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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > sgnsgn | Structured version Unicode version |
Description: Signum is idempotent. (Contributed by Thierry Arnoux, 2-Oct-2018.) |
Ref | Expression |
---|---|
sgnsgn |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 |
. 2
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2 | fveq2 5775 |
. . 3
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3 | id 22 |
. . 3
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4 | 2, 3 | eqeq12d 2471 |
. 2
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5 | fveq2 5775 |
. . 3
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6 | id 22 |
. . 3
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7 | 5, 6 | eqeq12d 2471 |
. 2
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8 | fveq2 5775 |
. . 3
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9 | id 22 |
. . 3
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10 | 8, 9 | eqeq12d 2471 |
. 2
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11 | sgn0 12666 |
. . 3
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12 | 11 | a1i 11 |
. 2
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13 | 1re 9472 |
. . . . 5
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14 | 13 | rexri 9523 |
. . . 4
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15 | 0lt1 9949 |
. . . 4
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16 | sgnp 12667 |
. . . 4
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17 | 14, 15, 16 | mp2an 672 |
. . 3
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18 | 17 | a1i 11 |
. 2
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19 | neg1rr 10513 |
. . . . 5
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20 | 19 | rexri 9523 |
. . . 4
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21 | neg1lt0 10515 |
. . . 4
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22 | sgnn 12671 |
. . . 4
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23 | 20, 21, 22 | mp2an 672 |
. . 3
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24 | 23 | a1i 11 |
. 2
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25 | 1, 4, 7, 10, 12, 18, 24 | sgn3da 27044 |
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 1592 ax-4 1603 ax-5 1671 ax-6 1709 ax-7 1729 ax-8 1759 ax-9 1761 ax-10 1776 ax-11 1781 ax-12 1793 ax-13 1944 ax-ext 2429 ax-sep 4497 ax-nul 4505 ax-pow 4554 ax-pr 4615 ax-un 6458 ax-cnex 9425 ax-resscn 9426 ax-1cn 9427 ax-icn 9428 ax-addcl 9429 ax-addrcl 9430 ax-mulcl 9431 ax-mulrcl 9432 ax-mulcom 9433 ax-addass 9434 ax-mulass 9435 ax-distr 9436 ax-i2m1 9437 ax-1ne0 9438 ax-1rid 9439 ax-rnegex 9440 ax-rrecex 9441 ax-cnre 9442 ax-pre-lttri 9443 ax-pre-lttrn 9444 ax-pre-ltadd 9445 ax-pre-mulgt0 9446 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3or 966 df-3an 967 df-tru 1373 df-ex 1588 df-nf 1591 df-sb 1702 df-eu 2263 df-mo 2264 df-clab 2436 df-cleq 2442 df-clel 2445 df-nfc 2598 df-ne 2643 df-nel 2644 df-ral 2797 df-rex 2798 df-reu 2799 df-rab 2801 df-v 3056 df-sbc 3271 df-csb 3373 df-dif 3415 df-un 3417 df-in 3419 df-ss 3426 df-nul 3722 df-if 3876 df-pw 3946 df-sn 3962 df-pr 3964 df-op 3968 df-uni 4176 df-br 4377 df-opab 4435 df-mpt 4436 df-id 4720 df-po 4725 df-so 4726 df-xp 4930 df-rel 4931 df-cnv 4932 df-co 4933 df-dm 4934 df-rn 4935 df-res 4936 df-ima 4937 df-iota 5465 df-fun 5504 df-fn 5505 df-f 5506 df-f1 5507 df-fo 5508 df-f1o 5509 df-fv 5510 df-riota 6137 df-ov 6179 df-oprab 6180 df-mpt2 6181 df-er 7187 df-en 7397 df-dom 7398 df-sdom 7399 df-pnf 9507 df-mnf 9508 df-xr 9509 df-ltxr 9510 df-le 9511 df-sub 9684 df-neg 9685 df-sgn 12664 |
This theorem is referenced by: signsvfn 27103 signsvfpn 27106 signsvfnn 27107 |
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