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Theorem cdleme16f 31017
 Description: Part of proof of Lemma E in [Crawley] p. 113, 3rd paragraph on p. 114, 3rd part of 3rd sentence. and represent f(s) and f(t) respectively. We show, in their notation, (s t) (f(s) f(t))=(f(s) f(t)) w. (Contributed by NM, 11-Oct-2012.)
Hypotheses
Ref Expression
cdleme12.l
cdleme12.j
cdleme12.m
cdleme12.a
cdleme12.h
cdleme12.u
cdleme12.f
cdleme12.g
Assertion
Ref Expression
cdleme16f

Proof of Theorem cdleme16f
StepHypRef Expression
1 simp11l 1068 . . . . 5
2 hllat 30098 . . . . 5
31, 2syl 16 . . . 4
4 simp21l 1074 . . . . 5
5 simp22l 1076 . . . . 5
6 eqid 2435 . . . . . 6
7 cdleme12.j . . . . . 6
8 cdleme12.a . . . . . 6
96, 7, 8hlatjcl 30101 . . . . 5
101, 4, 5, 9syl3anc 1184 . . . 4
11 simp11 987 . . . . . 6
12 simp12 988 . . . . . 6
13 simp13 989 . . . . . 6
14 simp21 990 . . . . . 6
15 simp23l 1078 . . . . . 6
16 simp31 993 . . . . . 6
17 cdleme12.l . . . . . . 7
18 cdleme12.m . . . . . . 7
19 cdleme12.h . . . . . . 7
20 cdleme12.u . . . . . . 7
21 cdleme12.f . . . . . . 7
2217, 7, 18, 8, 19, 20, 21cdleme3fa 30970 . . . . . 6
2311, 12, 13, 14, 15, 16, 22syl132anc 1202 . . . . 5
24 simp22 991 . . . . . 6
25 simp32 994 . . . . . 6
26 cdleme12.g . . . . . . 7
2717, 7, 18, 8, 19, 20, 26cdleme3fa 30970 . . . . . 6
2811, 12, 13, 24, 15, 25, 27syl132anc 1202 . . . . 5
296, 7, 8hlatjcl 30101 . . . . 5
301, 23, 28, 29syl3anc 1184 . . . 4
316, 17, 18latmle2 14498 . . . 4
323, 10, 30, 31syl3anc 1184 . . 3
3317, 7, 18, 8, 19, 20, 21, 26cdleme15 31012 . . 3
346, 18latmcl 14472 . . . . 5
353, 10, 30, 34syl3anc 1184 . . . 4
36 simp11r 1069 . . . . 5
376, 19lhpbase 30732 . . . . 5
3836, 37syl 16 . . . 4
396, 17, 18latlem12 14499 . . . 4
403, 35, 30, 38, 39syl13anc 1186 . . 3
4132, 33, 40mpbi2and 888 . 2
42 hlatl 30095 . . . 4
431, 42syl 16 . . 3
4417, 7, 18, 8, 19, 20, 21, 26cdleme16d 31015 . . 3
4517, 7, 18, 8, 19, 20, 21cdleme3 30971 . . . . 5
4611, 12, 13, 14, 15, 16, 45syl132anc 1202 . . . 4
4717, 7, 18, 8, 19, 20, 21, 26cdleme16b 31013 . . . 4
4817, 7, 18, 8, 19lhpat 30777 . . . 4
4911, 23, 46, 28, 47, 48syl122anc 1193 . . 3
5017, 8atcmp 30046 . . 3
5143, 44, 49, 50syl3anc 1184 . 2
5241, 51mpbid 202 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wb 177   wa 359   w3a 936   wceq 1652   wcel 1725   wne 2598   class class class wbr 4204  cfv 5446  (class class class)co 6073  cbs 13461  cple 13528  cjn 14393  cmee 14394  clat 14466  catm 29998  cal 29999  chlt 30085  clh 30718 This theorem is referenced by:  cdleme16g  31018 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-undef 6535  df-riota 6541  df-poset 14395  df-plt 14407  df-lub 14423  df-glb 14424  df-join 14425  df-meet 14426  df-p0 14460  df-p1 14461  df-lat 14467  df-clat 14529  df-oposet 29911  df-ol 29913  df-oml 29914  df-covers 30001  df-ats 30002  df-atl 30033  df-cvlat 30057  df-hlat 30086  df-llines 30232  df-lplanes 30233  df-lvols 30234  df-lines 30235  df-psubsp 30237  df-pmap 30238  df-padd 30530  df-lhyp 30722
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