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Mirrors > Home > MPE Home > Th. List > Mathboxes > cdlemftr0 | Structured version Visualization version GIF version |
Description: Special case of cdlemf 38993 showing existence of a non-identity translation. (Contributed by NM, 1-Aug-2013.) |
Ref | Expression |
---|---|
cdlemftr0.b | ⊢ 𝐵 = (Base‘𝐾) |
cdlemftr0.h | ⊢ 𝐻 = (LHyp‘𝐾) |
cdlemftr0.t | ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
Ref | Expression |
---|---|
cdlemftr0 | ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ∃𝑓 ∈ 𝑇 𝑓 ≠ ( I ↾ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cdlemftr0.b | . . 3 ⊢ 𝐵 = (Base‘𝐾) | |
2 | cdlemftr0.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
3 | cdlemftr0.t | . . 3 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
4 | eqid 2736 | . . 3 ⊢ ((trL‘𝐾)‘𝑊) = ((trL‘𝐾)‘𝑊) | |
5 | 1, 2, 3, 4 | cdlemftr1 38997 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ∃𝑓 ∈ 𝑇 (𝑓 ≠ ( I ↾ 𝐵) ∧ (((trL‘𝐾)‘𝑊)‘𝑓) ≠ I )) |
6 | simpl 483 | . . 3 ⊢ ((𝑓 ≠ ( I ↾ 𝐵) ∧ (((trL‘𝐾)‘𝑊)‘𝑓) ≠ I ) → 𝑓 ≠ ( I ↾ 𝐵)) | |
7 | 6 | reximi 3085 | . 2 ⊢ (∃𝑓 ∈ 𝑇 (𝑓 ≠ ( I ↾ 𝐵) ∧ (((trL‘𝐾)‘𝑊)‘𝑓) ≠ I ) → ∃𝑓 ∈ 𝑇 𝑓 ≠ ( I ↾ 𝐵)) |
8 | 5, 7 | syl 17 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ∃𝑓 ∈ 𝑇 𝑓 ≠ ( I ↾ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2941 ∃wrex 3071 I cid 5528 ↾ cres 5633 ‘cfv 6493 Basecbs 17075 HLchlt 37779 LHypclh 38414 LTrncltrn 38531 trLctrl 38588 |
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 2707 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7668 ax-riotaBAD 37382 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-iin 4955 df-br 5104 df-opab 5166 df-mpt 5187 df-id 5529 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7309 df-ov 7356 df-oprab 7357 df-mpo 7358 df-1st 7917 df-2nd 7918 df-undef 8200 df-map 8763 df-proset 18176 df-poset 18194 df-plt 18211 df-lub 18227 df-glb 18228 df-join 18229 df-meet 18230 df-p0 18306 df-p1 18307 df-lat 18313 df-clat 18380 df-oposet 37605 df-ol 37607 df-oml 37608 df-covers 37695 df-ats 37696 df-atl 37727 df-cvlat 37751 df-hlat 37780 df-llines 37928 df-lplanes 37929 df-lvols 37930 df-lines 37931 df-psubsp 37933 df-pmap 37934 df-padd 38226 df-lhyp 38418 df-laut 38419 df-ldil 38534 df-ltrn 38535 df-trl 38589 |
This theorem is referenced by: tendo0mul 39256 tendo0mulr 39257 tendo1ne0 39258 tendoconid 39259 cdleml4N 39409 erngdv 39423 erngdv-rN 39431 |
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