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Mirrors > Home > MPE Home > Th. List > Mathboxes > cdleme7 | Structured version Visualization version Unicode version |
Description: Part of proof of Lemma E in [Crawley] p. 113. and represent fs(r) and f(s) respectively. is the fiducial co-atom (hyperplane) that they call w. Here and in cdleme7ga 35535 above, we show that fs(r) W (top of p. 114), meaning it is an atom and not under w, which in our notation is expressed as . (Note that we do not have a symbol for their W.) Their proof provides no details of our cdleme7aa 35529 through cdleme7 35536, so there may be a simpler proof that we have overlooked. (Contributed by NM, 9-Jun-2012.) |
Ref | Expression |
---|---|
cdleme4.l | |
cdleme4.j | |
cdleme4.m | |
cdleme4.a | |
cdleme4.h | |
cdleme4.u | |
cdleme4.f | |
cdleme4.g |
Ref | Expression |
---|---|
cdleme7 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cdleme4.l | . . 3 | |
2 | cdleme4.j | . . 3 | |
3 | cdleme4.m | . . 3 | |
4 | cdleme4.a | . . 3 | |
5 | cdleme4.h | . . 3 | |
6 | cdleme4.u | . . 3 | |
7 | cdleme4.f | . . 3 | |
8 | cdleme4.g | . . 3 | |
9 | eqid 2622 | . . 3 | |
10 | 1, 2, 3, 4, 5, 6, 7, 8, 9 | cdleme7d 35533 | . 2 |
11 | simp11l 1172 | . . . . . 6 | |
12 | simp2ll 1128 | . . . . . 6 | |
13 | 1, 2, 3, 4, 5, 6, 7, 8 | cdleme7ga 35535 | . . . . . 6 |
14 | 1, 2, 4 | hlatlej2 34662 | . . . . . 6 |
15 | 11, 12, 13, 14 | syl3anc 1326 | . . . . 5 |
16 | 15 | biantrurd 529 | . . . 4 |
17 | hllat 34650 | . . . . . . 7 | |
18 | 11, 17 | syl 17 | . . . . . 6 |
19 | eqid 2622 | . . . . . . . 8 | |
20 | 19, 4 | atbase 34576 | . . . . . . 7 |
21 | 13, 20 | syl 17 | . . . . . 6 |
22 | 19, 2, 4 | hlatjcl 34653 | . . . . . . 7 |
23 | 11, 12, 13, 22 | syl3anc 1326 | . . . . . 6 |
24 | simp11r 1173 | . . . . . . 7 | |
25 | 19, 5 | lhpbase 35284 | . . . . . . 7 |
26 | 24, 25 | syl 17 | . . . . . 6 |
27 | 19, 1, 3 | latlem12 17078 | . . . . . 6 |
28 | 18, 21, 23, 26, 27 | syl13anc 1328 | . . . . 5 |
29 | simp11 1091 | . . . . . . 7 | |
30 | simp12l 1174 | . . . . . . 7 | |
31 | simp13l 1176 | . . . . . . 7 | |
32 | simp2l 1087 | . . . . . . 7 | |
33 | simp2r 1088 | . . . . . . 7 | |
34 | simp32 1098 | . . . . . . 7 | |
35 | 1, 2, 3, 4, 5, 6, 7, 8 | cdleme6 35528 | . . . . . . 7 |
36 | 29, 30, 31, 32, 33, 34, 35 | syl132anc 1344 | . . . . . 6 |
37 | 36 | breq2d 4665 | . . . . 5 |
38 | 28, 37 | bitrd 268 | . . . 4 |
39 | hlatl 34647 | . . . . . 6 | |
40 | 11, 39 | syl 17 | . . . . 5 |
41 | simp12 1092 | . . . . . 6 | |
42 | simp31 1097 | . . . . . 6 | |
43 | 1, 2, 3, 4, 5, 6 | lhpat2 35331 | . . . . . 6 |
44 | 29, 41, 31, 42, 43 | syl112anc 1330 | . . . . 5 |
45 | 1, 4 | atcmp 34598 | . . . . 5 |
46 | 40, 13, 44, 45 | syl3anc 1326 | . . . 4 |
47 | 16, 38, 46 | 3bitrd 294 | . . 3 |
48 | 47 | necon3bbid 2831 | . 2 |
49 | 10, 48 | mpbird 247 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wi 4 wb 196 wa 384 w3a 1037 wceq 1483 wcel 1990 wne 2794 class class class wbr 4653 cfv 5888 (class class class)co 6650 cbs 15857 cple 15948 cjn 16944 cmee 16945 clat 17045 catm 34550 cal 34551 chlt 34637 clh 35270 |
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 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 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-ral 2917 df-rex 2918 df-reu 2919 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-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-lines 34787 df-psubsp 34789 df-pmap 34790 df-padd 35082 df-lhyp 35274 |
This theorem is referenced by: cdleme18a 35578 cdleme22f2 35635 cdlemefs32sn1aw 35702 |
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