cdleme15c - Mathbox for Norm Megill

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Theorem cdleme15c 30758

Description: Part of proof of Lemma E in [Crawley] p. 113, 3rd paragraph on p. 114, showing, in their notation, ((p $\/$ s₁) $/\$ (q $\/$ s₁)) $\/$ ((p $\/$ t₁) $/\$ (q $\/$ t₁))=s₁ $\/$ t₁.

and

represent s₁ and t₁ respectively. The order of our operations is slightly different. (Contributed by NM, 10-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
cdleme15c	$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $./\$ $.\/$ $.\/$ $.\/$ $./\$ $.\/$ $.\/$

**Proof of Theorem cdleme15c**
Step	Hyp	Ref	Expression
1		simp11 987	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $/\$
2		simp12 988	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $/\$
3		simp13 989	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $/\$
4		simp22 991	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $/\$
5		simp21 990	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $/\$
6		simp23l 1078	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$
7		simp23r 1079	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$
8	7	necomd 2650	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$
9	6, 8	jca 519	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $/\$
10		simp32 994	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $.\/$
11		simp31 993	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $.\/$
12		simp33 995	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $.\/$
13		simp11l 1068	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$
14		simp21l 1074	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$
15		simp22l 1076	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$
16		cdleme12.j	. . . . . . 7 $.\/$
17		cdleme12.a	. . . . . . 7
18	16, 17	hlatjcom 29850	. . . . . 6 $/\$ $/\$ $.\/$ $.\/$
19	13, 14, 15, 18	syl3anc 1184	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $.\/$
20	19	breq2d 4184	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $.\/$
21	12, 20	mtbid 292	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $.\/$
22		cdleme12.l	. . . 4
23		cdleme12.m	. . . 4 $./\$
24		cdleme12.h	. . . 4
25		cdleme12.u	. . . 4 $.\/$ $./\$
26		cdleme12.g	. . . 4 $.\/$ $./\$ $.\/$ $.\/$ $./\$
27		cdleme12.f	. . . 4 $.\/$ $./\$ $.\/$ $.\/$ $./\$
28		cdleme15.x	. . . 4 $.\/$ $./\$
29		cdleme15.c	. . . 4 $.\/$ $./\$
30	22, 16, 23, 17, 24, 25, 26, 27, 28, 29	cdleme15b 30757	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $./\$ $.\/$
31	1, 2, 3, 4, 5, 9, 10, 11, 21, 30	syl333anc 1216	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $./\$ $.\/$
32	22, 16, 23, 17, 24, 25, 27, 26, 29, 28	cdleme15b 30757	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $./\$ $.\/$
33	31, 32	oveq12d 6058	1 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $./\$ $.\/$ $.\/$ $.\/$ $./\$ $.\/$ $.\/$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 359 $/\$ w3a 936

wceq 1649

wcel 1721

wne 2567 class class class wbr 4172

cfv 5413 (class class class)co 6040

cple 13491

cjn 14356

cmee 14357

catm 29746

chlt 29833

clh 30466

This theorem is referenced by: cdleme15 30760

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1552 ax-5 1563 ax-17 1623 ax-9 1662 ax-8 1683 ax-13 1723 ax-14 1725 ax-6 1740 ax-7 1745 ax-11 1757 ax-12 1946 ax-ext 2385 ax-rep 4280 ax-sep 4290 ax-nul 4298 ax-pow 4337 ax-pr 4363 ax-un 4660

This theorem depends on definitions: df-bi 178 df-or 360 df-an 361 df-3an 938 df-tru 1325 df-ex 1548 df-nf 1551 df-sb 1656 df-eu 2258 df-mo 2259 df-clab 2391 df-cleq 2397 df-clel 2400 df-nfc 2529 df-ne 2569 df-nel 2570 df-ral 2671 df-rex 2672 df-reu 2673 df-rab 2675 df-v 2918 df-sbc 3122 df-csb 3212 df-dif 3283 df-un 3285 df-in 3287 df-ss 3294 df-nul 3589 df-if 3700 df-pw 3761 df-sn 3780 df-pr 3781 df-op 3783 df-uni 3976 df-iun 4055 df-iin 4056 df-br 4173 df-opab 4227 df-mpt 4228 df-id 4458 df-xp 4843 df-rel 4844 df-cnv 4845 df-co 4846 df-dm 4847 df-rn 4848 df-res 4849 df-ima 4850 df-iota 5377 df-fun 5415 df-fn 5416 df-f 5417 df-f1 5418 df-fo 5419 df-f1o 5420 df-fv 5421 df-ov 6043 df-oprab 6044 df-mpt2 6045 df-1st 6308 df-2nd 6309 df-undef 6502 df-riota 6508 df-poset 14358 df-plt 14370 df-lub 14386 df-glb 14387 df-join 14388 df-meet 14389 df-p0 14423 df-lat 14430 df-clat 14492 df-oposet 29659 df-ol 29661 df-oml 29662 df-covers 29749 df-ats 29750 df-atl 29781 df-cvlat 29805 df-hlat 29834 df-psubsp 29985 df-pmap 29986 df-padd 30278 df-lhyp 30470

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