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| Mirrors > Home > ILE Home > Th. List > seq3f1o | Unicode version | ||
| Description: Rearrange a sum via an
arbitrary bijection on |
| Ref | Expression |
|---|---|
| iseqf1o.1 |
|
| iseqf1o.2 |
|
| iseqf1o.3 |
|
| iseqf1o.4 |
|
| iseqf1o.6 |
|
| iseqf1o.7 |
|
| iseqf1o.h |
|
| iseqf1o.8 |
|
| Ref | Expression |
|---|---|
| seq3f1o |
|
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | iseqf1o.4 |
. . 3
| |
| 2 | elfzle2 10149 |
. . . . . 6
| |
| 3 | 2 | iftrued 3577 |
. . . . 5
|
| 4 | 3 | adantl 277 |
. . . 4
|
| 5 | elfzuz 10142 |
. . . . 5
| |
| 6 | fveq2 5575 |
. . . . . . . 8
| |
| 7 | 6 | eleq1d 2273 |
. . . . . . 7
|
| 8 | iseqf1o.7 |
. . . . . . . . 9
| |
| 9 | 8 | ralrimiva 2578 |
. . . . . . . 8
|
| 10 | 9 | adantr 276 |
. . . . . . 7
|
| 11 | iseqf1o.6 |
. . . . . . . . . 10
| |
| 12 | f1of 5521 |
. . . . . . . . . 10
| |
| 13 | 11, 12 | syl 14 |
. . . . . . . . 9
|
| 14 | 13 | ffvelcdmda 5714 |
. . . . . . . 8
|
| 15 | elfzuz 10142 |
. . . . . . . 8
| |
| 16 | 14, 15 | syl 14 |
. . . . . . 7
|
| 17 | 7, 10, 16 | rspcdva 2881 |
. . . . . 6
|
| 18 | 4, 17 | eqeltrd 2281 |
. . . . 5
|
| 19 | breq1 4046 |
. . . . . . 7
| |
| 20 | 2fveq3 5580 |
. . . . . . 7
| |
| 21 | 19, 20 | ifbieq1d 3592 |
. . . . . 6
|
| 22 | eqid 2204 |
. . . . . 6
| |
| 23 | 21, 22 | fvmptg 5654 |
. . . . 5
|
| 24 | 5, 18, 23 | syl2an2 594 |
. . . 4
|
| 25 | iseqf1o.8 |
. . . 4
| |
| 26 | 4, 24, 25 | 3eqtr4rd 2248 |
. . 3
|
| 27 | iseqf1o.h |
. . 3
| |
| 28 | simpr 110 |
. . . . 5
| |
| 29 | fveq2 5575 |
. . . . . . . 8
| |
| 30 | 29 | eleq1d 2273 |
. . . . . . 7
|
| 31 | fveq2 5575 |
. . . . . . . . . . 11
| |
| 32 | 31 | eleq1d 2273 |
. . . . . . . . . 10
|
| 33 | 32 | cbvralv 2737 |
. . . . . . . . 9
|
| 34 | 9, 33 | sylib 122 |
. . . . . . . 8
|
| 35 | 34 | ad2antrr 488 |
. . . . . . 7
|
| 36 | 13 | ad2antrr 488 |
. . . . . . . . 9
|
| 37 | eluzel2 9652 |
. . . . . . . . . . . 12
| |
| 38 | 1, 37 | syl 14 |
. . . . . . . . . . 11
|
| 39 | 38 | ad2antrr 488 |
. . . . . . . . . 10
|
| 40 | eluzelz 9656 |
. . . . . . . . . . . 12
| |
| 41 | 1, 40 | syl 14 |
. . . . . . . . . . 11
|
| 42 | 41 | ad2antrr 488 |
. . . . . . . . . 10
|
| 43 | eluzelz 9656 |
. . . . . . . . . . 11
| |
| 44 | 43 | ad2antlr 489 |
. . . . . . . . . 10
|
| 45 | eluzle 9659 |
. . . . . . . . . . 11
| |
| 46 | 45 | ad2antlr 489 |
. . . . . . . . . 10
|
| 47 | simpr 110 |
. . . . . . . . . 10
| |
| 48 | elfz4 10139 |
. . . . . . . . . 10
| |
| 49 | 39, 42, 44, 46, 47, 48 | syl32anc 1257 |
. . . . . . . . 9
|
| 50 | 36, 49 | ffvelcdmd 5715 |
. . . . . . . 8
|
| 51 | elfzuz 10142 |
. . . . . . . 8
| |
| 52 | 50, 51 | syl 14 |
. . . . . . 7
|
| 53 | 30, 35, 52 | rspcdva 2881 |
. . . . . 6
|
| 54 | fveq2 5575 |
. . . . . . . . 9
| |
| 55 | 54 | eleq1d 2273 |
. . . . . . . 8
|
| 56 | uzid 9661 |
. . . . . . . . 9
| |
| 57 | 38, 56 | syl 14 |
. . . . . . . 8
|
| 58 | 55, 9, 57 | rspcdva 2881 |
. . . . . . 7
|
| 59 | 58 | ad2antrr 488 |
. . . . . 6
|
| 60 | 41 | adantr 276 |
. . . . . . 7
|
| 61 | zdcle 9448 |
. . . . . . 7
| |
| 62 | 43, 60, 61 | syl2an2 594 |
. . . . . 6
|
| 63 | 53, 59, 62 | ifcldadc 3599 |
. . . . 5
|
| 64 | breq1 4046 |
. . . . . . 7
| |
| 65 | 2fveq3 5580 |
. . . . . . 7
| |
| 66 | 64, 65 | ifbieq1d 3592 |
. . . . . 6
|
| 67 | 66, 22 | fvmptg 5654 |
. . . . 5
|
| 68 | 28, 63, 67 | syl2anc 411 |
. . . 4
|
| 69 | 68, 63 | eqeltrd 2281 |
. . 3
|
| 70 | iseqf1o.1 |
. . 3
| |
| 71 | 1, 26, 27, 69, 70 | seq3fveq 10622 |
. 2
|
| 72 | iseqf1o.2 |
. . 3
| |
| 73 | iseqf1o.3 |
. . 3
| |
| 74 | 66 | cbvmptv 4139 |
. . 3
|
| 75 | 70, 72, 73, 1, 11, 8, 74 | seq3f1oleml 10659 |
. 2
|
| 76 | 71, 75 | eqtrd 2237 |
1
|
| Colors of variables: wff set class |
| Syntax hints: |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1469 ax-7 1470 ax-gen 1471 ax-ie1 1515 ax-ie2 1516 ax-8 1526 ax-10 1527 ax-11 1528 ax-i12 1529 ax-bndl 1531 ax-4 1532 ax-17 1548 ax-i9 1552 ax-ial 1556 ax-i5r 1557 ax-13 2177 ax-14 2178 ax-ext 2186 ax-coll 4158 ax-sep 4161 ax-nul 4169 ax-pow 4217 ax-pr 4252 ax-un 4479 ax-setind 4584 ax-iinf 4635 ax-cnex 8015 ax-resscn 8016 ax-1cn 8017 ax-1re 8018 ax-icn 8019 ax-addcl 8020 ax-addrcl 8021 ax-mulcl 8022 ax-addcom 8024 ax-addass 8026 ax-distr 8028 ax-i2m1 8029 ax-0lt1 8030 ax-0id 8032 ax-rnegex 8033 ax-cnre 8035 ax-pre-ltirr 8036 ax-pre-ltwlin 8037 ax-pre-lttrn 8038 ax-pre-apti 8039 ax-pre-ltadd 8040 |
| This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1375 df-fal 1378 df-nf 1483 df-sb 1785 df-eu 2056 df-mo 2057 df-clab 2191 df-cleq 2197 df-clel 2200 df-nfc 2336 df-ne 2376 df-nel 2471 df-ral 2488 df-rex 2489 df-reu 2490 df-rab 2492 df-v 2773 df-sbc 2998 df-csb 3093 df-dif 3167 df-un 3169 df-in 3171 df-ss 3178 df-nul 3460 df-if 3571 df-pw 3617 df-sn 3638 df-pr 3639 df-op 3641 df-uni 3850 df-int 3885 df-iun 3928 df-br 4044 df-opab 4105 df-mpt 4106 df-tr 4142 df-id 4339 df-iord 4412 df-on 4414 df-ilim 4415 df-suc 4417 df-iom 4638 df-xp 4680 df-rel 4681 df-cnv 4682 df-co 4683 df-dm 4684 df-rn 4685 df-res 4686 df-ima 4687 df-iota 5231 df-fun 5272 df-fn 5273 df-f 5274 df-f1 5275 df-fo 5276 df-f1o 5277 df-fv 5278 df-riota 5898 df-ov 5946 df-oprab 5947 df-mpo 5948 df-1st 6225 df-2nd 6226 df-recs 6390 df-frec 6476 df-1o 6501 df-er 6619 df-en 6827 df-fin 6829 df-pnf 8108 df-mnf 8109 df-xr 8110 df-ltxr 8111 df-le 8112 df-sub 8244 df-neg 8245 df-inn 9036 df-n0 9295 df-z 9372 df-uz 9648 df-fz 10130 df-fzo 10264 df-seqfrec 10591 |
| This theorem is referenced by: summodclem3 11662 prodmodclem3 11857 |
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