Mathbox for Norm Megill < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cdleme15a Unicode version

Theorem cdleme15a 29630
 Description: Part of proof of Lemma E in [Crawley] p. 113, 3rd paragraph on p. 114, showing, in their notation, ((s p) (f(s) q)) ((t p) (f(t) q))=((p s1) (q s1)) ((p t1) (q t1)). We represent f(s), f(t), s1, and t1 with , , , and respectively. The order of our operations is slightly different. (Contributed by NM, 9-Oct-2012.)
Hypotheses
Ref Expression
cdleme12.l
cdleme12.j
cdleme12.m
cdleme12.a
cdleme12.h
cdleme12.u
cdleme12.f
cdleme12.g
cdleme15.c
cdleme15.x
Assertion
Ref Expression
cdleme15a

Proof of Theorem cdleme15a
StepHypRef Expression
1 simp11l 1071 . . . . 5
2 simp11r 1072 . . . . 5
3 simp12l 1073 . . . . 5
4 simp12r 1074 . . . . 5
5 simp22l 1079 . . . . 5
6 cdleme12.l . . . . . 6
7 cdleme12.j . . . . . 6
8 cdleme12.m . . . . . 6
9 cdleme12.a . . . . . 6
10 cdleme12.h . . . . . 6
11 cdleme15.x . . . . . 6
126, 7, 8, 9, 10, 11cdleme8 29606 . . . . 5
131, 2, 3, 4, 5, 12syl221anc 1198 . . . 4
147, 9hlatjcom 28724 . . . . 5
151, 3, 5, 14syl3anc 1187 . . . 4
1613, 15eqtr2d 2291 . . 3
17 simp11 990 . . . . . 6
18 simp12 991 . . . . . 6
19 simp13 992 . . . . . 6
20 simp22 994 . . . . . 6
21 simp23l 1081 . . . . . 6
22 simp32 997 . . . . . 6
23 cdleme12.u . . . . . . 7
24 cdleme12.g . . . . . . 7
256, 7, 8, 9, 10, 23, 24cdleme3fa 29592 . . . . . 6
2617, 18, 19, 20, 21, 22, 25syl132anc 1205 . . . . 5
27 simp13l 1075 . . . . 5
287, 9hlatjcom 28724 . . . . 5
291, 26, 27, 28syl3anc 1187 . . . 4
306, 7, 8, 9, 10, 23, 11, 23, 24cdleme11g 29621 . . . . 5
3117, 3, 19, 5, 21, 30syl131anc 1200 . . . 4
3229, 31eqtrd 2290 . . 3
3316, 32oveq12d 5810 . 2
34 simp21l 1077 . . . . 5
35 cdleme15.c . . . . . 6
366, 7, 8, 9, 10, 35cdleme8 29606 . . . . 5
371, 2, 3, 4, 34, 36syl221anc 1198 . . . 4
3837eqcomd 2263 . . 3
39 cdleme12.f . . . . 5
406, 7, 8, 9, 10, 23, 35, 23, 39cdleme11g 29621 . . . 4
4117, 3, 19, 34, 21, 40syl131anc 1200 . . 3
4238, 41oveq12d 5810 . 2
4333, 42oveq12d 5810 1
 Colors of variables: wff set class Syntax hints:   wn 5   wi 6   wa 360   w3a 939   wceq 1619   wcel 1621   wne 2421   class class class wbr 3997  cfv 4673  (class class class)co 5792  cple 13177  cjn 14040  cmee 14041  catm 28620  chlt 28707  clh 29340 This theorem is referenced by:  cdleme15  29634 This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-reu 2525  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3802  df-iun 3881  df-iin 3882  df-br 3998  df-opab 4052  df-mpt 4053  df-id 4281  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-1st 6056  df-2nd 6057  df-iota 6225  df-undef 6264  df-riota 6272  df-poset 14042  df-plt 14054  df-lub 14070  df-glb 14071  df-join 14072  df-meet 14073  df-p0 14107  df-p1 14108  df-lat 14114  df-clat 14176  df-oposet 28533  df-ol 28535  df-oml 28536  df-covers 28623  df-ats 28624  df-atl 28655  df-cvlat 28679  df-hlat 28708  df-lines 28857  df-psubsp 28859  df-pmap 28860  df-padd 29152  df-lhyp 29344
 Copyright terms: Public domain W3C validator