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Mirrors > Home > ILE Home > Th. List > fprodeq0 | Unicode version |
Description: Any finite product containing a zero term is itself zero. (Contributed by Scott Fenton, 27-Dec-2017.) |
Ref | Expression |
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fprodeq0.1 | |
fprodeq0.2 | |
fprodeq0.3 | |
fprodeq0.4 |
Ref | Expression |
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fprodeq0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eluzel2 9445 | . . . . . . 7 | |
2 | 1 | adantl 275 | . . . . . 6 |
3 | 2 | zred 9287 | . . . . 5 |
4 | 3 | ltp1d 8802 | . . . 4 |
5 | fzdisj 9955 | . . . 4 | |
6 | 4, 5 | syl 14 | . . 3 |
7 | fprodeq0.2 | . . . . . . . 8 | |
8 | eluzel2 9445 | . . . . . . . . 9 | |
9 | fprodeq0.1 | . . . . . . . . 9 | |
10 | 8, 9 | eleq2s 2252 | . . . . . . . 8 |
11 | 7, 10 | syl 14 | . . . . . . 7 |
12 | 11 | adantr 274 | . . . . . 6 |
13 | eluzelz 9449 | . . . . . . 7 | |
14 | 13 | adantl 275 | . . . . . 6 |
15 | 12, 14, 2 | 3jca 1162 | . . . . 5 |
16 | eluzle 9452 | . . . . . . . 8 | |
17 | 16, 9 | eleq2s 2252 | . . . . . . 7 |
18 | 7, 17 | syl 14 | . . . . . 6 |
19 | eluzle 9452 | . . . . . 6 | |
20 | 18, 19 | anim12i 336 | . . . . 5 |
21 | elfz2 9920 | . . . . 5 | |
22 | 15, 20, 21 | sylanbrc 414 | . . . 4 |
23 | fzsplit 9954 | . . . 4 | |
24 | 22, 23 | syl 14 | . . 3 |
25 | 12, 14 | fzfigd 10334 | . . 3 |
26 | elfzelz 9929 | . . . . . 6 | |
27 | 26 | adantl 275 | . . . . 5 |
28 | 12 | adantr 274 | . . . . 5 |
29 | 2 | adantr 274 | . . . . 5 |
30 | fzdcel 9943 | . . . . 5 DECID | |
31 | 27, 28, 29, 30 | syl3anc 1220 | . . . 4 DECID |
32 | 31 | ralrimiva 2530 | . . 3 DECID |
33 | elfzuz 9925 | . . . . . 6 | |
34 | 33, 9 | eleqtrrdi 2251 | . . . . 5 |
35 | fprodeq0.3 | . . . . 5 | |
36 | 34, 35 | sylan2 284 | . . . 4 |
37 | 36 | adantlr 469 | . . 3 |
38 | 6, 24, 25, 32, 37 | fprodsplitdc 11497 | . 2 |
39 | 7, 9 | eleqtrdi 2250 | . . . . . 6 |
40 | elfzuz 9925 | . . . . . . . 8 | |
41 | 40, 9 | eleqtrrdi 2251 | . . . . . . 7 |
42 | 41, 35 | sylan2 284 | . . . . . 6 |
43 | 39, 42 | fprodm1s 11502 | . . . . 5 |
44 | fprodeq0.4 | . . . . . . 7 | |
45 | 7, 44 | csbied 3077 | . . . . . 6 |
46 | 45 | oveq2d 5841 | . . . . 5 |
47 | eluzelz 9449 | . . . . . . . . . 10 | |
48 | 39, 47 | syl 14 | . . . . . . . . 9 |
49 | peano2zm 9206 | . . . . . . . . 9 | |
50 | 48, 49 | syl 14 | . . . . . . . 8 |
51 | 11, 50 | fzfigd 10334 | . . . . . . 7 |
52 | elfzuz 9925 | . . . . . . . . 9 | |
53 | 52, 9 | eleqtrrdi 2251 | . . . . . . . 8 |
54 | 53, 35 | sylan2 284 | . . . . . . 7 |
55 | 51, 54 | fprodcl 11508 | . . . . . 6 |
56 | 55 | mul01d 8269 | . . . . 5 |
57 | 43, 46, 56 | 3eqtrd 2194 | . . . 4 |
58 | 57 | adantr 274 | . . 3 |
59 | 58 | oveq1d 5840 | . 2 |
60 | 2 | peano2zd 9290 | . . . . 5 |
61 | 60, 14 | fzfigd 10334 | . . . 4 |
62 | 9 | peano2uzs 9496 | . . . . . . . . 9 |
63 | 7, 62 | syl 14 | . . . . . . . 8 |
64 | elfzuz 9925 | . . . . . . . 8 | |
65 | 9 | uztrn2 9457 | . . . . . . . 8 |
66 | 63, 64, 65 | syl2an 287 | . . . . . . 7 |
67 | 66 | adantrl 470 | . . . . . 6 |
68 | 67, 35 | syldan 280 | . . . . 5 |
69 | 68 | anassrs 398 | . . . 4 |
70 | 61, 69 | fprodcl 11508 | . . 3 |
71 | 70 | mul02d 8268 | . 2 |
72 | 38, 59, 71 | 3eqtrd 2194 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 DECID wdc 820 w3a 963 wceq 1335 wcel 2128 csb 3031 cun 3100 cin 3101 c0 3394 class class class wbr 3966 cfv 5171 (class class class)co 5825 cc 7731 cc0 7733 c1 7734 caddc 7736 cmul 7738 clt 7913 cle 7914 cmin 8047 cz 9168 cuz 9440 cfz 9913 cprod 11451 |
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 4080 ax-sep 4083 ax-nul 4091 ax-pow 4136 ax-pr 4170 ax-un 4394 ax-setind 4497 ax-iinf 4548 ax-cnex 7824 ax-resscn 7825 ax-1cn 7826 ax-1re 7827 ax-icn 7828 ax-addcl 7829 ax-addrcl 7830 ax-mulcl 7831 ax-mulrcl 7832 ax-addcom 7833 ax-mulcom 7834 ax-addass 7835 ax-mulass 7836 ax-distr 7837 ax-i2m1 7838 ax-0lt1 7839 ax-1rid 7840 ax-0id 7841 ax-rnegex 7842 ax-precex 7843 ax-cnre 7844 ax-pre-ltirr 7845 ax-pre-ltwlin 7846 ax-pre-lttrn 7847 ax-pre-apti 7848 ax-pre-ltadd 7849 ax-pre-mulgt0 7850 ax-pre-mulext 7851 ax-arch 7852 ax-caucvg 7853 |
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 3774 df-int 3809 df-iun 3852 df-br 3967 df-opab 4027 df-mpt 4028 df-tr 4064 df-id 4254 df-po 4257 df-iso 4258 df-iord 4327 df-on 4329 df-ilim 4330 df-suc 4332 df-iom 4551 df-xp 4593 df-rel 4594 df-cnv 4595 df-co 4596 df-dm 4597 df-rn 4598 df-res 4599 df-ima 4600 df-iota 5136 df-fun 5173 df-fn 5174 df-f 5175 df-f1 5176 df-fo 5177 df-f1o 5178 df-fv 5179 df-isom 5180 df-riota 5781 df-ov 5828 df-oprab 5829 df-mpo 5830 df-1st 6089 df-2nd 6090 df-recs 6253 df-irdg 6318 df-frec 6339 df-1o 6364 df-oadd 6368 df-er 6481 df-en 6687 df-dom 6688 df-fin 6689 df-pnf 7915 df-mnf 7916 df-xr 7917 df-ltxr 7918 df-le 7919 df-sub 8049 df-neg 8050 df-reap 8451 df-ap 8458 df-div 8547 df-inn 8835 df-2 8893 df-3 8894 df-4 8895 df-n0 9092 df-z 9169 df-uz 9441 df-q 9530 df-rp 9562 df-fz 9914 df-fzo 10046 df-seqfrec 10349 df-exp 10423 df-ihash 10654 df-cj 10746 df-re 10747 df-im 10748 df-rsqrt 10902 df-abs 10903 df-clim 11180 df-proddc 11452 |
This theorem is referenced by: (None) |
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