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Mirrors > Home > ILE Home > Th. List > seq3caopr2 | Unicode version |
Description: The sum of two infinite series (generalized to an arbitrary commutative and associative operation). (Contributed by Mario Carneiro, 30-May-2014.) (Revised by Jim Kingdon, 23-Apr-2023.) |
Ref | Expression |
---|---|
seqcaopr2.1 | |
seqcaopr2.2 | |
seqcaopr2.3 | |
seqcaopr2.4 | |
seq3caopr2.5 | |
seq3caopr2.6 | |
seq3caopr2.7 |
Ref | Expression |
---|---|
seq3caopr2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | seqcaopr2.1 | . 2 | |
2 | seqcaopr2.2 | . 2 | |
3 | seqcaopr2.4 | . 2 | |
4 | seq3caopr2.5 | . 2 | |
5 | seq3caopr2.6 | . 2 | |
6 | seq3caopr2.7 | . 2 | |
7 | eqid 2157 | . . . . 5 | |
8 | eluzel2 9445 | . . . . . . 7 | |
9 | 3, 8 | syl 14 | . . . . . 6 |
10 | 9 | adantr 274 | . . . . 5 ..^ |
11 | 5 | ralrimiva 2530 | . . . . . . 7 |
12 | 11 | adantr 274 | . . . . . 6 ..^ |
13 | fveq2 5469 | . . . . . . . 8 | |
14 | 13 | eleq1d 2226 | . . . . . . 7 |
15 | 14 | rspccva 2815 | . . . . . 6 |
16 | 12, 15 | sylan 281 | . . . . 5 ..^ |
17 | 1 | adantlr 469 | . . . . 5 ..^ |
18 | 7, 10, 16, 17 | seqf 10364 | . . . 4 ..^ |
19 | elfzouz 10054 | . . . . 5 ..^ | |
20 | 19 | adantl 275 | . . . 4 ..^ |
21 | 18, 20 | ffvelrnd 5604 | . . 3 ..^ |
22 | fzssuz 9968 | . . . . 5 | |
23 | fzofzp1 10130 | . . . . 5 ..^ | |
24 | 22, 23 | sseldi 3126 | . . . 4 ..^ |
25 | fveq2 5469 | . . . . . 6 | |
26 | 25 | eleq1d 2226 | . . . . 5 |
27 | 26 | rspccva 2815 | . . . 4 |
28 | 11, 24, 27 | syl2an 287 | . . 3 ..^ |
29 | 4 | ralrimiva 2530 | . . . . . . . 8 |
30 | fveq2 5469 | . . . . . . . . . 10 | |
31 | 30 | eleq1d 2226 | . . . . . . . . 9 |
32 | 31 | rspccva 2815 | . . . . . . . 8 |
33 | 29, 32 | sylan 281 | . . . . . . 7 |
34 | 33 | adantlr 469 | . . . . . 6 ..^ |
35 | 7, 10, 34, 17 | seqf 10364 | . . . . 5 ..^ |
36 | 35, 20 | ffvelrnd 5604 | . . . 4 ..^ |
37 | fveq2 5469 | . . . . . . 7 | |
38 | 37 | eleq1d 2226 | . . . . . 6 |
39 | 38 | rspccva 2815 | . . . . 5 |
40 | 29, 24, 39 | syl2an 287 | . . . 4 ..^ |
41 | seqcaopr2.3 | . . . . . . . 8 | |
42 | 41 | anassrs 398 | . . . . . . 7 |
43 | 42 | ralrimivva 2539 | . . . . . 6 |
44 | 43 | ralrimivva 2539 | . . . . 5 |
45 | 44 | adantr 274 | . . . 4 ..^ |
46 | oveq1 5832 | . . . . . . . 8 | |
47 | 46 | oveq1d 5840 | . . . . . . 7 |
48 | oveq1 5832 | . . . . . . . 8 | |
49 | 48 | oveq1d 5840 | . . . . . . 7 |
50 | 47, 49 | eqeq12d 2172 | . . . . . 6 |
51 | 50 | 2ralbidv 2481 | . . . . 5 |
52 | oveq1 5832 | . . . . . . . 8 | |
53 | 52 | oveq2d 5841 | . . . . . . 7 |
54 | oveq2 5833 | . . . . . . . 8 | |
55 | 54 | oveq1d 5840 | . . . . . . 7 |
56 | 53, 55 | eqeq12d 2172 | . . . . . 6 |
57 | 56 | 2ralbidv 2481 | . . . . 5 |
58 | 51, 57 | rspc2va 2830 | . . . 4 |
59 | 36, 40, 45, 58 | syl21anc 1219 | . . 3 ..^ |
60 | oveq2 5833 | . . . . . 6 | |
61 | 60 | oveq1d 5840 | . . . . 5 |
62 | oveq1 5832 | . . . . . 6 | |
63 | 62 | oveq2d 5841 | . . . . 5 |
64 | 61, 63 | eqeq12d 2172 | . . . 4 |
65 | oveq2 5833 | . . . . . 6 | |
66 | 65 | oveq2d 5841 | . . . . 5 |
67 | oveq2 5833 | . . . . . 6 | |
68 | 67 | oveq2d 5841 | . . . . 5 |
69 | 66, 68 | eqeq12d 2172 | . . . 4 |
70 | 64, 69 | rspc2va 2830 | . . 3 |
71 | 21, 28, 59, 70 | syl21anc 1219 | . 2 ..^ |
72 | 1, 2, 3, 4, 5, 6, 71 | seq3caopr3 10384 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1335 wcel 2128 wral 2435 cfv 5171 (class class class)co 5825 c1 7734 caddc 7736 cz 9168 cuz 9440 cfz 9913 ..^cfzo 10045 cseq 10348 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-coll 4080 ax-sep 4083 ax-nul 4091 ax-pow 4136 ax-pr 4170 ax-un 4394 ax-setind 4497 ax-iinf 4548 ax-cnex 7824 ax-resscn 7825 ax-1cn 7826 ax-1re 7827 ax-icn 7828 ax-addcl 7829 ax-addrcl 7830 ax-mulcl 7831 ax-addcom 7833 ax-addass 7835 ax-distr 7837 ax-i2m1 7838 ax-0lt1 7839 ax-0id 7841 ax-rnegex 7842 ax-cnre 7844 ax-pre-ltirr 7845 ax-pre-ltwlin 7846 ax-pre-lttrn 7847 ax-pre-ltadd 7849 |
This theorem depends on definitions: df-bi 116 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-reu 2442 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3774 df-int 3809 df-iun 3852 df-br 3967 df-opab 4027 df-mpt 4028 df-tr 4064 df-id 4254 df-iord 4327 df-on 4329 df-ilim 4330 df-suc 4332 df-iom 4551 df-xp 4593 df-rel 4594 df-cnv 4595 df-co 4596 df-dm 4597 df-rn 4598 df-res 4599 df-ima 4600 df-iota 5136 df-fun 5173 df-fn 5174 df-f 5175 df-f1 5176 df-fo 5177 df-f1o 5178 df-fv 5179 df-riota 5781 df-ov 5828 df-oprab 5829 df-mpo 5830 df-1st 6089 df-2nd 6090 df-recs 6253 df-frec 6339 df-pnf 7915 df-mnf 7916 df-xr 7917 df-ltxr 7918 df-le 7919 df-sub 8049 df-neg 8050 df-inn 8835 df-n0 9092 df-z 9169 df-uz 9441 df-fz 9914 df-fzo 10046 df-seqfrec 10349 |
This theorem is referenced by: seq3caopr 10386 ser3sub 10409 |
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