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Mirrors > Home > ILE Home > Th. List > prodfap0 | Unicode version |
Description: The product of finitely many terms apart from zero is apart from zero. (Contributed by Scott Fenton, 14-Jan-2018.) (Revised by Jim Kingdon, 23-Mar-2024.) |
Ref | Expression |
---|---|
prodfap0.1 | |
prodfap0.2 | |
prodfap0.3 | # |
Ref | Expression |
---|---|
prodfap0 | # |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prodfap0.1 | . . 3 | |
2 | eluzfz2 9815 | . . 3 | |
3 | 1, 2 | syl 14 | . 2 |
4 | fveq2 5421 | . . . . 5 | |
5 | 4 | breq1d 3939 | . . . 4 # # |
6 | 5 | imbi2d 229 | . . 3 # # |
7 | fveq2 5421 | . . . . 5 | |
8 | 7 | breq1d 3939 | . . . 4 # # |
9 | 8 | imbi2d 229 | . . 3 # # |
10 | fveq2 5421 | . . . . 5 | |
11 | 10 | breq1d 3939 | . . . 4 # # |
12 | 11 | imbi2d 229 | . . 3 # # |
13 | fveq2 5421 | . . . . 5 | |
14 | 13 | breq1d 3939 | . . . 4 # # |
15 | 14 | imbi2d 229 | . . 3 # # |
16 | eluzfz1 9814 | . . . 4 | |
17 | elfzelz 9809 | . . . . . . . 8 | |
18 | 17 | adantl 275 | . . . . . . 7 |
19 | prodfap0.2 | . . . . . . . 8 | |
20 | 19 | adantlr 468 | . . . . . . 7 |
21 | mulcl 7750 | . . . . . . . 8 | |
22 | 21 | adantl 275 | . . . . . . 7 |
23 | 18, 20, 22 | seq3-1 10236 | . . . . . 6 |
24 | fveq2 5421 | . . . . . . . . . 10 | |
25 | 24 | breq1d 3939 | . . . . . . . . 9 # # |
26 | 25 | imbi2d 229 | . . . . . . . 8 # # |
27 | prodfap0.3 | . . . . . . . . 9 # | |
28 | 27 | expcom 115 | . . . . . . . 8 # |
29 | 26, 28 | vtoclga 2752 | . . . . . . 7 # |
30 | 29 | impcom 124 | . . . . . 6 # |
31 | 23, 30 | eqbrtrd 3950 | . . . . 5 # |
32 | 31 | expcom 115 | . . . 4 # |
33 | 16, 32 | syl 14 | . . 3 # |
34 | elfzouz 9931 | . . . . . . . . 9 ..^ | |
35 | 34 | 3ad2ant2 1003 | . . . . . . . 8 ..^ # |
36 | 19 | 3ad2antl1 1143 | . . . . . . . 8 ..^ # |
37 | 21 | adantl 275 | . . . . . . . 8 ..^ # |
38 | 35, 36, 37 | seq3p1 10238 | . . . . . . 7 ..^ # |
39 | elfzofz 9942 | . . . . . . . . . 10 ..^ | |
40 | elfzuz 9805 | . . . . . . . . . . 11 | |
41 | eqid 2139 | . . . . . . . . . . . . 13 | |
42 | 1, 16, 17 | 3syl 17 | . . . . . . . . . . . . 13 |
43 | 41, 42, 19 | prodf 11310 | . . . . . . . . . . . 12 |
44 | 43 | ffvelrnda 5555 | . . . . . . . . . . 11 |
45 | 40, 44 | sylan2 284 | . . . . . . . . . 10 |
46 | 39, 45 | sylan2 284 | . . . . . . . . 9 ..^ |
47 | 46 | 3adant3 1001 | . . . . . . . 8 ..^ # |
48 | fzofzp1 10007 | . . . . . . . . . . 11 ..^ | |
49 | fveq2 5421 | . . . . . . . . . . . . . 14 | |
50 | 49 | eleq1d 2208 | . . . . . . . . . . . . 13 |
51 | 50 | imbi2d 229 | . . . . . . . . . . . 12 |
52 | elfzuz 9805 | . . . . . . . . . . . . 13 | |
53 | 19 | expcom 115 | . . . . . . . . . . . . 13 |
54 | 52, 53 | syl 14 | . . . . . . . . . . . 12 |
55 | 51, 54 | vtoclga 2752 | . . . . . . . . . . 11 |
56 | 48, 55 | syl 14 | . . . . . . . . . 10 ..^ |
57 | 56 | impcom 124 | . . . . . . . . 9 ..^ |
58 | 57 | 3adant3 1001 | . . . . . . . 8 ..^ # |
59 | simp3 983 | . . . . . . . 8 ..^ # # | |
60 | 49 | breq1d 3939 | . . . . . . . . . . . . 13 # # |
61 | 60 | imbi2d 229 | . . . . . . . . . . . 12 # # |
62 | 61, 28 | vtoclga 2752 | . . . . . . . . . . 11 # |
63 | 62 | impcom 124 | . . . . . . . . . 10 # |
64 | 48, 63 | sylan2 284 | . . . . . . . . 9 ..^ # |
65 | 64 | 3adant3 1001 | . . . . . . . 8 ..^ # # |
66 | 47, 58, 59, 65 | mulap0d 8422 | . . . . . . 7 ..^ # # |
67 | 38, 66 | eqbrtrd 3950 | . . . . . 6 ..^ # # |
68 | 67 | 3exp 1180 | . . . . 5 ..^ # # |
69 | 68 | com12 30 | . . . 4 ..^ # # |
70 | 69 | a2d 26 | . . 3 ..^ # # |
71 | 6, 9, 12, 15, 33, 70 | fzind2 10019 | . 2 # |
72 | 3, 71 | mpcom 36 | 1 # |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 962 wceq 1331 wcel 1480 class class class wbr 3929 cfv 5123 (class class class)co 5774 cc 7621 cc0 7623 c1 7624 caddc 7626 cmul 7628 # cap 8346 cz 9057 cuz 9329 cfz 9793 ..^cfzo 9922 cseq 10221 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 ax-cnex 7714 ax-resscn 7715 ax-1cn 7716 ax-1re 7717 ax-icn 7718 ax-addcl 7719 ax-addrcl 7720 ax-mulcl 7721 ax-mulrcl 7722 ax-addcom 7723 ax-mulcom 7724 ax-addass 7725 ax-mulass 7726 ax-distr 7727 ax-i2m1 7728 ax-0lt1 7729 ax-1rid 7730 ax-0id 7731 ax-rnegex 7732 ax-precex 7733 ax-cnre 7734 ax-pre-ltirr 7735 ax-pre-ltwlin 7736 ax-pre-lttrn 7737 ax-pre-apti 7738 ax-pre-ltadd 7739 ax-pre-mulgt0 7740 ax-pre-mulext 7741 |
This theorem depends on definitions: df-bi 116 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-po 4218 df-iso 4219 df-iord 4288 df-on 4290 df-ilim 4291 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-frec 6288 df-pnf 7805 df-mnf 7806 df-xr 7807 df-ltxr 7808 df-le 7809 df-sub 7938 df-neg 7939 df-reap 8340 df-ap 8347 df-inn 8724 df-n0 8981 df-z 9058 df-uz 9330 df-fz 9794 df-fzo 9923 df-seqfrec 10222 |
This theorem is referenced by: prodfrecap 11318 prodfdivap 11319 |
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