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Theorem ltrnid 39054
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 782 . . . . . . 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 2733 . . . . . . . . 9 (LAutβ€˜πΎ) = (LAutβ€˜πΎ)
4 ltrneq.t . . . . . . . . 9 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
52, 3, 4ltrnlaut 39042 . . . . . . . 8 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) β†’ 𝐹 ∈ (LAutβ€˜πΎ))
65ad2antrr 725 . . . . . . 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 486 . . . . . . 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 774 . . . . . . . . . . . . 13 (((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ 𝑝 ∈ 𝐴) ∧ 𝑝 ≀ π‘Š) β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))
9 simpllr 775 . . . . . . . . . . . . 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 38207 . . . . . . . . . . . . . 14 (𝑝 ∈ 𝐴 β†’ 𝑝 ∈ 𝐡)
1312ad2antlr 726 . . . . . . . . . . . . 13 (((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ 𝑝 ∈ 𝐴) ∧ 𝑝 ≀ π‘Š) β†’ 𝑝 ∈ 𝐡)
14 simpr 486 . . . . . . . . . . . . 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 39053 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑝 ∈ 𝐡 ∧ 𝑝 ≀ π‘Š)) β†’ (πΉβ€˜π‘) = 𝑝)
178, 9, 13, 14, 16syl112anc 1375 . . . . . . . . . . . 12 (((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ 𝑝 ∈ 𝐴) ∧ 𝑝 ≀ π‘Š) β†’ (πΉβ€˜π‘) = 𝑝)
1817ex 414 . . . . . . . . . . 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 3168 . . . . . . . . 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 408 . . . . . . . 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 482 . . . . . . 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 39014 . . . . . . 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 1374 . . . . . 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 7171 . . . . . . 7 (π‘₯ ∈ 𝐡 β†’ (( I β†Ύ 𝐡)β€˜π‘₯) = π‘₯)
2726adantl 483 . . . . . 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 2776 . . . . 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 3147 . . . 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 39043 . . . . . . 7 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) β†’ 𝐹:𝐡–1-1-onto→𝐡)
3130adantr 482 . . . . . 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 6835 . . . . . 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 6680 . . . . 5 ( I β†Ύ 𝐡) Fn 𝐡
35 eqfnfv 7033 . . . . 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 587 . . . 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 257 . . 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 414 . 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 483 . . . . . 6 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ 𝑝 ∈ 𝐴) β†’ 𝑝 ∈ 𝐡)
40 fvresi 7171 . . . . . 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 6891 . . . . . 6 (𝐹 = ( I β†Ύ 𝐡) β†’ (πΉβ€˜π‘) = (( I β†Ύ 𝐡)β€˜π‘))
4342eqeq1d 2735 . . . . 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 3155 . 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 397   = wceq 1542   ∈ wcel 2107  βˆ€wral 3062   class class class wbr 5149   I cid 5574   β†Ύ cres 5679   Fn wfn 6539  β€“1-1-ontoβ†’wf1o 6543  β€˜cfv 6544  Basecbs 17144  lecple 17204  Atomscatm 38181  HLchlt 38268  LHypclh 38903  LAutclaut 38904  LTrncltrn 39020
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-map 8822  df-proset 18248  df-poset 18266  df-plt 18283  df-lub 18299  df-glb 18300  df-join 18301  df-meet 18302  df-p0 18378  df-lat 18385  df-clat 18452  df-oposet 38094  df-ol 38096  df-oml 38097  df-covers 38184  df-ats 38185  df-atl 38216  df-cvlat 38240  df-hlat 38269  df-laut 38908  df-ldil 39023  df-ltrn 39024
This theorem is referenced by:  ltrnnid  39055
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