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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 6818 | . . . 4 β’ ( I βΎ β€):β€β1-1-ontoββ€ | |
4 | f1of 6780 | . . . 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 16151 | . 2 β’ (π β βπ₯ β π΄ (( I βΎ β€)βπ₯) β₯ βπ β π΄ (( I βΎ β€)βπ)) |
7 | 2 | sselda 3943 | . . . . . 6 β’ ((π β§ π₯ β π΄) β π₯ β β€) |
8 | fvresi 7114 | . . . . . 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 2744 | . . . 4 β’ ((π β§ π₯ β π΄) β π₯ = (( I βΎ β€)βπ₯)) |
11 | 2 | sseld 3942 | . . . . . . . . 9 β’ (π β (π β π΄ β π β β€)) |
12 | 11 | adantr 482 | . . . . . . . 8 β’ ((π β§ π₯ β π΄) β (π β π΄ β π β β€)) |
13 | 12 | imp 408 | . . . . . . 7 β’ (((π β§ π₯ β π΄) β§ π β π΄) β π β β€) |
14 | fvresi 7114 | . . . . . . 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 2744 | . . . . 5 β’ (((π β§ π₯ β π΄) β§ π β π΄) β π = (( I βΎ β€)βπ)) |
17 | 16 | prodeq2dv 15741 | . . . 4 β’ ((π β§ π₯ β π΄) β βπ β π΄ π = βπ β π΄ (( I βΎ β€)βπ)) |
18 | 10, 17 | breq12d 5117 | . . 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 3171 | . 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 257 | 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 397 = wceq 1542 β wcel 2107 βwral 3063 β wss 3909 class class class wbr 5104 I cid 5528 βΎ cres 5633 βΆwf 6488 β1-1-ontoβwf1o 6491 βcfv 6492 Fincfn 8817 β€cz 12433 βcprod 15723 β₯ cdvds 16071 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2709 ax-rep 5241 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7663 ax-inf2 9511 ax-cnex 11041 ax-resscn 11042 ax-1cn 11043 ax-icn 11044 ax-addcl 11045 ax-addrcl 11046 ax-mulcl 11047 ax-mulrcl 11048 ax-mulcom 11049 ax-addass 11050 ax-mulass 11051 ax-distr 11052 ax-i2m1 11053 ax-1ne0 11054 ax-1rid 11055 ax-rnegex 11056 ax-rrecex 11057 ax-cnre 11058 ax-pre-lttri 11059 ax-pre-lttrn 11060 ax-pre-ltadd 11061 ax-pre-mulgt0 11062 ax-pre-sup 11063 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-int 4907 df-iun 4955 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-se 5587 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6250 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6444 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7306 df-ov 7353 df-oprab 7354 df-mpo 7355 df-om 7794 df-1st 7912 df-2nd 7913 df-frecs 8180 df-wrecs 8211 df-recs 8285 df-rdg 8324 df-1o 8380 df-er 8582 df-en 8818 df-dom 8819 df-sdom 8820 df-fin 8821 df-sup 9312 df-oi 9380 df-card 9809 df-pnf 11125 df-mnf 11126 df-xr 11127 df-ltxr 11128 df-le 11129 df-sub 11321 df-neg 11322 df-div 11747 df-nn 12088 df-2 12150 df-3 12151 df-n0 12348 df-z 12434 df-uz 12697 df-rp 12845 df-fz 13354 df-fzo 13497 df-seq 13836 df-exp 13897 df-hash 14159 df-cj 14918 df-re 14919 df-im 14920 df-sqrt 15054 df-abs 15055 df-clim 15305 df-prod 15724 df-dvds 16072 |
This theorem is referenced by: absproddvds 16428 |
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