cmtvalN - Mathbox for Norm Megill

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Theorem cmtvalN 34498

Description: Equivalence for commutes relation. Definition of commutes in [Kalmbach] p. 20. (cmbr 28443 analog.) (Contributed by NM, 6-Nov-2011.) (New usage is discouraged.)

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

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

Proof of Theorem cmtvalN Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cmtfval.b	. . . . . 6
2		cmtfval.j	. . . . . 6 $.\/$
3		cmtfval.m	. . . . . 6 $./\$
4		cmtfval.o	. . . . . 6
5		cmtfval.c	. . . . . 6
6	1, 2, 3, 4, 5	cmtfvalN 34497	. . . . 5 $/\$ $/\$ $./\$ $.\/$ $./\$
7		df-3an 1039	. . . . . 6 $/\$ $/\$ $./\$ $.\/$ $./\$ $/\$ $/\$ $./\$ $.\/$ $./\$
8	7	opabbii 4717	. . . . 5 $/\$ $/\$ $./\$ $.\/$ $./\$ $/\$ $/\$ $./\$ $.\/$ $./\$
9	6, 8	syl6eq 2672	. . . 4 $/\$ $/\$ $./\$ $.\/$ $./\$
10	9	breqd 4664	. . 3 $/\$ $/\$ $./\$ $.\/$ $./\$
11	10	3ad2ant1 1082	. 2 $/\$ $/\$ $/\$ $/\$ $./\$ $.\/$ $./\$
12		df-br 4654	. . . 4 $/\$ $/\$ $./\$ $.\/$ $./\$ $/\$ $/\$ $./\$ $.\/$ $./\$
13		id 22	. . . . . 6
14		oveq1 6657	. . . . . . 7 $./\$ $./\$
15		oveq1 6657	. . . . . . 7 $./\$ $./\$
16	14, 15	oveq12d 6668	. . . . . 6 $./\$ $.\/$ $./\$ $./\$ $.\/$ $./\$
17	13, 16	eqeq12d 2637	. . . . 5 $./\$ $.\/$ $./\$ $./\$ $.\/$ $./\$
18		oveq2 6658	. . . . . . 7 $./\$ $./\$
19		fveq2 6191	. . . . . . . 8
20	19	oveq2d 6666	. . . . . . 7 $./\$ $./\$
21	18, 20	oveq12d 6668	. . . . . 6 $./\$ $.\/$ $./\$ $./\$ $.\/$ $./\$
22	21	eqeq2d 2632	. . . . 5 $./\$ $.\/$ $./\$ $./\$ $.\/$ $./\$
23	17, 22	opelopab2 4996	. . . 4 $/\$ $/\$ $/\$ $./\$ $.\/$ $./\$ $./\$ $.\/$ $./\$
24	12, 23	syl5bb 272	. . 3 $/\$ $/\$ $/\$ $./\$ $.\/$ $./\$ $./\$ $.\/$ $./\$
25	24	3adant1 1079	. 2 $/\$ $/\$ $/\$ $/\$ $./\$ $.\/$ $./\$ $./\$ $.\/$ $./\$
26	11, 25	bitrd 268	1 $/\$ $/\$ $./\$ $.\/$ $./\$

Colors of variables: wff setvar class

Syntax hints:

wi 4

wb 196 $/\$ wa 384 $/\$ w3a 1037

wceq 1483

wcel 1990

cop 4183 class class class wbr 4653

copab 4712

cfv 5888 (class class class)co 6650

cbs 15857

coc 15949

cjn 16944

cmee 16945

ccmtN 34460

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-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-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-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 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-iota 5851 df-fun 5890 df-fv 5896 df-ov 6653 df-cmtN 34464

This theorem is referenced by: cmtcomlemN 34535 cmt2N 34537 cmtbr2N 34540 cmtbr3N 34541

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