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Mirrors > Home > MPE Home > Th. List > fproddvdsd | Structured version Visualization version GIF version |
Description: A finite product of integers is divisible by any of its factors. (Contributed by AV, 14-Aug-2020.) (Proof shortened by AV, 2-Aug-2021.) |
Ref | Expression |
---|---|
fproddvdsd.f | β’ (π β π΄ β Fin) |
fproddvdsd.s | β’ (π β π΄ β β€) |
Ref | Expression |
---|---|
fproddvdsd | β’ (π β βπ₯ β π΄ π₯ β₯ βπ β π΄ π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fproddvdsd.f | . . 3 β’ (π β π΄ β Fin) | |
2 | fproddvdsd.s | . . 3 β’ (π β π΄ β β€) | |
3 | f1oi 6817 | . . . 4 β’ ( I βΎ β€):β€β1-1-ontoββ€ | |
4 | f1of 6779 | . . . 4 β’ (( I βΎ β€):β€β1-1-ontoββ€ β ( I βΎ β€):β€βΆβ€) | |
5 | 3, 4 | mp1i 13 | . . 3 β’ (π β ( I βΎ β€):β€βΆβ€) |
6 | 1, 2, 5 | fprodfvdvdsd 16150 | . 2 β’ (π β βπ₯ β π΄ (( I βΎ β€)βπ₯) β₯ βπ β π΄ (( I βΎ β€)βπ)) |
7 | 2 | sselda 3942 | . . . . . 6 β’ ((π β§ π₯ β π΄) β π₯ β β€) |
8 | fvresi 7113 | . . . . . 6 β’ (π₯ β β€ β (( I βΎ β€)βπ₯) = π₯) | |
9 | 7, 8 | syl 17 | . . . . 5 β’ ((π β§ π₯ β π΄) β (( I βΎ β€)βπ₯) = π₯) |
10 | 9 | eqcomd 2743 | . . . 4 β’ ((π β§ π₯ β π΄) β π₯ = (( I βΎ β€)βπ₯)) |
11 | 2 | sseld 3941 | . . . . . . . . 9 β’ (π β (π β π΄ β π β β€)) |
12 | 11 | adantr 481 | . . . . . . . 8 β’ ((π β§ π₯ β π΄) β (π β π΄ β π β β€)) |
13 | 12 | imp 407 | . . . . . . 7 β’ (((π β§ π₯ β π΄) β§ π β π΄) β π β β€) |
14 | fvresi 7113 | . . . . . . 7 β’ (π β β€ β (( I βΎ β€)βπ) = π) | |
15 | 13, 14 | syl 17 | . . . . . 6 β’ (((π β§ π₯ β π΄) β§ π β π΄) β (( I βΎ β€)βπ) = π) |
16 | 15 | eqcomd 2743 | . . . . 5 β’ (((π β§ π₯ β π΄) β§ π β π΄) β π = (( I βΎ β€)βπ)) |
17 | 16 | prodeq2dv 15740 | . . . 4 β’ ((π β§ π₯ β π΄) β βπ β π΄ π = βπ β π΄ (( I βΎ β€)βπ)) |
18 | 10, 17 | breq12d 5116 | . . 3 β’ ((π β§ π₯ β π΄) β (π₯ β₯ βπ β π΄ π β (( I βΎ β€)βπ₯) β₯ βπ β π΄ (( I βΎ β€)βπ))) |
19 | 18 | ralbidva 3170 | . 2 β’ (π β (βπ₯ β π΄ π₯ β₯ βπ β π΄ π β βπ₯ β π΄ (( I βΎ β€)βπ₯) β₯ βπ β π΄ (( I βΎ β€)βπ))) |
20 | 6, 19 | mpbird 256 | 1 β’ (π β βπ₯ β π΄ π₯ β₯ βπ β π΄ π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 396 = wceq 1541 β wcel 2106 βwral 3062 β wss 3908 class class class wbr 5103 I cid 5527 βΎ cres 5632 βΆwf 6487 β1-1-ontoβwf1o 6490 βcfv 6491 Fincfn 8816 β€cz 12432 βcprod 15722 β₯ cdvds 16070 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7662 ax-inf2 9510 ax-cnex 11040 ax-resscn 11041 ax-1cn 11042 ax-icn 11043 ax-addcl 11044 ax-addrcl 11045 ax-mulcl 11046 ax-mulrcl 11047 ax-mulcom 11048 ax-addass 11049 ax-mulass 11050 ax-distr 11051 ax-i2m1 11052 ax-1ne0 11053 ax-1rid 11054 ax-rnegex 11055 ax-rrecex 11056 ax-cnre 11057 ax-pre-lttri 11058 ax-pre-lttrn 11059 ax-pre-ltadd 11060 ax-pre-mulgt0 11061 ax-pre-sup 11062 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-int 4906 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5528 df-eprel 5534 df-po 5542 df-so 5543 df-fr 5585 df-se 5586 df-we 5587 df-xp 5636 df-rel 5637 df-cnv 5638 df-co 5639 df-dm 5640 df-rn 5641 df-res 5642 df-ima 5643 df-pred 6249 df-ord 6316 df-on 6317 df-lim 6318 df-suc 6319 df-iota 6443 df-fun 6493 df-fn 6494 df-f 6495 df-f1 6496 df-fo 6497 df-f1o 6498 df-fv 6499 df-isom 6500 df-riota 7305 df-ov 7352 df-oprab 7353 df-mpo 7354 df-om 7793 df-1st 7911 df-2nd 7912 df-frecs 8179 df-wrecs 8210 df-recs 8284 df-rdg 8323 df-1o 8379 df-er 8581 df-en 8817 df-dom 8818 df-sdom 8819 df-fin 8820 df-sup 9311 df-oi 9379 df-card 9808 df-pnf 11124 df-mnf 11125 df-xr 11126 df-ltxr 11127 df-le 11128 df-sub 11320 df-neg 11321 df-div 11746 df-nn 12087 df-2 12149 df-3 12150 df-n0 12347 df-z 12433 df-uz 12696 df-rp 12844 df-fz 13353 df-fzo 13496 df-seq 13835 df-exp 13896 df-hash 14158 df-cj 14917 df-re 14918 df-im 14919 df-sqrt 15053 df-abs 15054 df-clim 15304 df-prod 15723 df-dvds 16071 |
This theorem is referenced by: absproddvds 16427 |
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