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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 10053 | . . . . . . 7 | |
2 | 1 | ad2antll 483 | . . . . . 6 |
3 | fsum0diag2.3 | . . . . . . . . . 10 | |
4 | 3 | expr 373 | . . . . . . . . 9 |
5 | 4 | ralrimiv 2536 | . . . . . . . 8 |
6 | fsum0diag2.1 | . . . . . . . . . 10 | |
7 | 6 | eleq1d 2233 | . . . . . . . . 9 |
8 | 7 | cbvralv 2689 | . . . . . . . 8 |
9 | 5, 8 | sylibr 133 | . . . . . . 7 |
10 | 9 | adantrr 471 | . . . . . 6 |
11 | nfcsb1v 3073 | . . . . . . . 8 | |
12 | 11 | nfel1 2317 | . . . . . . 7 |
13 | csbeq1a 3049 | . . . . . . . 8 | |
14 | 13 | eleq1d 2233 | . . . . . . 7 |
15 | 12, 14 | rspc 2819 | . . . . . 6 |
16 | 2, 10, 15 | sylc 62 | . . . . 5 |
17 | fisum0diag2.n | . . . . 5 | |
18 | 16, 17 | fisum0diag 11368 | . . . 4 |
19 | 0zd 9194 | . . . . . . 7 | |
20 | 17 | adantr 274 | . . . . . . . 8 |
21 | elfzelz 9951 | . . . . . . . . 9 | |
22 | 21 | adantl 275 | . . . . . . . 8 |
23 | 20, 22 | zsubcld 9309 | . . . . . . 7 |
24 | nfcsb1v 3073 | . . . . . . . . . 10 | |
25 | 24 | nfel1 2317 | . . . . . . . . 9 |
26 | csbeq1a 3049 | . . . . . . . . . 10 | |
27 | 26 | eleq1d 2233 | . . . . . . . . 9 |
28 | 25, 27 | rspc 2819 | . . . . . . . 8 |
29 | 9, 28 | mpan9 279 | . . . . . . 7 |
30 | csbeq1 3043 | . . . . . . 7 | |
31 | 19, 23, 29, 30 | fisumrev2 11373 | . . . . . 6 |
32 | elfz3nn0 10040 | . . . . . . . . . . . 12 | |
33 | 32 | ad2antlr 481 | . . . . . . . . . . 11 |
34 | 21 | ad2antlr 481 | . . . . . . . . . . 11 |
35 | nn0cn 9115 | . . . . . . . . . . . 12 | |
36 | zcn 9187 | . . . . . . . . . . . 12 | |
37 | subcl 8088 | . . . . . . . . . . . 12 | |
38 | 35, 36, 37 | syl2an 287 | . . . . . . . . . . 11 |
39 | 33, 34, 38 | syl2anc 409 | . . . . . . . . . 10 |
40 | 39 | addid2d 8039 | . . . . . . . . 9 |
41 | 40 | oveq1d 5851 | . . . . . . . 8 |
42 | 41 | csbeq1d 3047 | . . . . . . 7 |
43 | 42 | sumeq2dv 11295 | . . . . . 6 |
44 | 31, 43 | eqtrd 2197 | . . . . 5 |
45 | 44 | sumeq2dv 11295 | . . . 4 |
46 | elfz3nn0 10040 | . . . . . . . . . 10 | |
47 | 46 | adantl 275 | . . . . . . . . 9 |
48 | addid2 8028 | . . . . . . . . 9 | |
49 | 47, 35, 48 | 3syl 17 | . . . . . . . 8 |
50 | 49 | oveq1d 5851 | . . . . . . 7 |
51 | 50 | oveq2d 5852 | . . . . . 6 |
52 | 50 | oveq1d 5851 | . . . . . . . . 9 |
53 | 52 | adantr 274 | . . . . . . . 8 |
54 | 46 | ad2antlr 481 | . . . . . . . . 9 |
55 | elfzelz 9951 | . . . . . . . . . 10 | |
56 | 55 | ad2antlr 481 | . . . . . . . . 9 |
57 | elfzelz 9951 | . . . . . . . . . 10 | |
58 | 57 | adantl 275 | . . . . . . . . 9 |
59 | zcn 9187 | . . . . . . . . . 10 | |
60 | sub32 8123 | . . . . . . . . . 10 | |
61 | 35, 59, 36, 60 | syl3an 1269 | . . . . . . . . 9 |
62 | 54, 56, 58, 61 | syl3anc 1227 | . . . . . . . 8 |
63 | 53, 62 | eqtrd 2197 | . . . . . . 7 |
64 | 63 | csbeq1d 3047 | . . . . . 6 |
65 | 51, 64 | sumeq12rdv 11300 | . . . . 5 |
66 | 65 | sumeq2dv 11295 | . . . 4 |
67 | 18, 45, 66 | 3eqtr4d 2207 | . . 3 |
68 | 0zd 9194 | . . . 4 | |
69 | 0zd 9194 | . . . . . 6 | |
70 | elfzelz 9951 | . . . . . . 7 | |
71 | 70 | adantl 275 | . . . . . 6 |
72 | 69, 71 | fzfigd 10356 | . . . . 5 |
73 | elfzuz3 9948 | . . . . . . . . . 10 | |
74 | 73 | adantl 275 | . . . . . . . . 9 |
75 | elfzuz3 9948 | . . . . . . . . . . 11 | |
76 | 75 | adantl 275 | . . . . . . . . . 10 |
77 | 76 | adantr 274 | . . . . . . . . 9 |
78 | elfzuzb 9945 | . . . . . . . . 9 | |
79 | 74, 77, 78 | sylanbrc 414 | . . . . . . . 8 |
80 | elfzelz 9951 | . . . . . . . . . 10 | |
81 | 80 | adantl 275 | . . . . . . . . 9 |
82 | 17 | ad2antrr 480 | . . . . . . . . 9 |
83 | 70 | ad2antlr 481 | . . . . . . . . 9 |
84 | fzsubel 9985 | . . . . . . . . 9 | |
85 | 81, 82, 83, 81, 84 | syl22anc 1228 | . . . . . . . 8 |
86 | 79, 85 | mpbid 146 | . . . . . . 7 |
87 | subid 8108 | . . . . . . . . 9 | |
88 | 81, 36, 87 | 3syl 17 | . . . . . . . 8 |
89 | 88 | oveq1d 5851 | . . . . . . 7 |
90 | 86, 89 | eleqtrd 2243 | . . . . . 6 |
91 | simpll 519 | . . . . . . 7 | |
92 | fzss2 9989 | . . . . . . . . 9 | |
93 | 76, 92 | syl 14 | . . . . . . . 8 |
94 | 93 | sselda 3137 | . . . . . . 7 |
95 | 91, 94, 9 | syl2anc 409 | . . . . . 6 |
96 | nfcsb1v 3073 | . . . . . . . 8 | |
97 | 96 | nfel1 2317 | . . . . . . 7 |
98 | csbeq1a 3049 | . . . . . . . 8 | |
99 | 98 | eleq1d 2233 | . . . . . . 7 |
100 | 97, 99 | rspc 2819 | . . . . . 6 |
101 | 90, 95, 100 | sylc 62 | . . . . 5 |
102 | 72, 101 | fsumcl 11327 | . . . 4 |
103 | oveq2 5844 | . . . . 5 | |
104 | oveq1 5843 | . . . . . . 7 | |
105 | 104 | csbeq1d 3047 | . . . . . 6 |
106 | 105 | adantr 274 | . . . . 5 |
107 | 103, 106 | sumeq12dv 11299 | . . . 4 |
108 | 68, 17, 102, 107 | fisumrev2 11373 | . . 3 |
109 | 67, 108 | eqtr4d 2200 | . 2 |
110 | vex 2724 | . . . . . 6 | |
111 | 110, 6 | csbie 3085 | . . . . 5 |
112 | 111 | a1i 9 | . . . 4 |
113 | 112 | sumeq2dv 11295 | . . 3 |
114 | 113 | sumeq2i 11291 | . 2 |
115 | 70 | adantr 274 | . . . . . 6 |
116 | 80 | adantl 275 | . . . . . 6 |
117 | 115, 116 | zsubcld 9309 | . . . . 5 |
118 | fsum0diag2.2 | . . . . . 6 | |
119 | 118 | adantl 275 | . . . . 5 |
120 | 117, 119 | csbied 3086 | . . . 4 |
121 | 120 | sumeq2dv 11295 | . . 3 |
122 | 121 | sumeq2i 11291 | . 2 |
123 | 109, 114, 122 | 3eqtr3g 2220 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1342 wcel 2135 wral 2442 csb 3040 wss 3111 cfv 5182 (class class class)co 5836 cc 7742 cc0 7744 caddc 7747 cmin 8060 cn0 9105 cz 9182 cuz 9457 cfz 9935 csu 11280 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4091 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-iinf 4559 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-mulrcl 7843 ax-addcom 7844 ax-mulcom 7845 ax-addass 7846 ax-mulass 7847 ax-distr 7848 ax-i2m1 7849 ax-0lt1 7850 ax-1rid 7851 ax-0id 7852 ax-rnegex 7853 ax-precex 7854 ax-cnre 7855 ax-pre-ltirr 7856 ax-pre-ltwlin 7857 ax-pre-lttrn 7858 ax-pre-apti 7859 ax-pre-ltadd 7860 ax-pre-mulgt0 7861 ax-pre-mulext 7862 ax-arch 7863 ax-caucvg 7864 |
This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rmo 2450 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-if 3516 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-iun 3862 df-disj 3954 df-br 3977 df-opab 4038 df-mpt 4039 df-tr 4075 df-id 4265 df-po 4268 df-iso 4269 df-iord 4338 df-on 4340 df-ilim 4341 df-suc 4343 df-iom 4562 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-isom 5191 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-recs 6264 df-irdg 6329 df-frec 6350 df-1o 6375 df-oadd 6379 df-er 6492 df-en 6698 df-dom 6699 df-fin 6700 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 df-sub 8062 df-neg 8063 df-reap 8464 df-ap 8471 df-div 8560 df-inn 8849 df-2 8907 df-3 8908 df-4 8909 df-n0 9106 df-z 9183 df-uz 9458 df-q 9549 df-rp 9581 df-fz 9936 df-fzo 10068 df-seqfrec 10371 df-exp 10445 df-ihash 10678 df-cj 10770 df-re 10771 df-im 10772 df-rsqrt 10926 df-abs 10927 df-clim 11206 df-sumdc 11281 |
This theorem is referenced by: mertensabs 11464 |
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