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Mirrors > Home > MPE Home > Th. List > Mathboxes > cdlemftr2 | Structured version Visualization version GIF version |
Description: Special case of cdlemf 40275 showing existence of non-identity translation with trace different from any 2 given lattice elements. (Contributed by NM, 25-Jul-2013.) |
Ref | Expression |
---|---|
cdlemftr.b | ⊢ 𝐵 = (Base‘𝐾) |
cdlemftr.h | ⊢ 𝐻 = (LHyp‘𝐾) |
cdlemftr.t | ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
cdlemftr.r | ⊢ 𝑅 = ((trL‘𝐾)‘𝑊) |
Ref | Expression |
---|---|
cdlemftr2 | ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ∃𝑓 ∈ 𝑇 (𝑓 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝑓) ≠ 𝑋 ∧ (𝑅‘𝑓) ≠ 𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cdlemftr.b | . . 3 ⊢ 𝐵 = (Base‘𝐾) | |
2 | cdlemftr.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
3 | cdlemftr.t | . . 3 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
4 | cdlemftr.r | . . 3 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊) | |
5 | 1, 2, 3, 4 | cdlemftr3 40277 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ∃𝑓 ∈ 𝑇 (𝑓 ≠ ( I ↾ 𝐵) ∧ ((𝑅‘𝑓) ≠ 𝑋 ∧ (𝑅‘𝑓) ≠ 𝑌 ∧ (𝑅‘𝑓) ≠ 𝑌))) |
6 | simpl 481 | . . . 4 ⊢ ((𝑓 ≠ ( I ↾ 𝐵) ∧ ((𝑅‘𝑓) ≠ 𝑋 ∧ (𝑅‘𝑓) ≠ 𝑌 ∧ (𝑅‘𝑓) ≠ 𝑌)) → 𝑓 ≠ ( I ↾ 𝐵)) | |
7 | simpr1 1191 | . . . 4 ⊢ ((𝑓 ≠ ( I ↾ 𝐵) ∧ ((𝑅‘𝑓) ≠ 𝑋 ∧ (𝑅‘𝑓) ≠ 𝑌 ∧ (𝑅‘𝑓) ≠ 𝑌)) → (𝑅‘𝑓) ≠ 𝑋) | |
8 | simpr2 1192 | . . . 4 ⊢ ((𝑓 ≠ ( I ↾ 𝐵) ∧ ((𝑅‘𝑓) ≠ 𝑋 ∧ (𝑅‘𝑓) ≠ 𝑌 ∧ (𝑅‘𝑓) ≠ 𝑌)) → (𝑅‘𝑓) ≠ 𝑌) | |
9 | 6, 7, 8 | 3jca 1125 | . . 3 ⊢ ((𝑓 ≠ ( I ↾ 𝐵) ∧ ((𝑅‘𝑓) ≠ 𝑋 ∧ (𝑅‘𝑓) ≠ 𝑌 ∧ (𝑅‘𝑓) ≠ 𝑌)) → (𝑓 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝑓) ≠ 𝑋 ∧ (𝑅‘𝑓) ≠ 𝑌)) |
10 | 9 | reximi 3074 | . 2 ⊢ (∃𝑓 ∈ 𝑇 (𝑓 ≠ ( I ↾ 𝐵) ∧ ((𝑅‘𝑓) ≠ 𝑋 ∧ (𝑅‘𝑓) ≠ 𝑌 ∧ (𝑅‘𝑓) ≠ 𝑌)) → ∃𝑓 ∈ 𝑇 (𝑓 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝑓) ≠ 𝑋 ∧ (𝑅‘𝑓) ≠ 𝑌)) |
11 | 5, 10 | syl 17 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ∃𝑓 ∈ 𝑇 (𝑓 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝑓) ≠ 𝑋 ∧ (𝑅‘𝑓) ≠ 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∧ w3a 1084 = wceq 1534 ∈ wcel 2099 ≠ wne 2930 ∃wrex 3060 I cid 5571 ↾ cres 5676 ‘cfv 6546 Basecbs 17208 HLchlt 39061 LHypclh 39696 LTrncltrn 39813 trLctrl 39870 |
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 5282 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7738 ax-riotaBAD 38664 |
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 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-iun 4995 df-iin 4996 df-br 5146 df-opab 5208 df-mpt 5229 df-id 5572 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-1st 7995 df-2nd 7996 df-undef 8280 df-map 8849 df-proset 18315 df-poset 18333 df-plt 18350 df-lub 18366 df-glb 18367 df-join 18368 df-meet 18369 df-p0 18445 df-p1 18446 df-lat 18452 df-clat 18519 df-oposet 38887 df-ol 38889 df-oml 38890 df-covers 38977 df-ats 38978 df-atl 39009 df-cvlat 39033 df-hlat 39062 df-llines 39210 df-lplanes 39211 df-lvols 39212 df-lines 39213 df-psubsp 39215 df-pmap 39216 df-padd 39508 df-lhyp 39700 df-laut 39701 df-ldil 39816 df-ltrn 39817 df-trl 39871 |
This theorem is referenced by: cdlemftr1 40279 cdlemk26b-3 40617 cdlemk29-3 40623 cdlemk38 40627 cdlemkid5 40647 cdlemkid 40648 cdlemk55b 40672 |
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