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Mirrors > Home > MPE Home > Th. List > Mathboxes > cdleme | Structured version Visualization version GIF version |
Description: Lemma E in [Crawley] p. 113. If p,q are atoms not under hyperplane w, then there is a unique translation f such that f(p) = q. (Contributed by NM, 11-Apr-2013.) |
Ref | Expression |
---|---|
cdleme.l | ⊢ ≤ = (le‘𝐾) |
cdleme.a | ⊢ 𝐴 = (Atoms‘𝐾) |
cdleme.h | ⊢ 𝐻 = (LHyp‘𝐾) |
cdleme.t | ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
Ref | Expression |
---|---|
cdleme | ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → ∃!𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑄) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cdleme.l | . . 3 ⊢ ≤ = (le‘𝐾) | |
2 | cdleme.a | . . 3 ⊢ 𝐴 = (Atoms‘𝐾) | |
3 | cdleme.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
4 | cdleme.t | . . 3 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
5 | 1, 2, 3, 4 | cdleme50ex 37088 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → ∃𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑄) |
6 | simp11 1183 | . . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑓 ∈ 𝑇 ∧ 𝑧 ∈ 𝑇) ∧ ((𝑓‘𝑃) = 𝑄 ∧ (𝑧‘𝑃) = 𝑄)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
7 | simp2l 1179 | . . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑓 ∈ 𝑇 ∧ 𝑧 ∈ 𝑇) ∧ ((𝑓‘𝑃) = 𝑄 ∧ (𝑧‘𝑃) = 𝑄)) → 𝑓 ∈ 𝑇) | |
8 | simp2r 1180 | . . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑓 ∈ 𝑇 ∧ 𝑧 ∈ 𝑇) ∧ ((𝑓‘𝑃) = 𝑄 ∧ (𝑧‘𝑃) = 𝑄)) → 𝑧 ∈ 𝑇) | |
9 | simp12 1184 | . . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑓 ∈ 𝑇 ∧ 𝑧 ∈ 𝑇) ∧ ((𝑓‘𝑃) = 𝑄 ∧ (𝑧‘𝑃) = 𝑄)) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) | |
10 | eqtr3 2795 | . . . . . 6 ⊢ (((𝑓‘𝑃) = 𝑄 ∧ (𝑧‘𝑃) = 𝑄) → (𝑓‘𝑃) = (𝑧‘𝑃)) | |
11 | 10 | 3ad2ant3 1115 | . . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑓 ∈ 𝑇 ∧ 𝑧 ∈ 𝑇) ∧ ((𝑓‘𝑃) = 𝑄 ∧ (𝑧‘𝑃) = 𝑄)) → (𝑓‘𝑃) = (𝑧‘𝑃)) |
12 | 1, 2, 3, 4 | cdlemd 36736 | . . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ 𝑇 ∧ 𝑧 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑓‘𝑃) = (𝑧‘𝑃)) → 𝑓 = 𝑧) |
13 | 6, 7, 8, 9, 11, 12 | syl311anc 1364 | . . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑓 ∈ 𝑇 ∧ 𝑧 ∈ 𝑇) ∧ ((𝑓‘𝑃) = 𝑄 ∧ (𝑧‘𝑃) = 𝑄)) → 𝑓 = 𝑧) |
14 | 13 | 3exp 1099 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → ((𝑓 ∈ 𝑇 ∧ 𝑧 ∈ 𝑇) → (((𝑓‘𝑃) = 𝑄 ∧ (𝑧‘𝑃) = 𝑄) → 𝑓 = 𝑧))) |
15 | 14 | ralrimivv 3134 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → ∀𝑓 ∈ 𝑇 ∀𝑧 ∈ 𝑇 (((𝑓‘𝑃) = 𝑄 ∧ (𝑧‘𝑃) = 𝑄) → 𝑓 = 𝑧)) |
16 | fveq1 6492 | . . . 4 ⊢ (𝑓 = 𝑧 → (𝑓‘𝑃) = (𝑧‘𝑃)) | |
17 | 16 | eqeq1d 2774 | . . 3 ⊢ (𝑓 = 𝑧 → ((𝑓‘𝑃) = 𝑄 ↔ (𝑧‘𝑃) = 𝑄)) |
18 | 17 | reu4 3630 | . 2 ⊢ (∃!𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑄 ↔ (∃𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑄 ∧ ∀𝑓 ∈ 𝑇 ∀𝑧 ∈ 𝑇 (((𝑓‘𝑃) = 𝑄 ∧ (𝑧‘𝑃) = 𝑄) → 𝑓 = 𝑧))) |
19 | 5, 15, 18 | sylanbrc 575 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → ∃!𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑄) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 387 ∧ w3a 1068 = wceq 1507 ∈ wcel 2048 ∀wral 3082 ∃wrex 3083 ∃!wreu 3084 class class class wbr 4923 ‘cfv 6182 lecple 16418 Atomscatm 35792 HLchlt 35879 LHypclh 36513 LTrncltrn 36630 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2745 ax-rep 5043 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 ax-riotaBAD 35482 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-ral 3087 df-rex 3088 df-reu 3089 df-rmo 3090 df-rab 3091 df-v 3411 df-sbc 3678 df-csb 3783 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-nul 4174 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-op 4442 df-uni 4707 df-iun 4788 df-iin 4789 df-br 4924 df-opab 4986 df-mpt 5003 df-id 5305 df-xp 5406 df-rel 5407 df-cnv 5408 df-co 5409 df-dm 5410 df-rn 5411 df-res 5412 df-ima 5413 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-riota 6931 df-ov 6973 df-oprab 6974 df-mpo 6975 df-1st 7494 df-2nd 7495 df-undef 7735 df-map 8200 df-proset 17386 df-poset 17404 df-plt 17416 df-lub 17432 df-glb 17433 df-join 17434 df-meet 17435 df-p0 17497 df-p1 17498 df-lat 17504 df-clat 17566 df-oposet 35705 df-ol 35707 df-oml 35708 df-covers 35795 df-ats 35796 df-atl 35827 df-cvlat 35851 df-hlat 35880 df-llines 36027 df-lplanes 36028 df-lvols 36029 df-lines 36030 df-psubsp 36032 df-pmap 36033 df-padd 36325 df-lhyp 36517 df-laut 36518 df-ldil 36633 df-ltrn 36634 df-trl 36688 |
This theorem is referenced by: ltrniotaval 37110 cdlemeiota 37114 cdlemksv2 37376 cdlemkuv2 37396 |
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