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Theorem ltrnid 38648
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 38636 . . . . . . . 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 37801 . . . . . . . . . . . . . 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 38647 . . . . . . . . . . . . 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 3161 . . . . . . . . 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 38608 . . . . . . 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 7123 . . . . . . 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 3140 . . . 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 38637 . . . . . . 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 6789 . . . . . 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 6634 . . . . 5 ( I β†Ύ 𝐡) Fn 𝐡
35 eqfnfv 6986 . . . . 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 7123 . . . . . 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 6845 . . . . . 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 247 . . . 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 3148 . 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 3061   class class class wbr 5109   I cid 5534   β†Ύ cres 5639   Fn wfn 6495  β€“1-1-ontoβ†’wf1o 6499  β€˜cfv 6500  Basecbs 17091  lecple 17148  Atomscatm 37775  HLchlt 37862  LHypclh 38497  LAutclaut 38498  LTrncltrn 38614
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 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
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 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7317  df-ov 7364  df-oprab 7365  df-mpo 7366  df-map 8773  df-proset 18192  df-poset 18210  df-plt 18227  df-lub 18243  df-glb 18244  df-join 18245  df-meet 18246  df-p0 18322  df-lat 18329  df-clat 18396  df-oposet 37688  df-ol 37690  df-oml 37691  df-covers 37778  df-ats 37779  df-atl 37810  df-cvlat 37834  df-hlat 37863  df-laut 38502  df-ldil 38617  df-ltrn 38618
This theorem is referenced by:  ltrnnid  38649
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