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Theorem dia0 40519
Description: The value of the partial isomorphism A at the lattice zero is the singleton of the identity translation i.e. the zero subspace. (Contributed by NM, 26-Nov-2013.)
Hypotheses
Ref Expression
dia0.b 𝐡 = (Baseβ€˜πΎ)
dia0.z 0 = (0.β€˜πΎ)
dia0.h 𝐻 = (LHypβ€˜πΎ)
dia0.i 𝐼 = ((DIsoAβ€˜πΎ)β€˜π‘Š)
Assertion
Ref Expression
dia0 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ (πΌβ€˜ 0 ) = {( I β†Ύ 𝐡)})
Proof of Theorem dia0
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 id 22 . . 3 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))
2 hlatl 38826 . . . . 5 (𝐾 ∈ HL β†’ 𝐾 ∈ AtLat)
3 dia0.b . . . . . 6 𝐡 = (Baseβ€˜πΎ)
4 dia0.z . . . . . 6 0 = (0.β€˜πΎ)
53, 4atl0cl 38769 . . . . 5 (𝐾 ∈ AtLat β†’ 0 ∈ 𝐡)
62, 5syl 17 . . . 4 (𝐾 ∈ HL β†’ 0 ∈ 𝐡)
76adantr 480 . . 3 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ 0 ∈ 𝐡)
8 dia0.h . . . . 5 𝐻 = (LHypβ€˜πΎ)
93, 8lhpbase 39465 . . . 4 (π‘Š ∈ 𝐻 β†’ π‘Š ∈ 𝐡)
10 eqid 2728 . . . . 5 (leβ€˜πΎ) = (leβ€˜πΎ)
113, 10, 4atl0le 38770 . . . 4 ((𝐾 ∈ AtLat ∧ π‘Š ∈ 𝐡) β†’ 0 (leβ€˜πΎ)π‘Š)
122, 9, 11syl2an 595 . . 3 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ 0 (leβ€˜πΎ)π‘Š)
13 eqid 2728 . . . 4 ((LTrnβ€˜πΎ)β€˜π‘Š) = ((LTrnβ€˜πΎ)β€˜π‘Š)
14 eqid 2728 . . . 4 ((trLβ€˜πΎ)β€˜π‘Š) = ((trLβ€˜πΎ)β€˜π‘Š)
15 dia0.i . . . 4 𝐼 = ((DIsoAβ€˜πΎ)β€˜π‘Š)
163, 10, 8, 13, 14, 15diaval 40499 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ( 0 ∈ 𝐡 ∧ 0 (leβ€˜πΎ)π‘Š)) β†’ (πΌβ€˜ 0 ) = {𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ∣ (((trLβ€˜πΎ)β€˜π‘Š)β€˜π‘“)(leβ€˜πΎ) 0 })
171, 7, 12, 16syl12anc 836 . 2 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ (πΌβ€˜ 0 ) = {𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ∣ (((trLβ€˜πΎ)β€˜π‘Š)β€˜π‘“)(leβ€˜πΎ) 0 })
182ad2antrr 725 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š)) β†’ 𝐾 ∈ AtLat)
193, 8, 13, 14trlcl 39631 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š)) β†’ (((trLβ€˜πΎ)β€˜π‘Š)β€˜π‘“) ∈ 𝐡)
203, 10, 4atlle0 38771 . . . . 5 ((𝐾 ∈ AtLat ∧ (((trLβ€˜πΎ)β€˜π‘Š)β€˜π‘“) ∈ 𝐡) β†’ ((((trLβ€˜πΎ)β€˜π‘Š)β€˜π‘“)(leβ€˜πΎ) 0 ↔ (((trLβ€˜πΎ)β€˜π‘Š)β€˜π‘“) = 0 ))
2118, 19, 20syl2anc 583 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š)) β†’ ((((trLβ€˜πΎ)β€˜π‘Š)β€˜π‘“)(leβ€˜πΎ) 0 ↔ (((trLβ€˜πΎ)β€˜π‘Š)β€˜π‘“) = 0 ))
223, 4, 8, 13, 14trlid0b 39645 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š)) β†’ (𝑓 = ( I β†Ύ 𝐡) ↔ (((trLβ€˜πΎ)β€˜π‘Š)β€˜π‘“) = 0 ))
2321, 22bitr4d 282 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š)) β†’ ((((trLβ€˜πΎ)β€˜π‘Š)β€˜π‘“)(leβ€˜πΎ) 0 ↔ 𝑓 = ( I β†Ύ 𝐡)))
2423rabbidva 3435 . 2 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ {𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ∣ (((trLβ€˜πΎ)β€˜π‘Š)β€˜π‘“)(leβ€˜πΎ) 0 } = {𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ∣ 𝑓 = ( I β†Ύ 𝐡)})
253, 8, 13idltrn 39617 . . 3 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ ( I β†Ύ 𝐡) ∈ ((LTrnβ€˜πΎ)β€˜π‘Š))
26 rabsn 4721 . . 3 (( I β†Ύ 𝐡) ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) β†’ {𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ∣ 𝑓 = ( I β†Ύ 𝐡)} = {( I β†Ύ 𝐡)})
2725, 26syl 17 . 2 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ {𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ∣ 𝑓 = ( I β†Ύ 𝐡)} = {( I β†Ύ 𝐡)})
2817, 24, 273eqtrd 2772 1 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ (πΌβ€˜ 0 ) = {( I β†Ύ 𝐡)})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   = wceq 1534   ∈ wcel 2099  {crab 3428  {csn 4624   class class class wbr 5142   I cid 5569   β†Ύ cres 5674  β€˜cfv 6542  Basecbs 17173  lecple 17233  0.cp0 18408  AtLatcal 38730  HLchlt 38816  LHypclh 39451  LTrncltrn 39568  trLctrl 39625  DIsoAcdia 40495
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-ral 3058  df-rex 3067  df-rmo 3372  df-reu 3373  df-rab 3429  df-v 3472  df-sbc 3776  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-map 8840  df-proset 18280  df-poset 18298  df-plt 18315  df-lub 18331  df-glb 18332  df-join 18333  df-meet 18334  df-p0 18410  df-p1 18411  df-lat 18417  df-clat 18484  df-oposet 38642  df-ol 38644  df-oml 38645  df-covers 38732  df-ats 38733  df-atl 38764  df-cvlat 38788  df-hlat 38817  df-lhyp 39455  df-laut 39456  df-ldil 39571  df-ltrn 39572  df-trl 39626  df-disoa 40496
This theorem is referenced by:  dib0  40631
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