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Mirrors > Home > MPE Home > Th. List > pmtrrn2 | Structured version Visualization version Unicode version |
Description: For any transposition there are two points it is transposing. (Contributed by SO, 15-Jul-2018.) |
Ref | Expression |
---|---|
pmtrrn.t |
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pmtrrn.r |
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Ref | Expression |
---|---|
pmtrrn2 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pmtrrn.t |
. . . . . . 7
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2 | pmtrrn.r |
. . . . . . 7
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3 | eqid 2471 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
4 | 1, 2, 3 | pmtrfrn 17177 |
. . . . . 6
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5 | 4 | simpld 466 |
. . . . 5
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6 | 5 | simp3d 1044 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
7 | en2 7825 |
. . . 4
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8 | 6, 7 | syl 17 |
. . 3
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9 | 5 | simp2d 1043 |
. . . . . . 7
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10 | 4 | simprd 470 |
. . . . . . 7
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11 | 9, 6, 10 | jca32 544 |
. . . . . 6
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12 | sseq1 3439 |
. . . . . . 7
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13 | breq1 4398 |
. . . . . . . 8
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14 | fveq2 5879 |
. . . . . . . . 9
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15 | 14 | eqeq2d 2481 |
. . . . . . . 8
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16 | 13, 15 | anbi12d 725 |
. . . . . . 7
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17 | 12, 16 | anbi12d 725 |
. . . . . 6
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18 | 11, 17 | syl5ibcom 228 |
. . . . 5
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19 | vex 3034 |
. . . . . . . 8
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20 | vex 3034 |
. . . . . . . 8
![]() ![]() ![]() ![]() | |
21 | 19, 20 | prss 4117 |
. . . . . . 7
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22 | 21 | bicomi 207 |
. . . . . 6
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23 | pr2ne 8454 |
. . . . . . . 8
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24 | 19, 20, 23 | mp2an 686 |
. . . . . . 7
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25 | 24 | anbi1i 709 |
. . . . . 6
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26 | 22, 25 | anbi12i 711 |
. . . . 5
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27 | 18, 26 | syl6ib 234 |
. . . 4
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28 | 27 | 2eximdv 1774 |
. . 3
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29 | 8, 28 | mpd 15 |
. 2
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30 | r2ex 2901 |
. 2
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31 | 29, 30 | sylibr 217 |
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 1677 ax-4 1690 ax-5 1766 ax-6 1813 ax-7 1859 ax-8 1906 ax-9 1913 ax-10 1932 ax-11 1937 ax-12 1950 ax-13 2104 ax-ext 2451 ax-rep 4508 ax-sep 4518 ax-nul 4527 ax-pow 4579 ax-pr 4639 ax-un 6602 |
This theorem depends on definitions: df-bi 190 df-or 377 df-an 378 df-3or 1008 df-3an 1009 df-tru 1455 df-ex 1672 df-nf 1676 df-sb 1806 df-eu 2323 df-mo 2324 df-clab 2458 df-cleq 2464 df-clel 2467 df-nfc 2601 df-ne 2643 df-ral 2761 df-rex 2762 df-reu 2763 df-rab 2765 df-v 3033 df-sbc 3256 df-csb 3350 df-dif 3393 df-un 3395 df-in 3397 df-ss 3404 df-pss 3406 df-nul 3723 df-if 3873 df-pw 3944 df-sn 3960 df-pr 3962 df-tp 3964 df-op 3966 df-uni 4191 df-iun 4271 df-br 4396 df-opab 4455 df-mpt 4456 df-tr 4491 df-eprel 4750 df-id 4754 df-po 4760 df-so 4761 df-fr 4798 df-we 4800 df-xp 4845 df-rel 4846 df-cnv 4847 df-co 4848 df-dm 4849 df-rn 4850 df-res 4851 df-ima 4852 df-ord 5433 df-on 5434 df-lim 5435 df-suc 5436 df-iota 5553 df-fun 5591 df-fn 5592 df-f 5593 df-f1 5594 df-fo 5595 df-f1o 5596 df-fv 5597 df-om 6712 df-1o 7200 df-2o 7201 df-er 7381 df-en 7588 df-dom 7589 df-sdom 7590 df-fin 7591 df-pmtr 17161 |
This theorem is referenced by: mdetunilem7 19720 |
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