hlmod1i - Mathbox for Norm Megill

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Theorem hlmod1i 35142

Description: A version of the modular law pmod1i 35134 that holds in a Hilbert lattice. (Contributed by NM, 13-May-2012.)

**Hypotheses**
Ref	Expression
hlmod.b
hlmod.l
hlmod.j	$.\/$
hlmod.m	$./\$
hlmod.f
hlmod.p

**Assertion**
Ref	Expression
hlmod1i	$/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$

**Proof of Theorem hlmod1i**
Step	Hyp	Ref	Expression
1		hlmod.b	. . 3
2		hlmod.l	. . 3
3		hllat 34650	. . . 4
4	3	3ad2ant1 1082	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$
5		simp21 1094	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$
6		simp22 1095	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$
7		hlmod.j	. . . . . 6 $.\/$
8	1, 7	latjcl 17051	. . . . 5 $/\$ $/\$ $.\/$
9	4, 5, 6, 8	syl3anc 1326	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$
10		simp23 1096	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$
11		hlmod.m	. . . . 5 $./\$
12	1, 11	latmcl 17052	. . . 4 $/\$ $.\/$ $/\$ $.\/$ $./\$
13	4, 9, 10, 12	syl3anc 1326	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $./\$
14	1, 11	latmcl 17052	. . . . 5 $/\$ $/\$ $./\$
15	4, 6, 10, 14	syl3anc 1326	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$
16	1, 7	latjcl 17051	. . . 4 $/\$ $/\$ $./\$ $.\/$ $./\$
17	4, 5, 15, 16	syl3anc 1326	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $./\$
18		simp1 1061	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$
19		eqid 2622	. . . . . . . . 9
20		hlmod.f	. . . . . . . . 9
21	1, 19, 20	pmapssat 35045	. . . . . . . 8 $/\$
22	18, 5, 21	syl2anc 693	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$
23	1, 19, 20	pmapssat 35045	. . . . . . . 8 $/\$
24	18, 6, 23	syl2anc 693	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$
25		eqid 2622	. . . . . . . . 9
26	1, 25, 20	pmapsub 35054	. . . . . . . 8 $/\$
27	4, 10, 26	syl2anc 693	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$
28		simp3l 1089	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$
29	1, 2, 20	pmaple 35047	. . . . . . . . 9 $/\$ $/\$
30	18, 5, 10, 29	syl3anc 1326	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$
31	28, 30	mpbid 222	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$
32		hlmod.p	. . . . . . . . 9
33	19, 25, 32	pmod1i 35134	. . . . . . . 8 $/\$ $/\$ $/\$
34	33	3impia 1261	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
35	18, 22, 24, 27, 31, 34	syl131anc 1339	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$
36	1, 11, 19, 20	pmapmeet 35059	. . . . . . . 8 $/\$ $.\/$ $/\$ $.\/$ $./\$ $.\/$
37	18, 9, 10, 36	syl3anc 1326	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $./\$ $.\/$
38		simp3r 1090	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$
39	38	ineq1d 3813	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$
40	37, 39	eqtrd 2656	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $./\$
41	1, 11, 19, 20	pmapmeet 35059	. . . . . . . 8 $/\$ $/\$ $./\$
42	18, 6, 10, 41	syl3anc 1326	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$
43	42	oveq2d 6666	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$
44	35, 40, 43	3eqtr4d 2666	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $./\$ $./\$
45	1, 7, 20, 32	pmapjoin 35138	. . . . . 6 $/\$ $/\$ $./\$ $./\$ $.\/$ $./\$
46	4, 5, 15, 45	syl3anc 1326	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $./\$
47	44, 46	eqsstrd 3639	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$
48	1, 2, 20	pmaple 35047	. . . . 5 $/\$ $.\/$ $./\$ $/\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$
49	18, 13, 17, 48	syl3anc 1326	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$
50	47, 49	mpbird 247	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$
51	1, 2, 7, 11	mod1ile 17105	. . . . 5 $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $./\$
52	51	3impia 1261	. . . 4 $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $./\$
53	4, 5, 6, 10, 28, 52	syl131anc 1339	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$
54	1, 2, 4, 13, 17, 50, 53	latasymd 17057	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$
55	54	3expia 1267	1 $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $./\$ $.\/$ $./\$

Colors of variables: wff setvar class

Syntax hints:

wi 4

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

wceq 1483

wcel 1990

cin 3573

wss 3574 class class class wbr 4653

cfv 5888 (class class class)co 6650

cbs 15857

cple 15948

cjn 16944

cmee 16945

clat 17045

catm 34550

chlt 34637

cpsubsp 34782

cpmap 34783

cpadd 35081

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-rep 4771 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-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-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-iin 4523 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-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-preset 16928 df-poset 16946 df-plt 16958 df-lub 16974 df-glb 16975 df-join 16976 df-meet 16977 df-p0 17039 df-lat 17046 df-clat 17108 df-oposet 34463 df-ol 34465 df-oml 34466 df-covers 34553 df-ats 34554 df-atl 34585 df-cvlat 34609 df-hlat 34638 df-psubsp 34789 df-pmap 34790 df-padd 35082

This theorem is referenced by: atmod1i1 35143 atmod1i2 35145 llnmod1i2 35146

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