Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > signsvtn | Structured version Visualization version Unicode version |
Description: Adding a letter of a different sign as the highest coefficient changes the sign. (Contributed by Thierry Arnoux, 12-Oct-2018.) |
Ref | Expression |
---|---|
signsv.p | |
signsv.w | |
signsv.t | Word ..^ g sgn |
signsv.v | Word ..^ |
signsvf.e | Word |
signsvf.0 | |
signsvf.f | ++ |
signsvf.a | |
signsvf.n | |
signsvt.b |
Ref | Expression |
---|---|
signsvtn |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | signsvf.f | . . . . . 6 ++ | |
2 | 1 | fveq2d 6195 | . . . . 5 ++ |
3 | signsvf.e | . . . . . 6 Word | |
4 | signsvf.0 | . . . . . 6 | |
5 | signsvf.a | . . . . . 6 | |
6 | signsv.p | . . . . . . 7 | |
7 | signsv.w | . . . . . . 7 | |
8 | signsv.t | . . . . . . 7 Word ..^ g sgn | |
9 | signsv.v | . . . . . . 7 Word ..^ | |
10 | 6, 7, 8, 9 | signsvfn 30659 | . . . . . 6 Word ++ |
11 | 3, 4, 5, 10 | syl21anc 1325 | . . . . 5 ++ |
12 | 2, 11 | eqtrd 2656 | . . . 4 |
13 | 12 | adantr 481 | . . 3 |
14 | signsvt.b | . . . . . . . 8 | |
15 | signsvf.n | . . . . . . . . . 10 | |
16 | 15 | oveq1i 6660 | . . . . . . . . 9 |
17 | 16 | fveq2i 6194 | . . . . . . . 8 |
18 | 14, 17 | eqtri 2644 | . . . . . . 7 |
19 | 18 | oveq1i 6660 | . . . . . 6 |
20 | 3 | adantr 481 | . . . . . . . . . . . . 13 Word |
21 | 20 | eldifad 3586 | . . . . . . . . . . . 12 Word |
22 | 6, 7, 8, 9 | signstf 30643 | . . . . . . . . . . . 12 Word Word |
23 | wrdf 13310 | . . . . . . . . . . . 12 Word ..^ | |
24 | 21, 22, 23 | 3syl 18 | . . . . . . . . . . 11 ..^ |
25 | eldifsn 4317 | . . . . . . . . . . . . . . 15 Word Word | |
26 | 3, 25 | sylib 208 | . . . . . . . . . . . . . 14 Word |
27 | 26 | adantr 481 | . . . . . . . . . . . . 13 Word |
28 | lennncl 13325 | . . . . . . . . . . . . 13 Word | |
29 | fzo0end 12560 | . . . . . . . . . . . . 13 ..^ | |
30 | 27, 28, 29 | 3syl 18 | . . . . . . . . . . . 12 ..^ |
31 | 6, 7, 8, 9 | signstlen 30644 | . . . . . . . . . . . . . 14 Word |
32 | 21, 31 | syl 17 | . . . . . . . . . . . . 13 |
33 | 32 | oveq2d 6666 | . . . . . . . . . . . 12 ..^ ..^ |
34 | 30, 33 | eleqtrrd 2704 | . . . . . . . . . . 11 ..^ |
35 | 24, 34 | ffvelrnd 6360 | . . . . . . . . . 10 |
36 | 18, 35 | syl5eqel 2705 | . . . . . . . . 9 |
37 | 36 | recnd 10068 | . . . . . . . 8 |
38 | 5 | adantr 481 | . . . . . . . . 9 |
39 | 38 | recnd 10068 | . . . . . . . 8 |
40 | 37, 39 | mulcomd 10061 | . . . . . . 7 |
41 | simpr 477 | . . . . . . 7 | |
42 | 40, 41 | eqbrtrd 4675 | . . . . . 6 |
43 | 19, 42 | syl5eqbrr 4689 | . . . . 5 |
44 | 43 | iftrued 4094 | . . . 4 |
45 | 44 | oveq2d 6666 | . . 3 |
46 | 13, 45 | eqtr2d 2657 | . 2 |
47 | 6, 7, 8, 9 | signsvvf 30656 | . . . . . 6 Word |
48 | 47 | a1i 11 | . . . . 5 Word |
49 | 1 | adantr 481 | . . . . . 6 ++ |
50 | 38 | s1cld 13383 | . . . . . . 7 Word |
51 | ccatcl 13359 | . . . . . . 7 Word Word ++ Word | |
52 | 21, 50, 51 | syl2anc 693 | . . . . . 6 ++ Word |
53 | 49, 52 | eqeltrd 2701 | . . . . 5 Word |
54 | 48, 53 | ffvelrnd 6360 | . . . 4 |
55 | 54 | nn0cnd 11353 | . . 3 |
56 | 48, 21 | ffvelrnd 6360 | . . . 4 |
57 | 56 | nn0cnd 11353 | . . 3 |
58 | 1cnd 10056 | . . 3 | |
59 | 55, 57, 58 | subaddd 10410 | . 2 |
60 | 46, 59 | mpbird 247 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 wceq 1483 wcel 1990 wne 2794 cdif 3571 c0 3915 cif 4086 csn 4177 cpr 4179 ctp 4181 cop 4183 class class class wbr 4653 cmpt 4729 wf 5884 cfv 5888 (class class class)co 6650 cmpt2 6652 cr 9935 cc0 9936 c1 9937 caddc 9939 cmul 9941 clt 10074 cmin 10266 cneg 10267 cn 11020 cn0 11292 cfz 12326 ..^cfzo 12465 chash 13117 Word cword 13291 ++ cconcat 13293 cs1 13294 sgncsgn 13826 csu 14416 cnx 15854 cbs 15857 cplusg 15941 g cgsu 16101 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-oi 8415 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-xnn0 11364 df-z 11378 df-uz 11688 df-rp 11833 df-fz 12327 df-fzo 12466 df-seq 12802 df-exp 12861 df-hash 13118 df-word 13299 df-lsw 13300 df-concat 13301 df-s1 13302 df-substr 13303 df-sgn 13827 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-clim 14219 df-sum 14417 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-plusg 15954 df-0g 16102 df-gsum 16103 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-mulg 17541 df-cntz 17750 |
This theorem is referenced by: signsvfnn 30663 |
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