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Mirrors > Home > ILE Home > Th. List > seqfeq3 | Unicode version |
Description: Equality of series under different addition operations which agree on an additively closed subset. (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by Mario Carneiro, 25-Apr-2016.) |
Ref | Expression |
---|---|
seqfeq3.m | |
seqfeq3.f | |
seqfeq3.cl | |
seqfeq3.id |
Ref | Expression |
---|---|
seqfeq3 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2139 | . . . 4 | |
2 | seqfeq3.m | . . . 4 | |
3 | seqfeq3.f | . . . 4 | |
4 | seqfeq3.cl | . . . 4 | |
5 | 1, 2, 3, 4 | seqf 10237 | . . 3 |
6 | 5 | ffnd 5273 | . 2 |
7 | seqfeq3.id | . . . . 5 | |
8 | 7, 4 | eqeltrrd 2217 | . . . 4 |
9 | 1, 2, 3, 8 | seqf 10237 | . . 3 |
10 | 9 | ffnd 5273 | . 2 |
11 | 5 | ffvelrnda 5555 | . . . 4 |
12 | fvi 5478 | . . . 4 | |
13 | 11, 12 | syl 14 | . . 3 |
14 | 4 | adantlr 468 | . . . 4 |
15 | 3 | adantlr 468 | . . . 4 |
16 | simpr 109 | . . . 4 | |
17 | 7 | adantlr 468 | . . . . 5 |
18 | fvi 5478 | . . . . . 6 | |
19 | 14, 18 | syl 14 | . . . . 5 |
20 | fvi 5478 | . . . . . . 7 | |
21 | 20 | ad2antrl 481 | . . . . . 6 |
22 | fvi 5478 | . . . . . . 7 | |
23 | 22 | ad2antll 482 | . . . . . 6 |
24 | 21, 23 | oveq12d 5792 | . . . . 5 |
25 | 17, 19, 24 | 3eqtr4d 2182 | . . . 4 |
26 | fvi 5478 | . . . . 5 | |
27 | 15, 26 | syl 14 | . . . 4 |
28 | 8 | adantlr 468 | . . . 4 |
29 | 14, 15, 16, 25, 27, 15, 28 | seq3homo 10286 | . . 3 |
30 | 13, 29 | eqtr3d 2174 | . 2 |
31 | 6, 10, 30 | eqfnfvd 5521 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1331 wcel 1480 cid 4210 cfv 5123 (class class class)co 5774 cz 9057 cuz 9329 cseq 10221 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 ax-cnex 7714 ax-resscn 7715 ax-1cn 7716 ax-1re 7717 ax-icn 7718 ax-addcl 7719 ax-addrcl 7720 ax-mulcl 7721 ax-addcom 7723 ax-addass 7725 ax-distr 7727 ax-i2m1 7728 ax-0lt1 7729 ax-0id 7731 ax-rnegex 7732 ax-cnre 7734 ax-pre-ltirr 7735 ax-pre-ltwlin 7736 ax-pre-lttrn 7737 ax-pre-ltadd 7739 |
This theorem depends on definitions: df-bi 116 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-iord 4288 df-on 4290 df-ilim 4291 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-frec 6288 df-pnf 7805 df-mnf 7806 df-xr 7807 df-ltxr 7808 df-le 7809 df-sub 7938 df-neg 7939 df-inn 8724 df-n0 8981 df-z 9058 df-uz 9330 df-seqfrec 10222 |
This theorem is referenced by: (None) |
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