oposlem - Mathbox for Norm Megill

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Theorem oposlem 34469

Description: Lemma for orthoposet properties. (Contributed by NM, 20-Oct-2011.)

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

Proof of Theorem oposlem Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oposlem.b	. . . . 5
2		eqid 2622	. . . . 5
3		eqid 2622	. . . . 5
4		oposlem.l	. . . . 5
5		oposlem.o	. . . . 5
6		oposlem.j	. . . . 5 $.\/$
7		oposlem.m	. . . . 5 $./\$
8		oposlem.f	. . . . 5
9		oposlem.u	. . . . 5
10	1, 2, 3, 4, 5, 6, 7, 8, 9	isopos 34467	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $./\$
11	10	simprbi 480	. . 3 $/\$ $/\$ $/\$ $.\/$ $/\$ $./\$
12		fveq2 6191	. . . . . . 7
13	12	eleq1d 2686	. . . . . 6
14	12	fveq2d 6195	. . . . . . 7
15		id 22	. . . . . . 7
16	14, 15	eqeq12d 2637	. . . . . 6
17		breq1 4656	. . . . . . 7
18	12	breq2d 4665	. . . . . . 7
19	17, 18	imbi12d 334	. . . . . 6
20	13, 16, 19	3anbi123d 1399	. . . . 5 $/\$ $/\$ $/\$ $/\$
21	15, 12	oveq12d 6668	. . . . . 6 $.\/$ $.\/$
22	21	eqeq1d 2624	. . . . 5 $.\/$ $.\/$
23	15, 12	oveq12d 6668	. . . . . 6 $./\$ $./\$
24	23	eqeq1d 2624	. . . . 5 $./\$ $./\$
25	20, 22, 24	3anbi123d 1399	. . . 4 $/\$ $/\$ $/\$ $.\/$ $/\$ $./\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $./\$
26		breq2 4657	. . . . . . 7
27		fveq2 6191	. . . . . . . 8
28	27	breq1d 4663	. . . . . . 7
29	26, 28	imbi12d 334	. . . . . 6
30	29	3anbi3d 1405	. . . . 5 $/\$ $/\$ $/\$ $/\$
31	30	3anbi1d 1403	. . . 4 $/\$ $/\$ $/\$ $.\/$ $/\$ $./\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $./\$
32	25, 31	rspc2v 3322	. . 3 $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $./\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $./\$
33	11, 32	mpan9 486	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $./\$
34	33	3impb 1260	1 $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $./\$

Colors of variables: wff setvar class

Syntax hints:

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

wceq 1483

wcel 1990

wral 2912 class class class wbr 4653

cdm 5114

cfv 5888 (class class class)co 6650

cbs 15857

cple 15948

coc 15949

cpo 16940

club 16942

cglb 16943

cjn 16944

cmee 16945

cp0 17037

cp1 17038

cops 34459

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 ax-nul 4789

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-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 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-dm 5124 df-iota 5851 df-fv 5896 df-ov 6653 df-oposet 34463

This theorem is referenced by: opoccl 34481 opococ 34482 oplecon3 34486 opexmid 34494 opnoncon 34495