hlsuprexch - Mathbox for Norm Megill

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Theorem hlsuprexch 34667

Description: A Hilbert lattice has the superposition and exchange properties. (Contributed by NM, 13-Nov-2011.)

**Hypotheses**
Ref	Expression
hlsuprexch.b
hlsuprexch.l
hlsuprexch.j	$.\/$
hlsuprexch.a

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

Distinct variable groups:

Allowed substitution hints: $.\/$ (

)

(

)

Proof of Theorem hlsuprexch Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		hlsuprexch.b	. . . . 5
2		hlsuprexch.l	. . . . 5
3		eqid 2622	. . . . 5
4		hlsuprexch.j	. . . . 5 $.\/$
5		eqid 2622	. . . . 5
6		eqid 2622	. . . . 5
7		hlsuprexch.a	. . . . 5
8	1, 2, 3, 4, 5, 6, 7	ishlat2 34640	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $/\$ $/\$
9		simprl 794	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$
10	8, 9	sylbi 207	. . 3 $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$
11		neeq1 2856	. . . . . 6
12		neeq2 2857	. . . . . . . 8
13		oveq1 6657	. . . . . . . . 9 $.\/$ $.\/$
14	13	breq2d 4665	. . . . . . . 8 $.\/$ $.\/$
15	12, 14	3anbi13d 1401	. . . . . . 7 $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$
16	15	rexbidv 3052	. . . . . 6 $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$
17	11, 16	imbi12d 334	. . . . 5 $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$
18		breq1 4656	. . . . . . . . 9
19	18	notbid 308	. . . . . . . 8
20		breq1 4656	. . . . . . . 8 $.\/$ $.\/$
21	19, 20	anbi12d 747	. . . . . . 7 $/\$ $.\/$ $/\$ $.\/$
22		oveq2 6658	. . . . . . . 8 $.\/$ $.\/$
23	22	breq2d 4665	. . . . . . 7 $.\/$ $.\/$
24	21, 23	imbi12d 334	. . . . . 6 $/\$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$
25	24	ralbidv 2986	. . . . 5 $/\$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$
26	17, 25	anbi12d 747	. . . 4 $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$
27		neeq2 2857	. . . . . 6
28		neeq2 2857	. . . . . . . 8
29		oveq2 6658	. . . . . . . . 9 $.\/$ $.\/$
30	29	breq2d 4665	. . . . . . . 8 $.\/$ $.\/$
31	28, 30	3anbi23d 1402	. . . . . . 7 $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$
32	31	rexbidv 3052	. . . . . 6 $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$
33	27, 32	imbi12d 334	. . . . 5 $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$
34		oveq2 6658	. . . . . . . . 9 $.\/$ $.\/$
35	34	breq2d 4665	. . . . . . . 8 $.\/$ $.\/$
36	35	anbi2d 740	. . . . . . 7 $/\$ $.\/$ $/\$ $.\/$
37		breq1 4656	. . . . . . 7 $.\/$ $.\/$
38	36, 37	imbi12d 334	. . . . . 6 $/\$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$
39	38	ralbidv 2986	. . . . 5 $/\$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$
40	33, 39	anbi12d 747	. . . 4 $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$
41	26, 40	rspc2v 3322	. . 3 $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$
42	10, 41	mpan9 486	. 2 $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$
43	42	3impb 1260	1 $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$

Colors of variables: wff setvar class

Syntax hints:

wn 3

wi 4 $/\$ wa 384 $/\$ w3a 1037

wceq 1483

wcel 1990

wne 2794

wral 2912

wrex 2913 class class class wbr 4653

cfv 5888 (class class class)co 6650

cbs 15857

cple 15948

cplt 16941

cjn 16944

cp0 17037

cp1 17038

ccla 17107

coml 34462

catm 34550

cal 34551

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-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602

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-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-iota 5851 df-fv 5896 df-ov 6653 df-cvlat 34609 df-hlat 34638

This theorem is referenced by: hlsupr 34672

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