dihfval - Mathbox for Norm Megill

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Theorem dihfval 36520

Description: Isomorphism H for a lattice

. Definition of isomorphism map in [Crawley] p. 122 line 3. (Contributed by NM, 28-Jan-2014.)

**Hypotheses**
Ref	Expression
dihval.b
dihval.l
dihval.j	$.\/$
dihval.m	$./\$
dihval.a
dihval.h
dihval.i
dihval.d
dihval.c
dihval.u
dihval.s
dihval.p

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

Distinct variable groups:

Allowed substitution hints:

(

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(

) $.\/$ (

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) $./\$ (

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Proof of Theorem dihfval Dummy variable

is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dihval.i	. . 3
2		dihval.b	. . . . 5
3		dihval.l	. . . . 5
4		dihval.j	. . . . 5 $.\/$
5		dihval.m	. . . . 5 $./\$
6		dihval.a	. . . . 5
7		dihval.h	. . . . 5
8	2, 3, 4, 5, 6, 7	dihffval 36519	. . . 4 $/\$ $.\/$ $./\$ $./\$
9	8	fveq1d 6193	. . 3 $/\$ $.\/$ $./\$ $./\$
10	1, 9	syl5eq 2668	. 2 $/\$ $.\/$ $./\$ $./\$
11		breq2 4657	. . . . 5
12		fveq2 6191	. . . . . . 7
13		dihval.d	. . . . . . 7
14	12, 13	syl6eqr 2674	. . . . . 6
15	14	fveq1d 6193	. . . . 5
16		fveq2 6191	. . . . . . . . 9
17		dihval.u	. . . . . . . . 9
18	16, 17	syl6eqr 2674	. . . . . . . 8
19	18	fveq2d 6195	. . . . . . 7
20		dihval.s	. . . . . . 7
21	19, 20	syl6eqr 2674	. . . . . 6
22		breq2 4657	. . . . . . . . . 10
23	22	notbid 308	. . . . . . . . 9
24		oveq2 6658	. . . . . . . . . . 11 $./\$ $./\$
25	24	oveq2d 6666	. . . . . . . . . 10 $.\/$ $./\$ $.\/$ $./\$
26	25	eqeq1d 2624	. . . . . . . . 9 $.\/$ $./\$ $.\/$ $./\$
27	23, 26	anbi12d 747	. . . . . . . 8 $/\$ $.\/$ $./\$ $/\$ $.\/$ $./\$
28	18	fveq2d 6195	. . . . . . . . . . 11
29		dihval.p	. . . . . . . . . . 11
30	28, 29	syl6eqr 2674	. . . . . . . . . 10
31		fveq2 6191	. . . . . . . . . . . 12
32		dihval.c	. . . . . . . . . . . 12
33	31, 32	syl6eqr 2674	. . . . . . . . . . 11
34	33	fveq1d 6193	. . . . . . . . . 10
35	14, 24	fveq12d 6197	. . . . . . . . . 10 $./\$ $./\$
36	30, 34, 35	oveq123d 6671	. . . . . . . . 9 $./\$ $./\$
37	36	eqeq2d 2632	. . . . . . . 8 $./\$ $./\$
38	27, 37	imbi12d 334	. . . . . . 7 $/\$ $.\/$ $./\$ $./\$ $/\$ $.\/$ $./\$ $./\$
39	38	ralbidv 2986	. . . . . 6 $/\$ $.\/$ $./\$ $./\$ $/\$ $.\/$ $./\$ $./\$
40	21, 39	riotaeqbidv 6614	. . . . 5 $/\$ $.\/$ $./\$ $./\$ $/\$ $.\/$ $./\$ $./\$
41	11, 15, 40	ifbieq12d 4113	. . . 4 $/\$ $.\/$ $./\$ $./\$ $/\$ $.\/$ $./\$ $./\$
42	41	mpteq2dv 4745	. . 3 $/\$ $.\/$ $./\$ $./\$ $/\$ $.\/$ $./\$ $./\$
43		eqid 2622	. . 3 $/\$ $.\/$ $./\$ $./\$ $/\$ $.\/$ $./\$ $./\$
44		fvex 6201	. . . . 5
45	2, 44	eqeltri 2697	. . . 4
46	45	mptex 6486	. . 3 $/\$ $.\/$ $./\$ $./\$
47	42, 43, 46	fvmpt 6282	. 2 $/\$ $.\/$ $./\$ $./\$ $/\$ $.\/$ $./\$ $./\$
48	10, 47	sylan9eq 2676	1 $/\$ $/\$ $.\/$ $./\$ $./\$

Colors of variables: wff setvar class

Syntax hints:

wn 3

wi 4 $/\$ wa 384

wceq 1483

wcel 1990

wral 2912

cvv 3200

cif 4086 class class class wbr 4653

cmpt 4729

cfv 5888

crio 6610 (class class class)co 6650

cbs 15857

cple 15948

cjn 16944

cmee 16945

clsm 18049

clss 18932

catm 34550

clh 35270

cdvh 36367

cdib 36427

cdic 36461

cdih 36517

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-rep 4771 ax-sep 4781 ax-nul 4789 ax-pr 4906

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-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-dih 36518

This theorem is referenced by: dihval 36521 dihf11lem 36555

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