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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 2117 | . . . 4 | |
2 | seqfeq3.m | . . . 4 | |
3 | seqfeq3.f | . . . 4 | |
4 | seqfeq3.cl | . . . 4 | |
5 | 1, 2, 3, 4 | seqf 10202 | . . 3 |
6 | 5 | ffnd 5243 | . 2 |
7 | seqfeq3.id | . . . . 5 | |
8 | 7, 4 | eqeltrrd 2195 | . . . 4 |
9 | 1, 2, 3, 8 | seqf 10202 | . . 3 |
10 | 9 | ffnd 5243 | . 2 |
11 | 5 | ffvelrnda 5523 | . . . 4 |
12 | fvi 5446 | . . . 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 5446 | . . . . . 6 | |
19 | 14, 18 | syl 14 | . . . . 5 |
20 | fvi 5446 | . . . . . . 7 | |
21 | 20 | ad2antrl 481 | . . . . . 6 |
22 | fvi 5446 | . . . . . . 7 | |
23 | 22 | ad2antll 482 | . . . . . 6 |
24 | 21, 23 | oveq12d 5760 | . . . . 5 |
25 | 17, 19, 24 | 3eqtr4d 2160 | . . . 4 |
26 | fvi 5446 | . . . . 5 | |
27 | 15, 26 | syl 14 | . . . 4 |
28 | 8 | adantlr 468 | . . . 4 |
29 | 14, 15, 16, 25, 27, 15, 28 | seq3homo 10251 | . . 3 |
30 | 13, 29 | eqtr3d 2152 | . 2 |
31 | 6, 10, 30 | eqfnfvd 5489 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1316 wcel 1465 cid 4180 cfv 5093 (class class class)co 5742 cz 9022 cuz 9294 cseq 10186 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-coll 4013 ax-sep 4016 ax-nul 4024 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-iinf 4472 ax-cnex 7679 ax-resscn 7680 ax-1cn 7681 ax-1re 7682 ax-icn 7683 ax-addcl 7684 ax-addrcl 7685 ax-mulcl 7686 ax-addcom 7688 ax-addass 7690 ax-distr 7692 ax-i2m1 7693 ax-0lt1 7694 ax-0id 7696 ax-rnegex 7697 ax-cnre 7699 ax-pre-ltirr 7700 ax-pre-ltwlin 7701 ax-pre-lttrn 7702 ax-pre-ltadd 7704 |
This theorem depends on definitions: df-bi 116 df-3or 948 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-nel 2381 df-ral 2398 df-rex 2399 df-reu 2400 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-iun 3785 df-br 3900 df-opab 3960 df-mpt 3961 df-tr 3997 df-id 4185 df-iord 4258 df-on 4260 df-ilim 4261 df-suc 4263 df-iom 4475 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 df-fv 5101 df-riota 5698 df-ov 5745 df-oprab 5746 df-mpo 5747 df-1st 6006 df-2nd 6007 df-recs 6170 df-frec 6256 df-pnf 7770 df-mnf 7771 df-xr 7772 df-ltxr 7773 df-le 7774 df-sub 7903 df-neg 7904 df-inn 8689 df-n0 8946 df-z 9023 df-uz 9295 df-seqfrec 10187 |
This theorem is referenced by: (None) |
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