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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > xrge00 | Structured version Visualization version Unicode version |
Description: The zero of the extended nonnegative real numbers monoid. (Contributed by Thierry Arnoux, 30-Jan-2017.) |
Ref | Expression |
---|---|
xrge00 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2462 |
. . 3
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2 | 1 | xrs1mnd 19055 |
. 2
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3 | xrge0cmn 19059 |
. . 3
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4 | cmnmnd 17494 |
. . 3
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5 | 3, 4 | ax-mp 5 |
. 2
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6 | mnflt0 11456 |
. . . . . . 7
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7 | mnfxr 11443 |
. . . . . . . 8
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8 | 0xr 9713 |
. . . . . . . 8
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9 | xrltnle 9727 |
. . . . . . . 8
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10 | 7, 8, 9 | mp2an 683 |
. . . . . . 7
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11 | 6, 10 | mpbi 213 |
. . . . . 6
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12 | 11 | intnan 930 |
. . . . 5
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13 | elxrge0 11770 |
. . . . 5
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14 | 12, 13 | mtbir 305 |
. . . 4
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15 | difsn 4119 |
. . . 4
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16 | 14, 15 | ax-mp 5 |
. . 3
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17 | iccssxr 11746 |
. . . 4
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18 | ssdif 3580 |
. . . 4
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19 | 17, 18 | ax-mp 5 |
. . 3
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20 | 16, 19 | eqsstr3i 3475 |
. 2
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21 | 0e0iccpnf 11772 |
. 2
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22 | difss 3572 |
. . . . 5
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23 | df-ss 3430 |
. . . . 5
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24 | 22, 23 | mpbi 213 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
25 | xrex 11328 |
. . . . . 6
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26 | difexg 4565 |
. . . . . 6
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27 | 25, 26 | ax-mp 5 |
. . . . 5
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28 | xrsbas 19033 |
. . . . . 6
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29 | 1, 28 | ressbas 15228 |
. . . . 5
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30 | 27, 29 | ax-mp 5 |
. . . 4
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31 | 24, 30 | eqtr3i 2486 |
. . 3
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32 | 1 | xrs10 19056 |
. . 3
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33 | ovex 6343 |
. . . . 5
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34 | ressress 15236 |
. . . . 5
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35 | 27, 33, 34 | mp2an 683 |
. . . 4
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36 | dfss 3431 |
. . . . . . 7
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37 | 20, 36 | mpbi 213 |
. . . . . 6
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38 | incom 3637 |
. . . . . 6
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39 | 37, 38 | eqtr2i 2485 |
. . . . 5
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40 | 39 | oveq2i 6326 |
. . . 4
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41 | 35, 40 | eqtr2i 2485 |
. . 3
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42 | 31, 32, 41 | submnd0 16615 |
. 2
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43 | 2, 5, 20, 21, 42 | mp4an 684 |
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-8 1900 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 ax-un 6610 ax-cnex 9621 ax-resscn 9622 ax-1cn 9623 ax-icn 9624 ax-addcl 9625 ax-addrcl 9626 ax-mulcl 9627 ax-mulrcl 9628 ax-mulcom 9629 ax-addass 9630 ax-mulass 9631 ax-distr 9632 ax-i2m1 9633 ax-1ne0 9634 ax-1rid 9635 ax-rnegex 9636 ax-rrecex 9637 ax-cnre 9638 ax-pre-lttri 9639 ax-pre-lttrn 9640 ax-pre-ltadd 9641 ax-pre-mulgt0 9642 |
This theorem depends on definitions: df-bi 190 df-or 376 df-an 377 df-3or 992 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-nel 2636 df-ral 2754 df-rex 2755 df-reu 2756 df-rmo 2757 df-rab 2758 df-v 3059 df-sbc 3280 df-csb 3376 df-dif 3419 df-un 3421 df-in 3423 df-ss 3430 df-pss 3432 df-nul 3744 df-if 3894 df-pw 3965 df-sn 3981 df-pr 3983 df-tp 3985 df-op 3987 df-uni 4213 df-int 4249 df-iun 4294 df-br 4417 df-opab 4476 df-mpt 4477 df-tr 4512 df-eprel 4764 df-id 4768 df-po 4774 df-so 4775 df-fr 4812 df-we 4814 df-xp 4859 df-rel 4860 df-cnv 4861 df-co 4862 df-dm 4863 df-rn 4864 df-res 4865 df-ima 4866 df-pred 5399 df-ord 5445 df-on 5446 df-lim 5447 df-suc 5448 df-iota 5565 df-fun 5603 df-fn 5604 df-f 5605 df-f1 5606 df-fo 5607 df-f1o 5608 df-fv 5609 df-riota 6277 df-ov 6318 df-oprab 6319 df-mpt2 6320 df-om 6720 df-1st 6820 df-2nd 6821 df-wrecs 7054 df-recs 7116 df-rdg 7154 df-1o 7208 df-oadd 7212 df-er 7389 df-en 7596 df-dom 7597 df-sdom 7598 df-fin 7599 df-pnf 9703 df-mnf 9704 df-xr 9705 df-ltxr 9706 df-le 9707 df-sub 9888 df-neg 9889 df-nn 10638 df-2 10696 df-3 10697 df-4 10698 df-5 10699 df-6 10700 df-7 10701 df-8 10702 df-9 10703 df-10 10704 df-n0 10899 df-z 10967 df-dec 11081 df-uz 11189 df-xadd 11439 df-icc 11671 df-fz 11814 df-struct 15172 df-ndx 15173 df-slot 15174 df-base 15175 df-sets 15176 df-ress 15177 df-plusg 15252 df-mulr 15253 df-tset 15258 df-ple 15259 df-ds 15261 df-0g 15389 df-xrs 15449 df-mgm 16537 df-sgrp 16576 df-mnd 16586 df-submnd 16632 df-cmn 17481 |
This theorem is referenced by: xrge0mulgnn0 28500 xrge0slmod 28656 xrge0iifmhm 28794 esumgsum 28915 esumnul 28918 esum0 28919 gsumesum 28929 esumsnf 28934 esumss 28942 esumpfinval 28945 esumpfinvalf 28946 esumcocn 28950 sitmcl 29233 |
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