![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > iprodn0 | Structured version Visualization version GIF version |
Description: Nonzero series product with an upper integer index set (i.e. an infinite product.) (Contributed by Scott Fenton, 6-Dec-2017.) |
Ref | Expression |
---|---|
zprodn0.1 | โข ๐ = (โคโฅโ๐) |
zprodn0.2 | โข (๐ โ ๐ โ โค) |
zprodn0.3 | โข (๐ โ ๐ โ 0) |
zprodn0.4 | โข (๐ โ seq๐( ยท , ๐น) โ ๐) |
iprodn0.5 | โข ((๐ โง ๐ โ ๐) โ (๐นโ๐) = ๐ต) |
iprodn0.6 | โข ((๐ โง ๐ โ ๐) โ ๐ต โ โ) |
Ref | Expression |
---|---|
iprodn0 | โข (๐ โ โ๐ โ ๐ ๐ต = ๐) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zprodn0.1 | . 2 โข ๐ = (โคโฅโ๐) | |
2 | zprodn0.2 | . 2 โข (๐ โ ๐ โ โค) | |
3 | zprodn0.3 | . 2 โข (๐ โ ๐ โ 0) | |
4 | zprodn0.4 | . 2 โข (๐ โ seq๐( ยท , ๐น) โ ๐) | |
5 | ssidd 4005 | . 2 โข (๐ โ ๐ โ ๐) | |
6 | iprodn0.5 | . . 3 โข ((๐ โง ๐ โ ๐) โ (๐นโ๐) = ๐ต) | |
7 | iftrue 4538 | . . . 4 โข (๐ โ ๐ โ if(๐ โ ๐, ๐ต, 1) = ๐ต) | |
8 | 7 | adantl 480 | . . 3 โข ((๐ โง ๐ โ ๐) โ if(๐ โ ๐, ๐ต, 1) = ๐ต) |
9 | 6, 8 | eqtr4d 2771 | . 2 โข ((๐ โง ๐ โ ๐) โ (๐นโ๐) = if(๐ โ ๐, ๐ต, 1)) |
10 | iprodn0.6 | . 2 โข ((๐ โง ๐ โ ๐) โ ๐ต โ โ) | |
11 | 1, 2, 3, 4, 5, 9, 10 | zprodn0 15923 | 1 โข (๐ โ โ๐ โ ๐ ๐ต = ๐) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 โง wa 394 = wceq 1533 โ wcel 2098 โ wne 2937 ifcif 4532 class class class wbr 5152 โcfv 6553 โcc 11144 0cc0 11146 1c1 11147 ยท cmul 11151 โคcz 12596 โคโฅcuz 12860 seqcseq 14006 โ cli 15468 โcprod 15889 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-inf2 9672 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-pre-sup 11224 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-isom 6562 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-er 8731 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-sup 9473 df-oi 9541 df-card 9970 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-div 11910 df-nn 12251 df-2 12313 df-3 12314 df-n0 12511 df-z 12597 df-uz 12861 df-rp 13015 df-fz 13525 df-fzo 13668 df-seq 14007 df-exp 14067 df-hash 14330 df-cj 15086 df-re 15087 df-im 15088 df-sqrt 15222 df-abs 15223 df-clim 15472 df-prod 15890 |
This theorem is referenced by: iprodefisum 35368 iprodfac 35374 |
Copyright terms: Public domain | W3C validator |