cvrexch - Mathbox for Norm Megill

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Theorem cvrexch 34706

Description: A Hilbert lattice satisfies the exchange axiom. Proposition 1(iii) of [Kalmbach] p. 140 and its converse. Originally proved by Garrett Birkhoff in 1933. (cvexchi 29228 analog.) (Contributed by NM, 18-Nov-2011.)

**Hypotheses**
Ref	Expression
cvrexch.b
cvrexch.j	$.\/$
cvrexch.m	$./\$
cvrexch.c

**Assertion**
Ref	Expression
cvrexch	$/\$ $/\$ $./\$ $.\/$

**Proof of Theorem cvrexch**
Step	Hyp	Ref	Expression
1		cvrexch.b	. . 3
2		cvrexch.j	. . 3 $.\/$
3		cvrexch.m	. . 3 $./\$
4		cvrexch.c	. . 3
5	1, 2, 3, 4	cvrexchlem 34705	. 2 $/\$ $/\$ $./\$ $.\/$
6		simp1 1061	. . . . 5 $/\$ $/\$
7		hlop 34649	. . . . . . 7
8	7	3ad2ant1 1082	. . . . . 6 $/\$ $/\$
9		simp3 1063	. . . . . 6 $/\$ $/\$
10		eqid 2622	. . . . . . 7
11	1, 10	opoccl 34481	. . . . . 6 $/\$
12	8, 9, 11	syl2anc 693	. . . . 5 $/\$ $/\$
13		simp2 1062	. . . . . 6 $/\$ $/\$
14	1, 10	opoccl 34481	. . . . . 6 $/\$
15	8, 13, 14	syl2anc 693	. . . . 5 $/\$ $/\$
16	1, 2, 3, 4	cvrexchlem 34705	. . . . 5 $/\$ $/\$ $./\$ $.\/$
17	6, 12, 15, 16	syl3anc 1326	. . . 4 $/\$ $/\$ $./\$ $.\/$
18		hlol 34648	. . . . . . 7
19	1, 2, 3, 10	oldmj1 34508	. . . . . . 7 $/\$ $/\$ $.\/$ $./\$
20	18, 19	syl3an1 1359	. . . . . 6 $/\$ $/\$ $.\/$ $./\$
21		hllat 34650	. . . . . . . 8
22	21	3ad2ant1 1082	. . . . . . 7 $/\$ $/\$
23	1, 3	latmcom 17075	. . . . . . 7 $/\$ $/\$ $./\$ $./\$
24	22, 15, 12, 23	syl3anc 1326	. . . . . 6 $/\$ $/\$ $./\$ $./\$
25	20, 24	eqtrd 2656	. . . . 5 $/\$ $/\$ $.\/$ $./\$
26	25	breq1d 4663	. . . 4 $/\$ $/\$ $.\/$ $./\$
27	1, 2, 3, 10	oldmm1 34504	. . . . . . 7 $/\$ $/\$ $./\$ $.\/$
28	18, 27	syl3an1 1359	. . . . . 6 $/\$ $/\$ $./\$ $.\/$
29	1, 2	latjcom 17059	. . . . . . 7 $/\$ $/\$ $.\/$ $.\/$
30	22, 15, 12, 29	syl3anc 1326	. . . . . 6 $/\$ $/\$ $.\/$ $.\/$
31	28, 30	eqtrd 2656	. . . . 5 $/\$ $/\$ $./\$ $.\/$
32	31	breq2d 4665	. . . 4 $/\$ $/\$ $./\$ $.\/$
33	17, 26, 32	3imtr4d 283	. . 3 $/\$ $/\$ $.\/$ $./\$
34	1, 2	latjcl 17051	. . . . 5 $/\$ $/\$ $.\/$
35	21, 34	syl3an1 1359	. . . 4 $/\$ $/\$ $.\/$
36	1, 10, 4	cvrcon3b 34564	. . . 4 $/\$ $/\$ $.\/$ $.\/$ $.\/$
37	8, 13, 35, 36	syl3anc 1326	. . 3 $/\$ $/\$ $.\/$ $.\/$
38	1, 3	latmcl 17052	. . . . 5 $/\$ $/\$ $./\$
39	21, 38	syl3an1 1359	. . . 4 $/\$ $/\$ $./\$
40	1, 10, 4	cvrcon3b 34564	. . . 4 $/\$ $./\$ $/\$ $./\$ $./\$
41	8, 39, 9, 40	syl3anc 1326	. . 3 $/\$ $/\$ $./\$ $./\$
42	33, 37, 41	3imtr4d 283	. 2 $/\$ $/\$ $.\/$ $./\$
43	5, 42	impbid 202	1 $/\$ $/\$ $./\$ $.\/$

Colors of variables: wff setvar class

Syntax hints:

wi 4

wb 196 $/\$ w3a 1037

wceq 1483

wcel 1990 class class class wbr 4653

cfv 5888 (class class class)co 6650

cbs 15857

coc 15949

cjn 16944

cmee 16945

clat 17045

cops 34459

col 34461

ccvr 34549

chlt 34637

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-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-preset 16928 df-poset 16946 df-plt 16958 df-lub 16974 df-glb 16975 df-join 16976 df-meet 16977 df-p0 17039 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

This theorem is referenced by: cvrat3 34728 2lplnmN 34845 2llnmj 34846 2llnm2N 34854 2lplnm2N 34907 2lplnmj 34908 lhpmcvr 35309

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