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Mirrors > Home > MPE Home > Th. List > Mathboxes > ltrniotaval | Structured version Visualization version GIF version |
Description: Value of the unique translation specified by a value. (Contributed by NM, 21-Feb-2014.) |
Ref | Expression |
---|---|
ltrniotaval.l | ⊢ ≤ = (le‘𝐾) |
ltrniotaval.a | ⊢ 𝐴 = (Atoms‘𝐾) |
ltrniotaval.h | ⊢ 𝐻 = (LHyp‘𝐾) |
ltrniotaval.t | ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
ltrniotaval.f | ⊢ 𝐹 = (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑄) |
Ref | Expression |
---|---|
ltrniotaval | ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝐹‘𝑃) = 𝑄) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltrniotaval.l | . . 3 ⊢ ≤ = (le‘𝐾) | |
2 | ltrniotaval.a | . . 3 ⊢ 𝐴 = (Atoms‘𝐾) | |
3 | ltrniotaval.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
4 | ltrniotaval.t | . . 3 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
5 | 1, 2, 3, 4 | cdleme 38230 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → ∃!𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑄) |
6 | ltrniotaval.f | . . . . . . 7 ⊢ 𝐹 = (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑄) | |
7 | nfriota1 7147 | . . . . . . 7 ⊢ Ⅎ𝑓(℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑄) | |
8 | 6, 7 | nfcxfr 2898 | . . . . . 6 ⊢ Ⅎ𝑓𝐹 |
9 | nfcv 2900 | . . . . . 6 ⊢ Ⅎ𝑓𝑃 | |
10 | 8, 9 | nffv 6697 | . . . . 5 ⊢ Ⅎ𝑓(𝐹‘𝑃) |
11 | 10 | nfeq1 2915 | . . . 4 ⊢ Ⅎ𝑓(𝐹‘𝑃) = 𝑄 |
12 | fveq1 6686 | . . . . 5 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑃) = (𝐹‘𝑃)) | |
13 | 12 | eqeq1d 2741 | . . . 4 ⊢ (𝑓 = 𝐹 → ((𝑓‘𝑃) = 𝑄 ↔ (𝐹‘𝑃) = 𝑄)) |
14 | 11, 6, 13 | riotaprop 7168 | . . 3 ⊢ (∃!𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑄 → (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) = 𝑄)) |
15 | 14 | simprd 499 | . 2 ⊢ (∃!𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑄 → (𝐹‘𝑃) = 𝑄) |
16 | 5, 15 | syl 17 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝐹‘𝑃) = 𝑄) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 ∧ w3a 1088 = wceq 1542 ∈ wcel 2114 ∃!wreu 3056 class class class wbr 5040 ‘cfv 6350 ℩crio 7139 lecple 16688 Atomscatm 36933 HLchlt 37020 LHypclh 37654 LTrncltrn 37771 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5242 ax-pr 5306 ax-un 7492 ax-riotaBAD 36623 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3402 df-sbc 3686 df-csb 3801 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-op 4533 df-uni 4807 df-iun 4893 df-iin 4894 df-br 5041 df-opab 5103 df-mpt 5121 df-id 5439 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-iota 6308 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7140 df-ov 7186 df-oprab 7187 df-mpo 7188 df-1st 7727 df-2nd 7728 df-undef 7981 df-map 8452 df-proset 17667 df-poset 17685 df-plt 17697 df-lub 17713 df-glb 17714 df-join 17715 df-meet 17716 df-p0 17778 df-p1 17779 df-lat 17785 df-clat 17847 df-oposet 36846 df-ol 36848 df-oml 36849 df-covers 36936 df-ats 36937 df-atl 36968 df-cvlat 36992 df-hlat 37021 df-llines 37168 df-lplanes 37169 df-lvols 37170 df-lines 37171 df-psubsp 37173 df-pmap 37174 df-padd 37466 df-lhyp 37658 df-laut 37659 df-ldil 37774 df-ltrn 37775 df-trl 37829 |
This theorem is referenced by: ltrniotacnvval 38252 ltrniotaidvalN 38253 ltrniotavalbN 38254 cdlemm10N 38788 cdlemn2 38865 cdlemn3 38867 cdlemn9 38875 dihmeetlem13N 38989 dih1dimatlem0 38998 dihjatcclem3 39090 |
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