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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 2170 | . . . 4 | |
2 | seqfeq3.m | . . . 4 | |
3 | seqfeq3.f | . . . 4 | |
4 | seqfeq3.cl | . . . 4 | |
5 | 1, 2, 3, 4 | seqf 10410 | . . 3 |
6 | 5 | ffnd 5346 | . 2 |
7 | seqfeq3.id | . . . . 5 | |
8 | 7, 4 | eqeltrrd 2248 | . . . 4 |
9 | 1, 2, 3, 8 | seqf 10410 | . . 3 |
10 | 9 | ffnd 5346 | . 2 |
11 | 5 | ffvelrnda 5629 | . . . 4 |
12 | fvi 5551 | . . . 4 | |
13 | 11, 12 | syl 14 | . . 3 |
14 | 4 | adantlr 474 | . . . 4 |
15 | 3 | adantlr 474 | . . . 4 |
16 | simpr 109 | . . . 4 | |
17 | 7 | adantlr 474 | . . . . 5 |
18 | fvi 5551 | . . . . . 6 | |
19 | 14, 18 | syl 14 | . . . . 5 |
20 | fvi 5551 | . . . . . . 7 | |
21 | 20 | ad2antrl 487 | . . . . . 6 |
22 | fvi 5551 | . . . . . . 7 | |
23 | 22 | ad2antll 488 | . . . . . 6 |
24 | 21, 23 | oveq12d 5869 | . . . . 5 |
25 | 17, 19, 24 | 3eqtr4d 2213 | . . . 4 |
26 | fvi 5551 | . . . . 5 | |
27 | 15, 26 | syl 14 | . . . 4 |
28 | 8 | adantlr 474 | . . . 4 |
29 | 14, 15, 16, 25, 27, 15, 28 | seq3homo 10459 | . . 3 |
30 | 13, 29 | eqtr3d 2205 | . 2 |
31 | 6, 10, 30 | eqfnfvd 5594 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1348 wcel 2141 cid 4271 cfv 5196 (class class class)co 5851 cz 9205 cuz 9480 cseq 10394 |
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 4102 ax-sep 4105 ax-nul 4113 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 ax-iinf 4570 ax-cnex 7858 ax-resscn 7859 ax-1cn 7860 ax-1re 7861 ax-icn 7862 ax-addcl 7863 ax-addrcl 7864 ax-mulcl 7865 ax-addcom 7867 ax-addass 7869 ax-distr 7871 ax-i2m1 7872 ax-0lt1 7873 ax-0id 7875 ax-rnegex 7876 ax-cnre 7878 ax-pre-ltirr 7879 ax-pre-ltwlin 7880 ax-pre-lttrn 7881 ax-pre-ltadd 7883 |
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 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-int 3830 df-iun 3873 df-br 3988 df-opab 4049 df-mpt 4050 df-tr 4086 df-id 4276 df-iord 4349 df-on 4351 df-ilim 4352 df-suc 4354 df-iom 4573 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-res 4621 df-ima 4622 df-iota 5158 df-fun 5198 df-fn 5199 df-f 5200 df-f1 5201 df-fo 5202 df-f1o 5203 df-fv 5204 df-riota 5807 df-ov 5854 df-oprab 5855 df-mpo 5856 df-1st 6117 df-2nd 6118 df-recs 6282 df-frec 6368 df-pnf 7949 df-mnf 7950 df-xr 7951 df-ltxr 7952 df-le 7953 df-sub 8085 df-neg 8086 df-inn 8872 df-n0 9129 df-z 9206 df-uz 9481 df-seqfrec 10395 |
This theorem is referenced by: (None) |
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