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Mirrors > Home > ILE Home > Th. List > fisum0diag2 | Unicode version |
Description: Two ways to express "the sum of over the triangular region , , ". (Contributed by Mario Carneiro, 21-Jul-2014.) |
Ref | Expression |
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fsum0diag2.1 | |
fsum0diag2.2 | |
fsum0diag2.3 | |
fisum0diag2.n |
Ref | Expression |
---|---|
fisum0diag2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fznn0sub2 10098 | . . . . . . 7 | |
2 | 1 | ad2antll 491 | . . . . . 6 |
3 | fsum0diag2.3 | . . . . . . . . . 10 | |
4 | 3 | expr 375 | . . . . . . . . 9 |
5 | 4 | ralrimiv 2547 | . . . . . . . 8 |
6 | fsum0diag2.1 | . . . . . . . . . 10 | |
7 | 6 | eleq1d 2244 | . . . . . . . . 9 |
8 | 7 | cbvralv 2701 | . . . . . . . 8 |
9 | 5, 8 | sylibr 134 | . . . . . . 7 |
10 | 9 | adantrr 479 | . . . . . 6 |
11 | nfcsb1v 3088 | . . . . . . . 8 | |
12 | 11 | nfel1 2328 | . . . . . . 7 |
13 | csbeq1a 3064 | . . . . . . . 8 | |
14 | 13 | eleq1d 2244 | . . . . . . 7 |
15 | 12, 14 | rspc 2833 | . . . . . 6 |
16 | 2, 10, 15 | sylc 62 | . . . . 5 |
17 | fisum0diag2.n | . . . . 5 | |
18 | 16, 17 | fisum0diag 11417 | . . . 4 |
19 | 0zd 9238 | . . . . . . 7 | |
20 | 17 | adantr 276 | . . . . . . . 8 |
21 | elfzelz 9995 | . . . . . . . . 9 | |
22 | 21 | adantl 277 | . . . . . . . 8 |
23 | 20, 22 | zsubcld 9353 | . . . . . . 7 |
24 | nfcsb1v 3088 | . . . . . . . . . 10 | |
25 | 24 | nfel1 2328 | . . . . . . . . 9 |
26 | csbeq1a 3064 | . . . . . . . . . 10 | |
27 | 26 | eleq1d 2244 | . . . . . . . . 9 |
28 | 25, 27 | rspc 2833 | . . . . . . . 8 |
29 | 9, 28 | mpan9 281 | . . . . . . 7 |
30 | csbeq1 3058 | . . . . . . 7 | |
31 | 19, 23, 29, 30 | fisumrev2 11422 | . . . . . 6 |
32 | elfz3nn0 10085 | . . . . . . . . . . . 12 | |
33 | 32 | ad2antlr 489 | . . . . . . . . . . 11 |
34 | 21 | ad2antlr 489 | . . . . . . . . . . 11 |
35 | nn0cn 9159 | . . . . . . . . . . . 12 | |
36 | zcn 9231 | . . . . . . . . . . . 12 | |
37 | subcl 8130 | . . . . . . . . . . . 12 | |
38 | 35, 36, 37 | syl2an 289 | . . . . . . . . . . 11 |
39 | 33, 34, 38 | syl2anc 411 | . . . . . . . . . 10 |
40 | 39 | addid2d 8081 | . . . . . . . . 9 |
41 | 40 | oveq1d 5880 | . . . . . . . 8 |
42 | 41 | csbeq1d 3062 | . . . . . . 7 |
43 | 42 | sumeq2dv 11344 | . . . . . 6 |
44 | 31, 43 | eqtrd 2208 | . . . . 5 |
45 | 44 | sumeq2dv 11344 | . . . 4 |
46 | elfz3nn0 10085 | . . . . . . . . . 10 | |
47 | 46 | adantl 277 | . . . . . . . . 9 |
48 | addid2 8070 | . . . . . . . . 9 | |
49 | 47, 35, 48 | 3syl 17 | . . . . . . . 8 |
50 | 49 | oveq1d 5880 | . . . . . . 7 |
51 | 50 | oveq2d 5881 | . . . . . 6 |
52 | 50 | oveq1d 5880 | . . . . . . . . 9 |
53 | 52 | adantr 276 | . . . . . . . 8 |
54 | 46 | ad2antlr 489 | . . . . . . . . 9 |
55 | elfzelz 9995 | . . . . . . . . . 10 | |
56 | 55 | ad2antlr 489 | . . . . . . . . 9 |
57 | elfzelz 9995 | . . . . . . . . . 10 | |
58 | 57 | adantl 277 | . . . . . . . . 9 |
59 | zcn 9231 | . . . . . . . . . 10 | |
60 | sub32 8165 | . . . . . . . . . 10 | |
61 | 35, 59, 36, 60 | syl3an 1280 | . . . . . . . . 9 |
62 | 54, 56, 58, 61 | syl3anc 1238 | . . . . . . . 8 |
63 | 53, 62 | eqtrd 2208 | . . . . . . 7 |
64 | 63 | csbeq1d 3062 | . . . . . 6 |
65 | 51, 64 | sumeq12rdv 11349 | . . . . 5 |
66 | 65 | sumeq2dv 11344 | . . . 4 |
67 | 18, 45, 66 | 3eqtr4d 2218 | . . 3 |
68 | 0zd 9238 | . . . 4 | |
69 | 0zd 9238 | . . . . . 6 | |
70 | elfzelz 9995 | . . . . . . 7 | |
71 | 70 | adantl 277 | . . . . . 6 |
72 | 69, 71 | fzfigd 10401 | . . . . 5 |
73 | elfzuz3 9992 | . . . . . . . . . 10 | |
74 | 73 | adantl 277 | . . . . . . . . 9 |
75 | elfzuz3 9992 | . . . . . . . . . . 11 | |
76 | 75 | adantl 277 | . . . . . . . . . 10 |
77 | 76 | adantr 276 | . . . . . . . . 9 |
78 | elfzuzb 9989 | . . . . . . . . 9 | |
79 | 74, 77, 78 | sylanbrc 417 | . . . . . . . 8 |
80 | elfzelz 9995 | . . . . . . . . . 10 | |
81 | 80 | adantl 277 | . . . . . . . . 9 |
82 | 17 | ad2antrr 488 | . . . . . . . . 9 |
83 | 70 | ad2antlr 489 | . . . . . . . . 9 |
84 | fzsubel 10030 | . . . . . . . . 9 | |
85 | 81, 82, 83, 81, 84 | syl22anc 1239 | . . . . . . . 8 |
86 | 79, 85 | mpbid 147 | . . . . . . 7 |
87 | subid 8150 | . . . . . . . . 9 | |
88 | 81, 36, 87 | 3syl 17 | . . . . . . . 8 |
89 | 88 | oveq1d 5880 | . . . . . . 7 |
90 | 86, 89 | eleqtrd 2254 | . . . . . 6 |
91 | simpll 527 | . . . . . . 7 | |
92 | fzss2 10034 | . . . . . . . . 9 | |
93 | 76, 92 | syl 14 | . . . . . . . 8 |
94 | 93 | sselda 3153 | . . . . . . 7 |
95 | 91, 94, 9 | syl2anc 411 | . . . . . 6 |
96 | nfcsb1v 3088 | . . . . . . . 8 | |
97 | 96 | nfel1 2328 | . . . . . . 7 |
98 | csbeq1a 3064 | . . . . . . . 8 | |
99 | 98 | eleq1d 2244 | . . . . . . 7 |
100 | 97, 99 | rspc 2833 | . . . . . 6 |
101 | 90, 95, 100 | sylc 62 | . . . . 5 |
102 | 72, 101 | fsumcl 11376 | . . . 4 |
103 | oveq2 5873 | . . . . 5 | |
104 | oveq1 5872 | . . . . . . 7 | |
105 | 104 | csbeq1d 3062 | . . . . . 6 |
106 | 105 | adantr 276 | . . . . 5 |
107 | 103, 106 | sumeq12dv 11348 | . . . 4 |
108 | 68, 17, 102, 107 | fisumrev2 11422 | . . 3 |
109 | 67, 108 | eqtr4d 2211 | . 2 |
110 | vex 2738 | . . . . . 6 | |
111 | 110, 6 | csbie 3100 | . . . . 5 |
112 | 111 | a1i 9 | . . . 4 |
113 | 112 | sumeq2dv 11344 | . . 3 |
114 | 113 | sumeq2i 11340 | . 2 |
115 | 70 | adantr 276 | . . . . . 6 |
116 | 80 | adantl 277 | . . . . . 6 |
117 | 115, 116 | zsubcld 9353 | . . . . 5 |
118 | fsum0diag2.2 | . . . . . 6 | |
119 | 118 | adantl 277 | . . . . 5 |
120 | 117, 119 | csbied 3101 | . . . 4 |
121 | 120 | sumeq2dv 11344 | . . 3 |
122 | 121 | sumeq2i 11340 | . 2 |
123 | 109, 114, 122 | 3eqtr3g 2231 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 104 wb 105 wceq 1353 wcel 2146 wral 2453 csb 3055 wss 3127 cfv 5208 (class class class)co 5865 cc 7784 cc0 7786 caddc 7789 cmin 8102 cn0 9149 cz 9226 cuz 9501 cfz 9979 csu 11329 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1445 ax-7 1446 ax-gen 1447 ax-ie1 1491 ax-ie2 1492 ax-8 1502 ax-10 1503 ax-11 1504 ax-i12 1505 ax-bndl 1507 ax-4 1508 ax-17 1524 ax-i9 1528 ax-ial 1532 ax-i5r 1533 ax-13 2148 ax-14 2149 ax-ext 2157 ax-coll 4113 ax-sep 4116 ax-nul 4124 ax-pow 4169 ax-pr 4203 ax-un 4427 ax-setind 4530 ax-iinf 4581 ax-cnex 7877 ax-resscn 7878 ax-1cn 7879 ax-1re 7880 ax-icn 7881 ax-addcl 7882 ax-addrcl 7883 ax-mulcl 7884 ax-mulrcl 7885 ax-addcom 7886 ax-mulcom 7887 ax-addass 7888 ax-mulass 7889 ax-distr 7890 ax-i2m1 7891 ax-0lt1 7892 ax-1rid 7893 ax-0id 7894 ax-rnegex 7895 ax-precex 7896 ax-cnre 7897 ax-pre-ltirr 7898 ax-pre-ltwlin 7899 ax-pre-lttrn 7900 ax-pre-apti 7901 ax-pre-ltadd 7902 ax-pre-mulgt0 7903 ax-pre-mulext 7904 ax-arch 7905 ax-caucvg 7906 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1459 df-sb 1761 df-eu 2027 df-mo 2028 df-clab 2162 df-cleq 2168 df-clel 2171 df-nfc 2306 df-ne 2346 df-nel 2441 df-ral 2458 df-rex 2459 df-reu 2460 df-rmo 2461 df-rab 2462 df-v 2737 df-sbc 2961 df-csb 3056 df-dif 3129 df-un 3131 df-in 3133 df-ss 3140 df-nul 3421 df-if 3533 df-pw 3574 df-sn 3595 df-pr 3596 df-op 3598 df-uni 3806 df-int 3841 df-iun 3884 df-disj 3976 df-br 3999 df-opab 4060 df-mpt 4061 df-tr 4097 df-id 4287 df-po 4290 df-iso 4291 df-iord 4360 df-on 4362 df-ilim 4363 df-suc 4365 df-iom 4584 df-xp 4626 df-rel 4627 df-cnv 4628 df-co 4629 df-dm 4630 df-rn 4631 df-res 4632 df-ima 4633 df-iota 5170 df-fun 5210 df-fn 5211 df-f 5212 df-f1 5213 df-fo 5214 df-f1o 5215 df-fv 5216 df-isom 5217 df-riota 5821 df-ov 5868 df-oprab 5869 df-mpo 5870 df-1st 6131 df-2nd 6132 df-recs 6296 df-irdg 6361 df-frec 6382 df-1o 6407 df-oadd 6411 df-er 6525 df-en 6731 df-dom 6732 df-fin 6733 df-pnf 7968 df-mnf 7969 df-xr 7970 df-ltxr 7971 df-le 7972 df-sub 8104 df-neg 8105 df-reap 8506 df-ap 8513 df-div 8603 df-inn 8893 df-2 8951 df-3 8952 df-4 8953 df-n0 9150 df-z 9227 df-uz 9502 df-q 9593 df-rp 9625 df-fz 9980 df-fzo 10113 df-seqfrec 10416 df-exp 10490 df-ihash 10724 df-cj 10819 df-re 10820 df-im 10821 df-rsqrt 10975 df-abs 10976 df-clim 11255 df-sumdc 11330 |
This theorem is referenced by: mertensabs 11513 |
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