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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > psubclsetN | Structured version Visualization version Unicode version |
Description: The set of closed projective subspaces in a Hilbert lattice. (Contributed by NM, 23-Jan-2012.) (New usage is discouraged.) |
Ref | Expression |
---|---|
psubclset.a |
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psubclset.p |
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psubclset.c |
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Ref | Expression |
---|---|
psubclsetN |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3066 |
. 2
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2 | psubclset.c |
. . 3
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3 | fveq2 5888 |
. . . . . . . 8
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4 | psubclset.a |
. . . . . . . 8
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5 | 3, 4 | syl6eqr 2514 |
. . . . . . 7
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6 | 5 | sseq2d 3472 |
. . . . . 6
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7 | fveq2 5888 |
. . . . . . . . 9
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8 | psubclset.p |
. . . . . . . . 9
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9 | 7, 8 | syl6eqr 2514 |
. . . . . . . 8
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10 | 9 | fveq1d 5890 |
. . . . . . . 8
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11 | 9, 10 | fveq12d 5894 |
. . . . . . 7
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12 | 11 | eqeq1d 2464 |
. . . . . 6
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13 | 6, 12 | anbi12d 722 |
. . . . 5
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14 | 13 | abbidv 2580 |
. . . 4
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15 | df-psubclN 33545 |
. . . 4
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16 | fvex 5898 |
. . . . . . 7
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17 | 4, 16 | eqeltri 2536 |
. . . . . 6
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18 | 17 | pwex 4600 |
. . . . 5
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19 | selpw 3970 |
. . . . . . . 8
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20 | 19 | anbi1i 706 |
. . . . . . 7
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21 | 20 | abbii 2578 |
. . . . . 6
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22 | ssab2 3525 |
. . . . . 6
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23 | 21, 22 | eqsstr3i 3475 |
. . . . 5
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24 | 18, 23 | ssexi 4562 |
. . . 4
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25 | 14, 15, 24 | fvmpt 5971 |
. . 3
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26 | 2, 25 | syl5eq 2508 |
. 2
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27 | 1, 26 | syl 17 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1680 ax-4 1693 ax-5 1769 ax-6 1816 ax-7 1862 ax-9 1907 ax-10 1926 ax-11 1931 ax-12 1944 ax-13 2102 ax-ext 2442 ax-sep 4539 ax-nul 4548 ax-pow 4595 ax-pr 4653 |
This theorem depends on definitions: df-bi 190 df-or 376 df-an 377 df-3an 993 df-tru 1458 df-ex 1675 df-nf 1679 df-sb 1809 df-eu 2314 df-mo 2315 df-clab 2449 df-cleq 2455 df-clel 2458 df-nfc 2592 df-ne 2635 df-ral 2754 df-rex 2755 df-rab 2758 df-v 3059 df-sbc 3280 df-dif 3419 df-un 3421 df-in 3423 df-ss 3430 df-nul 3744 df-if 3894 df-pw 3965 df-sn 3981 df-pr 3983 df-op 3987 df-uni 4213 df-br 4417 df-opab 4476 df-mpt 4477 df-id 4768 df-xp 4859 df-rel 4860 df-cnv 4861 df-co 4862 df-dm 4863 df-iota 5565 df-fun 5603 df-fv 5609 df-psubclN 33545 |
This theorem is referenced by: ispsubclN 33547 |
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