Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > sge0lempt | Structured version Visualization version Unicode version |
Description: If all of the terms of sums compare, so do the sums. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
Ref | Expression |
---|---|
sge0lempt.xph | |
sge0lempt.a | |
sge0lempt.b | |
sge0lempt.c | |
sge0lempt.le |
Ref | Expression |
---|---|
sge0lempt | Σ^ Σ^ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sge0lempt.a | . 2 | |
2 | sge0lempt.xph | . . 3 | |
3 | sge0lempt.b | . . 3 | |
4 | eqid 2622 | . . 3 | |
5 | 2, 3, 4 | fmptdf 6387 | . 2 |
6 | sge0lempt.c | . . 3 | |
7 | eqid 2622 | . . 3 | |
8 | 2, 6, 7 | fmptdf 6387 | . 2 |
9 | nfv 1843 | . . . . . 6 | |
10 | 2, 9 | nfan 1828 | . . . . 5 |
11 | nfcv 2764 | . . . . . . 7 | |
12 | 11 | nfcsb1 3548 | . . . . . 6 |
13 | nfcv 2764 | . . . . . 6 | |
14 | 11 | nfcsb1 3548 | . . . . . 6 |
15 | 12, 13, 14 | nfbr 4699 | . . . . 5 |
16 | 10, 15 | nfim 1825 | . . . 4 |
17 | eleq1 2689 | . . . . . 6 | |
18 | 17 | anbi2d 740 | . . . . 5 |
19 | csbeq1a 3542 | . . . . . 6 | |
20 | csbeq1a 3542 | . . . . . 6 | |
21 | 19, 20 | breq12d 4666 | . . . . 5 |
22 | 18, 21 | imbi12d 334 | . . . 4 |
23 | sge0lempt.le | . . . 4 | |
24 | 16, 22, 23 | chvar 2262 | . . 3 |
25 | simpr 477 | . . . . 5 | |
26 | simpl 473 | . . . . . 6 | |
27 | 12 | nfel1 2779 | . . . . . . . 8 |
28 | 10, 27 | nfim 1825 | . . . . . . 7 |
29 | 19 | eleq1d 2686 | . . . . . . . 8 |
30 | 18, 29 | imbi12d 334 | . . . . . . 7 |
31 | 28, 30, 3 | chvar 2262 | . . . . . 6 |
32 | 26, 25, 31 | syl2anc 693 | . . . . 5 |
33 | 11, 12, 19, 4 | fvmptf 6301 | . . . . 5 |
34 | 25, 32, 33 | syl2anc 693 | . . . 4 |
35 | nfcv 2764 | . . . . . . . . 9 | |
36 | 14, 35 | nfel 2777 | . . . . . . . 8 |
37 | 10, 36 | nfim 1825 | . . . . . . 7 |
38 | 20 | eleq1d 2686 | . . . . . . . 8 |
39 | 18, 38 | imbi12d 334 | . . . . . . 7 |
40 | 37, 39, 6 | chvar 2262 | . . . . . 6 |
41 | 26, 25, 40 | syl2anc 693 | . . . . 5 |
42 | 11, 14, 20, 7 | fvmptf 6301 | . . . . 5 |
43 | 25, 41, 42 | syl2anc 693 | . . . 4 |
44 | 34, 43 | breq12d 4666 | . . 3 |
45 | 24, 44 | mpbird 247 | . 2 |
46 | 1, 5, 8, 45 | sge0le 40624 | 1 Σ^ Σ^ |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 wceq 1483 wnf 1708 wcel 1990 csb 3533 class class class wbr 4653 cmpt 4729 cfv 5888 (class class class)co 6650 cc0 9936 cpnf 10071 cle 10075 cicc 12178 Σ^csumge0 40579 |
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-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-z 11378 df-uz 11688 df-rp 11833 df-ico 12181 df-icc 12182 df-fz 12327 df-fzo 12466 df-seq 12802 df-exp 12861 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-clim 14219 df-sum 14417 df-sumge0 40580 |
This theorem is referenced by: sge0iunmptlemre 40632 sge0xadd 40652 meaiunlelem 40685 hoicvrrex 40770 ovnsubaddlem1 40784 sge0hsphoire 40803 hoidmv1lelem1 40805 hoidmv1lelem2 40806 hoidmv1lelem3 40807 hoidmvlelem1 40809 hoidmvlelem2 40810 hoidmvlelem4 40812 hspmbllem2 40841 ovolval5lem1 40866 |
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