Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ishlat1 | Structured version Visualization version Unicode version |
Description: The predicate "is a Hilbert lattice," which is orthomodular ( ), complete ( ), atomic and satisfying the exchange (or covering) property ( ), satisfies the superposition principle, and has a minimum height of 4. (Contributed by NM, 5-Nov-2012.) |
Ref | Expression |
---|---|
ishlat.b | |
ishlat.l | |
ishlat.s | |
ishlat.j | |
ishlat.z | |
ishlat.u | |
ishlat.a |
Ref | Expression |
---|---|
ishlat1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6191 | . . . . . 6 | |
2 | ishlat.a | . . . . . 6 | |
3 | 1, 2 | syl6eqr 2674 | . . . . 5 |
4 | fveq2 6191 | . . . . . . . . . . . 12 | |
5 | ishlat.l | . . . . . . . . . . . 12 | |
6 | 4, 5 | syl6eqr 2674 | . . . . . . . . . . 11 |
7 | 6 | breqd 4664 | . . . . . . . . . 10 |
8 | fveq2 6191 | . . . . . . . . . . . . 13 | |
9 | ishlat.j | . . . . . . . . . . . . 13 | |
10 | 8, 9 | syl6eqr 2674 | . . . . . . . . . . . 12 |
11 | 10 | oveqd 6667 | . . . . . . . . . . 11 |
12 | 11 | breq2d 4665 | . . . . . . . . . 10 |
13 | 7, 12 | bitrd 268 | . . . . . . . . 9 |
14 | 13 | 3anbi3d 1405 | . . . . . . . 8 |
15 | 3, 14 | rexeqbidv 3153 | . . . . . . 7 |
16 | 15 | imbi2d 330 | . . . . . 6 |
17 | 3, 16 | raleqbidv 3152 | . . . . 5 |
18 | 3, 17 | raleqbidv 3152 | . . . 4 |
19 | fveq2 6191 | . . . . . 6 | |
20 | ishlat.b | . . . . . 6 | |
21 | 19, 20 | syl6eqr 2674 | . . . . 5 |
22 | fveq2 6191 | . . . . . . . . . . . 12 | |
23 | ishlat.s | . . . . . . . . . . . 12 | |
24 | 22, 23 | syl6eqr 2674 | . . . . . . . . . . 11 |
25 | 24 | breqd 4664 | . . . . . . . . . 10 |
26 | fveq2 6191 | . . . . . . . . . . . 12 | |
27 | ishlat.z | . . . . . . . . . . . 12 | |
28 | 26, 27 | syl6eqr 2674 | . . . . . . . . . . 11 |
29 | 28 | breq1d 4663 | . . . . . . . . . 10 |
30 | 25, 29 | bitrd 268 | . . . . . . . . 9 |
31 | 24 | breqd 4664 | . . . . . . . . 9 |
32 | 30, 31 | anbi12d 747 | . . . . . . . 8 |
33 | 24 | breqd 4664 | . . . . . . . . 9 |
34 | 24 | breqd 4664 | . . . . . . . . . 10 |
35 | fveq2 6191 | . . . . . . . . . . . 12 | |
36 | ishlat.u | . . . . . . . . . . . 12 | |
37 | 35, 36 | syl6eqr 2674 | . . . . . . . . . . 11 |
38 | 37 | breq2d 4665 | . . . . . . . . . 10 |
39 | 34, 38 | bitrd 268 | . . . . . . . . 9 |
40 | 33, 39 | anbi12d 747 | . . . . . . . 8 |
41 | 32, 40 | anbi12d 747 | . . . . . . 7 |
42 | 21, 41 | rexeqbidv 3153 | . . . . . 6 |
43 | 21, 42 | rexeqbidv 3153 | . . . . 5 |
44 | 21, 43 | rexeqbidv 3153 | . . . 4 |
45 | 18, 44 | anbi12d 747 | . . 3 |
46 | df-hlat 34638 | . . 3 | |
47 | 45, 46 | elrab2 3366 | . 2 |
48 | elin 3796 | . . . . 5 | |
49 | 48 | anbi1i 731 | . . . 4 |
50 | elin 3796 | . . . 4 | |
51 | df-3an 1039 | . . . 4 | |
52 | 49, 50, 51 | 3bitr4ri 293 | . . 3 |
53 | 52 | anbi1i 731 | . 2 |
54 | 47, 53 | bitr4i 267 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 w3a 1037 wceq 1483 wcel 1990 wne 2794 wral 2912 wrex 2913 cin 3573 class class class wbr 4653 cfv 5888 (class class class)co 6650 cbs 15857 cple 15948 cplt 16941 cjn 16944 cp0 17037 cp1 17038 ccla 17107 coml 34462 catm 34550 clc 34552 chlt 34637 |
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-hlat 34638 |
This theorem is referenced by: ishlat2 34640 ishlat3N 34641 hlomcmcv 34643 |
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