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Mirrors > Home > MPE Home > Th. List > Mathboxes > lvolset | Structured version Visualization version Unicode version |
Description: The set of 3-dim lattice volumes in a Hilbert lattice. (Contributed by NM, 1-Jul-2012.) |
Ref | Expression |
---|---|
lvolset.b | |
lvolset.c | |
lvolset.p | |
lvolset.v |
Ref | Expression |
---|---|
lvolset |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3212 | . 2 | |
2 | lvolset.v | . . 3 | |
3 | fveq2 6191 | . . . . . 6 | |
4 | lvolset.b | . . . . . 6 | |
5 | 3, 4 | syl6eqr 2674 | . . . . 5 |
6 | fveq2 6191 | . . . . . . 7 | |
7 | lvolset.p | . . . . . . 7 | |
8 | 6, 7 | syl6eqr 2674 | . . . . . 6 |
9 | fveq2 6191 | . . . . . . . 8 | |
10 | lvolset.c | . . . . . . . 8 | |
11 | 9, 10 | syl6eqr 2674 | . . . . . . 7 |
12 | 11 | breqd 4664 | . . . . . 6 |
13 | 8, 12 | rexeqbidv 3153 | . . . . 5 |
14 | 5, 13 | rabeqbidv 3195 | . . . 4 |
15 | df-lvols 34786 | . . . 4 | |
16 | fvex 6201 | . . . . . 6 | |
17 | 4, 16 | eqeltri 2697 | . . . . 5 |
18 | 17 | rabex 4813 | . . . 4 |
19 | 14, 15, 18 | fvmpt 6282 | . . 3 |
20 | 2, 19 | syl5eq 2668 | . 2 |
21 | 1, 20 | syl 17 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wceq 1483 wcel 1990 wrex 2913 crab 2916 cvv 3200 class class class wbr 4653 cfv 5888 cbs 15857 ccvr 34549 clpl 34778 clvol 34779 |
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-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-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-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-lvols 34786 |
This theorem is referenced by: islvol 34859 |
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