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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > cdlemg1ci2 | Structured version Visualization version GIF version |
Description: Any function of the form of the function constructed for cdleme 40269 is a translation. TODO: Fix comment. (Contributed by NM, 18-Apr-2013.) |
Ref | Expression |
---|---|
cdlemg1c.l | ⊢ ≤ = (le‘𝐾) |
cdlemg1c.a | ⊢ 𝐴 = (Atoms‘𝐾) |
cdlemg1c.h | ⊢ 𝐻 = (LHyp‘𝐾) |
cdlemg1c.t | ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
Ref | Expression |
---|---|
cdlemg1ci2 | ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝐹 = (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑄)) → 𝐹 ∈ 𝑇) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 483 | . 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝐹 = (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑄)) → 𝐹 = (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑄)) | |
2 | cdlemg1c.l | . . . 4 ⊢ ≤ = (le‘𝐾) | |
3 | cdlemg1c.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
4 | cdlemg1c.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
5 | cdlemg1c.t | . . . 4 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
6 | eqid 2726 | . . . 4 ⊢ (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑄) = (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑄) | |
7 | 2, 3, 4, 5, 6 | ltrniotacl 40288 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑄) ∈ 𝑇) |
8 | 7 | adantr 479 | . 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝐹 = (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑄)) → (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑄) ∈ 𝑇) |
9 | 1, 8 | eqeltrd 2826 | 1 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝐹 = (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑄)) → 𝐹 ∈ 𝑇) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 394 ∧ w3a 1084 = wceq 1534 ∈ wcel 2099 class class class wbr 5143 ‘cfv 6543 ℩crio 7368 lecple 17265 Atomscatm 38971 HLchlt 39058 LHypclh 39693 LTrncltrn 39810 |
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 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7735 ax-riotaBAD 38661 |
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 5144 df-opab 5206 df-mpt 5227 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 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-1st 7992 df-2nd 7993 df-undef 8277 df-map 8846 df-proset 18312 df-poset 18330 df-plt 18347 df-lub 18363 df-glb 18364 df-join 18365 df-meet 18366 df-p0 18442 df-p1 18443 df-lat 18449 df-clat 18516 df-oposet 38884 df-ol 38886 df-oml 38887 df-covers 38974 df-ats 38975 df-atl 39006 df-cvlat 39030 df-hlat 39059 df-llines 39207 df-lplanes 39208 df-lvols 39209 df-lines 39210 df-psubsp 39212 df-pmap 39213 df-padd 39505 df-lhyp 39697 df-laut 39698 df-ldil 39813 df-ltrn 39814 df-trl 39868 |
This theorem is referenced by: cdlemg1cN 40296 cdlemg1cex 40297 |
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