omllaw - Mathbox for Norm Megill

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Theorem omllaw 34530

Description: The orthomodular law. (Contributed by NM, 18-Sep-2011.)

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

Proof of Theorem omllaw Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		omllaw.b	. . . . 5
2		omllaw.l	. . . . 5
3		omllaw.j	. . . . 5 $.\/$
4		omllaw.m	. . . . 5 $./\$
5		omllaw.o	. . . . 5
6	1, 2, 3, 4, 5	isoml 34525	. . . 4 $/\$ $.\/$ $./\$
7	6	simprbi 480	. . 3 $.\/$ $./\$
8		breq1 4656	. . . . 5
9		id 22	. . . . . . 7
10		fveq2 6191	. . . . . . . 8
11	10	oveq2d 6666	. . . . . . 7 $./\$ $./\$
12	9, 11	oveq12d 6668	. . . . . 6 $.\/$ $./\$ $.\/$ $./\$
13	12	eqeq2d 2632	. . . . 5 $.\/$ $./\$ $.\/$ $./\$
14	8, 13	imbi12d 334	. . . 4 $.\/$ $./\$ $.\/$ $./\$
15		breq2 4657	. . . . 5
16		id 22	. . . . . 6
17		oveq1 6657	. . . . . . 7 $./\$ $./\$
18	17	oveq2d 6666	. . . . . 6 $.\/$ $./\$ $.\/$ $./\$
19	16, 18	eqeq12d 2637	. . . . 5 $.\/$ $./\$ $.\/$ $./\$
20	15, 19	imbi12d 334	. . . 4 $.\/$ $./\$ $.\/$ $./\$
21	14, 20	rspc2v 3322	. . 3 $/\$ $.\/$ $./\$ $.\/$ $./\$
22	7, 21	syl5com 31	. 2 $/\$ $.\/$ $./\$
23	22	3impib 1262	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

cfv 5888 (class class class)co 6650

cbs 15857

cple 15948

coc 15949

cjn 16944

cmee 16945

col 34461

coml 34462

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-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-oml 34466

This theorem is referenced by: omllaw2N 34531 omllaw3 34532 omllaw4 34533