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Mirrors > Home > MPE Home > Th. List > Mathboxes > cdlemftr0 | Structured version Visualization version GIF version |
Description: Special case of cdlemf 40273 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 2726 | . . 3 ⊢ ((trL‘𝐾)‘𝑊) = ((trL‘𝐾)‘𝑊) | |
5 | 1, 2, 3, 4 | cdlemftr1 40277 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ∃𝑓 ∈ 𝑇 (𝑓 ≠ ( I ↾ 𝐵) ∧ (((trL‘𝐾)‘𝑊)‘𝑓) ≠ I )) |
6 | simpl 481 | . . 3 ⊢ ((𝑓 ≠ ( I ↾ 𝐵) ∧ (((trL‘𝐾)‘𝑊)‘𝑓) ≠ I ) → 𝑓 ≠ ( I ↾ 𝐵)) | |
7 | 6 | reximi 3074 | . 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 394 = wceq 1534 ∈ wcel 2099 ≠ wne 2930 ∃wrex 3060 I cid 5570 ↾ cres 5675 ‘cfv 6544 Basecbs 17206 HLchlt 39059 LHypclh 39694 LTrncltrn 39811 trLctrl 39868 |
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 2697 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7736 ax-riotaBAD 38662 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rmo 3365 df-reu 3366 df-rab 3421 df-v 3465 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4907 df-iun 4996 df-iin 4997 df-br 5145 df-opab 5207 df-mpt 5228 df-id 5571 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 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 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-1st 7993 df-2nd 7994 df-undef 8278 df-map 8847 df-proset 18313 df-poset 18331 df-plt 18348 df-lub 18364 df-glb 18365 df-join 18366 df-meet 18367 df-p0 18443 df-p1 18444 df-lat 18450 df-clat 18517 df-oposet 38885 df-ol 38887 df-oml 38888 df-covers 38975 df-ats 38976 df-atl 39007 df-cvlat 39031 df-hlat 39060 df-llines 39208 df-lplanes 39209 df-lvols 39210 df-lines 39211 df-psubsp 39213 df-pmap 39214 df-padd 39506 df-lhyp 39698 df-laut 39699 df-ldil 39814 df-ltrn 39815 df-trl 39869 |
This theorem is referenced by: tendo0mul 40536 tendo0mulr 40537 tendo1ne0 40538 tendoconid 40539 cdleml4N 40689 erngdv 40703 erngdv-rN 40711 |
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