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Mirrors > Home > MPE Home > Th. List > seqcaopr | Structured version Visualization version 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 Mario Carneiro, 30-May-2014.) |
Ref | Expression |
---|---|
seqcaopr.1 | |
seqcaopr.2 | |
seqcaopr.3 | |
seqcaopr.4 | |
seqcaopr.5 | |
seqcaopr.6 | |
seqcaopr.7 |
Ref | Expression |
---|---|
seqcaopr |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | seqcaopr.1 | . . 3 | |
2 | 1 | caovclg 6826 | . 2 |
3 | simpl 473 | . . . . . . 7 | |
4 | simprrl 804 | . . . . . . 7 | |
5 | simprlr 803 | . . . . . . 7 | |
6 | seqcaopr.2 | . . . . . . . 8 | |
7 | 6 | caovcomg 6829 | . . . . . . 7 |
8 | 3, 4, 5, 7 | syl12anc 1324 | . . . . . 6 |
9 | 8 | oveq1d 6665 | . . . . 5 |
10 | simprrr 805 | . . . . . 6 | |
11 | seqcaopr.3 | . . . . . . 7 | |
12 | 11 | caovassg 6832 | . . . . . 6 |
13 | 3, 4, 5, 10, 12 | syl13anc 1328 | . . . . 5 |
14 | 11 | caovassg 6832 | . . . . . 6 |
15 | 3, 5, 4, 10, 14 | syl13anc 1328 | . . . . 5 |
16 | 9, 13, 15 | 3eqtr3d 2664 | . . . 4 |
17 | 16 | oveq2d 6666 | . . 3 |
18 | simprll 802 | . . . 4 | |
19 | 1 | caovclg 6826 | . . . . 5 |
20 | 3, 5, 10, 19 | syl12anc 1324 | . . . 4 |
21 | 11 | caovassg 6832 | . . . 4 |
22 | 3, 18, 4, 20, 21 | syl13anc 1328 | . . 3 |
23 | 1 | caovclg 6826 | . . . . 5 |
24 | 23 | adantrl 752 | . . . 4 |
25 | 11 | caovassg 6832 | . . . 4 |
26 | 3, 18, 5, 24, 25 | syl13anc 1328 | . . 3 |
27 | 17, 22, 26 | 3eqtr4d 2666 | . 2 |
28 | seqcaopr.4 | . 2 | |
29 | seqcaopr.5 | . 2 | |
30 | seqcaopr.6 | . 2 | |
31 | seqcaopr.7 | . 2 | |
32 | 2, 2, 27, 28, 29, 30, 31 | seqcaopr2 12837 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 w3a 1037 wceq 1483 wcel 1990 cfv 5888 (class class class)co 6650 cuz 11687 cfz 12326 cseq 12801 |
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-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 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 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 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-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-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-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-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-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-fzo 12466 df-seq 12802 |
This theorem is referenced by: seradd 12843 prodfmul 14622 mulgnn0di 18231 lgsdir 25057 lgsdi 25059 |
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