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Mirrors > Home > MPE Home > Th. List > Mathboxes > cdleme | Structured version Visualization version Unicode 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 | |
cdleme.a | |
cdleme.h | |
cdleme.t |
Ref | Expression |
---|---|
cdleme |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cdleme.l | . . 3 | |
2 | cdleme.a | . . 3 | |
3 | cdleme.h | . . 3 | |
4 | cdleme.t | . . 3 | |
5 | 1, 2, 3, 4 | cdleme50ex 35847 | . 2 |
6 | simp11 1091 | . . . . 5 | |
7 | simp2l 1087 | . . . . 5 | |
8 | simp2r 1088 | . . . . 5 | |
9 | simp12 1092 | . . . . 5 | |
10 | eqtr3 2643 | . . . . . 6 | |
11 | 10 | 3ad2ant3 1084 | . . . . 5 |
12 | 1, 2, 3, 4 | cdlemd 35494 | . . . . 5 |
13 | 6, 7, 8, 9, 11, 12 | syl311anc 1340 | . . . 4 |
14 | 13 | 3exp 1264 | . . 3 |
15 | 14 | ralrimivv 2970 | . 2 |
16 | fveq1 6190 | . . . 4 | |
17 | 16 | eqeq1d 2624 | . . 3 |
18 | 17 | reu4 3400 | . 2 |
19 | 5, 15, 18 | sylanbrc 698 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wi 4 wa 384 w3a 1037 wceq 1483 wcel 1990 wral 2912 wrex 2913 wreu 2914 class class class wbr 4653 cfv 5888 cple 15948 catm 34550 chlt 34637 clh 35270 cltrn 35387 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-riotaBAD 34239 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-undef 7399 df-map 7859 df-preset 16928 df-poset 16946 df-plt 16958 df-lub 16974 df-glb 16975 df-join 16976 df-meet 16977 df-p0 17039 df-p1 17040 df-lat 17046 df-clat 17108 df-oposet 34463 df-ol 34465 df-oml 34466 df-covers 34553 df-ats 34554 df-atl 34585 df-cvlat 34609 df-hlat 34638 df-llines 34784 df-lplanes 34785 df-lvols 34786 df-lines 34787 df-psubsp 34789 df-pmap 34790 df-padd 35082 df-lhyp 35274 df-laut 35275 df-ldil 35390 df-ltrn 35391 df-trl 35446 |
This theorem is referenced by: ltrniotaval 35869 cdlemeiota 35873 cdlemksv2 36135 cdlemkuv2 36155 |
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