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Mirrors > Home > MPE Home > Th. List > deg1val | Structured version Visualization version Unicode version |
Description: Value of the univariate degree as a supremum. (Contributed by Stefan O'Rear, 29-Mar-2015.) (Revised by AV, 25-Jul-2019.) |
Ref | Expression |
---|---|
deg1leb.d |
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deg1leb.p |
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deg1leb.b |
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deg1leb.y |
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deg1leb.a |
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Ref | Expression |
---|---|
deg1val |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | deg1leb.d |
. . . 4
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2 | 1 | deg1fval 23108 |
. . 3
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3 | eqid 2471 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
4 | deg1leb.p |
. . . 4
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5 | eqid 2471 |
. . . 4
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6 | deg1leb.b |
. . . 4
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7 | 4, 5, 6 | ply1bas 18865 |
. . 3
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8 | deg1leb.y |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
9 | psr1baslem 18855 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
10 | tdeglem2 23089 |
. . 3
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11 | 2, 3, 7, 8, 9, 10 | mdegval 23091 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
12 | fvex 5889 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
13 | 8, 12 | eqeltri 2545 |
. . . . . . . 8
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14 | suppimacnv 6944 |
. . . . . . . 8
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15 | 13, 14 | mpan2 685 |
. . . . . . 7
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16 | 15 | imaeq2d 5174 |
. . . . . 6
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17 | imaco 5347 |
. . . . . 6
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18 | 16, 17 | syl6eqr 2523 |
. . . . 5
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19 | deg1leb.a |
. . . . . . . . 9
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20 | df1o2 7212 |
. . . . . . . . . 10
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21 | nn0ex 10899 |
. . . . . . . . . 10
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22 | 0ex 4528 |
. . . . . . . . . 10
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23 | eqid 2471 |
. . . . . . . . . 10
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24 | 20, 21, 22, 23 | mapsncnv 7536 |
. . . . . . . . 9
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25 | 19, 6, 4, 24 | coe1fval2 18880 |
. . . . . . . 8
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26 | 25 | cnveqd 5015 |
. . . . . . 7
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27 | cnvco 5025 |
. . . . . . . 8
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28 | cocnvcnv1 5353 |
. . . . . . . 8
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29 | 27, 28 | eqtri 2493 |
. . . . . . 7
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30 | 26, 29 | syl6req 2522 |
. . . . . 6
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31 | 30 | imaeq1d 5173 |
. . . . 5
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32 | 18, 31 | eqtrd 2505 |
. . . 4
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33 | fvex 5889 |
. . . . . 6
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34 | 19, 33 | eqeltri 2545 |
. . . . 5
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35 | suppimacnv 6944 |
. . . . . 6
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36 | 35 | eqcomd 2477 |
. . . . 5
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37 | 34, 13, 36 | mp2an 686 |
. . . 4
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38 | 32, 37 | syl6eq 2521 |
. . 3
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39 | 38 | supeq1d 7978 |
. 2
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40 | 11, 39 | eqtrd 2505 |
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 ax-inf2 8164 ax-cnex 9613 ax-resscn 9614 ax-1cn 9615 ax-icn 9616 ax-addcl 9617 ax-addrcl 9618 ax-mulcl 9619 ax-mulrcl 9620 ax-mulcom 9621 ax-addass 9622 ax-mulass 9623 ax-distr 9624 ax-i2m1 9625 ax-1ne0 9626 ax-1rid 9627 ax-rnegex 9628 ax-rrecex 9629 ax-cnre 9630 ax-pre-lttri 9631 ax-pre-lttrn 9632 ax-pre-ltadd 9633 ax-pre-mulgt0 9634 ax-addf 9636 ax-mulf 9637 |
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-nel 2644 df-ral 2761 df-rex 2762 df-reu 2763 df-rmo 2764 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-int 4227 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-se 4799 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-pred 5387 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-isom 5598 df-riota 6270 df-ov 6311 df-oprab 6312 df-mpt2 6313 df-of 6550 df-om 6712 df-1st 6812 df-2nd 6813 df-supp 6934 df-wrecs 7046 df-recs 7108 df-rdg 7146 df-1o 7200 df-oadd 7204 df-er 7381 df-map 7492 df-en 7588 df-dom 7589 df-sdom 7590 df-fin 7591 df-fsupp 7902 df-sup 7974 df-oi 8043 df-card 8391 df-pnf 9695 df-mnf 9696 df-xr 9697 df-ltxr 9698 df-le 9699 df-sub 9882 df-neg 9883 df-nn 10632 df-2 10690 df-3 10691 df-4 10692 df-5 10693 df-6 10694 df-7 10695 df-8 10696 df-9 10697 df-10 10698 df-n0 10894 df-z 10962 df-dec 11075 df-uz 11183 df-fz 11811 df-fzo 11943 df-seq 12252 df-hash 12554 df-struct 15201 df-ndx 15202 df-slot 15203 df-base 15204 df-sets 15205 df-ress 15206 df-plusg 15281 df-mulr 15282 df-starv 15283 df-sca 15284 df-vsca 15285 df-tset 15287 df-ple 15288 df-ds 15290 df-unif 15291 df-0g 15418 df-gsum 15419 df-mgm 16566 df-sgrp 16605 df-mnd 16615 df-grp 16751 df-mulg 16754 df-cntz 17049 df-cmn 17510 df-mgp 17802 df-ring 17860 df-cring 17861 df-psr 18657 df-mpl 18659 df-opsr 18661 df-psr1 18850 df-ply1 18852 df-coe1 18853 df-cnfld 19048 df-mdeg 23083 df-deg1 23084 |
This theorem is referenced by: deg1mul3 23143 deg1mul3le 23144 |
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