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Mirrors > Home > ILE Home > Th. List > seq3caopr | Unicode version |
Description: The sum of two infinite series (generalized to an arbitrary commutative and associative operation). (Contributed by NM, 17-Mar-2005.) (Revised by Jim Kingdon, 23-Apr-2023.) |
Ref | Expression |
---|---|
seqcaopr.1 |
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seqcaopr.2 |
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seqcaopr.3 |
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seqcaopr.4 |
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seq3caopr.5 |
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seq3caopr.6 |
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seq3caopr.7 |
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Ref | Expression |
---|---|
seq3caopr |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | seqcaopr.1 |
. . 3
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2 | 1 | caovclg 6017 |
. 2
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3 | simpl 109 |
. . . . . . 7
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4 | simprrl 539 |
. . . . . . 7
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5 | simprlr 538 |
. . . . . . 7
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6 | seqcaopr.2 |
. . . . . . . 8
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7 | 6 | caovcomg 6020 |
. . . . . . 7
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8 | 3, 4, 5, 7 | syl12anc 1236 |
. . . . . 6
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9 | 8 | oveq1d 5880 |
. . . . 5
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10 | simprrr 540 |
. . . . . 6
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11 | seqcaopr.3 |
. . . . . . 7
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12 | 11 | caovassg 6023 |
. . . . . 6
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13 | 3, 4, 5, 10, 12 | syl13anc 1240 |
. . . . 5
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14 | 11 | caovassg 6023 |
. . . . . 6
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15 | 3, 5, 4, 10, 14 | syl13anc 1240 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
16 | 9, 13, 15 | 3eqtr3d 2216 |
. . . 4
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17 | 16 | oveq2d 5881 |
. . 3
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18 | simprll 537 |
. . . 4
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19 | 1 | caovclg 6017 |
. . . . 5
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20 | 3, 5, 10, 19 | syl12anc 1236 |
. . . 4
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21 | 11 | caovassg 6023 |
. . . 4
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22 | 3, 18, 4, 20, 21 | syl13anc 1240 |
. . 3
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23 | 1 | caovclg 6017 |
. . . . 5
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24 | 23 | adantrl 478 |
. . . 4
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25 | 11 | caovassg 6023 |
. . . 4
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26 | 3, 18, 5, 24, 25 | syl13anc 1240 |
. . 3
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27 | 17, 22, 26 | 3eqtr4d 2218 |
. 2
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28 | seqcaopr.4 |
. 2
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29 | seq3caopr.5 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
30 | seq3caopr.6 |
. 2
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31 | seq3caopr.7 |
. 2
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32 | 2, 2, 27, 28, 29, 30, 31 | seq3caopr2 10450 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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-addcom 7886 ax-addass 7888 ax-distr 7890 ax-i2m1 7891 ax-0lt1 7892 ax-0id 7894 ax-rnegex 7895 ax-cnre 7897 ax-pre-ltirr 7898 ax-pre-ltwlin 7899 ax-pre-lttrn 7900 ax-pre-ltadd 7902 |
This theorem depends on definitions: df-bi 117 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-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-pw 3574 df-sn 3595 df-pr 3596 df-op 3598 df-uni 3806 df-int 3841 df-iun 3884 df-br 3999 df-opab 4060 df-mpt 4061 df-tr 4097 df-id 4287 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-riota 5821 df-ov 5868 df-oprab 5869 df-mpo 5870 df-1st 6131 df-2nd 6132 df-recs 6296 df-frec 6382 df-pnf 7968 df-mnf 7969 df-xr 7970 df-ltxr 7971 df-le 7972 df-sub 8104 df-neg 8105 df-inn 8891 df-n0 9148 df-z 9225 df-uz 9500 df-fz 9978 df-fzo 10111 df-seqfrec 10414 |
This theorem is referenced by: ser3add 10473 prod3fmul 11515 |
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