 𝑝)
4139, 40syl 17 . . . . 5 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ 𝑝 ∈ 𝐴) β†’ (( I β†Ύ 𝐡)β€˜π‘) = 𝑝)
42 fveq1 6890 . . . . . 6 (𝐹 = ( I β†Ύ 𝐡) β†’ (πΉβ€˜π‘) = (( I β†Ύ 𝐡)β€˜π‘))
4342eqeq1d 2734 . . . . 5 (𝐹 = ( I β†Ύ 𝐡) β†’ ((πΉβ€˜π‘) = 𝑝 ↔ (( I β†Ύ 𝐡)β€˜π‘) = 𝑝))
4441, 43syl5ibrcom 246 . . . 4 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ 𝑝 ∈ 𝐴) β†’ (𝐹 = ( I β†Ύ 𝐡) β†’ (πΉβ€˜π‘) = 𝑝))
4544a1dd 50 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ 𝑝 ∈ 𝐴) β†’ (𝐹 = ( I β†Ύ 𝐡) β†’ (Β¬ 𝑝 ≀ π‘Š β†’ (πΉβ€˜π‘) = 𝑝)))
4645ralrimdva 3154 . 2 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) β†’ (𝐹 = ( I β†Ύ 𝐡) β†’ βˆ€π‘ ∈ 𝐴 (Β¬ 𝑝 ≀ π‘Š β†’ (πΉβ€˜π‘) = 𝑝)))
4738, 46impbid 211 1 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) β†’ (βˆ€π‘ ∈ 𝐴 (Β¬ 𝑝 ≀ π‘Š β†’ (πΉβ€˜π‘) = 𝑝) ↔ 𝐹 = ( I β†Ύ 𝐡)))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061   class class class wbr 5148   I cid 5573   β†Ύ cres 5678   Fn wfn 6538  β€“1-1-ontoβ†’wf1o 6542  β€˜cfv 6543  Basecbs 17143  lecple 17203  Atomscatm 38128  HLchlt 38215  LHypclh 38850  LAutclaut 38851  LTrncltrn 38967
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7364  df-ov 7411  df-oprab 7412  df-mpo 7413  df-map 8821  df-proset 18247  df-poset 18265  df-plt 18282  df-lub 18298  df-glb 18299  df-join 18300  df-meet 18301  df-p0 18377  df-lat 18384  df-clat 18451  df-oposet 38041  df-ol 38043  df-oml 38044  df-covers 38131  df-ats 38132  df-atl 38163  df-cvlat 38187  df-hlat 38216  df-laut 38855  df-ldil 38970  df-ltrn 38971
This theorem is referenced by:  ltrnnid  39002
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