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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 2170 | . . . . 5 | |
8 | eluzel2 9492 | . . . . . . 7 | |
9 | 3, 8 | syl 14 | . . . . . 6 |
10 | 9 | adantr 274 | . . . . 5 ..^ |
11 | 5 | ralrimiva 2543 | . . . . . . 7 |
12 | 11 | adantr 274 | . . . . . 6 ..^ |
13 | fveq2 5496 | . . . . . . . 8 | |
14 | 13 | eleq1d 2239 | . . . . . . 7 |
15 | 14 | rspccva 2833 | . . . . . 6 |
16 | 12, 15 | sylan 281 | . . . . 5 ..^ |
17 | 1 | adantlr 474 | . . . . 5 ..^ |
18 | 7, 10, 16, 17 | seqf 10417 | . . . 4 ..^ |
19 | elfzouz 10107 | . . . . 5 ..^ | |
20 | 19 | adantl 275 | . . . 4 ..^ |
21 | 18, 20 | ffvelrnd 5632 | . . 3 ..^ |
22 | fzssuz 10021 | . . . . 5 | |
23 | fzofzp1 10183 | . . . . 5 ..^ | |
24 | 22, 23 | sselid 3145 | . . . 4 ..^ |
25 | fveq2 5496 | . . . . . 6 | |
26 | 25 | eleq1d 2239 | . . . . 5 |
27 | 26 | rspccva 2833 | . . . 4 |
28 | 11, 24, 27 | syl2an 287 | . . 3 ..^ |
29 | 4 | ralrimiva 2543 | . . . . . . . 8 |
30 | fveq2 5496 | . . . . . . . . . 10 | |
31 | 30 | eleq1d 2239 | . . . . . . . . 9 |
32 | 31 | rspccva 2833 | . . . . . . . 8 |
33 | 29, 32 | sylan 281 | . . . . . . 7 |
34 | 33 | adantlr 474 | . . . . . 6 ..^ |
35 | 7, 10, 34, 17 | seqf 10417 | . . . . 5 ..^ |
36 | 35, 20 | ffvelrnd 5632 | . . . 4 ..^ |
37 | fveq2 5496 | . . . . . . 7 | |
38 | 37 | eleq1d 2239 | . . . . . 6 |
39 | 38 | rspccva 2833 | . . . . 5 |
40 | 29, 24, 39 | syl2an 287 | . . . 4 ..^ |
41 | seqcaopr2.3 | . . . . . . . 8 | |
42 | 41 | anassrs 398 | . . . . . . 7 |
43 | 42 | ralrimivva 2552 | . . . . . 6 |
44 | 43 | ralrimivva 2552 | . . . . 5 |
45 | 44 | adantr 274 | . . . 4 ..^ |
46 | oveq1 5860 | . . . . . . . 8 | |
47 | 46 | oveq1d 5868 | . . . . . . 7 |
48 | oveq1 5860 | . . . . . . . 8 | |
49 | 48 | oveq1d 5868 | . . . . . . 7 |
50 | 47, 49 | eqeq12d 2185 | . . . . . 6 |
51 | 50 | 2ralbidv 2494 | . . . . 5 |
52 | oveq1 5860 | . . . . . . . 8 | |
53 | 52 | oveq2d 5869 | . . . . . . 7 |
54 | oveq2 5861 | . . . . . . . 8 | |
55 | 54 | oveq1d 5868 | . . . . . . 7 |
56 | 53, 55 | eqeq12d 2185 | . . . . . 6 |
57 | 56 | 2ralbidv 2494 | . . . . 5 |
58 | 51, 57 | rspc2va 2848 | . . . 4 |
59 | 36, 40, 45, 58 | syl21anc 1232 | . . 3 ..^ |
60 | oveq2 5861 | . . . . . 6 | |
61 | 60 | oveq1d 5868 | . . . . 5 |
62 | oveq1 5860 | . . . . . 6 | |
63 | 62 | oveq2d 5869 | . . . . 5 |
64 | 61, 63 | eqeq12d 2185 | . . . 4 |
65 | oveq2 5861 | . . . . . 6 | |
66 | 65 | oveq2d 5869 | . . . . 5 |
67 | oveq2 5861 | . . . . . 6 | |
68 | 67 | oveq2d 5869 | . . . . 5 |
69 | 66, 68 | eqeq12d 2185 | . . . 4 |
70 | 64, 69 | rspc2va 2848 | . . 3 |
71 | 21, 28, 59, 70 | syl21anc 1232 | . 2 ..^ |
72 | 1, 2, 3, 4, 5, 6, 71 | seq3caopr3 10437 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1348 wcel 2141 wral 2448 cfv 5198 (class class class)co 5853 c1 7775 caddc 7777 cz 9212 cuz 9487 cfz 9965 ..^cfzo 10098 cseq 10401 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4104 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-iinf 4572 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-addcom 7874 ax-addass 7876 ax-distr 7878 ax-i2m1 7879 ax-0lt1 7880 ax-0id 7882 ax-rnegex 7883 ax-cnre 7885 ax-pre-ltirr 7886 ax-pre-ltwlin 7887 ax-pre-lttrn 7888 ax-pre-ltadd 7890 |
This theorem depends on definitions: df-bi 116 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-tr 4088 df-id 4278 df-iord 4351 df-on 4353 df-ilim 4354 df-suc 4356 df-iom 4575 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-1st 6119 df-2nd 6120 df-recs 6284 df-frec 6370 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-le 7960 df-sub 8092 df-neg 8093 df-inn 8879 df-n0 9136 df-z 9213 df-uz 9488 df-fz 9966 df-fzo 10099 df-seqfrec 10402 |
This theorem is referenced by: seq3caopr 10439 ser3sub 10462 |
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