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Mirrors > Home > ILE Home > Th. List > fprodrev | Unicode version |
Description: Reversal of a finite product. (Contributed by Scott Fenton, 5-Jan-2018.) |
Ref | Expression |
---|---|
fprodshft.1 | |
fprodshft.2 | |
fprodshft.3 | |
fprodshft.4 | |
fprodrev.5 |
Ref | Expression |
---|---|
fprodrev |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fprodrev.5 | . 2 | |
2 | fprodshft.1 | . . . 4 | |
3 | fprodshft.3 | . . . 4 | |
4 | 2, 3 | zsubcld 9285 | . . 3 |
5 | fprodshft.2 | . . . 4 | |
6 | 2, 5 | zsubcld 9285 | . . 3 |
7 | 4, 6 | fzfigd 10323 | . 2 |
8 | eqid 2157 | . . 3 | |
9 | 2 | adantr 274 | . . . 4 |
10 | elfzelz 9921 | . . . . 5 | |
11 | 10 | adantl 275 | . . . 4 |
12 | 9, 11 | zsubcld 9285 | . . 3 |
13 | 2 | adantr 274 | . . . 4 |
14 | elfzelz 9921 | . . . . 5 | |
15 | 14 | adantl 275 | . . . 4 |
16 | 13, 15 | zsubcld 9285 | . . 3 |
17 | simprr 522 | . . . . . 6 | |
18 | simprl 521 | . . . . . . 7 | |
19 | 5 | adantr 274 | . . . . . . . 8 |
20 | 3 | adantr 274 | . . . . . . . 8 |
21 | 2 | adantr 274 | . . . . . . . 8 |
22 | 10 | ad2antrl 482 | . . . . . . . 8 |
23 | fzrev 9979 | . . . . . . . 8 | |
24 | 19, 20, 21, 22, 23 | syl22anc 1221 | . . . . . . 7 |
25 | 18, 24 | mpbid 146 | . . . . . 6 |
26 | 17, 25 | eqeltrd 2234 | . . . . 5 |
27 | oveq2 5829 | . . . . . . 7 | |
28 | 27 | ad2antll 483 | . . . . . 6 |
29 | 2 | zcnd 9281 | . . . . . . . 8 |
30 | 29 | adantr 274 | . . . . . . 7 |
31 | 10 | zcnd 9281 | . . . . . . . 8 |
32 | 31 | ad2antrl 482 | . . . . . . 7 |
33 | 30, 32 | nncand 8185 | . . . . . 6 |
34 | 28, 33 | eqtr2d 2191 | . . . . 5 |
35 | 26, 34 | jca 304 | . . . 4 |
36 | simprr 522 | . . . . . 6 | |
37 | simprl 521 | . . . . . . 7 | |
38 | 5 | adantr 274 | . . . . . . . 8 |
39 | 3 | adantr 274 | . . . . . . . 8 |
40 | 2 | adantr 274 | . . . . . . . 8 |
41 | 14 | ad2antrl 482 | . . . . . . . 8 |
42 | fzrev2 9980 | . . . . . . . 8 | |
43 | 38, 39, 40, 41, 42 | syl22anc 1221 | . . . . . . 7 |
44 | 37, 43 | mpbid 146 | . . . . . 6 |
45 | 36, 44 | eqeltrd 2234 | . . . . 5 |
46 | oveq2 5829 | . . . . . . 7 | |
47 | 46 | ad2antll 483 | . . . . . 6 |
48 | 29 | adantr 274 | . . . . . . 7 |
49 | 14 | zcnd 9281 | . . . . . . . 8 |
50 | 49 | ad2antrl 482 | . . . . . . 7 |
51 | 48, 50 | nncand 8185 | . . . . . 6 |
52 | 47, 51 | eqtr2d 2191 | . . . . 5 |
53 | 45, 52 | jca 304 | . . . 4 |
54 | 35, 53 | impbida 586 | . . 3 |
55 | 8, 12, 16, 54 | f1od 6020 | . 2 |
56 | oveq2 5829 | . . 3 | |
57 | simpr 109 | . . 3 | |
58 | 2 | adantr 274 | . . . 4 |
59 | elfzelz 9921 | . . . . 5 | |
60 | 59 | adantl 275 | . . . 4 |
61 | 58, 60 | zsubcld 9285 | . . 3 |
62 | 8, 56, 57, 61 | fvmptd3 5560 | . 2 |
63 | fprodshft.4 | . 2 | |
64 | 1, 7, 55, 62, 63 | fprodf1o 11478 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1335 wcel 2128 cmpt 4025 (class class class)co 5821 cc 7724 cmin 8040 cz 9161 cfz 9905 cprod 11440 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-coll 4079 ax-sep 4082 ax-nul 4090 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4495 ax-iinf 4546 ax-cnex 7817 ax-resscn 7818 ax-1cn 7819 ax-1re 7820 ax-icn 7821 ax-addcl 7822 ax-addrcl 7823 ax-mulcl 7824 ax-mulrcl 7825 ax-addcom 7826 ax-mulcom 7827 ax-addass 7828 ax-mulass 7829 ax-distr 7830 ax-i2m1 7831 ax-0lt1 7832 ax-1rid 7833 ax-0id 7834 ax-rnegex 7835 ax-precex 7836 ax-cnre 7837 ax-pre-ltirr 7838 ax-pre-ltwlin 7839 ax-pre-lttrn 7840 ax-pre-apti 7841 ax-pre-ltadd 7842 ax-pre-mulgt0 7843 ax-pre-mulext 7844 ax-arch 7845 ax-caucvg 7846 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-reu 2442 df-rmo 2443 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-if 3506 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-int 3808 df-iun 3851 df-br 3966 df-opab 4026 df-mpt 4027 df-tr 4063 df-id 4253 df-po 4256 df-iso 4257 df-iord 4326 df-on 4328 df-ilim 4329 df-suc 4331 df-iom 4549 df-xp 4591 df-rel 4592 df-cnv 4593 df-co 4594 df-dm 4595 df-rn 4596 df-res 4597 df-ima 4598 df-iota 5134 df-fun 5171 df-fn 5172 df-f 5173 df-f1 5174 df-fo 5175 df-f1o 5176 df-fv 5177 df-isom 5178 df-riota 5777 df-ov 5824 df-oprab 5825 df-mpo 5826 df-1st 6085 df-2nd 6086 df-recs 6249 df-irdg 6314 df-frec 6335 df-1o 6360 df-oadd 6364 df-er 6477 df-en 6683 df-dom 6684 df-fin 6685 df-pnf 7908 df-mnf 7909 df-xr 7910 df-ltxr 7911 df-le 7912 df-sub 8042 df-neg 8043 df-reap 8444 df-ap 8451 df-div 8540 df-inn 8828 df-2 8886 df-3 8887 df-4 8888 df-n0 9085 df-z 9162 df-uz 9434 df-q 9522 df-rp 9554 df-fz 9906 df-fzo 10035 df-seqfrec 10338 df-exp 10412 df-ihash 10643 df-cj 10735 df-re 10736 df-im 10737 df-rsqrt 10891 df-abs 10892 df-clim 11169 df-proddc 11441 |
This theorem is referenced by: (None) |
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