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Mirrors > Home > ILE Home > Th. List > seq3clss | Unicode version |
Description: Closure property of the recursive sequence builder. (Contributed by Jim Kingdon, 28-Sep-2022.) |
Ref | Expression |
---|---|
seq3clss.n | |
seq3clss.ft | |
seq3clss.fs | |
seq3clss.scl | |
seq3clss.t | |
seq3clss.tcl |
Ref | Expression |
---|---|
seq3clss |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | seq3clss.n | . . 3 | |
2 | eluzfz2 9958 | . . 3 | |
3 | 1, 2 | syl 14 | . 2 |
4 | fveq2 5481 | . . . . 5 | |
5 | 4 | eleq1d 2233 | . . . 4 |
6 | 5 | imbi2d 229 | . . 3 |
7 | fveq2 5481 | . . . . 5 | |
8 | 7 | eleq1d 2233 | . . . 4 |
9 | 8 | imbi2d 229 | . . 3 |
10 | fveq2 5481 | . . . . 5 | |
11 | 10 | eleq1d 2233 | . . . 4 |
12 | 11 | imbi2d 229 | . . 3 |
13 | fveq2 5481 | . . . . 5 | |
14 | 13 | eleq1d 2233 | . . . 4 |
15 | 14 | imbi2d 229 | . . 3 |
16 | eluzel2 9463 | . . . . . . 7 | |
17 | 1, 16 | syl 14 | . . . . . 6 |
18 | seq3clss.ft | . . . . . 6 | |
19 | seq3clss.tcl | . . . . . 6 | |
20 | 17, 18, 19 | seq3-1 10386 | . . . . 5 |
21 | fveq2 5481 | . . . . . . 7 | |
22 | 21 | eleq1d 2233 | . . . . . 6 |
23 | seq3clss.fs | . . . . . . 7 | |
24 | 23 | ralrimiva 2537 | . . . . . 6 |
25 | eluzfz1 9957 | . . . . . . 7 | |
26 | 1, 25 | syl 14 | . . . . . 6 |
27 | 22, 24, 26 | rspcdva 2831 | . . . . 5 |
28 | 20, 27 | eqeltrd 2241 | . . . 4 |
29 | 28 | a1i 9 | . . 3 |
30 | elfzouz 10077 | . . . . . . . . 9 ..^ | |
31 | 30 | ad2antlr 481 | . . . . . . . 8 ..^ |
32 | 18 | adantlr 469 | . . . . . . . . 9 ..^ |
33 | 32 | adantlr 469 | . . . . . . . 8 ..^ |
34 | 19 | adantlr 469 | . . . . . . . . 9 ..^ |
35 | 34 | adantlr 469 | . . . . . . . 8 ..^ |
36 | 31, 33, 35 | seq3p1 10388 | . . . . . . 7 ..^ |
37 | seq3clss.scl | . . . . . . . . . 10 | |
38 | 37 | adantlr 469 | . . . . . . . . 9 ..^ |
39 | 38 | adantlr 469 | . . . . . . . 8 ..^ |
40 | simpr 109 | . . . . . . . 8 ..^ | |
41 | fveq2 5481 | . . . . . . . . . 10 | |
42 | 41 | eleq1d 2233 | . . . . . . . . 9 |
43 | 24 | ad2antrr 480 | . . . . . . . . 9 ..^ |
44 | fzofzp1 10153 | . . . . . . . . . 10 ..^ | |
45 | 44 | ad2antlr 481 | . . . . . . . . 9 ..^ |
46 | 42, 43, 45 | rspcdva 2831 | . . . . . . . 8 ..^ |
47 | 39, 40, 46 | caovcld 5987 | . . . . . . 7 ..^ |
48 | 36, 47 | eqeltrd 2241 | . . . . . 6 ..^ |
49 | 48 | ex 114 | . . . . 5 ..^ |
50 | 49 | expcom 115 | . . . 4 ..^ |
51 | 50 | a2d 26 | . . 3 ..^ |
52 | 6, 9, 12, 15, 29, 51 | fzind2 10165 | . 2 |
53 | 3, 52 | mpcom 36 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1342 wcel 2135 wral 2442 wss 3112 cfv 5183 (class class class)co 5837 c1 7746 caddc 7748 cz 9183 cuz 9458 cfz 9936 ..^cfzo 10068 cseq 10371 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4092 ax-sep 4095 ax-nul 4103 ax-pow 4148 ax-pr 4182 ax-un 4406 ax-setind 4509 ax-iinf 4560 ax-cnex 7836 ax-resscn 7837 ax-1cn 7838 ax-1re 7839 ax-icn 7840 ax-addcl 7841 ax-addrcl 7842 ax-mulcl 7843 ax-addcom 7845 ax-addass 7847 ax-distr 7849 ax-i2m1 7850 ax-0lt1 7851 ax-0id 7853 ax-rnegex 7854 ax-cnre 7856 ax-pre-ltirr 7857 ax-pre-ltwlin 7858 ax-pre-lttrn 7859 ax-pre-ltadd 7861 |
This theorem depends on definitions: df-bi 116 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2724 df-sbc 2948 df-csb 3042 df-dif 3114 df-un 3116 df-in 3118 df-ss 3125 df-nul 3406 df-pw 3556 df-sn 3577 df-pr 3578 df-op 3580 df-uni 3785 df-int 3820 df-iun 3863 df-br 3978 df-opab 4039 df-mpt 4040 df-tr 4076 df-id 4266 df-iord 4339 df-on 4341 df-ilim 4342 df-suc 4344 df-iom 4563 df-xp 4605 df-rel 4606 df-cnv 4607 df-co 4608 df-dm 4609 df-rn 4610 df-res 4611 df-ima 4612 df-iota 5148 df-fun 5185 df-fn 5186 df-f 5187 df-f1 5188 df-fo 5189 df-f1o 5190 df-fv 5191 df-riota 5793 df-ov 5840 df-oprab 5841 df-mpo 5842 df-1st 6101 df-2nd 6102 df-recs 6265 df-frec 6351 df-pnf 7927 df-mnf 7928 df-xr 7929 df-ltxr 7930 df-le 7931 df-sub 8063 df-neg 8064 df-inn 8850 df-n0 9107 df-z 9184 df-uz 9459 df-fz 9937 df-fzo 10069 df-seqfrec 10372 |
This theorem is referenced by: fsumcl2lem 11329 |
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